(![arneU Utttuecaitg Htbtarg ROLLIN ARTHUR HARRIS MATHEMATICAL LIBRARY THE aiFT OF EMILY DOTY HARRIS 1919 MATHEMATICS Cornell University Library QC 518.G77 A treatise on magnetism and electricity; 3 1924 001 081 763 Cornell University Library The original of tiiis book is in tine Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924001081763 A TREATISE ON MAGNETISM AND ELECTRICITY A TREATISE ON MAGNETISM AND ELECTRICITY BY ANDREW GRAY, LL.D., F.R.S. PROFESSOR OF PHYSICS IN THE UNIVERSITY COLLEGE OF NORTH WALES 'AH true and fruitful natural philosophy hath a double scale or ladder, an ascendent and descendent, ascending from experiments to the invention of causes and descending from causes to the invention of new experiments." Bacon, Advancement of Leai-ning^ IN TWO VOLUMES VOL. I ILonlron MACMILLAN AND CO., Limited NEW YORK : THE MACMILLAN COMPANY 1898 All 7 iffhts reserved Richard Clay and Sons, Limited, london and buncay. PEEFACE During the preparation of the second volume of my Treatise on Absolute Mcasfurements, I became strongly impressed with the desira- bility of a re-discussion of the whole subject of Electricity and Magnet- ism from the point of view of action in a medium. Since the publication of that work I have therefore tried to put together a statement which, from the beginning, should regard electric and magnetic forces as existing in a space-pervading medium in which the electric and magnetic energies are stored, and by which they are handed on from one place to another with a finite velocity of propagation. Of coarse it is impossible to avoid abstractions. We cannot explain the electric and magnetic inductions in the sense of giving the rationale of their production by a mechanical system, but this may not be because- we know less of thiags electric and magnetic than we do of the ordinary dynamical action qf material systems, but because the explanation of electrical phenomena must be sought in the solution of those very difficulties which have to be faced when we try to account for the- inertia of matter and ordinary forces between bodies, for example, gravitational attraction. The electromagnetic theory of light was established by Maxwell's dynamical discussion of the electro- magnetic field, and verified by. the experimental and theoretical researches of Hertz; but such dynamical theories seem to be possible only through the remarkable property of the Lagrangian method, by which the behaviour of the system may be qualitatively described and' expressed in terms of generalised co-ordinates and their velocities,, although we have no means of defining the connections between them and the co-ordinates and velocities of the particles of the system. The conception of a system of conductors carrying currents as a dynamical system has been rightly regarded as one of the great steps in advance which Maxwell took, and has been of great service in vi PEEFACE enabling mechanical explanations, or the construction of mechanical analogies of electrical action to be attempted with something of success, to the great profit of electrical science. As to the mode of treatment I have adopted, a few words are necessary. The book is not a treatise on the mathematical theory of electricity merely, but is, I hope, to some extent successful in bringing the theory and practice together. Thus while in general assuming some elementary acquaintance on the part of the reader with electrical phenomena and their laws, I have endeavoured first to look at the phenomena as they present themselves, and then to show how they fall into their places in the general scheme of electrical action, and to point out the consequences to which they lead. As stated in the words of Bacon I have placed on the title-page, it is a double process by which natural philosophy advances; we ascend from experiments to causes, and descend from causes to experiments, and it is " most requisite that 1 hese two parts be severally considered and handled." There are two chapters in the book the presence of which requires a word of explanation — the chapter on General Dynamical Theory, and the chapter on Fluid Motion. The former was written to provide in promptly accessible form, with references, all that the student could require for an intelligent apprehension of the special dynamical methods ■of the book, and so save him from havmg to disentangle what he wanted from a web of connected matter in a treatise on higher dynamics. I trust, however, that what I have written may lead to a wish to pursue the study of dynamics in the treatises and memoirs of the great masters of the subject. It is to be remembered that much of the ■chapter, especially the thermokinetic part, will find its application in Vol. II. The same thing has to be said for the discussion of Fluid Motion. It is rather long, but it is, as far as seemed to me possible without ■disturbing the order and natural mode of demonstration, an account of those theorems which will be required in the discussion of electrical phenomena considered as manifestations of the motion of the ether. Besides, the general theorems of irrotational and vortex motion aru those of the potential of electric and magnetic forces, and of the fields ■of distributions of currents, and are directly available in the electrical applications without further demonstration. A large range of applica- tions will be found for them in Vol. II., and the chapter will there save much space. PREFACE vii In the preparation of these chapters I have consulted many original papers and authorities, but I must specially acknowledge the great help I have obtained from the papers of Dr. Larmor and the treatise of Professor Horace Lamb. With regard to the mathematical treatment, I have, after consider- ation, decided not to use the vector analysis, but to endeavour to insist as far as possible on the physical meaning of' the quantities symbolised. Some brevity of expression is no doubt lost by this process ; but the discussion is, on the other hand, within the reach of a greater number of readers. I am, of course, very deeply indebted to the writings of Lord Kelvin, Lord Rayleigh, and Clerk Maxwell ; and, in connection with the Electro- magnetic Theory of Light, I wish to express my great obligations to the papers of Mr. Oliver Heaviside. No other writer on Maxwell's Theory has done so much to elucidate and render consistent its various parts, or contributed so much to the practical solution of problems of wave pro- pagation. Of the importance of Mr. Heaviside's views on general theory I am strongly convinced, and no adequate presentment of Maxwell's Theory can be obtained in which they are not largely adopted. Thus, Volume II., which will deal more with general discussions, will be affected to a still greater degree by the results of Mr. Heaviside's work. Mr. Heaviside has strongly urged a recasting of our system of units which would get rid of the 47r factor, which appears in formulse for the relations of quantities measured in the units now internationally adopted. There can be no doubt that such a reform would, on the whole, materially simplify mathematical expressions in the theory of electricity ; but I have not felt justified in adopting it in the present volume. In Volume II. " rational units " will be employed in the more general theoretical discussions, and where the results of these come into com- parison with those of quantitative experiments, the expressions will, if necessary, be modified to suit the C.G.S. units as at present defined. As to notation, I have employed Clarendon type in general for ■directed quantities, and block type for some quantities such as energy, total magnetic induction through a circuit, and the like, which are merely scalar. In one point I have deviated from ordinary usage : k and //. here denote the electric and magnetic inductivities of a medium, while K and vs denote its specific inductive capacity and magnetic permeability, that is, the ratio of the inductivity, electric or magnetic as the case may be, to that of the standard medium. viii PREFACE This procL'diuc is in accordance with the oi-iginal signification of specific inductive capacity and magnetic permeability, as defined by Faraday and Lord Kelvin respectively ; and whether or not the electric magnetic inductivity of the standard medium (ether par excellence) is taken as unity, it seems desirable, for the sake of clearness in con- siderations regarding units, to represent always the inductivity by its appropriate symbol where it properly occurs. Of course when in a particular range of applications of formula a certain assumed value of the inductivity is used throughout, as in Chapters III. and IV. of the present volume, where fi is taken as unity for air, the symbol may be conveniently omitted. A good deal here and there of theoretical matter has been taken from what I have formerly written on electrical subjects, but it has all been thoroughly revised and corrected, as far as lay in my power, in the light of teaching experience and the advance of knowledge. In the correction of proofs I have been aided in the most devoted manner throughout by Mr. Alfred Hay, B.Sc, of Liverpool College, by Dr. Maclean, of Glasgow University, and in the final revision of the latter part of the book I have had the great advantage of the help and criticism of my friend and former colleague, Mr. G. B. Mathews, F.R.S. Volume II. will contain among other subjects an account of Electro- lysis, of Magnetic Research, of General Theories of the Electromagnetic Field, of Distribution of Electricity on conductors at rest and in motion, and of recent work in the theory and observation of Electro- magnetic Waves. ANDREW GRAY. Univehsitt College of North Wales, Bangor, Feh. 23, 1898. CONTENTS CHAPTER I PERMANENT MAGNETISM Section T. — Magnetic Phenomena and Elementary Theoretical FrojMsition^ Elementary Facts. — Magnetic Field and Lines of Force. — Magnetization by In- duction. — Law of Magnetic Attraction and Repulsion. — Hypothesis of Magnetic Matter. — Magnetic Field of Uniformly Magnetized Magnet. Unit Quantity of Magnetism. — Definition of Magnetic Force or Field-Intensity. — Graphical Description of Lmes of Force. — Lines of Force of a Very Small Magnet. — Magnetic Potential. — Equipotential Surface. Equipotential Curves. — Actual Magnets. Equivalent Surface Distribution. — Magnetic Moment. Axis of a, Magnet. — Couple on Magnet in Magnetic Field. — Potential Energy of a Magnet. — Magnetic Poles. — Actions between Magnets. Simple Cases Pages 1—28- Section' I] — Synopsis of Elementary Theory of Magnetism. Magnetic Potential of any given Distribution Potential of a Magnetic Shell. — Lamellar Distribution of Magnetism. — Complex Lamellar "Distribution Pages 29 — 34 CHAPTER n MAGNETIC INTENSITY AND MAGNETIC INDUCTION Influence of Medium occupying the Field. — Total Magnetic Induction over Closed Surfaces in Field. — Solenoidal Condition Fulfilled by Magnetic Induction. — Particular Cases of Magnetization. — Mutual Energy of Two Pomt-Charges. Apparent Force of Repulsion between them. — Magnetic Inductivity. Super- position of Magnetic Effects. — Field Containing Difierent Media. — Magnetic Intensity and Induction in Cavity cut in Magnetized Body. — Magnetic Inductivity, Permeability and Susceptibility. — Surface Conditions Fulfilled by Magnetic Intensity. — Magnetic Potential. Equations of Laplace and Poisson. — Method of 'Vector Potential. — Specification of Vector Potential. — Uniformly Magnetized Ellipsoid. — Uniformly Magnetized Ellipsoid of Revolution. Demagnetizing Forces. — Uniformly Magnetized Sphere. — Field- External to LTniformly Magnetized Sphere Pages 35 — 57" X CONTENTS CHAPTER III TERRESTRIAL MAGNETISM Directive Force on Compass Needle. Magnetic Dip. Magnetic Equator. —Terrestrial Magnetic Poles.— True and False Magnetic Poles.— Magnetic Potential at Earth's Surface. —Horizontal Terrestrial Magnetic Force at Earth's Surface.— Determination of Surface Potential from Horizontal Force Relations between Horizontal Components.— Expression of Magnetic Potential at an External Point in Series of Spherical Harmonics. —The Magnetic Moment of the Earth.— Locality of Cause of Terrestrial Magnetic Phenomena.— Are the Terrestrial Magnetic Forces derivable from a Potential? Question of Current Per- pendicular to Earth's Surface.— The Magnetic Elements.— Time Changes of the Magnetic Elements and their Causes.— Diurnal Changes of Intensity at Points on same Parallel of Latitude. Representation by Vector-Diagrams.— Diurnal Changes in Different Latitudes. Difference of Amount in Summer and Winter. Results of Observations on "Quiet Days. "—Question of Derivation of Diumal Changes of Intensity from a Potential.— Eijuipotential Lines of Diurnal Changes. — Magnetic Storms . . . Parjes 58 — 84 CHAPTER IV MAGNETISM OF AN IRON SHIP AND COMPENSATION OF THE COMPASS Ship's Magnetism. — Soft Iron of Ship Represented by Iron Rods.— Expressions for Total Deviation of the Compass. Analysis of Deviations. — The Quadrantal Error. — The Semicircular Error. — Graphical Representation of Deviations. Determination of the Coefficients. The Dygogram. — The Heeling Error. — Compensation of the Compass .... Pages 85 — 100 CHAPTER V ELEMENTARY PHENOMEMA AND THEORY OF ELECTROSTATICS Section I. — Experimental Results and Action of Medium Elementary Notions. — Location and Transfer of Energy. Current in a Wire. — Electric Induction and Electric Intensity. — "Electrics" and "Non-Electrics." Conductors and Insulators. Electric Attraction and Repulsion. — Forces on Electrified Bodies regarded as due to Action of a Medium. — Faraday's Ice-PaO Experiment. — Division of Electric Field into Two Parts by Conducting Screen. Genesis of Field External to Closed Conductor. — Hypothesis of Incompressible Fluid. — Specification of Electric Induction and Electric Intensity. Energy of Field.— Surface Integral of Electric Induction. — Energy in Case of .^olotropic Medium. — Charged Spherical Conductor in Uniform Dielectric. Energy of System. — Spherical Condenser. — Tubes of Electric Induction. Unit Tubes. — Tension along Lines of Induction in an Electric Field. Traction on Surface of a Conductor. — Constancy of I EcZ.v along any Path from one Conductor to another. Potential. Difi'erenoe of Potential. Equipotential Surfaces. — General Problem of Electrostatics . Parjes 101 — 119 CONTENTS Section II. — Electrostatic Capacity and Electric Energy of Charged Conductors Electric Condensers or Leydens. — Energy in Terms of Electrostatic Capacity. Energy of a System of Charged Conductors. — Reciprocal Relation between two States of a. System of Conductors. Applications. — Coefficients of Potential and Induction and Electrostatic Capacities of a System of Conductors. Reciprocal Theorem. — Properties of the Coefficients of Potential and Charge of a System of Conductors. — Energy of a System of Charged Conductors Expressed as a Quadratic Function of Potentials or Charges. — Reciprocal Relations. Explora- tion of an Electric Field. — Nature of Charges on Conductors on Different Cases. — Energy Change due to Relative Displacement of Conductors under Different Conditions. — Characteristic Equation of the Potential. — Electric Induction and Potential in Particular Cases. — Approximate Values of Coefficients of Potential iind Induction. — Determination of Field within and without a Conductor by Potential Method. — Surface Distributions consistent with Surface Values of Potential. Green's Problem. — Green's Function. — Green's Function for a Spherical Conductor. Induced Distribution on a Spherical Conductor and on an Infinite Plane under the Influence of a Point-Charge. Electric Images. — Mutual Force between Two Charged Spheres. Explanation of an Apparent Anomaly. — Method of Inversion. Geometrical Inversion. — Electrical Inversion. Derivation of Induced Distribution from Equilibrium Distribution and Vii-n- rersa. — Inversion of Uniform Spherical Distribution. Problem of Two Parallel Infinite Conducting Planes with Point-Charge between them. — Distribution on Two Spheres in Contact Obtained by Inverting Induced Distribution on Two Parallel Planes. — Case of Two Equal Spheres. Electric Kaleidoscope. Pafje^ 120—157 HAPTER VI STEADY FLOW OF ELECTRICITY IN LINEAR CONDUCTORS Process of Change from the State of Equilibrium to Another. Electric Current. — Steady Currents. — Analogue of an Electric Current. — Ohm's Law. Resistance. Direction of Current. Electromotive Force. — Flow in a Conductor containing an Electromotive Force. Heterogeneous Conductors and Circuits. — Meaning of Resistance. Rate of Production of Heat in Conductors. Joule's Laws. — Arrangements of Electric Generators in Series and in Parallel. — Arrangement of given Battery to produce Maximum Current through given External Resistance. — Arrangement for Maximum Current not that of' Greatest Efficiency. — Theory of a Network of Conductors — Two Fundamental Principles : (1) Principle of Continuity. — (2) Sum of Electromotive Forces in any Circuit of Network equal to Sum of Products of Currents round Circuit into Resistances of Conductors. — Examples. (1) Two Points connected by Conductors in Parallel. Conductance. Resistance and Conductance of Parallel Conductors. Specific Resistance and Conductivity. — (2) Bridge Arrangement. — Analytical Treatment of a General Network. — Solution of Equations. First Reciprocal Theorem. Conjugate Conductors. Second Reciprocal Theorem. — Cycle Method for a Network. — Activity in a Network of Conductors. — Currents fulfilling Ohm's Law give Minimum Dissipation of Energy in Heat. — Elementary Discussion of Network of Conductors. — Reduction of Network to Bridge Arrangement. First Reciprocal Theorem. Conjugate Conductors. Second Reciprocal Theorem. — Effect of Joining a Wire between Two Points of a Network of Conductors. Pmjes 158—179 xii CONTENTS CHAPTER VII GENERAL DYNAMICAL THEORY Generalised Co-ordinates and Velocities. Kinetic Energy.— Theory of Action.— Lagrange's Dynaniical Equations. — Generalised Momenta. Reciprocal Theoreris. — Ignoration of Co-ordinates. Modified Lagrangian Function. — Lagrange's Equations with Gyrostatic Terms.— Application of Lagrange's Equation to Theory of Gyrostatic Pendulum. —Generalised Forces. Applied Forces, and Forces of Constraint. Lagrange's Equations deduced from Cartesian Equations of Motion of Set of Particles. — Work done by Forces of Con- straint in Variation of Kinematical Conditions. — Hamilton's Form of the Equations of Motion. — Impulsive Forces. Work done by System of Impulses. — Motional Forces. Dissipative Forces. Dissipation Function. — Controllable and Uncontrollable Co-ordinates. Thermokinetic Principle. — Dynamical Ana- logues of Thermodynamic Relations.— v. Helmholtz's Thermokinetic Theorems of Cyclic Systems ... . . Pages 180—211 CHAPTER VIII MOTION OF A FLUID Section I. — General Kinematical Theory Fundamental Assumptions. Lagrangian and Eulerian Methods. — Calculation of Acceleration of Fluid. — Velocity Potential. — Equation of Continuity. Theorem of Divergence. — Rotational and Irrotational Motion. Angular Velocity at Point in a Fluid. — Permanence of Velocity Potential. — Flow and Circulation. Circulation round Curve expressed as Surface Integral of Normal Spin. — Reducible and Irreducible Circuits. — Analysis of Motion at any Point in a Fluid. — Constancy of Flow along any Path moving with the Fluid. Pages 212—225- Section II. — Dynamical Theory Equations ot Motion of a Fluid. — Kinetic Energy. Rate of Variation of Energy in Given Space. — Stream-lines. — Motion in Two Dimensions. Conjugate Func- tions. — General Theorems for Incompressible Fluid in Irrotational Motion. Integral Equation and Continuity. 'Tubes of Flow. — Periphractic Spaces. — Mean Potential over Spherical Surface . . . Pages 226 — 235- Section III. — Green's Theorem Proof of (ireen's Theorem. Surfaces of Discontinuity. — Existence Theorem for Potential Function. Motion of Minimum Kinetic Energy. Uniqueness of N'alue of Potential Function for Given Space and Given Values at Surface. — Deductions from Green's Theorem. Sources and Sinks. — Kinetic Energy of Irrotixtional Motion. Expression as a Surface Integral. Cases in which Motion cannot exist. — Multiple Connection of Spaces. Reduction, of Con- nectivity by Diaphragms. Number of Irreconcilable Circuits. — Cyclic Motion in Multiply Connected Space. Cyclic Constants. —Green's Theorem in a Multiply Connected Space. —Kinetic Energy of Cyclic Motion of Fluid. — Uniqueness of Motion in Multiply Connected Space.— Perforated Solids Moving in a Liquid. Ignoration of Co-ordinates . . . Pages 235—255. CONTENTS Section IV. — Vortex Motion Yortex Lines and Vortex Tubes.— Permanence of a Vortex Filament.— Vortex Surface. — Determination of Velocities for Given Spin and Expansion of Fluid. — Auxiliary Function called Vector Potential. "Curling." — Velocity due to Long Straight Filament. Velocity due to Element of any Filament.— Velocity due to Closed Vortex Filament. — Velocity Potential of System of Vortices.— Electromagnetic Analogy. — Vortex Sheets. Cyclic Irrotational Motion regarded as due to Vortex Sheets on Bounding Surfaces. Electromagnetic Analogy. — Kinetic Energy of System of Vortices. Action of Rectilineal Vortices Pages 255 — 275 CHAPTEE IX ELEIIENTAEY FAC'JS AND THEORY OF ELECTEOMAGNETISM Magnetic Fields of Currents. Electromagnebic.Forces.— Ursted's Experiment. — Ampfere's Theorem of Equivalence of n, Current and a Magnetic Shell. — Definition of Unit Current. Proof of Ampfere's Theorem. — Theorem of Work done in carrying a Unit Pole round Closed Path in Field of Current. — Effect of Medium occupying the Field. — Experiments Confirmatory of Ampere's Theory. — Theorem of Biot and Savart. — Magnetic Potential of a Current. — Theorem of Work done in Field of Current. — Equations of Currents. — Applications of Theorem of Work m Closed Path. Solenoidal Current. Cylindrical Distribu- tion of Currents Parallel to Axis. — Examples of the Theorem of Equivalence of a Circuit and a Magnetic Shell. — Elementary Theory of Tangent Galvanometer. — Energy of Current-Carrying Circuit. — Mutual Energy of Two Current- Carrying Circuits. — Electrokinetic Energy. — Electromagnetic Force on Element of Circuit. Most General Specification of Current. — Mutual Energy of a Current and Magnetic Distribution. — Reaction of a Current-Element on a Magnetic System. — Magnetic Litensity of Straiglit Current Imbedded in Infinite Conductiag Medium. — Magnetic Action of Radial Cui-rent in Infinite Medium. — Vector-Potential of Current. — Magnetic Intensity for (1) Case of Straight Vertical Conductor with One End on Surface of Earth, (2) for Plane Current Sheet with Lines of Flow Circles Round Point in it, and Current Inversely as Square of Radius of Circle. — Experimental Illustrations of Electromagnetic Action. — Expression of the Electrokinetic Energy of any System by means of the Vector- Potential. — Self and Mutual Inductance. — Rational Current Elements. Mutual Energy of Two Current Elements. — Forces between Two Current Elements. — Explanation of Actions on Conductors by Stresses in the Field. — Ampere's Experiment. — Weber's Experiments . Pages 276 — 323 CHAPTER X INDUCTION OF CUEEENTS Section I. — Experimental Basis of Theory of Current Induction Faraday's Experiments.- — Felici's Experiments. — Induced Currents are due to Change of Magnetic Induction. — Faraday's Theory of Lines of Force (Induction). — Faraday's Experiments on " Unipolar Induction." — Numerical Estimation of Liductive Electromotive Force. — Electromagnetic Forces due to Induced Currents. Law of Lenz. — Self Induction. — Faraday's Experiments on Self Induction. — Faraday's Theory of Self Induction. — Henry's Experiments. — Application of Principle of Energy . Pwjes 324 — 336 xiv CONTENTS Skctiox II. — Dynamical Theory of Current Induction Eluctrodynamics. Electrical Co-ordinates. — Electrokinetic Energy. Current or Electrokinetic iMomentum.— Case of Two Mutually Influencing Circuits.— Unit Electromotive Force and Unit Resistance — Volt, Ohm, Ampere, itc. — Maxwell's Dynamical Illustration. — Lord Rayleigh's Dynamical Illustration. — Mutual Action of Two Invariable Circuits. — Single Circuit with Self- Inductance. — Theory of a Network of Conductors. — Primary with Secondary as Derived Circuit, — Oscillatory and Non-Oscillatory Discharge of a Condenser. — Con- denser and Inductance-Coil in Series with Simple Harmonic Electromotive Force. — Inductance Counteracted by Capacity. Arrangement for Maximum Current. Impedance, Effective Impedance. Graphical Representation of Results. — Effect of Inductance. on Signalling through a Cable or in Telephony. — Differ- ence of Potential between Terminals of Condenser. Resonance. —Maxwell's Dynamical Analogies of Inductance and Capacity. Resonance. — Primary and Secondary. Inductance and Capacity in Primary. with Harmonic Electromotive Force. — Conductors in Parallel, and Containing Resistance, Inductance, and Cajjaeity. Equivalent Resistance and Inductance of Parallel Circuits. — Action of Induction Coil, or Inductorium, with Condenser across Break in Primary. Case I. Secondary Closed. — Action of Condenser in Induction Coil. Case II. Secondary Open. — Mechanical Illustration of Action of Induction Coil with Condenser. — Dynamical Theory of Vibrating System, having only Kinetic Energy and Acted on by Dissipative Forces. Effective "Resistance" and "Inductance." Illustration by Mutually Influencing Circuits Pages 337 — 384 CHAPTER XI GENERAL ELECTROSIAGNETIC THEORY Section I. — Electromagnetic Theory of Light Electromotive Intensity at Element of Moving C(jnductor. Total Electromotive Force. — Flow of Energy in the Insulating Medium, and Dissipation in a Conductor. Displacement Currents. — Impressed Electric Force. lUusti'ation by Ideal Magneto-electric Machine. Energy is received by System at Seat of Impressed Forces. — Complete Specification of Current in Imperfect Conductor or Imperfect Insulator. Fundamental Circuital Equations of Field. — Propagation (if a Plane Electromagnetic Wave. — Plane Wave in .iEolotropic Dielectric, Principal Wave- Velocities. — Question of Relation of Electrical Vibrations to Elastic Vibrations of Ether. — Determination of Velocity of Propagation. — Ratio ( if Units. — Electromagnetic Radiation. Hertz's Solution of Maxwell's Equations. Vibrating Electric Doublets. — Electric and Magnetic Intensities iu Field of Doublet. — Direction of Vibration. "Longitudinal Light." Rate of Pro- pagation. Different Directions and at Different Distances. — Electromagnetic Theory of the Blue Sky. — Propagation of Electrical Potential. — Graphical Representation of Field of Vibrator. — Experimental Verification of Theory of Vibrator. — Approximate Theory of Hertzian Receiver. — Experiments with Different Positions of the Receiver. — Exploration of Electric Field by Receiver. —Period of Vibrator. Determination of Wave Length and Velocity of Pro- pagation in Rate of Radiation of Energy from Vibrator. — Reflection of Waves in Air from Metallic Surfaces. Standing Waves. — Multiple Resonance. — Effect of Size of Reflecting Surface. — Reflection of Electric Waves by Mirrors. Refraction by Prisms. — Polarization of Electromagnetic Beam. Relation of Plane of Polarization to Direction of Electric Vibration. — Transparency of Ordinarily Opaque Substances to Electromagnetic Waves . Pages 385 — 421 CONTENTS SECTIO^f IT. — Flow of Energy in the Electromagnetic Field Motion of Energy across Bounding Surface of a Closed Space. Poynting's Theorem.— £x«Hipics; 1. Straight Wire with Steady Current. Energy Stream- Lines. — 2. Discharge of a Condenser. — 3. Radiation of Energy from a Hertzian Yibrator. — Distribution of Currents in Cross Section of Cylindrical Conductor. Pages 421—430 Section III. — Moving Electric Charges Convection Currents. — Radiation in a Magnetic Field. The Zeemann Effect. Theory. — Experimental Verification Pcujes 430 — 435> CHAPTER XII THE VOLTAIC CELL "\'olta's Experiments on Contact Electricity. — Interpretation of Volta's Results. — Objections to Volta's Method. His further Experiments. — Result for Chain containing Liquid. — Experiments of Kohlrausch. — Hankel's Experiments. — Lord Kelvin's Experiments. — Lord Kelvin's Copper and Zinc Electric Machine. — Lord Kelvin's Induction Electric Machine, Founded on Contact Electrical Action. — Experiments of Ayrton and Perry. Clifton's Experiments. — Com- parison of Results. — Pellat's Experiments. — Experiments on Metals Immersed in Diflferent Gases. — Later Views on Contact Electricity. Differences of Potential Inferred from Flow of Energy. — Energy value of Electromotive Force of a Cell. — Explanation of Volta Effects as Air Metal — Metal- Air Differences of Potential. Thermoelectric Measure of Difference of Potential. — Energy Cri- tei'ion of Existence of an Electromotive Force. — Summary of Results of Later Theory. — Appendix to Chapter XII . . . Pages 440 — 457 CHAPTER XQI 1 HERMOELECTRICIT Y Elementary Phenomena. — Thermoelectric Inversion. — Thermoelectric Power. — Total Electromotive Force. — Peltier Effect. — Source of Energy in Thermo- electric Circuit. — Peltier Effect is Zero at Neutral Temperature. — Thomson Effect — Electric Convection of Heat. — Thermodynamic Theory. — Thermoelectric Diagram. — Appendix to Chapter XIII . . . . Pages 458 — 473 Index ... . . ... Pages 475— 47S> LIST OF PLATES Pl. I. The Earth's Magnetism as showx by the Distribution op Lines op equal TOTAL Force in absolute i\lEASDRE (British units, foot-grain-second-system) ; THE Position op the Magnetic Poles and Equator (Approximately for 1875). Pl. II. Lines op Equal Horizontal Force (Horizontal Force at Greenwioli = 1). From the Admiralty Manual of Deviations of the Compass, 1893 Edition. Pl. IIL Lines of Equal Vertical Force, 1 895 (Horizontal Force at Greenwich = ] ). From the Admiralty 3faimal of Deviations of the Compass, 1893 Edition. Pl. IV. Lines op Equal Variation (1895). From the Admiralty ilanval of Deviations of the Compass, 1893 Edition. Pl. V. Secular Curves for Dipferent Latitudes (Bauer). From Nature, Dec. 23, 1897. Pl. VI. Diurnal Curves (Bauer) for Epochs 1780, 1885'. From Xature, Dec. 23, 1897. Pl. VII. Vector Diagrams por Different Latitudes. Pl. VIII. Curves of Equal Potential foe Diurnal Changes. COERIGENDA Page 60, last line,yo9' Fig. 32 read Fig. 31 61, line 10 from ioot,foi- Fig. 33 read Fig. 32 84, second last line, ybr annular read annual 197, line 8 from loot, for — G read — 328, line 4 from top, /or Force Induction read Force (Induction) 332, in Fig. 116 reverse the direction of motion of the slider 359, equation (46),yor e Hi+i-,) ' read e Ur+r-,) 2ni_ Sn-i ^ ., ,, (48),/.^^M'>Mmr^. 'l'« " ,l.'OI.,lMll.'»/l» >t/wwjfiifWifvw»j>mffMi»mm}f7 rru. ;i. Fig. 4. similarly the south-pointing end of one repels the south-pointing end of the other, and attracts the north -pointing end. It is usual to call the north-pointing end the north end or " pole " of the magnet,- and B 2 Us "S Fig. 5. 4 MAGNETISM AND ELECTRICITY chap. the rsouth-pointing end the south end or "pole." This statement is expressed shortly by the rule "Like ends or poles of magnets repel one another, and unlike ends or poles attract one another." 6. The experiments also illustrate the fact that as a rule _ each magnet has at least two regions or poles at which magnetic action is most intense, which in a bar magnet are near its ends, and that one of these regions is characterised by a tendency to move towards the south, the other towards the north, when the magnet is placed in A B C ordinary circumstances at a dis- A^ S| tt* §1 t'^ ®i tance from other magnets. '' 'i 'i '■ ■* It is to be observed that the forces acting between the magnets are mutual. For example the two magnets, if suspended horizontally at not too great a distance apart, would be found to come finally to rest in stable equilibrium with their unhke ends turned the same way, and their suspension fibres deflected from the vertical towards one another. The term " poles " has been used in this section in a somewhat vague sense to designate certain regions or parts of a magnet. An attempt is frequently made to use the term more definitely to designate certain points in the magnet. We shall deal with this usage of the term later. (See Art. 40.) Magnetic Field and Lines of Porce 7. Much of our knowledge of the elementary facts of magnetism is due to Dr. Gilbert, Physician in Ordinary to Queen Elizabeth. With him the modern science of magnetism may be said to begin. He first explained the directional tendency of a freely suspended magnet by supposing the earth to be a great magnet, and illustrated his theory by means of a terrella or "little earth" (Fig. 6) made from lodestone, which acted on a small needle, about the size of a grain of barley, placed in any desired position near it. Dr. Gilbert's researches are contained in a Latin treatise De Magnete} published in 1600, which abounds in acute observations and valuable results of experiment. 8. One of Dr Gilbert's most important investigations was that which he made of the mMs virtutis, or, as we now call it, the field of force surrounding a magnet. His method was beautifully simple. Iron filings were dusted over a sheet of paper or cardboard placed above the mag- net, then the paper was tapped so as to raise the filings from the ' An English translation has recently been made by Mr. Mottelay, and is published by Wiley and Son, New York. A translation with Notes is also in preparation by the Gilbert Club. PERMANENT MAGNETISM Fis. 6. paper for an instant, and enable them to take up freely the positions in which the magnetic forces tended to place them. In Fig. 7 is given a copy of a photograph of curves made by Faraday.^ The filings were first strewed on a sheet of dry gummed' paper placed over the magnet, then fixed by softening the gum temporarily by means of a jet of steam. The nature of the magnet or magnets used in each case is described in the notes attached to the diagrams. It will be seen that the filings arrange themselves in distinct lines running round from one end of the magnet to the other. These curves indicate what Faraday called the lines of force of the magnetic field. We sha,ll consider their exact meaning and their geometrical form in some simple cases presently. Magnetization by Induction 9. That the small pieces of iron attracted by a magnet become magnetized can be made clear in a number of ways. They have the power while in the field of the magnet of attracting other pieces, which also become magnets, and so on. Thus to one end of a bar magnet of moderate size it is possible to hang a succession of small nails, each clinging to its neighbour, and so on back to the bar. Also such pieces of iron have the power of deflecting suspended magnets, as may be proved by prolonging a steel magnet by a bar of iron, and presenting it to a test-magnet suspended in the- earth's field of force. If the bar is removed from a series of small pieces of iron thus clinging to one another, their magnetization disappears in great measure. Again pieces of iron become magnetized while resting in various positions on the earth's surface. For example bars of iron standing in a corner of a smithy, and the iron of a ship's hull while the ship is on the stocks in a ship-building yard, are magnetized by the action of the earth. 10. Inductive magnetization thus produced in the earth's field, and indeed inductive magnetization in general, is facilitated by jarring the iron by striking it with a mallet, or otherwise. A common form of lecture-room experiment consists in taking an ordinary kitchen poker, holding it in the direction of the dipping needle, and striking it on the upper end with a wooden mallet. The iron becomes magnetized and generall}' retains to a considerable degree its magnetization ; which, however, can be nearly all removed by placing the poker at right angles to the line of dip and jarring it in that position repeatedly with the 1 These curves are the property of Lord Kelvin, P.E.S., wko has kindly permitted their reproduction here. MAGNETISM AND ELECTRICITY CHAP. Lines of force of an ordinary bar magnet. Line.s of force of a spherical or disk-magnet (with "poles" at the extremities of a diameter), placed in a uniform field the lines of force of which are oppositely directed to those of the disk. Lines of magnetic force for a system of three spherical or disk-magnets with like "poles" turned inwards towards the centre of the system. Lines of magnetic force round three straight wires can-ying currents, at right angles to the plane of the paper, and passing through it at the points 1, 2, 3. The currents at 1 and 2 are in the same direction, the current at 3 is in the op- posite direction. Fig. 7. I PERMANENT MAGNETISM 7 mallet. But the bar after having been magnetized in one direction can have its magnetization at once reversed by simply turning the bar €nd for end from its first position, and striking it again a few times "with the mallet. 11. It is found in this experiment that the lower end of the poker xepels the north-pointing end of a freely suspended magnet, and is therefore itself a north-pointing pole of the magnet which the poker has now become. Similarly the upper end is found to be south-point- ing. The general rule of inductive magnetization is that the piece of iron, if placed in the field of force in the direction which a freely sus- pended magnet would assume at the place, is magnetized with the same polarity as the magnet. Again in a common process of magnetization by stroking a steel bar Avith one end of a magnet, as shown in Fig. 8, always beginning and «nding each stroke at the same ends of the magnet, the ends of S "the bar take the polarity shown, that is the end of the bar which is finally in contact with the stroking bar at the end of each stroke has the opposite polarity to that of the stroking end. 12. The small pieces of iron in the experiment with filings afford another example of in- ■ductive magnetization. Each little piece of iron becomes in Fio. 8. the field a small magnet, and "vvhen raised from the paper momentarily by the tapping, takes the ■direction which a small permanent magnet would take there, except •of course in so far as it may be disturbed by other magnetized particles. Thus the lines of force are given by little chains of small magnets, each successive pair in a chain having opposite poles adjoining, 13. The most powerful mode which has yet been devised of induc- tively magnetizing a bar of iron or steel is by surrounding the bar with a helix of wire, and causing a strong current of electricity to flow through the helix. Pieces of iron such as nails can be made to adhere to the extremities of the bar, and can be piled on one another, in positions in which they could not remain if the bar were unmagnetized. On the withdrawal of the current the demagnetization of the iron core . ■of the helix is shown by the pieces of iron falling off. In the older treatises on magnetism will be found elaborate accounts of various methods of magnetizing bars of steel, by what was called ■" touch." Only one of these, that called " single touch," is described in Section 10. Since the invention of electro-magnets these methods have all become obsolete. 14. A greater or less amount of magnetism is always retained by a 8 MAGNETISM AND ELECTRICITY chap. piece of iron after it has been inductively magnetized. The amount depends, other things being the same, on the nature of the iron. Iron which is mechanically very soft retains in general only a small amount of magnetization, mechanically hard iron a considerably greater amount. Thus iron in which residual magnetism is small is commonly called soft iron. Mechanical softness and magnetic softness do not, however, uni- formly coincide. Steel has great retentiveness for magnetism, and a specimen, after being magnetized by any process, say that here described,, cannot be demagnetized or have its magnetization reversed except by being placed in a sufficiently intense field oppositely directed (relatively to the specimen) to that by which it was magnetized. In their power of taking and retaining magnetization different kinds of steel dififer to a degree which can hardly be accounted for by differences in composition, but apparently depending on molecular constitution. The property of retaining residual magnetism has been called coercive force. We shall find later in the chapter on Magnetic Measurements a numerical reckoning for coercive force, and see how to determine its amount. Law of Magnetic Attraction and Repulsion 15. The natural philosophy stage of magnetic science may be said to have begun with the researches of Coulomb towards the end of last century. By means of his torsion balance he tested approximately the repulsion between two poles of the same name belonging 'to two different magnets. The balance as arranged for magnetic experiments is shown in Fig. 9. A wide glass case or box is prolonged upwards by a tube fixed over the centre of the cover, and carrying at its top a graduated torsion head for sustaining a fine silver wire. To the lower end of this- wire is attached a stirrap in which a bar magnet can be placed hori- zontally. Another magnet can be inserted vertically through the cover so that one end shall be on a level with the suspended bar, and opposite- one of its ends, . 16. It was found by Coulomb, by careful experiments, that to turn the upper end of the silver wire through any angle relatively to the lower end required the application of a couple proportional to the angle- Thus it was possible to determine a couple turning the suspended ma,g- net. This couple was produced by the action of the fixed magnet, which was introduced through the cover so that like poles of the two magnets should be opposed, and should have their poles on the circle in which that of the suspended magnet turned. First of all the suspended magnet only was placed in position, and the balance arranged so that the magnet rested in the earth's field of force without any disturbance from the torsion of the wire. The torsion head was then turned so as to deflect the magnet from that position through a small angle measured on the scale- surrounding the case. It was found in one experiment that to turn the- magnet 1° from its equilibrium position in the earth's field required a I PEEMAXEXT MAGNETISM 9 twist of 35° between the two ends of the wire. The fixed magnet was then placed in position and gave a deflection of -24°, which, takino- the couple required to produce b}' magnetic repulsion alone a deflectfon of 24 as twenty-tour times the couple required for a deflection of 1° cor- responded to a twist of -H X 35^\ But the twisting was resisted by the -i twist given to the fibre by the deflection, so that the total couple on the bar exerted by the magnet mavbe reckoned as that correspondmo- to 8G4° of torsion. ' i ^ The torsion head was now turned so as to bring down the deflection to l-2\ when it was found that the head had to be turned just eight Fir;. il. times round, that is through 2SS0°. Thus the couple on the wire was that due to 2SS0 + 12 x"3.5 + 12, or 8312, degrees of torsion. This is a little less than four times the former couple. From such experiments Coulomb obtained the result that two poles of ditierent magnets repel one another with a mutual force in\erselv proportional to the square of the distance between them. By findino- the couples necessary to keep the magnet deflected through difterent angles when unlike poles were opposed, the magnetic attraction between the poles could also be measured. 17. But while such experiments gave a rough approximation to a law of im'erse squares between two ordinary magnetic poles, it must be understood that for many reasons they could yield no exact result. The poles of the magnets could not be regarded as points, and hence no 10 MAGNETISM AND ELECTRICITY chap. inverse square law could possibly hold for them. Moreover the remote poles of the suspended and fixed magnets could not have been without effect; in fact the two bar magnets as wholes acted on one another, although the principal action was between the adjacent ends. Again, no account was taken of the earth's field in determining the forces acting on the maofnet, or of the mutual inductive action of the two magnets in diminishing their magnetization. A vibrational method used by Coulomb for determining the action of a magnet on a small needle placed at different distances from it will be described in the chapter on Magnetic Measurements. Hypothesis of Magnetic Matter 18. A theory of magnetism was put forward at an early date in the later developments of the subject. It was supposed that there existed two imponderable magnetic fluids which pervaded the apparently active regions of a magnet, which were such that a portion of either had the property of repelling a portion of the same fluid, and of attracting a portion of the other kind. A hypothesis of imaginary magnetic matter has been applied with great advantage by Lord Kelvin to the discussion of the phenomena of magnetization. It gives a language for the ex- pression of the theoretical results which have been arrived at, and enables an analogy to the probable constitution of a magnet to be pictured to the mind and usefully employed, without any danger of misunderstanding or pre-judgment of the actual facts of the case, if the investigator is careful not to be Jed by the mere words he uses to assign a reality to this imaginary matter which it does not possess. 19. It is of great importance also to remember that this hypothesis is only a short way of expressing certain facts of magnetism as they directly appear to us, and must not be permitted to lead to the conclu- sion that magnetic action is really action at a distance. There is nothing more certain than that magnetic action is propagated by means of a medium, and that that which appears to us the action of one magnet on another is really the action on each of the medium in contact with them. In describing magnetic phenomena it is convenient, however, to use language more or less descriptive of what is directly perceived. When we come to a discussion of theory we shall endeavour to keep action at a distance, and its methods and expressions, in their proper place as short cuts to results, or descriptions of the outcome of the more complete theory. A magnetized body, we shall see, probably consists of some kind or arrangement of molecules in motion, producing also in the surrounding medium a motion which is the propagating cause of the apparent action at a distance of one magnet on another. 20. We shall make use at present of this hypothesis of imaginary magnetic matter, which we shall call for shortness simply magnetism, for the deduction of some results, useful in what follows. In the first I PERMANENT MAGNETISM 11 place let us suppose a long thin bar, whether straight or curved, to be made up of slices which have magnetism distributed on their ends as indicated in Fig. 10 by the shading. Then if we suppose each slice exceedingly thin, and to have equal quantities of the two kinds of mag- netism at its ends as shown, the opposite magnetisms in contact at any junction of two adjoining slices will annul the action of one another on external magnetism, and there will be left only the unbalanced magnetism at their ends to produce external effect. This distribution of magnetic matter corresponds to the molecular arrangement which has place in an actual magnet. Each molecule has a polarity directed in the same way in all the particles, or nearly so, corresponding to the two kinds of magnetism on the ends. If, as may possibly be the case, each molecule be in rota- i^^A/i tion round its axis, the polarity consists in the two aspects of the rotation, according as the pai-ticle is regarded from one end or the other. Whatever the nature of the molecule may 'he, the two kinds of magnetic matter must if used be taken as symbolising two aspects of one thing, neither of which can be regarded as existing alone, any more than either aspect of the rotation of a fly-wheel can be regarded as existing apart from the other. That one kind of magnetism cannot exist without the other is shown by the experimental fact stated above, that if a bar magnet is broken each portion is a magnet having in general the same polarity as the original magnet. This can be easily verified by tempering glass-hard a straightened piece of clock-spring, magnetizing it, then breaking it into pieces, and testing them by means of a small horizontally suspended needle. 21. The magnetic distribution described above and illus- ^°" ' trated in Fig. 1 is that of a uniformly magnetized magnet, and its poles are thus exactly at its ends. They may in this case, if the bar be thin, be taken as points coincident with the ends of the bar. Such a magnet may be approximated to very closely by carefully mag- netizing a long thin knitting needle, by stroking it in the manner above described with another magnet. magnetic Field of Uniformly Kagnetized Kagnet. Unit Quantity of Kagnetism 22. We propose now to investigate the lines of force of such a magnet on the supposition that the quantities of magnetism at the ends of the bar are equal and opposite, and are there concentrated at points. Also we shall suppose that two like quantities repel one another, and two unlike quantities attract one another, in each case with a force inversely as the square of the distance between the points at which the quantities are supposed concentrated, and directly as the product of the two quan- 12 MAGNETISM AND ELECTRICITY chap. titles. Any multiple of a quantity of magnetism may be imagined as produced by placing that number of equal thin bars together so that their like poles coincide. Thus we assume that the force, F, between two quantities m, m', of magnetism at points at a distance r apart in a medium of uniform constitution is given by the equation F-^^ (1) where /4 is a constant depending upon the medium. According to this specification F is taken positive if m and m' have the same sign, and negative if they have opposite signs. In the first case, the force is a repulsion, in the latter, an attraction, for we take one kind of magnetism, namely, that of the north-pointing end of a magnet, as positive, the other kind as negative. Later we shall obtain this result from a theory of displacement or motion in a medium filling the field. 23. Equation (1) defines unit quantity of magnetism as that which concentrated at a point in a medium for which fi is unity repels with unit force an equal quantity also concentrated at a point at unit distance from the former. If the force is a force of one dyne, that is to say the force which acting on a gramme of matter for one second gives it a velocity of one centimetre per second, and the distance be one centimetre, the quantity of magnetism is one unit in the centimetre-gramme-second (C.G.S.) system of units. 24. The value of /a is very generally taken as unity for air. If we wish to be quite definite we may take air at standard atmospheric pressure and temperature 0° C, as that substance for which fi is unity ; but the alteration of the magnetic properties of air produced by any ordinary change of pressure or temperature is imperceptible. The value of /A however varies from medium to medium on account of some physical property which is different in different media ; and when we know more of the inner mechanism of magnetic bodies and media generally we shall no doubt find some natural measure of /* depending on that property. At present all we can do is to find the ratio of the values of /^ for any two different media. It is usual to call jm the magnetic inductive capacity, or magnetic inductivity, of the medium, or its magnetic permeability. We shall however use the term permeability to designate the ratio of the value of /i for a given medium to the value for some standard medium. In the remainder of the present chapter we shall suppose /i=l. Definition of Magnetic Force or Field-Intensity 25. By the force at a point in the field of a magnet is meant the force which a unit quantity of positive magnetism would experience if placed at that point. This is also called the intensity of the field at the point. We shall denote it in what follows by the symbol H. It is I PERMANENT MAGNETISM 13 clearly a directed quantity of such dimensions that when multiplied by a quantity m of magnetism it gives a product ?JiH which is a force in the ordinary dynamical sense. (See the Chapter on the Units and Dimensions of Physical Quantities.) If the unit of force here chosen is again the dyne, and the unit quantity of magnetism is ] C.G.S. unit, H is measured in C.G.S. units of field intensity. A uniform field is one for which H has the same value at every point. The reader will easily see after the dis- cussion of tubes of magnetic force that if H has the same numerical value at every ql- point of any region, it has the same ^.-■' |\ ; direction at every point, and conversely. x' ' ^- 26. A line of force is a curve so ^' , drawn in a magnetic field that the -' ,''' direction of the tangent at any point is ^ ' , ^ the direction of the magnetic force, or ,' ^'' field-intensity, at that point. , ; ' ' Consider the field of a uniformly ^ .>*^ magnetized filament. As we have seen the filament may be replaced by a quan- ^"^ ^^• tity of magnetism -|- m at ^, Fig 11, and another — m at B. Let FQ be two points on a line of force at a distance ds apart, and let AP = r, BP = r, IPAB = 6, LPBA = & . Then LQAB =0 + de, IQBA = 6' + d0'. Also sinBQP = - rW/ds, so that if QN be a normal drawn to QP at Q, cosBQN = — r'dffjds. Similarly, cos AQN = rdd'/ds. The forces on a positive unit of magnetism at Q are, neglecting small quantities, and taking ft, as unity everywhere, a repulsion mjr^ along AQ, and an attraction m/r'^ along QB. Thus since there can be no component normal to a line of force, we get resolving along the normal QN m 'dO m , d6' — r — + — . »• = r^ ds r"^ ds or 1 dO 1 dff -t: + - jT = r a* r ds Multiplied by the perpendicular from Q on AB this is wa.ed6 + sin^W =0. Hence integrating we obtain cos^ + cosfl' = c (2) where c is a parameter, constant for each line of force, but changing in value from line to line. 14 MAGNETISM AND ELECTRICITY chap. Graphical Description of Lines of Force 27. To describe these lines graphically we may proceed as follows. Draw a circle (Fig. 12) having unit diameter AG along AB, and draw- any chord BA. Produce if necessary DA to H so that DU is made equal to the parameter c. From ^with AE as radius describe an arc cutting the circle in F, then the angles DAG, GAP are angles 6, 6,' for which cos 6 + cos 6' = c. Thus it is only necessary to draw a line through B parallel to FA meet- ing AD produced if necessary in P. P is a point on the curve. If the point E falls between A and D, the distance AF Fig. 13 is made equal to ^^. A line through B drawn parallel to AF v^iW meet '/P Iio 13. AD ia a point on the curve. The angle 0' is now obtuse, and is the supplement of FA-B, so that cos d' is negative. In this way different points on the curve can be found. In the neighbourhood of A or P the curve must be drawn from its inclination to AB. This is given by the equations cos 6 — c — 1, cos 6' = c - 1. Fig. 14 shows curves for different values of c drawn for the magnetic filament and also illustrates the following other method of drawing the curves.^ Describe a circle on ^5 as diameter, and lay off a distance AM such that AM = c. AB. Then from A draw any line to cut the circle in Q and lay off Aq — AQ along AB. From B as centre with radius Mq describe a circle cutting the former circle in B. The point in which AQ and BB intersect is a point on the curve. The proof of the con- struction is left to the reader. A comparison of the form of these curves with the curves given by iron filings affords a rough confirmation of the law of magnetic force of which the curves of the Figure are a consequence. 1 This cut and the construction here given are taken from Constructive Oeonetry of Plane Curves, by T. H, Eagles, M.A. [London, Macmillan and Co.],. PERMANENT MAGNETISM 15 Lines of Force of a Very Small Magnet 28. The equation of these curves when AB is taken very small, or, which is the same thing, the equation of a curve given by (2) when the parameter c is very small, is important. Let the origin of co-ordinates be taken at the middle point of AB, and the axis x along it. Let the length of AB be denoted by 2Sa so that the co-ordinates of ^ are — Sa,0, Fig. 14, and of B, -f- Sa,0. Then if x, y, be the rectangular co-ordinates of P we have instead of (2) a; 4- 8a x — ha Jix'+haf + 2/2 V(i^^SaT2 + y"^ ~ If Sa be small this may be written X + ha X - ha or Jx^ V 2xha + y2 Jx^ - 2xha + y^ which reduced, and with 1/C put for c/2Sa becomes 2,2 1 {x^ + y^f- V ' (3) 16 MAGNETISM AND ELECTRICITY CHAP. where C is a parameter constant for any one curve. By varying G the whole family of curves is obtained. This is the integral equation of the lines of force of a small magnet. It will be found of great importance in the theory of electrical radiation discussed later, as it is the equation also of the lines of electrical force in the neighbourhood of (but not too near) a dumb-bell electrical vibrator of the kind generally used in the study of the propagation of electrical waves. 29. If r be the distance of any point from the origin and Q its inclination to the axis of x, the equation can obviously be written in the form r = C sin ^ (4) From this expression the curve can be described with great facility. For draw a line OA (Fig. 15) making an angle 6 with OX and meeting a circle described from as centre with radius C in the point A. Then let fall a perpendicular 45 on the axis OY, and a second perpendicular from B on OA meeting it in P. P is a point on the curve. For OB = sin e, and OP = OB sin 6 = G sin^^. This construction gives points on the curve with great ease except near the summit G. The part of the curve including G, can however be filled in sufficiently exactly by drawing a short circular arc with the proper radius of curvature for the point G. The expression for the radius of curvature at any point is Csin^ (sin^^ + 4cos02) 4/3 (sin^^ + 2cos20). For G, where = 7r/2, this is C/3. Hence the arc is to be drawn from Q, where GQ = C/3. With respect to the point on the axis, the equation of course does not apply as the magnet is there situated. Points fairly close to the origin are given quite well by the construction if carefully made. The family of curves for dififerent values of G is shown in Fig. 16. PERMANENT MAGNETISM 17 Magnetic Potential 30. Let a unit of positive magnetism he supposed to be so far off from a given magnetic distribution that it may be regarded as being outside the field of force, and let it then be brought against magnetic repulsion to any point P of the field. Work must be spent on the unit in so doing, and the amount of work thus spent is the measure of the poteutial at P. If work on the whole is done by the magnetic system Fig. 16. on the unit, while it is being brought to P, the work sjpent is negative that is the potential at P is negative. For any distribution whatever, the work D, done in bringing the unit from an infinite distance from the distribution to P is given by n - f H cos eds (5) where H is the field-intensity at any element ds of the path, the angle between the direction of H and that of ds (taken positive in the direction from P) and the integral is taken along the path from an infinite distance from any point of the distribution to the point P. If the distribution be a quantity m of magnetism situated at a point -d, in a field of unit magnetic inductivity, and D be the distance of .4 from any element ds of the path along which the unit is brought, H = m/D^ and we have cos = dDjds. Thus Q. l^dD = m IIP m r ■ (6) The work done therefore in carrying a unit of magnetism from the point P to another point Q at distance / from A is ' O =m\—r 18 MAGNETISM AND ELECTRICITY chap. This is called the difference of potential between P and Q. If we have any number of point distributions m-^, m^, etc., at distances r-y, r^, . . ., i\, ?•.,', from P and Q respectively the difference of potential be- tween P and Q is v-l) <«) O = 2»i where 2 denotes summation of all the quantities my{l/r\ — l/r^), m^{l/r^ — l/j'g), which are given by the different distributions. 31. If the distribution is a continuous one on a surface or throughout a given volume the summation becomes an integration throughout the distribution, thus n =Hk-^) (''> where dm is any element of the distribution, and r-^, r^, are the distances of P and Q from the element dm. For a surface distribution of density a per unit of area (that is amount of magnetism per unit of area) at an element dS of the surface, this is " = Wf-^) (9) and for a volume distribution of amount p per unit of volume (volume- density) at an element dv it is Q. h[h'T) ('°> the integrals being taken over the surface and throughout the volume in the respective cases. 32. As a simple example consider the potential at P, Fig. 11, due to a uniform filament AB ( - m- at A, + m at B). Clearly we have, if AP = ry,BP = r^, "=K^-i) (i^> that is the potential at P is equal to the work which would have to be spent if, there being a quantity m of magnetism at P, a unit of positive magnetism were canried from A to B. As another example we shall calculate the potential at a point P of a magnetic doublet consisting of two equal and opposite point charges say + m a,t B and — m at ^, Fig. 11. The potential of the doublet is Q = ,n(l _ 1) =,n ''-'-^" (12) \r^ rj r^r^ '^ ' But clearly if AB be small r^ — r^ = AB cos PAB=ds cos 0, and I PERMANENT MAGNETISM 19 '"i = ''2 = '■> iiearly. Hence if m denote the magnetic moment of the doublet we get ^ m cos 6 " = —^- (13) If the centre of the doublet be at the point {x,y,z), the co-ordinates of P be f.T/.f, and the direction cosines of the axis of the doublet be \iJ;v, the last equation becomes ^^^ Hi-=c) + ,.{v-y) + v{i-z ) ^^^^ since cos 6 = {\i^ — x) + fi {7} — y) + v {^- z) }/r. Clearly m may be taken as the moment of any combination of doublets at (x,y,z), if the resultant have its axis in the direction {\,/j.,v). Equipotential Surface. Equipotential Carves 33. The locus of an assemblage of points for every one of which H has the same value is called an equipotential surface. Any curve drawn on such a surface is called an equipotential line or cui've. Clearly there can be no component of field intensity tangential to an equipotential surface, and hence lines of force cut equipotential surfaces at right angles. Also the work done in carrying a unit of magnetism from any point of an equipotential surface to any other point of the surface along any path whatever is zero. Also if & be an element of a line of force between two equipotential surfaces, the potentials of which differ by 8X1, the work done in carrying a unit of positive magnetism from the surface of less to the surface of greater potential is 8TI. But this is numerically equal to HSs, if H be the resultant magnetic force at Ss. Hence if Ss be taken positive in the direction from the surface of greater potential towards that of less H 8s = - 8n or in the limit H--^ (15) that is, H is equal at any point to the rate of diminution of the potential along an element rfs of a line of force drawn through the point. From this we have for the components a,^,y, of magnetic force parallel to rectangular axes x, y, s, the values da ?n ?n ''= - d^'^^ -d^'l'- -d^ ■ ■ ■ (^^) c 2 20 MAGNETISM AND ELECTRICITY CHAP. 34. As an example we may consider again the uniform magnetic filament AB. Since by (11) the potential is n the equation of an equipotential surface is (17) Taking the section of this surface by the plane of the paper in Fig. 14, it is easy to see that the curve of section cuts normally the family of curves of which the typical equation is cos ^1 + cos 02 '= "• 35. To describe an equipotential curve of parjameter c' — see (17) — draw two rectangular axes OX, OY, Fig. 17, and . lay down the point G the co- ordinates of which are c', c'. Then draw lines through C cutting off intercepts OD, OE, the first from the axis of x on the positive side of the origin, the other from the axis of y on the nega- tive side of the origin. If r^, r^, be the numerical lengths of these intercepts we have by the Figure rjr^ = c'l{c' + r^) or Fig. 17. J_ Jl L the equation of the equipotential surface, and also of its curve of inter- section with the plane of the paper. All values of I'j, r^, thus obtained for a given value of c' will not belong to the equipotential curve which it is desired to draw. Only those are to be taken which lie between the two lines CD^E^ GD^E^ which give respectively OE^ + OD^ = AB, and OE^ - OD^ = AB, where AB is the distance between the extremities of the magnet. These lines can be drawn very exactly by calculating their inclinations to OX. If m = tan ODE, we have in both of the limiting positions of the line m = r^lr^ = (c' + r^ jc'. In the first case we have besides (putting A B = a) r.^ + r^ = a, in the second r^- r-^ = a. Eliminating r^, r^, from the equations in the two cases we get for the first and for the second (wi^ - I)c'— ma = (?n - 1)V — ma = 0. PERMANENT MAGNETISM 21 The greater roots of these equations are values of m which enable the limiting lines GB^E^, OD^E^, to be drawn. The lines ruled betweeit them through G, as ODE, give intercepts which form the radii from A,B of points situated on an equipotential curve but on one side of the axis. To complete the curve round one pole of the magnet it is only necessary to lay down with the same pairs of radii the points on the other side of the axis.. The same radii are of course available for the curve round: the other pole. Rotation of the whole diagram of equipotential curves round the axis of the magnet traces out the above family of equipotential surfaces be- FiG. 18. longing to the magnet. Fig. 14 shows lines of force and equipotential surfaces for the magnetic filament, Fig. 18 for an infinitely short mag- net. Fig. 19 shows the nature of the lines of force in the field between two magnets with equal like poles turned towards one another. It will be noticed that, at points on a line drawn at right angles to the common axis of the magnets and through the centre of the space between them, the resultant force is along that line and from the magnet. Actual Magnets. Equivalent Surface Distribution 36. Actual magnets ■ are not in general uniformly magnetized, but have throughout their substance a volume density of magnetization representing unbalanced polarity. This is called /ree magnetism. What- ever the distribution of this may be, it is easy to prove experimentally. 22 MAGNETISM AND ELECTRICITY chap. what has been assumed above, that the quantity of free positive mag- netism present in a magnet is exactly equal to the quantity of free negative magnetism. It is only necessary to suspend the magnet in the field of, but at some considerable distance from, another magnet. It is then found that the magnet experiences no sensible translational force, but a very con- siderable resultant directional couple, unless it happens to be so placed that that couple vanishes. The magnet above referred to as suspended in the field of the earth is a case in point. The most careful observation cannot detect any displacement of the magnet as a whole due to the earth's field, while the couple acting on it is very sensible, as much, perhaps for a square bar 60 cms. long and 1 cm. in diameter, hung at right angles to the earth's field-intensity, as nearly 20 grammes weight acting at an arm of 1 cm., or the weight of rather more than a quarter of an ounce acting at an arm of 1 inch. For external points the action of a magnet, whatever its distribution, can be imitated by a certain distribution of positive and negative mag- FiG. 19. netism on the surface of the magnet. Grounds for this conclusion will be found in Chapter VI. with other general theorems regarding surface dis- tributions. 'This surface distribution may be regarded as due to. the poles of uniform magnetic filaments running through the magnet and having their ends on the surface of the magnet. It must however be clearly understood that we can obtain no knowledge of the actual distribution of magnetism in a magnet ; all that can be obtained by the best of the so-called methods of determining magnetic distribution is a knowledge more or less accurate of the equivalent magnetization here referred to. Magnetic Moment. Axis of a Magnet 37. Consider first a long thin bar uniformly magnetized in the direction of its length. The magnetic moment of such a bar is the couple which it would experience if it were placed with the line joining its poles at right angles to the lines of force of a uniform field of unit intensity. If C.G.S. units are used the couple gives the moment in C.G.S. units. In the case of any other magnet there will be a position in which in a uniform field the couple acting on the magnet is a maximum. I PERMANENT MAGNETISM 23 The axis of the magnet is then at right angles to the lines of force of the field. The axis may be found as follows. Let the magnet be freely •suspended by its centre of gravity in a field uniform over the region •occupied by the magnet. The magnet will set itself so that the couple acting on it is zero. Let then a line in the magnet have its position in space marked, or, which is the same, let the positions of two points in the magnet be marked. Now let the magnet be resuspended by its •centre of gravity so that it rests in equilibrium in another position, and let the new positions of the same two points be marked. Draw then two planes bisecting at right angles the lines joining the two positions of each point. The line of intersection of these planes is the direction ■of the axis of the magnet. For, the axis of the magnet must have had the same direction in both cases, and hence it must have been possible to have turned the magnet from one position to a parallel one by simply turning it round an axis parallel to this direction, which is clearly given by the process described. This construction is not convenient in practice, but instead the magnet may be hung in the field, and its position marked, and then he removed and a long thin needle hung in its place. If this be also •suspended by its centre of gravity it will take the direction of the mag- netic force, that is, the direction of the axis of the more complex magnet •of considerable cross-section. Determinations of magnetic axis have however very seldom to be made. We shall see later how uncertainty arising from want of exact knowledge of the position of the axis of the needle of a dip-circle is •eliminated. Couple on Magnet in Magnetic Eield 38. If a magnet having taken up a position of equilibrium in a -uniform field be turned through an angle of 180° round an axis at right angles to the magnetic axis it will again be in a position of t ■equilibrium. It is clear from Fig. — e-^c 20 that in one of these positions DirccUcnj/MiU the equilibrium of the magnet is ->-j^^------ -^-:r-----rrrrrrr=r.-.-:-^<-- stable, in the other unstable. Any H angular displacement of the mag- I'ig- 20. net not compoiinded of a rotation through any angle round its own axis, and a rotation of 180° round a perpendicular axis, leaves it under the influence of a couple the moment •of which depends on (1) the magnet itself, (2) the angle which the new direction of the magnetic axis makes with its direction of stable equilibrium, (3) the intensity of the magnetic field. If the magnet be placed in a uniform field of intensity H so that its axis makes an angle 6 with the position of stable equilibrium, that is with the direction of H, the moment of the couple is MH sin 6 where 24 MAGNETISM AND ELECTRICITY chap. M is the magnetic moment of the magnet. Since the amount of free positive magnetism in the magnet is equal to the amount of free negative, the magnet may be regarded as made up of a very great number of uniformly magnetized filaments having their ends within or on the sur- face of the magnet. If oi)i be the free magnetism at either pole of one of these filaments the force exerted on that magnetism by the field is 8mH, and the forces on the poles are equal and in opposite directions. Hence if / be the distance between the poles, Si the angle between the filament and the direction of H, the moment of the couple on the fila- ment is H / sin •&. Bm. Hence summing for all the filaments and putting Ii for the resultant couple we have L = HS(^sin^Sm) (18) if the magnet be a thin plate with its breadth parallel to H, so that the- axes of the elementary couples are all parallel. But there is a position of the magnet and a corresponding value d' of ^ for the particular filament considered for which 2 (^ sin y 8m) = (19) Now let 6 be the angle through which the magnet must be turned from this position to attain its actual position in the field. Then A = + ^' and 2 (^ sin & Sot) = % \l sin {&' + 6) hm) = cos e 2 {I hn sin ^') + sin 6 2 {I Bm cos ^'). The first tei-m on the right is zero by (19). The second term gives L = H sin e 2 (^ Sot cos S') ..... (20) Thus the moment M of the magnet is given by the equation M = 2 (^ Sot cosy) (21) Similarly any other more complicated case may be treated. Potential Energy of a Magnet 39. If a magnet is placed in a given position in a magnetic field, the field and magnet have mutual potential energy measured by the work which must be done against magnetic forces in bringing the- magnet to the given position from another, defined as that for which the- mutual potential energy is zero. We shall assume that this potential energy is zero when the axis of the magnet is at right angles to the- lines of force of the field. If the magnet be so small that throughout it the field may be taken as one of the same intensity, no work will be- done in bringing it from outside the field to any given position, if it be kept always at right angles to the lines of force ; and therefore no work will be done in bringing it from outside the field in any manner what- ever, and leaving it with its axis at right angles to the lines of force. It is to be understood in this connection that the magnetization of th& PERMANENT MAGNETISM 25 magnet remains imclianged when it is moved in the field, otherwise the statements here made will not be correct. If now without change of position of the magnet as a whole it be turned round imtil the positive direction of its axis (from the south- pointing to the north-pointing end) makes an angle 6 with the position of stable equilibrium of this axis, the couple due to magnetic forces acting on the magnet is M H sin d for the angle 6, and this tends to diminish 6. The work done hy magnetic forces in bringing the magnet from the zero position to the final one is thus - j MHsin^rf^ = MHcose Hence the work done against magnetic forces is — M H cos 6 and if E denote the potential energy according to the specification E = - MHcos e (22) If the components of the magnetic force H referred to three rect- angular axes, X, y, z. Fig. 21, drawn in the true north, the east, and Fig. 21. the vertically downward directions respectively, be a, ^, 7, and the direc- tion cosines of the magnetic axis referred to the same axes be /, m, w, we have instead of (22) E = - M(Za + mp + ny) (23) Also substituting the value of sin in terms of a, j8, 7, 1, m, n, we have L = M{(my - n^Y + {na - lyf + (l^ - »ia)2}i . . (24) This is plainly the resultant of three couples ]II(wiy — K)8), M(na - ly), M(^j8 — ma) round the axes of a^, y, z, respectively. 26 MAGNETISM AND ELECTRICITY chap. Using polar co-ordinates, putting (see Fig. 21) ? for the angle which H makes with a horizontal plane, for the angle between the plane of X, z, and a vertical plane containing H, and r) and i/r for the corre- sponding angles for the magnetic axis we get a = H cos f cos ^, ^ = H cos ^ sin if>, y = S. sin f I = cos ■^ cos i/f, m = cos iy sin \j/, n = sin i;, and instead of (23) E= — MH|cos^cos7;cos(^ - i/?; -H sin^sinr;} .... (25) Also for the component couples we obtain 'M.{lp - ma) = MH cos ^ cos tj sm { - \f) . . . . (26) with two similar equations for the other components. Magnetic Poles 40. The determination of the positions of the " poles " of an ordinary bar magnet has been the object of much experimental research. Properly speaking, there are no definite poles in an ordinary magnet, if by poles are meant points at which the whole of the free magnetisms of the bar may be supposed concentrated, the negative at one, the positive at the other, so as to produce the actually existing external field. They exist only in the ideal case of an infinitely thin and uniformly magnetized filament, in which case they are the extremities of the bar. As a matter of approximation, however, the positions of such points can be found ; and one or two examples will be given in the next Chapter. 41. When a magnet is hung in a uniform field, it may be regarded as acted on by two sets of parallel forces, one set acting on the positive the other on the negative magnetism. The resultants of the two systems of parallel forces give the couple acting on the magnet ; and the centres of these systems, or, what is the same thing, the " centres of mass " of the two distributions of magnetism, may be regarded as poles. But this idea of pole is not of any use, as all we are concerned with is the moment of the couple on the magnet, which, as we have seen, is the product MH sin 6, where M is the magnetic moment, H the field intensity, and ff the angle between the direction of H and the magnetic axis. The term " pole " in the sense of a quantity of magnetism concen- trated at a point is also frequently used in specifying magnetic field intensity, or when discussing the mutual action between a magnet and a field, as, for example, when we speak of the force on a unit magnetic pole (that is, unit quantity of magnetism) placed at a point in the field. PERMANENT MAGNETISM Actions between Magnets. Simple Cases 42. We calculate first the field intensity produced at any point by a straight magnetic filament AB {- m s,t A, + m a.tB). Let the centre of the filament be taken as origin of co-ordinates, and x, y, as indicated ^Y -'' in Fig. 22, for the co-ordinates of the r-";<' point P at which the intensity is to ^j-'' \^ ioe found, and 21 the length of the filament. Then AF^ = r^^ = (x + If + y^, BF" = r^s = (a; - Z)'' + f'. The forces at P due to the ends A and P re- A O B spectively ai-e numerically m'r^, tnjr.^, Fig. 22. the former acting in the direction towards A, the latter in the direction from B. The direction cosines of the lines in which they act are i:hMs — {x + T)h\,-yji\, (x — 1)^t.-,, y />•.;,, respectively. The total component X along the magnet is thus given by and the component I' in the direction of y by Hence if . This may be seen in the following manner. Let the shell be made a simple closed shell by a cap fitting the boundary. The potential at the point outside will be made zero, that of the point inside ± 4 •jr •!>. Let ^tOi be the original potential at the latter point. Then W; has been changed by the amount ± 4 tt — to, which is the solid angle subtended at the internal point by the cap. But the solid angle subtended by the cap is the same at both points, and hence if w, be the solid angle subtended by the shell at the external point we have oJe + 47r — ui( = or (Oe - Wi = + 47r . . . (46) Lamellar Distribution of Magnetism 53. When the magnetism of the body may be regarded as made up of simple magnetic shells, either closed or having their edges on the surface of the body, the magnetization is said to be lamellar. Let, in this case, denote the sum of the strengths of the shells passed through by a point carried within the magnet from any chosen zero position to any other position {x, y, z) : then <^ = 1 1 cos 6ds. where cos6ds denotes the thickness of any shell passed through, I the intensity of magnetization, and 6 the angle between ds and the direction (X, fi, v) of I. But since ^ dx dii dz eosO = X-7- + a-j- + V y-, as as ■ d^ the equation for ^ may be written I PERMANENT MAGNETISM 33 and therefore d dx dy dz d dx d \-r-= rrr = Stt — sin 6 9\cos^d with the positive direction of magnetization. The direction of the vector- potential is at right angles to the plane of the angle , and, in accordance with the direction chosen as that of integration, appears to an eye regarding the path of integration of A in the direction opposed to that of magnetization of the element to be directed round the curve in the counter-clockwise direction. We shallsee that the specification of the vector-potential thus given corresponds precisely to that of the direction of the magnetic force at F, due to an element of a circuit replacing Idv so that the direction of flow of current is the same as that of magnetization. 81. To verify the specification let p, q, r be the direction cosines of I, X, y, z the coordinates of Idv, ^, rj, ^ those of the point considered, then the r inside the brackets only being a direction cosine. This gives dG = ^-^{r{i -X)- p(^ - z)}, Idv,_ ' ff = {(a~ - B—) J\ ct/ dxj (76) 'el{ , du du „3m', , ex oy D^' - \ AS7^-udv . (77) Hence we have since cu/d^= - du/dx, &c _ aff" aff _ a " ~ 81; ~ 3f "" ~ "ai. with similar expressions for h and c. The first term of the expression on the right is the value of fi^a, where a is the magnetic force at the point considered, due to the mag- netic distribution elsewhere, and fi^ is the inductive capacity of the medium with respect to which the values A, B, G of the components of magnetization are taken. The remaining term is zero unless the point considered falls within the limits of integration. If the latter is the case the value is 4i-n-A', if A' is the value of A at the point considered. For it is clear that AudvQ is the potential at the point considered of a volume element of attracting or repelling matter of density A, and therefore that the potential at the same point, U say, of the whole distribution is given by taking the volume integral, so that U = I Audv. Hence d'^U dW d^U , ,, „ where A' is the density at the point considered. But f< SJ'iU = \A\7'hidv, and therefore LiV^"*^" = ~ ^ttA'. Thus we obtain finally fiQa + i-irA' E 2 52 MAGNETISM AND ELECTRICITY chap. by equation (77). Similarly we should find ?F dll 3 (a;2 + 2/2)^ 2 where (7 is a parameter variable only from line to line. Either the positive or the negative sign may be given to the radical on the left. With the positive sign the equation suits the case of a paramagnetic sphere magnetized by the field, with the negative sign it expresses the form of an external line of force when the sphere is diamagnetized by the action of the field. The equation may be written in the form y' = i'-T^^, (91) (a;2 + y-Y by putting 2m/F/j,() = a^, and iCfFfig = H^. From this form of the equa- tion the curves given in Figs. 28 and 29 are plotted. These curves are taken from Lord Kelvin's paper on Lines of Force {Beprint of Papers on Electrostatics and Magnetism, § 632), in which the theory stated above was first given and illustrated. CHAPTER III TERRESTRIAL MAGXETISM Directive Force on Compass Needle. Magnetic Sip. Magnetic Equator 87. The tendency of a suspended lodestone or magnet, or compass needle to set itself at a given place in a particular direction, was at a very early date attributed to some action of the earth. It was recognized after the discovery of the magnetic dip by Hartmann about 1544 and Norman in 1576, that at every point of the earth's surface there is a magnetic force in a definite direction generally inclined to the horizontal, and further observation showed that the direction and magnitude of this force varied from point to point on the surface of the earth. The whole matter was discussed in the clearest manner by the famous Dr. Gilbert, of Colchester, Physician in Ordinary to Queen Eliza- beth, in his Latin treatise, Dc ilagncte magneticisgiie corporibus. There he put forward the extremely important idea that the earth is a great magnet, and that therefore, in the language of Faraday, there exists a terrestrial magnetic field (or oriis virtutis, as Gilbert called it), in which a small needle, except in so far as it is disturbed by gravitational or other forces, sets itself with its axis in the direction of the resultant force at the place where it is situated. The different positions of a small needle freely suspended at different parts of the earth's surface are illustrated by the drawing of a terrella, Fig. 6 above, which is taken from the second edition of Gilbert's book. It is instructive to compare this with the lines of force due to a uniformly magnetized sphere, which we have seen are at external points coincident with those due to a small magnet at the centre. Here it is clear that at the points N. and S. the needle would stand vertical, and at points on the circle midway between these horizontal. This corresponds roughly to what is found by magnetic surveys to take place at the earth's surface. At different points on a sinuous line surrounding the earth in the region of the equator the dipping needle (that is a bar magnet suspended by its centre of gravity) remains horizontal. This line is called the magnetic equator. The theory that terrestrial magnetic phenomena are due to a small CHAP. Ill TERRESTRIAL MAGNETISM 59 magnet at the earth's centre seems to have been held by Tobias Mayer, who flourished between 1723 and 1762. He worked out according to this theory the dip and force on a small needle at different places on the earth's surface. The results do not agree with observation ; but it was a noteworthy attempt to explain on a simple hypothesis the facts of terrestrial magnetism. Terrestrial Magnetic Poles 88. The conclusion which has been arrived at from a study of observations of terrestrial magnetism made at different parts of the earth's surface is that there are two distinct points, one in each hemi- sphere, at which the horizontal component of magnetic force vanishes and the dipping needle rests vertical. These are called magnetic poles. T'hey are marked on the chart, Plate I (at the end of this volume), showing lines of equal total magnetic force for the date 1875. It will be seen that the magnetic poles are not diametrically opposite and do not coincide with the extremities of the earth's axis. According to the calculations of Gauss, who investigated this subject, they should lie at latitude 73° 35' N., longitude 264° 21' E., and latitude 72° 35' S., longitude 152° 30' E., respectively. The north magnetic pole was, how- ever, reached in 1831 by Sir James Ross, and found to be at latitude 70° 5' N. and longitude 263° 17' E. Again, on the Erebus and Terror expedition of 1839-43, a dip of 88° 56' was attained, and it was inferred from the observations that the south magnetic pole was situated in latitude 73° 5' S. and longitude 147° 5' E. These positions agree very closely with those given by Gauss. As will be seen later, there is reason to believe that the poles are gradually changing their positions. It is to be nbted that the line joining these points is not parallel to the magnetic axis of the earth, that is the axis of greatest magnetic moment. This will appear later in the sketch we propose now to give of Gauss's great memoir on Terrestrial Magnetism.^ True and False Magnetic Poles 89. It has been held by many people that there are two north poles and two south poles of terrestrial magnetism. It is easy to show that if there are more north poles or more south poles than one, there must be an odd number of each. For consider surfaces of equal magnetic potential. A north or south pole must be a place where such a surface touches the surface of the earth. If then there are two north poles, these must be due either to the fact that two equipotential surfaces touch the earth, or that one touches in two points. If two surfaces touch, then, since equipotential surfaces cannot cut one another, they must each have two protuberances, as shown by the dotted lines in the diagram (Fig. 30). The intersection of the horizontal surface of the ' 1 Allgemeine Theorie des Erdviagnetismus. liemlt. d. Magnetiachen Vereins, Leipzig, 1839. Wyke, 5terBd., S. 121, 60 MAGNETISM AND ELECTRICITY CHAP. earth with the equipotential surfaces must therefore give the other curves represented in the diagram, namely, two series of closed curves passing into a figure of 8 curve, and then into larger curves inclosing the , \ \ \ ^5 Fig 30. others. Thus the points a and c' are true poles in the sense of the definition, that is, the horizontal coiiaponent of the magnetic intensity- vanishes at these points. But the point P is also a point at which an equipotential surface touches the surface of the earth at a point of depression, and is therefore also a magnetic pole. There is, however, a distinct difference between the pole P and either of the other two. Supposing the latter to be true north poles, that is points towards which the north pointing end of a magnet turns, it will be seen that a north pointing pole near P, but within the looped figure, will point away from P, while if it be outside it will point towards P. Hence such a pole has been called, though not quite appropriately, a false north pole. It is easy thus to see that counting both true and false poles, NORTH SOUTH Fig. 31. there must be an odd number if there are more than one pole of either kind. This result is also due to Gauss. 90. A false pole may also be produced by the presence in the earth's strata, near the surface, of a quantity of magnetic iron ore. The illustration of such a case given in Fig. 32 is due to Gauss. The Ill TERRESTRIAL MAGXETISM 61 diagram shows the effect, on the east and west equipotential lines, of a magnet buried under the surface with its north pointing pole upper- most. It will be seen at once that this is a particular case of Fig. 31, in which one of the two poles there shown is at an infinite distance from the other. Since the horizontal forces of the earth and magnet cancel one another at the double point P, it is here again a so-called false pole of the same kind as the earth's south pole for a needle placed outside the looped figure, that is north of the equipotential line through P, of the opposite kind for a needle placed elsewhere. Magnetic Potential at Earth's Surface 91. The question as to whether the magnetic intensities in the case of terrestrial magnetism are derivable from a potential is one not yet quite settled, and we shall return to it in Art. 105 below. Assuming that they are so derivable, let XI be the magnetic potential, and consider the force at a point on the earth's surface. If H be the horizontal component and H the total force we have H =13. cos -^ where ■y^ is the dip. If now ds be any element of a line drawn on the earth's surface making an angle 6 with the direction of H, that is with the magnetic meridian, the force along ds is H cos -<|r cos d. Hence = H cos \!r cos 6 = H cos 6 . . . . (1) ds Thus if O, n' be the values of O at the beginning and end of any line drawn on the surface of the earth €1 - n' =\H cose ds (2) where the integral is taken from end to end of the line. This integral has the same value along whatever path the integral is taken from a given initial to a given final point, and therefore, on the assumption made above, the integi-al taken round a closed curve is zero. Horizontal Terrestrial Magnetic Force at Earth's Surface 92. Let now P^ Pj, Pg, . . . . Fig. 33, be the angular points of a polygon dra^vn on the surface of the earth, the sides being arcs of great circles of the surface regarded as spherical, each small in length in comparison with the circumference of a great circle. We may take as the integral along each side the length of the side multiplied into the mean of the values of H cos 6 for the beginning and end. Thus if ^j be the inclination of the first side P„ P^ at its initial end, to the magnetic meridian, and 6\, the angle for the other end, we have for the integral along the side the approximate value. i(//o cos 6ii + //i cos 6\) P^Fy 62 MAGNETISM A^D ELECTRICITY CHAP. Denote hj ^ the variation at anj' place, that is the angle in azimuth between the astronomical north and the magnetic north, reckoning it positive when toward the west and negative when toward the east. Then if ff,^, H.^, H^, .... be the horizontal forces at P^, P^, P^,..., f^, (fi^^, the variation and the azimuth of the line P^ P^ at P^, ^^, 0'^^, the cor- Fig. 32. responding angles at P-^, and so on, the integral along P^ P^ is nearly -^ { i^ocos (^„i + Q + H^ cos (^'„i + Q\. Again the integral along P^ P^ is ?^- 1 ZTj cos (^1 + .^12) + ^2 cos (4 + .^'12) } , and so on. These added round a closed polygon ought to give a zero result provided, as we shall see, that there is on the whole no electric current flowing across the enclosed area; and therefore if the polygon be a triangle for which the lengths of the sides, the angles specified, and the horizontal forces at two of the angular points, are known, we are able to calculate the horizontal force at the third point. Gauss performed this calculation for a geodesic triangle having its vertices at Gottingen, Milan, and Paris. From the latitudes and longitudes of the places it was easy to calculate the angular distances of the places from one another along great circles on the earth's surface, and hence from the known radius of the earth to find their actual distances. The value Ill TEREESTMAL MAGNETISM 63 of H for Paris was thus found from those for Gottingen and Milan to within -J- per cent, of the observed value. 93. We may express the horizontal magnetic force at the earth's surface in terms of two components, one along the astronomical meridian of the place, the other at right angles to the meridian. Let X denote the longitude of the place, u the complement of the geographical latitude, and let the meridian be an ellipse of semi-major axis E, and semi-minor axis iJ (1 — e). Then the length ds of an arc of the meridian is Rdu (1 - e) [1 -h {(1- e)* - 1} sin^ztji/jl - {^.e - e^)sui^ u\ and an ele- ment of the parallel of latitude has the length dcr = R (1 — e) sin ud\j {1 — (26 — 6^) sin^ Mr. Taking now the component force X in the direc- tion of the astronomical meridian and Y that along the parallel of latitude in the westward, here the positive, direction of X, we have rfn _ {1 - (2e - 6^)sin2M}i 3n > cb " ie(l-€)[H-{(l-€)*-]}sin2M]i3^ I da- E (1 - c) sin u dX J The horizontal force is therefore VJT^-f Y^, and if f be the variation tan ? = Y/X. If 6 be put = 0, that is if the surface be regarded as spherical, jr = i- Y=-^—- (4) a du Esin u dk are force components at the surface. If the point (supposed external) be at distance r from the centre, JS in these expressions must be replaced by r. Determination of Surface Potential from Horizontal Force Kelations between Horizontal Components 94. We can now prove a number of theorems of great interest. First of all, if the northward component of force is known all over the surface of the earth, and the magnetic potential at one point, the potential at any other point can be found. For if D, be the potential at the given point, and Oj that at any other point of a latitude Wg, then by integrating along the meridian through the latter point to the north pole, and thence to the given point along its meridian, we find the required potential, thus n = Oq - RXdu + RXdu. J Wo J The first two terms give the potential at the'north pole, 0„ say. Thus n = n„ -H ^^RXdu (5) 64 MAGNETISM AND ELECTRICITY chap. From this we can find the westward component of the magnetic force at the given place. For 1 an I f»8X r=--l_'^=-4-r^c^. ... (6) K sin u o\ sin mJ ^ oA. 95. Again, if Y be given for all points the northward component at each point can also be found, provided the value of X along any line running from the north pole to the south is known, say along a meri- dian. For, integrating along the meridian first from the north pole, then along a parallel of latitude to the given point, we find » Q. ^ R\ Xdu - R\ rsinM(fX + n,j ... (7) Jo Jao Thus O is known along the earth's surface and therefore so also is X. The value of the constant fl„ so far as X is concerned is of no importance. 96. Consider now a closed path on the earth's surface, joining along parallels of latitude and meridians, four points defined by their longitude and colatitude as follows, \,u, X^fli, X-^tU', \u'. The line-integral along this path must vanish, and, therefore, putting X,X.^ for the forces along the meridians of longitude X, \j, and Y, Y' for the forces along the parallels of colatitude %i, v! , respectively, we get sinw ; Yd\ - X^du - sinii' Y'd\ + Xdu = 0. Ja ju Ja jii If we take the parallels of latitude very close together, so that 'u! = u — du, we have Y'=Y—dY/diLdu,Sind the equation just written becomes X^-X+{ ^{Ysinu)dX = .... (7') which is of course derivable from (7). From this it follows that if the northerly component of the horizontal magnetic force is known for any meridian and colatitude u, and the values of I^and dY/du are known along the parallel of latitude from that meridian to another, the value of the northerly component can be found for that other meridian. If the meridians difier in longitude by the amount d\{ = \ — X) only, (7') becomes Tx-'l^'''''^-^-' (H -which is only (6) in another form. If the longitude be expressed in terms of local time t, measured, say, from the meridian of Greenwich where the time is ifg, we have Ill TERRESTRIAL MAGNETISM 65 '\. = 2'n-(tQ~t)/T where T is the length of the day. Thus we may write (7'), (7") in the alternative forms Expression of Magnetic Potential at an External Point in Series of Spherical Harmonics 97. Whatever the distribution of magnetism may be to which terrestrtal ihagnetic phenomena are due, if it is wholly within the earth we can express the magnetic potential in a series of spherical harmonics. Thus if (Sq, S.^, S2, . . . .he spherical surface harmonics the potential at a point P distant r from the centre is given by ^ = ^oj + S.^ + S,^+ (8) which is convergent for E Are the Terrestrial Magnetic Porces derivable from a Potential ? Question of Current Perpendicular to Earth's Surface. 105. The question whether the magnetic intensity external to the earth's surface is derivable from a potential, as the theory of Gauss supposes, has engaged the attention of several investigators, among others Schuster, Schmidt, von Bezold, Rucker, and V. Carlheim-Gyllen- skiold. The matter was dealt with by Schmidt as follows.^ It has been shown by Gauss ^ that Y can be developed in a series of the form ? + rcos\+Z'sinA,+ . . . . Hence if the line integral of ^T round the earth along a parallel of latitude vanishes we must have 2Trl sin u = 0. But this of course is (flx+2n- — iix)/-5. As far as available data would allow this line integral has been computed by Schmidt for the epoch ] 885, and for every fifth parallel of latitude from 60° N. to 60° S. As a result it has been found that 2'7rfein It is at latitude 45° N. or S., apparently about 8 per cent, of the difference between the maximum and minimum values of 0/i2 for the parallel of latitude in question. At the equator it is less than 1 per cent, of this difference. Nearer the poles than 45° N. or S. it seems again to diminish. The question has also been tested by EtLcker, and by V. Carlheim- Gyllenskiold, who have determined the value of the line integral of magnetic intensity round closed circuits in regions for which exact 1 Abh. d. K. B. AkcuJ,. II. CI. xix. Bd. i. 1895. '■^ Gauss u. Weber, SesuUate u.s.w, im Jahre 1838. Ill TERRESTRIAL MAGNETISM 7T magnetic surveys have been naade, but without obtaining for any circuit a value sensibly different from zero. The same thing has been done also by von Bezold ^ for a trapezium bounded by the meridians 4° W. and 34° E., and the parallels of latitude 35° N. and 65° N., using numbers given by Neumayer in the collection of tables published by Landolt and Bornstein; but again the result has been a nearly zero value of the integral. If these line integrals were of sensible amount we should, as we shall see later, be forced to the conclusion that electric currents flow between external space and the surface enclosed by the path of integration. That the theory of Gauss is approximately correct seems certain, but its exactness can only be settled when accurate observations made at a greater number of stations are available. Also it would be of great interest to have for all the magnetic obser- vations in Europe, observations taken simultaneously, especially at some instant during a magnetic storm as suggested by Eschenhagen.^ It has also been suggested by von Bezold that electric currents may exist over inland seas and lakes, and that it might be worth while to obtain accurate values of the line integral round the Black Sea and the Caspian. The Magnetic Elements 106. The horizontal force, the dip (or inclination), and the variation (or declination) are called the magnetic elements, and are regularly observed and recorded at magnetic observatories. The methods of determining them will be fully discussed in the chapter on Magnetic Measurements. We shall here only deal with the distribution of their values over the surface of the earth, and their secular and other changes. Lines of equal horizontal force drawn on a Mercator's chart of the earth's surface are shown in PI. II., at the end of this volume. Lines of equal vertical force in PI. III., and lines of equal variation in PI. IV. These charts are taken from the Admiralty Manual of Deviations of the Compass, 1893 edition. The numbers at the right- hand side of the diagram opposite the corresponding lines are the values of the horizontal forces in terms of that at London taken as unity. To reduce to C.G.S. units they must be multiplied by "1832. By an inaccurate analogy the lines of equal magnetic dip have been called lines of magnetic latitude. The term however more properly belongs to lines of equal magnetic potential, since these are cut at right angles by the magnetic meridians, or lines of horizontal force. The chart of lines of equal magnetic variation is of great use to the navigator, as it enables him in the absence of sights of sun or stars, when he knows his approximate locality, to tell the direction in which ^ Zwr Th. d. Erdmag. Math. u. Saturw. Mitth. K. Phy. Akad. 2., Berlin. April, 1897. ^ L. A. Bauer, Terr. Magnetism. April, 1896. 72 ELECTRICITY AND MAGNETISM chap. his compass needle (supposed freed from the effect of the magnetism of the ship) actually points. The chart of lines of equal total magnetic force, shown in PI. I. at the end, has some interesting features. It will be seen that in both the Northern and Southern hemispheres, the lines bend in towards one another in the middle on the two sides, until the two sides meet to form a two-looped or figure of 8 curve. Then within each loop the lines become .small closed curves surrounding points of maximum or minimum total force. These points, it is to be observed, do not coincide with the magnetic poles, but fulfil an entirely different definition, namely, that of being points of maximum or minimum total force ; while a magnetic pole is a place at which the horizontal component of the magnetic force vanishes. There are of course irregularities in these curves due to local ■disturbing forces which are hardly taken account of in the charts. For example in parts of the British Islands where there are large masses of basalt in the underground strata, there are deflections of the north pointing pole towards such places from distances of as much as fifty miles. There seems little doubt that local deviations of the compass needle oa board ship have been thus produced. It is possible, as pointed out by Rticker {Rede Lecture, Nature, Dec. 23, 1897), that the surface strata containing magnetic matter are indications of much gre,ater masses of similar rock beneath, from which the strata have been extruded. Again there is an attraction of the north pointing pole towards places where rocks of the carboniferous or precarboniferous period have been thrust out through the newer strata, as at various coalfields. Thus a ridge line of magnetic matter runs along the Pennine Range, the so-called " Backbone of England," and another runs from Nuneaton to Dudley, and then south to Reading. Thence lines run in different directions ; one of these passes southwards to the Channel at Chichester, and appears again in France near Dieppe, whence it has been traced by M. Moureaux to a point fifty miles south of Paris (See Riicker, loc. cit.). Some remarkable irregularities have lately been observed by M. Moureaux in South Russia. At a village called Kotchetovka, in lat. 57° N, and long. 6° 8' east of Poulkowa, the extreme values of the elements at fifteen stations scattered over an area of about a square kilometre were Declination -|- 58° to - 43°, Dip 79° to 48°, and Hori- zontal Intensity "166 to '589 C.G.S. Thus the horizontal intensity is actually greater at some stations than at the equator, and since the dip does not fall below 48° there must be at such places an extremely high value of the total intensity. Time Changes of the Magnetic Elements and their Causes 107. Perhaps the most mysterious and perplexing of the phenomena of terrestrial magnetism are the secular and other changes which they undergo. Chief of these is the secular periodic change in the variation. An unbroken series of observations of the variation at Paris made nearly Ill TERRESTRIAL MAGNETISM 73 every year from 1680 to the present time, and at longer intervals for some time previously, shows that in 1666 the variation was zero there, and acquiring a westerly value. This westerly value continued to grow up until in 1814 it attained a maximum of 22° 34', when the needle began again to turn toward the east. In 1892 it was 15° 26'"9 W. The same changes have been going on throughout Europe. In 1659 the compass pointed north at London, and was gradually moving round towards the west. The westerly variation continued to increase until in 1823 it had attained a maximum of 24° 30', and had begun to diminish. In 1894 it was 17° 23' W. at Kew. While this change in the variation has been going on, the mag- netic dip at London has no doubt been gradually diminishing. At the present time the decrease is about 1''5 annually. This gradual diminution of the dip has been carefully observed at Paris. In 1671 the dip was about 75°, in 1776 it was 72° 25', in 1876 65° 37', and in 1892 it was 65° 9''2. Along with this diminution of the dip has taken place, as was to be expected, an increase' in the horizontal magnetic force. According to Thorpe and Riicker,^ there is a mean yearly increase of about '00022 C.G.S. for England, "00018 for Scotland, and "00020 for Ireland. It appears at first sight as if the whole terrestrial magnetic system were slowly turning round the earth's axis in a period of nearly 1000 years,^ and in the direction opposed to the earth's rotation. In a model made by Mr. Henry Wilde one of two sets of currents contained within a globe, which represents the earth, rotates round the polar axis and produces changes in fair agreement with those observed at several places.^ There are however great difficulties in the way of accepting any such theory as that here indicated, 108. The secular changes in variation and dip have been graphically represented by L. A. Bauer * by drawing the curve which the north pointing end of a magnet freely suspended at its centre of gravity would to an observer situated at the centre of the magnet seem to describe. Thus the curve shows the change of dip as well as that of declination. The curve for London, from 1540 to 1890, is shown LONDON ^^ ^^S- ^^> ^^^ ^°^ comparison, the curves for London, Rome, Ascension j'jc,_ 33, Island, and Cape Town, places within a range of longitude of about 30°, but of latitudes varying from 52° N. to 35° S., are shown at the end of this volume in Plate V. It will be seen that the direction of motion of I Magnetic Survey of the British Isles. Phil. Trans., 1890. ^ See Lord Kelvin's Popular Lectures and Addresses, Vol. III. p. 25. 3 Proe. S,.S. June 19, 1890. See also Riicker, B.A. Address, 1894. * Beitr. z. Kenntn. d. Wesens d. Hdcidar Variation d. Erdmagn. (Inaug. Dissert, Univ. of Berlin.) 74 MAGNETISM AND ELECTRICITY CHAP. OCCLINATION the pole is clockwise, and this is found to be the case at nearly all the places for which observations have been recorded. From the dates marked on the curve it will be seen that the speed of the pole in the secular orbit is not constant. Let us suppose, as was done by Mr. Wilde, that the changes are due to the superposition on a perfectly constant magnetic system, rotating with the earth, of a second magnetic system, also turning with the earth but at the same time describing a secular orbit round the earth's axis. If a needle could be suspended so that the earth turned beneath it, successive nearly iden- tical sets of daily changes of the position of the needle would be observed,, and each of these might be represented by a curve like that just described. The curve thus obtained for a 1780 ■■^^" single day ought, if the second system were 1885 '•^— invariable and its successive positions were symmetrical about the earth's axis, to agree with the secular variation curve so far as known for past time for the latitude of ob- servation, and could then be used to predict the remainder of the secular curve. The daily curves would be exactly re- peated, but the longitudes associated with various positions of the pole of the needle would continuously change. Without sym- metry of position and constancy of the disturbing system the curves would slowly change in form as well, which Fig. 34 and PI. VI. show is to some extent the case. Fig. 34 shows such curves drawn by Bauer for the epochs 1780, 1885, and a place in latitude 40° N., from recorded observations. Fig. 34. A comparison of curves for latitudes 40° N., 0°, and 40° S. is given in PI. VI.' The longitudes for dififerent positions of the needle are marked on the curves. It will be seen from Pis. V. and VI. that these diurnal curves resemble the secular curves in form and in respect of their difference of size for equal north and south latitudes. The hypothesis must however still be regarded as very uncertain. 109. Besides these secular changes there are changes of comparatively short periods both in the variation and in the other magnetic elements. The annual change in variation in the northern hemisphere is easterly from May to October, and westerly during the remaining six months, with a range at Greenwich of 2' 25". The maximum easterly deviation from the mean takes place in August, the maximum westerly in February. In the southern hemisphere the changes are simultaneous- but in the opposite direction. The daily march of the variation at Greenwich is shown in Fig. 35> LATITUDE 40° N. HI TERRESTRIAL MAGNETISM 75 The straight line running along the diagram and marked denotes the mean position of the needle ; the curve shows the deviation from the mean positions at the hours marked along the top of the figure beginning at midnight. It will be seen that at midnight the needle is I'J to the 2 14 le 18 20 22 2 A e 8 [0 1! \, . ^ \ > -^ ■^ V y \ / \ / Vi / Fig. 35. east of its mean position, and remains there until about 2 A.M. when it begins to move still further towards the east, reaching a maximum devia- tion of 4' about 8 A.M. Then it moves rapidly towards the west and attains a maximum westerly deviation at 1 P.M., after which it returns through zero towards the initial deviation of 1'^ east, reaching it about midnight. The cause of this daily change of variation is of course the changes which take place in the forces towards the geographical north and west. The changes in the latter during twenty-four hours are shown for Greenwich and Lisbon in the curves of Figs. 36 and 37, which are taken from Professor Schuster's recent paper on the Diurnal Varia- 4U0 320 -~" — "~- ;/ "^ ISO 80 / \ ' ^ \ *S£ -< IGO s / 1 s / 320 400 12 14 16 IS 20 22N00rl2 4 6 8 10 12 greenwich Fig. 36. 320 ^ "> . 160 80 ' /'' \ j ' \ ji \ N. 1 1 ^ 160 240 320 ^i i ' ^; / «n= 100, 2 1 0. 1 S 1 a 2 2 L 2 NO ISE ONE 301 . A £ ■ i S 1 a 12 Fig. 37. change is a tion of Terrestrial Magnetism.^ It will be seen that the combination mainly of a diurnal and a semidiurnal constituent. The dotted curves in the figures were drawn by Professor Schuster from results tabulated from the variable part of the magnetic potential. 1 Fhil. Trans. M.S., Part A., 1889. 76 MAGNETIS^I AND ELECTEICITY CHAP. as found by him after the manner of Gauss, from observations of mag- netic force at different parts of the earth's surface. It will be seen that the two sets of curves in each case very nearly coincide. 110. Professor Schuster also investigated the changes in the vertical force. His results for Greenwich and Lisbon are given in Figs. 38 and 39, and illustrate fairly his results. The object of the research was to decide whether the cause of the diurnal variation of terrestrial magnetism was external or internal to the earth's surface. The answer obtained was that the exciting cause is external to the earth, but is accompanied 400 480 •'s, / \ \ ,- \ I ■■ ... "'>/ \ / \ \ "^ / s s / ^\ / \ ! y' '< ! \ , \ / \ '^ V / soo 720 640 560 480 400 320 240 16 \ \ / \ / \ / ,^ / ^ \ \ ; ,- j \ ■/ •N \ '. / 1; / \ \ ■■^ / ^;; 1 \ / N ' \ /.'I \ I A / ' .' \ / ', ^ J ; \ / ', \ / '. / \ / / ../ '.,. 12 14 16 18 20 22 NOON 2 4 GREENWICH Fig. 38. 6 8 10 12 14 16 18 20 22NO0N2 LISBON Fig, 39. 4 6 8 10 12 Comparison between calculated and observed curve of vertical force. Abscissae denote astronomical time, ordinates denote vertical forces, to unit 10-' C.G.S. The full line is the observed curve ; the dotted line is the calculated curve on the hypothesis of out- side force ; the chain dotted line is the curve calculated on the hypothesis of internal force. bj' an internal source of change to which it stands in fixed relationship. This is shown very well in Figs. 38 and 39. The full curves show the observed variation of vertical force, the lightly dotted curves the changes of vertical force on the supposition of an external cause, the chain dotted curves the same thing on the supposition that the cause is in- ternal. It will be recognized at once that in each case (and the same holds for curves given in the paper for Bombay and St. Petersburg) the curve calculated on the former supposition agrees in character with the observed while the other calculated curves do not. The difference in' the amounts of the observed and calculated effects is due to the internal agency already alluded to as connected with the external cause Ill TEREESTEIAL MAGNETISM 77 of change. Such an internal source of periodic change of magnetic force was of course to be expected : for such changes must necessarily give rise to induced currents within the substance of the earth which will react on the magnetic field intensities. 111. The currents induced in a conducting sphere by periodic changes in magnetic potential are investigated by Professor Horace Lamb in an appendix to Professor Schuster's paper. The theory shows that if the whole earth were of good conducting material, the effect of the induced currents would tend to equality with the primary effect, and the phase of the resultant would approximate to a difference of 45° from that of the primary disturbance. The results given ia the curves show that there is good agreement of phase between the actual effect and the computed primary effect r so that the secondary action of the induced currents is only effective in reducing the amplitude. It has been suggested by Professor Lamb that this result would be produced if the currents circulated mainly in the internal parts of the earth, and were only slight in comparison in the outer strata. This is very likely to be the case, as the high internal temperature will undoubtedly have the effect of reducing the resistance of many of the materials of which the earth is composed. Besides reducing the amplitude of the variable part of the vertical force, the currents reduced in the earth's substance must be effective in increasing the amplitude of the periodic changes of horizontal force. 112. The following is Professor Schuster's own summary of the results of his investigation. " 1. The principal part of the diurnal variation is due to causes outside the earth's surface, and probably to electric currents in our atmosphere. " 2. Currents are induced in the earth by the diurnal variation, which produce a sensible effect chiefly in reducing the amplitude of the vertical component and increasing the amplitude of the horizontal components. " 3. As regards the currents produced by the diurnal variation, the earth does not behave as a uniformly conducting sphere, but the upper layers must conduct less than the inner layers. "4. The horizontal movements in the atmosphere which must accompany a tidal action of the sun or moon, or any periodic variation of the barometer such as is actually observed, would produce electric currents in the atmosphere having magnetic effects similar in character to the observed daily variation. " 5. If the variation is actually produced by the suggested cause the atmosphere must be in that sensitive state in which, according to the author's experiments, there is no lower limit to the electromotive force producing a current." Diurnal Changes of Intensity at Points on same Parallel of Latitude. Representation by Vector-Diagrams 113. The daily variations in terrestrial magnetism have been dis- cussed to some extent recently in connection with Schuster's theory by 78 MAGNETISM AND ELECTRICITY CHAP. V. Bezold. This writer first deals with the assumption that the mean normal daily changes, as they are obtained from days free from irregular disturbances, are equivalent to a system of magnetic forces which may be regarded as revolving round the earth, without alteration of itself in a period of twenty-four hours ; then he offers some observations on the desirability and method of testing whether this system of forces is derivable from a potential function. According to the assumption referred to, if Xg,, Yd,Za be the com- ponent magnetic intensities which express the daily variations, each of them must be a periodic function of \, that is of the local time of the place at which observations are made. Thus if t be that local time, t^ the corresponding time at Greenwich, and T the period of revolution (twenty-four hours expressed in terms of the chosen unit of time) we have Xd=/xlM, y(«-«o)}- with similar equations for Yd,Za. To compare this result with observation, the diagram first used for the representation of X^.Ta, by Airy is of great importance. This V. Bezold calls a vector-diagram. In it the change of the horizontal force in direction and magnitude is laid down for each hour of the day, by drawing straight lines from an origin (see Fig. 40), so that each line represents by its length the magnitude of \IX£'-\- Yi, and makes an angle tan -'^Y^IXa with the axis of the Y forces. The extremities of these lines or vectors lie on a closed curve from which the amount of change fur any interval during the day can be at once obtained. The successive hours are marked round the curve, and of course occur at different intervals de- pending on the varying rate of change during the day. Clearly the radii-vectores in such a \\ J/^ diagram are the directions in which a \ /f horizontally Suspended needle freed from the mean action of the earth would suc- cessively place itself in consequence of the diurnal change of horizontal inten- sity. Airy and Lloyd, who both used this •diagram, took as components of the change those along and per- pendicular to the magnetic meridian. For the present purpose this is not so convenient, and the axes are taken as indicated by the component Xa along the astronomical meridian at any place, and Y^ along the parallel of latitude there. When this is done it is found that the supposition as to its being the same succession of changes that occur at TERRESTRIAL MAGNETISM 79 ;all places along a parallel of latitude is confirmed. The vector-diagram at all such places is the same for the same local time at each. Fig. 40 is the vector-diagram for lat. 60° N., and the diagrams for latitiides differing successively by 20°, from 80° N. to 80° S., are shown in PI. VII. as obtained for the summer months of the northern hemi- sphere. These diagrams are taken from v. Bezold's paper, as deduced by him from Schuster's values of the potential. It will be seen that for .a place in lat. 60° N., where the magnetic meridian is astronomically north and south, the daily change in H, the magnetic intensity along the magnetic meridian, has its maxima at noon and 9 p.m., while the other ■component has its maxima at about 6 a.m. and 3 p.m. The component along OX is zero at the times corresponding to the intersections of the axis YOY with the edge of the curve, that is at 6 p.m. and a little before 4 p.m. 50 w. 35 E. 50 W. 35 E. Fig. 41.. To find the components along and perpendicular to the magnetic meridian at any other place, it is only necessary to draw the meridian on the diagram, making the proper angle with the axis of X. Thus let MM be the magnetic meridian. The maximum daily variations along MM, that is the variations in H, are obtained by drawing tangents to the vector-diagram at right angles to MM. Thus on the diagram they occur about 9 a.m. and 6 p.m. The components along MM are zero for the times corresponding to the intersection of ,a line perpendicular to MM with the curve, that is for Fig. 40 about 3.30 a.m. and 2 p.m. The maximum magnetic east and west variations of intensity are obtained by drawing tangents to the curve parallel to MM, and accord- ingly have place about midday and 9 p.m. 80 MAGNETISM AND ELECTRICITY chap. It is clear that from this curve the march of the change in H, that is of the component along MM, can be obtained and compared with that observed. Thus Fig. 41 gives such derived curves for the latitude 60° N. and the magnetic meridians for which the declinations are 50° W. and 0°. The vector-diagram, however, shows the whole phenomenon of which the periodic curves in this figure show only part. As that has different forms for places of different values of the mean declination its graphical representation does not clearly display the progressive march of the changes along the parallel of latitude. Diurnal Changes in Different Latitudes. Difference of Amount in Summer and Winter. Results of Observations on " Quiet Days " 114. An examination of PL VII. brings some curious facts to view. First there is the great difference between the summer and the winter results as regards the daily changes, and this difference as it alternates with summer and winter from hemisphere to hemisphere, suggests a cause connected with meteorological change. The difference would have been still more marked if the results had not been a mean for the half year, but had been taken for the time of the summer solstice, or for the months of June and July. It would be well also to obtain results for the winter solstice, or the month of December, and likewise for the equinoctial months. Another point of considerable interest is the comparative smallness of the north and south components of the daily change of magnetic intensity at 40° N. and S. Above and below that latitude in both hemi- spheres the two components give an irregular oval curve. The direction of rotation of the radius-vector round the diagram is negative (or clock- wise) in higher northern latitudes than 40°, positive in latitudes between 40° N., and some parallel south of the equator, thea again negative to lat. 40° S., and finally once more apparently positive in higher southern latitudes. 115. It is clear that for a more complete discussion of the daily variations of terrestrial magnetism a set of careful observations at places on parallels of latitude every 5° or 10° apart would be of great value; and though the daily progression of the phenomena along each parallel has been established, the degree of exactness of this repetition of the phenomena as \ varies would be tested by observations made for the same interval of time at different places on each of a sufficient number of parallels. The results of such observations, so far as regards horizontal force, should be for ease of interpretation represented by means of vector-diagrams, and to exhibit the variations in dip as well as in dechnation, Bauer's mode of representation (Art. 109) should be employed. Observations are now made of the daily changes at English Obser- vatories on five days of each month selected by the Astronomer-Royal. These days are chosen for their freedom from irregular disturbances, and in TERRESTRIAL MAGNETISM 81 aa-e therefore called " quiet days." It has been found however by Chree {B. A. Beps. 1895, 1896) that when the changes of direction of the needle during a solai- day are represented in the manner suggested by Bauer (Ai-t. 108 above) the needle does not come back at the end of the day to its original direction, that is the path is not closed. It is to be remembered that the path of the pole of the needle while showing the diurnal change must be affected by the secular change, which would carry the pole in the opposite direction, and so prevent the path from being closed. But this is not sufficient to account for the whole effect, and thus it appears that on the quiet days the effect of magnetic storms is to check the secular motion of the needle. Qaestion of Derivation of Diurnal Changes of Intensity from a Potential 116. With regard to the derivations of J[^, Y^, Z^, from a potential flrf, it is obvious of course that the daily revolution of the changes round the parallels of latitude makes Xl^ a periodic function of period T, that is : — Q.i = F{\, u, t^ + uT), if change of the revolving value of O^ with longitude, is permitted, and Art = f{u, X + ^) if the revolving system is invariable. The vanishing of / Yg, dt, that is of the line integral of Yg, round a parallel of latitude is necessary, but is not sufficient to establish that Y^ is derivable from a potential function. To settle the question of the existence of a potential, what is needed is an evaluation of the line integral round a sufficient number of closed paths of other forms, drawn on the surface of the earth. We have already given above the expression for the line integTal round a path consisting of two arcs of meridians joined by parallels of latitude, and found (equations (7'), (T") above) what the expression becomes when (1) the distance along the meridians is made infinitesimal, (2) when the distance along the parallels is also made infinitesimal. These formulas could easily be testedatplacesatwhichthedaily changes are observed, and the question settled. For this purpose simultaneous observations at a number of places properly placed would as in the case of the ordinary more slowly changing intensity be of great value. Eqnipotential Lines of Diurnal Changes 117. Von Bezold has also thrown Schuster's recalculated values of the potential Xld into a system of lines of equal potential drawn on a Mercator's projection of the earth's surface. "This chart is reproduced as G 82 MAGNETISM AND ELECTRICITY chap. PI. VIII. at the end of the present volume. It will be seen that these lines show four well marked poles of the system of daily magnetic intensities for noon at Greenwich. The chain-dotted line shows the boundaiy between the illuminated and non-illuminated pai-ts of the earth's surface, and it is to be observed that one of the poles lies in lat. 40° N. on the meridian of the place for which the non-circle is 11 h. 20 m., or thereabouts, while an opposite pole, in the same latitude nearly, is west of Greenwich almost on the chain-dotted line. The two other poles lie on the south of the equator, one of opposite sign to that first mentioned nearly on the same longitude and in about 30° south latitude; the fourth lies about 10° further south, 7 h. 30 m. west of Greenwich, or nearly in the same longitude as the second mentioned pole, and is of the opposite sign to the latter. This system of lines of course travels round the earth in a solar day. Von Bezold points out that these poles cannot be produced by a system of currents inside the earth, though of course the magnetic intensities may be modified by such a system. This he shows by considering the vertical and horizontal forces, and pointing out that an internal system, which accounted for the observed vertical component^ would give horizontal forces opposite to those observed, and vice mrsd. The difference between the phenomena in the summer and winter hemispheres is here again brought out very markedly, and suggests as before that the daily changes are greatly modified by meteorological influences. It is suggested by v. Bezold that currents depending on the great atmospheric currents, the cyclones, and the anticyclones, would produce these poles in or near 40° of north and south latitude. Magnetic Storms 118. Besides these regularly periodic variations in the magnetic elements there are irregular and violent changes which take place from time to time, and are well described as magnetic storms. Such dis^ turbances have been found to be simultaneous, often with auroral displays, a fact which seems to point to electric discharge in the upper regions of the atmosphere as a cause of magnetic storms, as well as of the regular changes. It has also been observed, though the correspon- dence has not been unmistakably made out, that magnetic storms seem to be more frequent at times of great solar activity as shown by the outburst of sun-spots. Hence some physicists have been led to credit magnetic action exerted by the sun with such magnetic disturb- ances and also with the annual and diurnal variations of the magnetic elements. In this theory there is at the outset a very serious difficulty. It may be tme that the sun is a powerful permanent or electro-magnet, exerting a steady effect, but to produce suddenly changes of the mag- netic forces experienced on the earth, comparable with the total amount Ill TEREESTEUL MAGNETISM 83 of these forces, this magnet must be subject to enormous changes of intensity rapidly produced and rapidly dying away. The subject has been discussed recently by Lord Kelvin in his Presidential Address (St. Andrew's Day, 1892) to the Royal Society, and the conclusion arrived at is adverse to the hypothesis of. a direct solar origin of these magnetic changes. The following is a short summary of his treatment of the subject. (See also Lloyd, Magnetism, p. 233, and G. J. Stoney, Phil. Mag., Oct., 1861.) 119. Supposing the sun magnetized to the same mean intensity a& is the earth, the magnetic force produced by it at an external point can easily be calculated. Let the direction of magnetization be at right angles to the plane of the ecliptic, which it roughly is no doubt, if the sun's magnetization is connected with its rotation, as Lord Kelvin thinks is the case with the earth. The force then at a distance D from the sun at a point in the ecliptic is ^irB^lj D^, if I be the intensity of magnetization, that is the magnetic moment per unit of volume, and R be the radius of the sun. If D be equal to the earth's distance from the sun so that B^ = 228^ iJ^, we get for the force the vg,lue |7rI/228^. The force at the earth's surface along a meridian at the equator is f ttI. Hence the force produced by the sun would be about l/228'3, or 1/12,000,000, of the earth's force. It is clear therefore that the sun must be something like 12,000 times as intensely magnetized as the earth in order to produce a force perceptible with ordinary instruments used in a magnetic laboratory. This according to the estimate given above (Art. 102) would be about the intensity of magnetization of well magnetized hard steel. Since the moon's apparent diameter is approximately the same as the sun's, the ratio of the moon's radius to the distance of the earth from the moon is nearly equal to the corresponding ratio for the sun, and so the estimate just made for the sun holds also for the moon. 120. Now considering the earth at the vernal equinox, the magnetic force due to the sun may be resolved into two components, one, '92 of the whole amount, parallel to the earth's axis, the other, '4 of the whole, at right angles to the axis. The former contains a constant part, and a part varying with the sun's distance and having therefore a period of a year. The latter component would affect the variation, supposing the magnetisms in the northern and southern hemispheres of the sun to be of the same sign as those on the corresponding hemi- spheres of the earth, as if the south- pointing ends of needles on the earth were attracted towards a star in the plane of the equator and at 270° Eight Ascension. This would give rise to a change in the declina- tion of period one sidereal day. This diurnal constituent, it is to be observed, is distinct from that discussed by Schuster, which had for period a solar day, and is smaller in amount than the latter. So far it has not been sought for by the harmonic analysis ; but even if a change in the declination were found, it would not follow that it was due to the action of the sun as a magnet. G 2 84 MAGNETISM AND ELECTRICITY chap, iii The whole effect might be due to effects of currents circulating in the earth's atmosphere, and therefore depending on the period of rotation, and moreover containing solar diurnal terms depending on difference of temperature produced by the sun's radiation. 121. But the chief objection urged by Lord Keh'in to solar agency as the cause of magnetic storms is the enormous expenditure of electri- cal energy necessary to produce by the action of the sun the oscillations of magnetic force observed in a magnetic storm. Thus in a magnetic storm which took place on June 25, 1885, the horizontal force at the following eleven places: St. Petersburg, Stonyhurst, Wilhelmshaven, Utrecht, Kew, Vienna, Lisbon, San Fernando, Colaba, Batavia, and Melbourne, increased considerably from 2 to 2.10 p.m., and fell from 2.10 to 3 P.M. with irregular changes in the interval. The mean value at all these places was '0005 above par at 2.10 and "005 below par at 3 p.m. The changes as shown by the photographic records were simultaneous at the different places. Assuming these electrical oscillations of the sun. Lord Kelvin estimates that the electrical activity of the sun during the storm, which lasted about eight hours, must have been about 160x10^* horse power, or about 12x10'° ergs per second ; that is about 364 times the activity of the total solar radiation, which is estimated at about 3x10'^ ergs per second. The electrical energy thus given out by the sun in such a storm would supply, if transformed to the electrical vibrations of shorter period con- cerned in its ordinary radiation, the whole light and heat radiated during a period of four months. This, as Lord Kelvin remarks, is con- clusive against the hypothesis that these violent magnetic disturbances are due to direct action of the sun. It is of course conceivable that the earth might be immersed in a ray of abnormally great solar energy ; but it is extremely unlikely that unless the radiation were abnormally great all round, the earth would remain in such a ray for a period of several hours. The cause of magnetic storms therefore remains undiscovered. But it is only one of the mysteries of terrestrial magnetism, for the great fact of the earth's magnetism itself, not to speak of all its wonderful periodic changes, secular, annular, and diurnal, remains unaccounted for by any satisfactory theory. CHAPTER IV MAGNETISM OF AN IRON SHIP AND COMPENSATION OF THE COMPASS Ship's Magnetism 122. It is not possible to discuss fully here the magnetism of an iron ship, and the compensation of a mariner's compass for use on board such a vessel; but these subjects are so important, both practically and from the point of view of pure science, that it seems desirable to give a short account of them. The magnetization of an iron ship is (1) permanent or sub-permanent, and (2) transient. The permanent magnetization is independent of the change of place of the ship or the lapse of time, except in so far as the iron is subject to strains or shocks due to unusual stress of weather or other cause. It is set up mainly by terrestrial magnetizing force during the building of the ship, and its distribution depends on the position of the ship while being built. The sub-permanent magnetism slowly changes with time, and its variations show effects only observable after the lapse of intervals of days or even weeks. The transient magnetization is induced by the varying magnetic force to which the iron is subject, and follows the changes of these forces, caused by the varying position of the ship. The deviation of the compass needle in a ship is the angle which its direction makes with that which it would have if the needle were under the earth's force alone. It may be regarded as made up mainly of two parts, called the semicircular deviation or error, and the quadrantal deviation or error. How these arise and the manner of their compensa- tion will form the first part of the present discussion. 123. The ship forms a large magnet or rather combination of magnets. It has (a) longitudinal magnetization, (&) transverse magnetization, and (c) vertical magnetization. The first must obviously exist, as the ship's- hull is a great iron or steel girder, with its length in the fore and aft direction ; the second is principally the longitudinal magnetization of the transverse beams and bulkheads ; the third component is mainly due to the ribs and vertical beams and bulkheads. These produce magnetic 86 MAGNETISM AND ELECTRICITY CHAP. forces within the ship, to which must be added also those produced by the magnetization of iron or steel masts or spars, or of iron or steel carried as cargo. Soft Iron of Ship Represented by Iron Rods 124. The magnetic forces at any point in the ship may be referred to as fixed in the ship, and the action on the compass needle for any position of the ship's head ascertained. Let the axes be drawn from the centre of the compass needle as origin, that of x from the stern to the head of the vessel, that of y from port to starboard, and that of z from the compass needle downwards, as shown in Fig. 42. Also let the compon- ents of magnetic force due to the earth be denoted by X, Y, Z, the forces due to the permanent magnetism by P, Q, R, and the total components by X', Y', Z'. The earth's force will magnetize the iron of the vessel tem- FlG. 42. porarily, and every one of the three components X, Y, Z, will produce its own part of each of the three total components JC, Y', Z'. For consider the first component X. It will produce at the compass needle, in con- sequence of the induced magnetization, a force which experience shows may be taken as proportional to X, and which may therefore be denoted by aX, where a is a constant. If a be positive this force will produce the same effect upon the needle as would a long soft iron rod placed in the ship in a fore and aft direction, beginning at a point in front of the needle and running towards the bow (as shown at a in Fig. 42), or begin- ning behind the needle and running towards the stern, and magnetized "by the force X. If a be negative, the force is equivalent to that which would be produced by a short rod beginning in front of the binnacle, passing under the binnacle, and ending behind. The same force X will produce, in consequence of unsymmetrical distribution of the iron of the ship about the longitudinal plane of IV MAGNETISM OF AN IRON SHIP 87 symmetry of figure (that is, the plane of the keel), an effect similar to that of a long bar beginning at a point, say, to port of the needle and running towards the stern, or to that of two bars, one situated as just specified, and the. other beginning at a point to starboard of the needle ■and running towards the bow {d in Fig. 42). This will give a force dJC in the direction of y, where d is positive. If the force dX due to this want of symmetry be negative, the rod or rods producing an equivalent eifect must be reversed in direction. Also there will similarly be produced a force gX in the direction of z, which will be equivalent to that produced by two rods, one beginning above {g of Fig. 42), the other below the needle, and running, the first towards the bow, the other towards the stern of the vessel, or vice versd, according as g is positive or negative. Of course, instead of the pair of rods here specified, either rod singly may be substituted. In like manner the forces produced by Y and Z can be accounted for by three iron rods thwart-ship, and three vertical, according to the schemes shown in Fig. 42. Thus we have the equations A" = A" + aX + bY+ cZ+ P \ r ^ Y + dX + eY+/Z+Q I .... (1) Z' = Z + gX + hY + kZ+ R ) where a, b, c ... are constants. These equations were first given by Poisson. We now proceed to consider the effect of the forces thus specified in producing deviations of the compass. Expressions for Total Deviation of the Compass. Analysis of Deviations 125. Let the ship be on even keel, and the direction of the head be towards a point at angular distance f eastward from the magnetic north, and ^ eastward from the north point as indicated by the com- pass. Then f shows the " magnetic course," f ' the " compass course." Clearly f — ^ i^ ^^^ deviation, B say, of the compass needle from the true magnetic north, that is, the compass error. fif ^be the total horizontal force of the earth, JET that in the ship, and i/r the dip, we have A' = ZTcos^, Y= - Hsia^, Z = ZTtani/r and A" = //' cos t, ¥' = - E' sin t, since the direction of the needle in the ship is that of IF. Thus from equations (1) we have H'cos^' = (I + a) H cos ^-bH sin ^ + cHtrnxj/ + F ■ ■ - (2) - H'sint; = (- sin^ + dcos^I^- eHsin^ +/HUinl, + Q (3) 88 MAGNETISM AND ELECTRICITY chap. Multiplying the first of these by sin f, the second by cos f, cadding and reducing we get -sin 8 = — r-^ + fctunxl, + — ^ sin^ + f/tani/r + — j cos^ + -2— sin2^+ -^-008 2^ . (4) Again, multiplying the first equation by cos f, the second by sin f, and subtracting we find ■H'' „,« + «/ P\ — cos 8 = 1 H ^j— + ( c tan \[f + — ) cos t, - (/tan^+|)sin^+^%os2^-^'^sin2f . . . (5) Denoting 1 + (a + 6)/2 by X, and writing \A = — - — •, XB = cteinxj/ + —, A.C = /tani/< + — , 2 ' II X2 „ a - e ,„ b + d XD = — , XE^ ^' we get A + i?sing + C'cosg + i)sin2g + .gcos2g ^■^ 1 + .Scos^ -Csin^ + Z)cos2^ - ^sin2^ . . . U which enables the deviation to be calculated in terms of the magnetic coiirse. Substituting f ' + S for ^, and sin S/ cos S for tan S in the last equation, and simplifying, we obtain sinS = ^cosS + .Bsint + Ccos^' + i)sin(2^' + 8) + jS-cos (2^' + 8) (7) or . . 5 sin t + C cos t + (^ + -D sin 2^' + .^^cos 2^) cos 8 ,., \ — D cos 2^ + ii sm 2^ The coefficient .E", it will be seen presently, is comparatively small, and we may neglect the term E sin 2^ in the denominator. If the denominator be then expanded, and S be not too great, so that we may put S = sin S, cos S = 1, we may write the above equation in the form 8 = ^1 + ^1 sin t + Cj cos ^ + D^ sin 2^ + E^ cos 2^ + F^ sin 3^' + G^ cos 3^ + «fec. . . . (9) The higher terms in the second line may generally be neglected. 126. This result could of course have been reached by noticing that if IV MAGNETISM OF AN IRON SHIP m the ship's head were turned round to successive points of the compass, the same deviation, S, of the needle virould recur every time the ship's head took a given direction, except in so far as the changes in tlie ship's magnetism lagged behind the change in the magnetic force. Hence 8 is a periodic function of ^', and is expressible in the manner indicated. The coefficients, A^, B.^, C^, &c., can be found from the preceding A,B, C, &c., by expanding the denominator (1 — Z* cos 2 ^)'"i and substituting for the powers of cos 2f' cosines of multiples of 2f'. Provided the compass error is not more than about 20°, which it hardly ever is, as the deviation is either less in itself, or is reduced to a smaller value by partial compensation, we may take only the first five terms of the series above, and put ^ = sin^j, B = sinB^il+\s,mD^, C = sinCj(l - |sinZ)j), D = wxD^, 2? = sin^j where A, B, G, &c., are the values given above. (See Admiralty Manual of Deviations of the Compass, 1893 edition, p. 132, et seq.) Considering the different terms which make up the deviation we see first that the constant term, A = (d — 6)/2X, is the effect of the unsym- metrical distribution of the soft iron of the ship with respect to the plane of the keel. This is the deviation when f ' = 0, that is, when the ship's head is on the north point as shown by the compass. The Quadrantal Error 127. The constants i and d enter also into the coefficient E, the term which gives the variable part of the deviation depending on this want of symmetry. It is to be noticed that since the distribution of the iron of the ship is very nearly symmetrical with respect to the plane referred to, the constants b and d are very small numerically, and there- fore the terms A^ + JS^ cos 2f ' are relatively small parts of the total deviation. The term D-^^ sin2 f ' depends on a and e, and represents the effect of the fore and aft, and thwart-ship components of induced magnetiza- tion on the corresponding components of disturbing force at the compass needle. Since the longitudinal effect is greater than the thwart-ship effect, (a — e)/2X, or D, is always positive. The pair of terms Bj^sm2t,' + ^iCos2^' make up what is called the quadrantal deviation. If we put in these i>i / Jn^^ + E-^^ = cos 20, E^ I sjBfVE} = sin 2<^, so that Z)i sin 2^ + E^ cos 2^' = jBf+~E} sin 2 (^ + <^), we see that this part of the deviation has a maximum when 2(f'-|-0) = 9O°, and again when 2(f '-|-^) = 360°-|-90°, that is, when 90 MAGNETISM AND ELECTRICITY chap. 5' + ^ = 45°, and f' + ^ = 225°. Hence if ^' + (f> is changed from to 360° (that is, since <^ is a small angle, when f' is changed through 360°) this part of the deviation has two equal maxima. It has also two minima of numerical value equal to the maxima, namely when 2(f + ^) = 270°, and when 2(5- ' + (/>) = 630°, that is for g-' + <^=135 °, and f ' + ^ = 31 5°. Thus a numerical maximum of amount, \/l>-^^ + E-^, or D.^ approximately, is obtained on four compass courses, successively a quadrant distant from one another. Hence the name "quadrantal error." This part of the compass error does not in general amount to more than from 5° to 10° in iron vessels, unless the compass has been badly placed, or soft iron carried as cargo is stowed too near the binnacle. The Semicircular Error 128. Considering now the remaining pair of terms, namely, 5j sin ^' + Cj cos ^' we see that, since approximately sin B, = B = ^- , sm C/j = (7 = ^-r — A A {■\jr being the dip), the coefficients of these terms depend partly on the effect produced by vertically induced magnetism in consequence of unsym- metrical arrangement of the soft iron of the ship, (1) with reference to the cross section of the hull at the compass needle, which gives the constant c, (2) with reference to the plane of the keel, which gives the constant /. The latter constant is small in comparison with c. The remaining parts of the coefficients i?^, G^, depend upon the per- manent magnetism of the ship, since 4" + f //tani/r = BJI, f + ^ Htanxj, = Off where ^ = sin B.^ C=sin Cj approximately. We shall see presently how these parts can be determined. The sum B^ sin t! + Cj cos 4' is what is called the semicircular deviation. For writing it in the form JB^TO^ Bin {I' + 4>') where cos <^' = 5i / V^i^ + (7i«, sin <^' = Cj / JB^^ + C^' we see that its numerical value is a maximum for 5'' + 0' = 90°, and f +^' = 270°. Thus when the compass course is changed through 360° IV MAGNETISM OF AN IRON SHIP 91 this part of the deviation attains a maximum numerical value on two courses 180° apart. The maximum semicircular error of the compass may amount to 30° or 40° in an armour-clad vessel, and to over 20° in an ordinary iron vessel. But as pointed out above, the total error can be kept down to less than 20° by partial compensation, in cases in which no attempt is made to completely annul it. -S'o'l- BUILT HEAD NORTH r NORTH BUILT HEAD EAST NORTH T Fig. 43. Graphical Representation of Deviations. Determination of the Coefficients. The Dygogrami 129. The semicircular and quadrantal errors for two ships, one built head north, the other built head east, are shown in Fig. 43. The successive points of the compass are laid down at equal distances apart along a straight line, and the errors on the various courses, as taken directly from the compass which is being observed, are shown by ' This somewhat unintelligible-looking word is a contraction of dynamogoniogram, that is, the diagram of force and of angle. The curve is well known as the Lima^on of Pascal. 92 MAGNETISM AND ELECTEICITY CHAP. ordinates dl•a^^^l at right angles to this line. The quadrantal errors are the same in both cases, and have a maximum of 5° on the N.E., S.E., S.W., and N.W. courses. This is shown by the diagram on the left. The other two diagrams give the semicircular errors, which are shown by the lighter full lines for the cases specified. It will be seen that the maximum semicircular error is here about 10°. The dark full lines in these diagrams show the resultant error obtained by compounding the quadrantal and semicircular errors. 130. Fig. 44 illustrates a mode of representing the errors graphically devised by the late Mr. J. R. Napier, F.R.S. Instead of laying off the deviations observed on given courses, at right angles to the line of courses, they are laid off along lines drawn at angles of 60° to this line. These lines are shown ^rj"'-- dotted in Fig. 44. Westerly deviations are laid off towards the left, easterly deviations towards the right of the line of courses. It is to be noticed that in drawing this diagram the courses taken are courses as shown by the compass on board, and are not correct magnetic courses. The convenience of Napier's diagram lies in this, that the curve of deviations having been thus drawn, the correct magnetic course to be laid down on the chart as corresponding to the compass course which has been steered or vice versd can be at once found. A series of full lines intersecting the dotted lines at angles of 60°, and cutting the line of courses at the same points are drawn on the diagram. The rule (reversed for the other problem) is then as follows : — ^s« > Take on the line of courses the compass course actually steered, then pass from that point parallel to the diagonal dotted lines to the curve, then from that point parallel to the diagonal full lines, back to the line of courses. The point arrived at is the true magnetic course to be used on the chart. This rule is obviously true. The distance from the starting point of the final point arrived at on the line of courses is equal to the deviation, since the three points are the vertices of an equilateral triangle. Fig. 44. The curves shown in Fig. 44 show the quadrantal and semicircular deviations, and the resultant error for H.M.S. Achilles,'^ a now obsolete ironclad. The maximum quad- rantal error is about 6°'9', the maximum semicircular error 21° 15'. 131. By determining the semicircular error on different compass courses the values of Bj^, Cj, can obviously be found to any necessary degree of accuracy. If the observations are made at positions of ' Elementary Manual of the Deviations of the Compass, Edition 1870. IV ■ MAGNETISM OF AN IRON SHIP 93 the ship sufficiently far apart in magnetic latitude, where the dip and the horizontal component of the earth's field intensity are known, the different parts of the exact coefficients B, G, can be calculated. For let B^, B^ be values of the former coefficient at two such places where Ey, H^, "^y yjr^ are the values of If and yjr. Then we have F G - + - ^1 tan i/^i = B^ H^ which give P G - + - ZTj tan 1/^2 = B^H^, P _ -H^ZTg (-5^ tan 1/^2 - B^t&'O-f^ i) A. H^t&rwp^ - jffjtani^j c B^H^ - B^H^ Similarly X ^j tan ^j^ - H^ tan yp^ A, J?2 ^^^ "/'2 ~ -^1 ^^^ i'l A, H-^ tan i/f j - H,^ tan i/fg (10) (11) (12) (13) The quantity \ (=! + (« + «)/2) is easily interpreted as the average value of S cos S/lf for all possible courses. To prove this we recall the equation 1 H' - — cosS = 1 + 5cos^- Csin^ + Decs 2^ - (7sin2^. \Bl Multiplying, both sides by d\^, and integrating from i, — 0, to 5"= Stt, we obtain 2rr COS 8 d^ = 2'7r, H' f 2ir 1 H'[ that is " 2.flr''^ ^'') which proves the proposition. Hence H\ = 5' cos 8 (15) where the bar denotes the mean value of the product. But H' is the component of the local horizontal force, and this is in the direction of the needle. Hence 5''cos S is the component horizontal magnetic force 94 MAGNETISM AND ELECTRICITY CHAP. in the direction of the head of the vessel. Thus HX denotes the mean value of this component for all possible directions of the vessel's head. ^^'e are unable here to go into particulars regarding the mode of observing compass deviations in actual cases. Full information will be found in the Admiralty Manual of Deviations of the Compass, and in treatises on Compass Adjustment.^ It must suffice to state that the different coefficients can be found by harmonic analysis if a sufficient number of values of 8 are obtained for different compass courses. 132. When the coefficients, A, B, C, &c., have been found, the deviations on different courses are capable of a very elegant graphical representation invented by Archibald Smith,^ to whom the complete working out of the theory of compass deviation is in great measure due. Draw first (Fig. 45) two lines op, or, to represent the magnetic north and the magnetic east directions. Then take a length op as great as is convenient to represent the value of \H; and from p take a length pa eastward or westward to represent a positive or negative value of A . \ir on the same scale, and a second length ae to represent H. Next draw from the point e a line ed northward or south- ward to represent D . \H, according as D is positive or negative, and a second length dh from d to represent B . \S. Lastly, from b draw to the east or west a line hn to represent G . XII according as it is positive Fig. 45. or negative. The angle nop is the deviation for f = 0, that is, when the ship's head is due north. For we have op + db + ed = \5'(1 + B + n), pa + bn -i- ae = \H{A + C + E). Thus if we take op as vinity, the first length represents 1 + B+D, the second A -{■ G + E. Now by equations (4) and (5) XH sinS --= A + C + E ^'''^ 1 + B + D (16) ■ See TraiU de la Rigulation et de la Compensation des Compos,, par A. Collet. Paris ; Challamel Ain^. ° Other forms of dygogram are also employed in practice. For their construction see the Admiralty Maniuil. IV MAGNETISM OF AN lEON SHIP 95 when ^= 0. Therefore for this value of i; tan no]} = tan 8g and II' To find tlie value of S for any other value of f, from a as centre and with ad as radius, describe a circle. Join nd and produce the line to cut the circle again in q. If now a slip of paper with straight edge be made of a length 2nd, and be placed in different positions so that the middle point of its straight edge always lies on the circle, and the edge passes through q, the extremities for each position will give two points on a curve, called by Archibald Smith the dygogram. Thus n, s are two positions, Jc, m other two, and so on. 133. Taking the angle kqn as representing f, we can easily show that the value of H'jXH is represented by ok, and the deviation for the corresponding magnetic course f by the angle pok. For each of the points n, k we have diametrically opposite points s, in, corresponding to course f + 180°, for which the dygogram gives the .deviation and mag- netic force. To prove these statements it is only necessary to observe that if /_ kqn = f, the projection of ok on the line op is op + projection of ad' + projection of dik. The second of these is the radius of the circle ad turned through 2^; hence its projection on op is the sum of the pro- jections of ae and ed on op, after each has been turned through an angle 2f in the same direction, Hence the projection is edcos2}^ — aesm 2f. Thus the projection is op + edcos, 2^— ae sin 2f -f- projection of d'k. But d'k is dn turned through the angle ^, and its projection on op is equal to the sum of the projection of the components db and hn when turned through the same angle, that is, is equal to dh cos f — hn sin ^. Thus the total projection sought is op + dh cos ^ - hn sin ^ + ed cos 2^ - ae sin 2^, which since op = \H, dh = B. XH, In = C. \H, ed = D. XH, ae = E. XIT, may be written ^^■(l + .Scos^ - Csin^ + i)cos2^ - ^sin2^). which we have seen (p. 88 above) is the value of H'cos S. Similarly it can be shown that the projection of ok upon or is \ff{A + ^sin^-t- Ccos^;-!- Z)sin2f -l-^cos2^), which we have seen is .ff'sin S. Hence L pok = 8, ok = H'. 96 MAGNETISM AND ELECTRICITY chap. Since angles at the circumference of a circle standing upon the same arc are equal, the angle ngh is equal to the angle dfd'. Hence since the angle nqk represents f, and the ship's head, when 5"= 0, is in the direction fd, when the compass course is f the direction is fd', and all the lines showing the direction of the ship's head pass through /. Thus the diagrammatic plan of the hull in Fig. 45 shows the position of the ship's head, d, d', the positions of the compass. It is important to remark that a kpowledge of the five exact coefficients A, B, G, B, E permits the dygogram to be traced. Then a single determination of the ratio ITjII of the horizontal force within the ship, at the place of the compass, to the value of the earth's horizontal force gives the value of XH, the mean value of the component of horizontal force in the direction of the ship's head. 134. There are also what are called sextantal and octantal errors which are sensible in compasses of which the needles are long, and are represented by the group of terms — Fsin 3^' + G cos 3^' + Ksin 4^' + L cos 4^. The two first terms are the sextantal error, and arise from too near approach of the needle to permanent magnets. It attains six equal numerical maxima, positive and negative alternately when the compass course is changed through 360°. These are at successive distances of 60° apart. The other two terms constitute the octantal error, so called because it passes through eight alternately positive and negative maxima, which occur at successive angular distances of 45°. These terms arise from the too near approach to the needle of pieces of soft iron, which are magnetized by and re-act on the needle. With modern compasses, such as that of Lord Kelvin, which have short needles, these terms are not of importance and may be neglected. The Heeling Error 135. Hitherto the ship has been supposed to be on even keel. When however she is heeled over to one side or the other the quantities a, I, e, &c., require modification. For the effect of the inclination is to raise one side of the ship and depress the other, hence altering generally the positions of the equivalent soft iron rods relatively to the compass. Supposing the ship heeled over through an angle i, the fore and "aft soft iron bar which gives a does not have its effect on the compass altered, but the pair of fore and aft bars on the port and starboard sides do not act in the same way as before. The bar formerly producing the effect dX along y has now the effect d cos i. X. On the other hand, the fore and aft soft iron bars above and below the compass which produced the effect gX are displaced by the heel, one to port the other to starboard of the compass, so that they produce a force in the IV MAGNETISM OF AN IRON SHIP 97 direction of y of amount - ^ sin i . JtT, supposing the heel is to starboard Thus we have to replace c? by a quantity Sj given by 'bi = d cos i - g sin i If the heel is in the opposite direction i is reckoned negative. In a similar manner it can be shown that the other quantities, h, c, &c., are replaced by those given in the following table : — bi = b cos i — c sin i, d-, = d cos i — g sin i | Ci = c cos i + b sin i, g, = g cos i -\- d sin i J 6; = e - (y + h) cos I sin i - (e — k) sin^ * fi=f+{c -h) cos i sin i - (/ + h) sin^ i /tj = A + (e - h) cos i sin i - {f + h) aiv? { ki = k+ {/ + h) cos i sin i + (e - A) sin^ i (17) (18) If i be so small that we can put sin i = i, cos i = 1 and neglect sin^ v and the iron be symmetrical about the fore and aft midship line, these give for the coefficients when the angle of keel is i, jr. = J' + - ( e - A - - j itan f = H + iJ. Then the deviation is S, = 8 + ^-2J^ + *Vcosr-^*cos2Z;' . . . (19) 136. The terms here added to S constitute the heeling error. The most important part is the term Ji cos t,', where j^ J(« - A- |)tan,/.= - (^Z> + ^ - l)tan./r (20) where /* = 1 + A + ^/^, and D has the value given above, namely l_(e+l)/X. The total heeling error can be written in the form in which it is convenient to consider it when discussing its correction : — 8j - S = J^cost + ^tsin2^ - ^icos2^' . . . (21) A A A smaller error due to pitching is not of sufficient importance to be taken into account. H 8 MAGNETISM AND ELECTRICITY chap. Compensation of the Compass 137. With regard to compensation of the compass a great deal might be said. We shall not here enter into any discussion of how the various coefficients of the expression for the deviation are determined but merely describe shortly the process followed in the adjustment of Fig. 46. Lord Kelvin's compass, which is now very generally adopted on board large iron ships in this country. The card of this compass is shown in Fig. 46. It consists of a paper ring, on which are marked the points and degrees in the ordinary manner, attached to a light rim of aluminium which keeps it in shape. Radial threads connect the ring to a central boss containing a sapphire cap, by which the compass is supported on an iridium point fixed below to the compass bowl. Below the card, strung like the steps of a rope ladder on two silk threads attached to the radial threads, are eight small magnets of glass-hard steel, which form the compass needle. These vary IV MAGNETISM OF AN IRON SHIP 99 in length from 3J inches to 2 inches, and are symmetrically arranged in the manner shown in the diagram. The entire weight of the card, including needles, is l70Jr grains. This extreme lightness, combined with the relatively great moment of inertia obtained by the distribution of mass, ensures a long period of free vibration and therefore great steadiness. It also enables the frictional error on the supporting point to be made very slight. 138. The semicircular error is corrected by placing steel magnets under the needle in the binnacle, so as to annul the error on the north and south and east and west courses, due to the two horizontal components of disturbing magnetic force, arising principally from the permanent mag- netism of the ship. In the correction of these errors two sets of per- manent magnets are used in the binnacle, with their centres vertically imder that of the needle, one set placed with their lengths in the fore and aft, the other with their lengths in the thwart-ship direction. In the process of correction, when marks on the shore are available the true bearings of which from the ship are known, the ship is first placed with her head in the magnetic north direction, and the thwart-ship magnets are moved so as to bring the compass needle to the north point on the card. For it is clear that the fore and aft magnetic force cannot have any effect on the needle when the ship's head and needle are both in the same direction. When this is the case we have, since ^=0, where Cj denotes the value of Cj as modified by the presence of the correcting magnets. The ship's head is now placed on the magnetic east (or west) point, and after an interval of about five minutes has been allowed to elapse, the fore and aft magnets are placed so that the compass needle points also due magnetic east. This gives ^1 + ^3 - ^1 = 0, where B^ ^^ ^'-^^ value of ^^ as modified by the correcting magnets. If A and JE are negligible the semicircular error has been corrected. 139. The ship's head is now changed to one of the quadrantal points, say the S.E. point (magnetic). If on this course, or the N.W. course, there is a deviation of the compass needle to the west (see Fig. 44), the coefficient D is positive, and a pair of equal spheres of soft iron are placed one on each side of the compass, at equal distances from the centre, and so that the line joining their centres is at right angles to the plane of the keel and passes through the centre of the needle. The distance of the spheres apart is adjusted until the deviation has disappeared. If, on the other hand, the deviation before the spheres are placed in position is found to be easterly on N.W. or S.E. courses, or westerly on N.E. or S.W. courses, then the coefficient D is negative, and the spheres must be placed in the plane of the keel, that is, afore and abaft the compass. This is, however, an altogether exceptional case. H -2 100 MAGNETISM AXD ELECTRICITV chap, iv When tlie ship is new the permanent magnetism of the iron will slowly change, and it will be necessary to alter from time to time the position of the compensating magnets correcting the semicircular error. When, however, the quadrantal error has been annulled the adjustment remains correct to whatever part of the world the ship may go. The induced magnetism of the ship, after long sailing in one direction, say east or west, generally lags behind a change 'of course. Thus when the ship's head is changed to north or south, from east say, a temporary error arises which the reader can easily trace the nature of. This (which is known as Gaussin's error) cannot conveniently be corrected, and must be allowed for. 140. The vertical force of the earth induces magnetism in the ship's iron, which gives a horizontal component of force on the compass needle, namely, the part depending on c in the term B^ sin ^. This varies as the ship goes from one latitude to another, and no provision for a corre- sponding soft iron compensator is made in the arrangement described above. Lord Kelvin has adopted in his compass the method, proposed long ago for the correction of this error by Captain Flinders, of placing an upright bar of soft iron exactly (in the case of a distribution symme- trical relatively to the plane of the keel) in front of or behind the binnacle, with its top about 2 inches above the needles. The bar used is round and about 3 inches' in diameter, and of length varying from 6 to 24 inches, according to the requirements of the case. The Flinders bar corrects the term of the heeling error ci sin^ f /A,, which is due to alteration of the positions of the equivalent soft iron rods representing cZ caused by the list given to the ship, and has its maximum when the ship's head is east or west. It also corrects partially the error Ji cos ^ by correcting the part — hi tan i/r cos ^JX of this term. The part — Ri tan ■^ cos i^ jZX is corrected by a vertical magnet placed in a tube immediately below the compass needle. The strength and distance of magnet required is determined by a comparison of the vertical force within the ship to the vertical force on shore. 141. Lord Kelvin has also shown how by means of an ingenious instrument called a deflector, which he has perfected, a comparison can be made of the directive forces on different courses. The adjustment then consists essentially in equalizing the directive force of the ship on a sufficient number of courses to make sure that it has as nearly as possible the same value on all courses, when it is certain that the compass is correct. This mode of adjustment is useful when sights of sun or stars or bearings of terrestrial objects are not available, and it can be carried out with great accuracy. For details the reader is referred to Captain Collet's treatise referred to on p. 94 above, or to Lord Kelvin's Instructioiis for Adjusting the Compass, to be obtained of James White, optician, Glasgow. On the whole subject of compass error and adjustment, the reader should consult, in addition to the treatise mentioned, the Admiralty Manual of Deviations of the Compass. CHAPTEE V ELEMENTARY PHENOMENA AND THEORY OF ELECTROSTATICS Section I. — Experimental liesuUs and Action of Medium Elementary Notions 142. Before proceeding to a discussion of electrical and electro- magnetic theory it will be convenient to give here a short account of the elementary phenomena of electrostatics and the steady flow of electricity. These phenomena are capable of being regarded from two different points of view, one in which the charge or the current of elec- tricity is the property only of the conductor, and another in which the charged conductor is simply part of the boundary of a state of strain in the dielectric surrounding it, and the wire along which a current flows is merely a guide for the transference of energy through the dielectric, from which also it receives in general a portion of energy to be dissipated in heat. The first electrical phenomenon observed was the attraction which substances, such as glass and sealing-wax, exerted on light bodies, as small feathers, the dried pith of elder, and the like. This, on the face of it, is action at a distance ; but it will be seen on consideration that such apparent attractions may be due to a medium in which the bodies are immersed, and which acts in such a manner that the two bodies are brought closer together, unless they are prevented from approaching by forces applied to the bodies by some 'other system, which, it may be noticed, must be some material system extending from one body to another. In fact the bodies are pushed towards one another in conse- quence of a state of stress existing in the medium surrounding the two bodies, and this can only be prevented from having any effect in dis- placing the bodies by another stress, set up in a material system by which the bodies are, so to speak, connected. 143. According to the old idea a piece of smooth glass rubbed with silk had developed upon it a certain fluid, which it was agreed to call electricity. This fluid had the properties of repelling other portions of the same fluid, and attracting portions of another fluid which was at the same time developed on the silk rubber. That idea is now replaced by 102 MAGXETIS.^r AND ELEGTEICITY chap. the conception, much more in accordance with the jDhenomena which have been observed, that by the rubbing a state of strain accompanied bj- internal stresses is set up in the non-conducting medium, or dielectric, as we shall call it, between the glass and the rubber ; that what were regarded as the electric charges on the glass and silk, and to which the apparent attractions or repulsions of the electrified bodies were attributed, are simply the surface manifestations of this state of the medium at the surfaces of the bodies immersed in it ; and that the attractions and repul- sions observed are really the result of a system of internal stresses in the medium, which are the natural accompaniment of the state of strain. 144. Of this system of strains and stresses we shall endeavour later to give some more detailed account, though it is unfortunately the case that a completely, satisfactory explanation, or even specification, of it is as yet impossible. Nevertheless it will be found to conduce to clearness of statement, and the prevention of false ideas, for example that of the real existence of a material something developed on the surface of an electrified body, or flowing through the substance of a wire carrying a current of electricity. The latter phenomenon, according to this more modern view of the subject, is an accompaniment of a progressive change in the state of the non-conducting medium, a change which in certain very important cases is continually made good in virtue of certain other changes going on in the state of a material system, so that the medium remains to our observation in a steady state, and what we call a steady flow of electricity, or a steady current, goes on along the conductor. Location and Transfer of Energy. Current in a Wire 145. The setting up of this state of the dielectric involves the expenditure of work, part of which has its equivalent in energy, which, has its seat in the medium while the state lasts, is conveyed to the medium as the state grows up, and leaves it as the state evanesces, in a manner which we shall seek to investigate. As a very important part of the new view to which reference has been made, a battery which furnishes a current to actuate a telegraph instrument, or an electric motor, or a voltameter or secondary battery in which electro-chemical' change is effected, or other arrangement in which useful work is done, does not transmit energy along the wire, but sends it out into the medium, from which it flows again upon the arrangement in which the energy is utilised, and at the same time upon the wire to supply the energy which in all cases is dissipated in the conductor. The medium has its state changed, and the energy required for that is thrown out into the medium from the battery. In consequence of the presence of the conductor completing the circuit the state of the medium is continually breaking down, through the passage of energy from the medium to the conductor and included instrument ; so that for the maintenance of the steady state energy is continually being thrown out into the medium by V ELEMENTARY PHENOMENA OF ELECTROSTATICS 103 the battery. Thus the wire and connecting conductors merely act as a guide, and the instrument or machine included in the circuit receives the energy which is given out by the battery, diminished by the energy which flows back upon the connecting conductors and is converted into heat. It is thus the medium, not the conductor, which acts as the carrier of energy; for example, in the case of a submarine cable, the vehicle of energy is the layer of gutta-percha which separates the conducting wire from the sea water in which the cable is laid. 146. What according to the newer theory goes on in the passage of a current along a wire will be more clearly understood by a short con- sideration of the discharge of a Leyden jar. According to the old and still common statement, the jar (which consists of a glass bottle coated inside and outside, about three-fourths of the way up, with tinfoil) is charged by allowing positive electricity (see below, p. 105) to flow into its inner coating from an electric machine ; this induces negative electricity on the inner side of the outer coating supposed connected with the earth, the negative electricity thus induced enables more positive to flow into the inner coating, this induces further negative electricity on the outer coating, and so on. Electric Induction and Electric Intensity 147. What really happens is that a state of strain is set up within the glass separator of the two coatings ; the opposite surface manifesta- tions of this are the two opposite charges of electricity. The state of strain is in a certain sense measured by a quantity called electric induc- tion, directed from the inner coating towards the outer in the case sup- posed. This quantity we shall specify completely later. At each point in the dielectric the induction has a definite direction, and a line drawn so that its direction at each point is the direction of the induction is called a line of induction. A line of induction thus starts from the coat- ing which we say is positively charged and ends in the negatively charged inner surface of the outer coating. A part of the dielectric bounded laterally by lines of induction is called a tube of induction. Thus in the Leyden jar tubes of induction start from the inner electrified surface and end on the outer, and the course of induction is such that each tube, whatever its scope on the surfaces may be, has quantities of electricity at its ends which are complementary not only in amount but in actual fact, that is the portion of the surface aspect of the strain which is enclosed by the tube at one end physically corresponds in the state of strain to that at the other end. The positive charge from which the tube starts is thus equal in amount and opposite in sign to that in which the tube ends. 148. The energy stored up in the jar thus exists within the dielectric while the system is in equilibrium. Now let the coatings be connected by a wire. The tubes of induction move outwards sideways from the condenser with their ends on the wire, so that the positive and 104 MAC4NETISM AND ELECTRICITY chap. negative electricities move along facing one another (thus moving in opposite directions round the circuit), while each tube gradually shortens as it advances, being swallowed up at its ends in the wire, thereby yield- ing up its energy to the conductor, until it becomes infinitely short and disappears. If a telegraph instrument or other arrangement in which energy is utilised is actuated, the tubes are only partially absorbed by the joining conductors, and there is a finite remainder of each tube which is absorbed by the arrangement. If the condenser have its terminals connected to earth, the earth forms in no sense a reservoir into which the electricity of the condenser is discharged, but only part of the guiding conductor. In the case of a battery the tubes are thrown out from the battery laterally into the medium, and then they move along as in the case just described with their ends on the conductor, being absorbed wholly or partially in it as they proceed. (See Chapter XI. below.) 149. When an electrified conductor is in electrical equilibrium the tubes of induction start normally from its surface, and exert on every element of it an outward pull, which is balanced, not by electrical strain of the conductor, for no such strain can be set up in perfectly conducting material, but by force due to ordinary mechanical strain of its material. For example, an electrified soap bubble is pulled everywhere normally outwards by the medium outside it, and the outward force on each element of the surface is balanced by part of the inward force on the same element due to the contractile force in the curved film. A perfect conductor is in this theory a body which . cannot endure strain of the kind which exists in the dielectric, and at which therefore the state of strain to which we have referred suddenly terminates. Conductors, however, may be more or less imperfect, and the state of strain capable of temporarily existing within them to some extent. 150. Dielectrics as well as ordinary conductors are, we have reason to believe, merely material systems imbedded in the ether which permeates their structure as it does all space, and the ether is itself the standard or ultimate dielectric medium to the action of which all electric and magnetic phenomena are to be referred. This view of the matter will become clearer as we proceed ; the preceding discussion will serve to introduce the ideas, and at present we go on to a short sketch of elementary electrical phenomena, bringing them as far as possible into relation with the ideas which have just been explained. It is to be clearly understood that at present we consider only the electrification of bodies which are at rest, relatively to the insulating medium in which they are immersed. Also it is to be supposed, unless the contrary is stated, that when conductors are referred to as brought into contact with one another, they are of the same material, so that there is no question of contact difference of potential. Further it is assumed that no change of the internal physical state of any of the bodies such as temperature, volume, or the like, accompanies the electrical actions or changes considered. ELEMENTARY PHENOMENA OF ELECTEOSTATICS 105 " Electrics" and "Non-Electrics." Conductors and Insulators. Electric Attraction and Bepulsion 151. When a rod of glass has been rubbed with silk it is found by suspending the silk and glass side by side that they apparently attract one another. Again, if two small pith-balls be hung by silk threads so as to rest as nearly as may be at the same level a short distance apart, and one of them be touched with the rubbed glass, the other with the silk rubber, they will appear to attract one another. Again, when a piece of sealing-wax is rubbed with a dry woollen cloth and then made to touch one of the balls, while the other is touched by the glass rod, attraction between the balls is observed, just as when the silk rubber and glass were used to touch the balls. Also when the balls are touched by the rubbed sealing-wax they seem to repel one another, and moreover one touched by the silk rubber repels one touched by the sealing-wax. The result obtained by rubbing other substances is always in a similar way either a repulsion or an attraction, which would have been obtained by rubbing smooth glass, or sealing-wax, or both, the glass with silk, the sealing-wax with wool. 152. From these facts, which were long ago observed, arose the idea of two kinds of electricity, that of smooth glass rubbed with silk, and that of sealing-wax rubbed with a woollen cloth. The former of these was called positive the latter negative electricity. A portion of one of these electricities was regarded as having the property of repelling another portion of the same kind, and of attracting a portion of the other kind. This qualitative result, which is still given in many elementary treatises on electricity as " the law of electrical attraction and repulsion," was practically all that was known before the forces between electrified bodies were quantitatively investigated by Coulomb. In the meantime it had been discovered by Stephen Gray that certain bodies such as silk, glass, sealing-wax, &c., acted as insulators, that is, did not allow electricity to pass off from themselves, or from bodies held or supported by them and excited by rubbing, and that certain other bodies, for example all metals, acted as conductors, that is when used as supports for electrified bodies allowed the electricity to escape to other bodies with which the supports were connected. This broke down the old distinction between electrics and non-electrics, or bodies which could be electrified by rub- bing, and those which apparently could not ; for it was immediately found that all bodies could be electrified, provided the body were held by a proper support to prevent the excitation from being dissipated as fast as it was produced. 153. The quantitative result obtained by Coulomb is in part ex- pressed by the statement that the repulsion between two small similarly electrified conducting spheres is inversely proportional to the square of 106 MAGXETLSM AND ELECTRICITY chap. the distance between their centres.^ Here spheres are considered small if the ratio of the distance of the centres apart to the radius is in each case large compared with unity ; for example, Coulomb's result would apply Avithout sensible error to a pair of pith-balls, each 5 millimetres in radius with their centres 20 centimetres apart. 154. When a conducting sphere is charged with electricity and, placed at a distance from other conductors great in comparison with its radius and the linear dimensions of those other conductors if they are charged, the distribution of electricity on the sphere is found to be symmetrical round the centre. This is proved experimentally by apply- ing to the surface of the sphere a small disk of thin metal held by a glass insulating stem so as to coincide with the surface of the sphere, and then removing it and observing the repulsion between the charge on the disk when held at a fixed distance from a small insulated charged sphere, say a charged pith-ball, hung by a single silk fibre. It is found that wherever the disk is applied to the sphere, the charge removed is the same, inasmuch as the force, as measured by the deflection of the pith-ball pendulum against the action of gravity, is the same. For if the sphere were more intensely electrified at one part of its surface than another, the disk when applied at such a place would be more intensely electrified, and a greater repulsion would be produced. Further, if such a proof-plane (as this insulated disk is called) is applied to the interior of a hollow conducting sphere, within which no electrified bodies are insulated, no charge is taken by it, showing that there is no electrification on the inner surface of such a sphere. Also, on the inner surface of a closed hollow of any shape within a conductor no electrification is found if there be no electrified bodies within the hollow. Forces on Electrified Bodies regarded as due to Action of a Medium 155. These results are consistent with the theory of the action of a medium referred to above. When the spherical conductor is at a great distance from other conductors, the state of the medium near the sphere is quite symmetrical all round it. The sphere, being a conductor, sup- ports no electrical strain within its substance, and none is transmitted through its substance to the medium existing within it, that is, no surface aspect of the state of strain in the dielectric is found to exist anywhere except at the external surface of the sphere, unless there are electrified bodies insulated in the hollow space within it. If however the sphere experience force from the field, this arises from dissymmetry of induction round its surface, due to the presence of other conductors. 156. Let us now suppose that we have a conductor. A, charged with electricity, and that it is connected by means of a wire with another conductor, B. B will also become charged with electricity. The state of the field is, in fact, not one of equilibrium with A charged and B not, 1 For an account of tlie torsion balance experiments by which this result was established see the author's Absolute Meamrementj m EledriciMj and Magnetism, Vol. I., p. 254, or any good elementaiy treatise on electricity. ELEMENTARY PHENOMENA OF ELECTROSTATICS 107 and the two in contact through a conductor, inasmuch as the region of strain in the field which abuts against the charged conductor, A, tends to spread itself out laterally, in a manner to be considered later, with the wire and the surface of the other conductor as guide, until this tendency is balanced by a distribution of the strain all round ; so that the lateral action on the sides of each element of a tube is balanced. According to Maxwell the dielectric is affected by stresses consisting of a tension along the tubes and an equal pressure in all perpendicular directions. The error, however, is to be avoided of identifying the strains above referred to with those in an elastic solid. We shall deal with this subject later. The electrification is thus extended over the surfaces of both A and B, and if, as we suppose, the wire be very thin, it may be neglected or removed without disturbing the distribution on either conductor. We shall show that in a certain sense there is the same total quantity of electricity on the two conductors that there was originally on A. Faraday's Ice-Pail Experiment 1-57. To prove this we make use of a celebrated experiment of Faraday, called his ice-pail experi- ment. A nearly closed vessel, such as a deep metal vessel, F, like that represented in the diagram, (Fig. 47), is hung by silk threads, or more conveniently supported, as shown in the diagram, by a block of solid paraffin or other non-conducting material, and has, in conducting connection with it as shown, two pith-ball pendu- lums, which are supported from one point by thin wires of metal. When the outside of the vessel is charged with electricity the balls become charged also and separate in consequence of their apparent repulsion. (This repul- sion will find its explanation in a pull exerted on the surface of a charged conductor by the sur- rounding medium, as more fully stated below.) Any change in the electrification of the vessel naturally leads to a corresponding change in that of the balls and an alteration of their equilibrium distance apart. Now if the vessel be ioitially uncharged, and a conducting ball be attached to a silk thread and charged, and then lowered within the vessel P, the pith-balls, which originally hung with the threads vertical. 1 Fig. 47. Ice-pail on paraffin block, con- nected by thin wire with two pith-balls- hnng from a point by wires. A charged ball can be lowered by a silk thread to touch the' bottom of the pail. A lid, with a hole in it through which the thread passes, may be supposed to close the pail. lOS MAGNETISM AXD ELECTRICITY chap. move further apart, showing the acquisition of a charge by the exterior of the vessel. It is found, as the charged body descends lower and lower in P, that the deflection of the balls goes on increasing more and more slowly until, if P be fairly deep, it becomes practically constant. If now the conductor be made to touch the bottom of P and be then removed, no change will take place in the deflection of the pendulums, and the body mil be found to be completely discharged. 158. Now let the experiment be varied by first lowering the ball near the bottom of the cylinder, then lowering by a silk thread another ball uncharged beside it. No change of the deflected position of the pith-baUs will take place. Next bring the two balls into contact ; it wiU be found that again there is no alteration of the deflection of the pendulums. Now withdraw the second ball, and test it for electrification by bringing it near a pith-ball pendulum, and it will be found to be electrified as was the first ball. It will also be found that the pendulums attached to P do not diverge so far as before, and that the former deflection is restored by reinserting the second ball. Further, if the two balls be equal, and when brought into contact be similarly placed with respect to the interior surface of P, it will be found that the pendulum deflection is the same for either ball left charged alone within the vessel, showing that the charge on the first has, so far as the effect on P is concerned, been equally divided between the two. The same result will be obtained more conveniently by bringing the two balls into contact outside the vessel and at a distance from other conductors, after which each will be found to have the same effect on the pith-ball pendulum outside. 159. Again, if the first ball with the original charge be kept at a certain distance from a similarly charged pith-ball forming the bob of a pendulum, and the force of repulsion be measured by the deflection of the ball from the vertical, then the same experiment be repeated with each of the balls after division of the charge, it will be found that the force in each of the latter experiments is half that observed in the first. Similarly, if the original charge be shared between three equal conducting spheres, the force shown by the pith-ball pendulum for the same distance of its bob from the charged ball will be one-third of that observed in the case of the ball with the original charge. The ice-pail experiment will assure us, however, that the total charge is unaltered. 160. We thus see how and in what sense we can subdivide charges into equal amounts while the total charge remains unaltered. We can also give by means of the same apparatus any multiple of a given charge to a conductor. Take an ice-pail, F, so small as to be capable of being placed entirely within P, and mount it on insulating supports so as to be easily moved about. Charge two balls with electricity, and introduce them together into P, and note the deflection of the pith-ball pendulum. Then lower them in succession into P, and bring them into contact Avith it at the bottom, and note that each, when withdrawn, is com- pletely discharged. Lower now F into P and note that the effect on P is the same as when the two charged balls were placed in it. The V ELEMENTARY PHENOMENA OF ELECTROSTATICS 109 charges on the balls have thus been transferred to P -without change of their aggregate amount. 161. Now perform the following experiment. Insert a charged ball within P and note the deflection of the pith-ball pendulums. Then withdraw the ball, taking care not to discharge it. Insulate P well within P, and provide a piece of wire held by a handle of vulcanite or glass to make contact between the outer surface of P and the inner surface of P. Now insert the charged ball within P, without bringing it into contact, and note that the deflection of the pith-ball pendulums is the same as before. Then while the ball is withia P make the connection indicated between P and P. The deflection of the pendulums will not be affected. Withdraw both the charged ball and P ; still the deflection of the pith-balls is unchanged. Thus the deflection is the same as if the charged ball had been at once brought into P, and discharged by being brought into contact with its interior surface. The same series of operations can be repeated as often as may be desired, and each time the same quantity of electricity is given to the vessel, as may be verified by comparing the result of a number of these operations, made with a single charged ball, jvith that of placing the same number of equally charged balls together within the vessel. It is also to be noticed that when uncharged conductors are insulated within P their presence does not affect the distribution on P or its external field, nor is there found any electrification or any induction whatever at any point within P. Division of Electric Field into Two Parts by Conducting Screen. Genesis of Field External to Closed Conductor 162. These experiments, besides illustrating the idea of quantity of electricity, show that the field of electric force, when a closed conductor contains electrified bodies, is divided into two parts, the region internal and the region external to the conductor, and that no matter how the connections among the electrified bodies in the interior may be varied, the external field undergoes no change ; that is to say, the external field is quite independent of the arrangement of the tubes of induction in the interior, so long as the same number starts from the internal conductors. 163. The mode in which the external field arises will appear from the consideration of a single charge first insulated on a sphere at a distance from other conductors, and then introduced within the closed conductor. At first the lines of induction are directed outwards along the radii of the sphere produced, curving round, however, at a distance from it so as to terminate on the other conductors, whatever these may be. Now let an insulated and uncharged hollow sphere be brought into the field, into the position shown in Fig. 48. Hardly any of the tubes of induction will enter the shell by the opening o; but their arrangement will be altered, and a number will be intercept by the external surface of the shell, as shown. Where these terminate 110 1IAGXETIS3I AND ELECTRICITY CHAP. are places of negative electrification on the sphere. By the bringing in of the hollo-w sphere, the radial direction of the tubes of mduction has been disturbed, and a considerable number of them severed each into two parts, of which one extends from the ball to the outside or inside of the hollow conductor, the other from the hollow con- ductor outwards to other conductors. For, consider a tube passing very close to the boUow conductor. At the surface of the conductor the resistance to lateral motion of the tubes does not exist, and hence a tube close to the surface is moved nearer to the surface. As soon as it comes into contact it breaks into two parts, having their ends, one positive, the other negative, on the surface of the hollow conductor. These shorten at once, one runs down to its shortest length, for example, c, between the charcred ball and the conductor; the other contracts into the part d, o Fig. 48. Fig. 49. running from the hollow conductor to the original termination' of the tube ; and so with other tubes until a distribution on both ball and conductor is arrived at in which there is equilibrium of lateral action of the tubes. This is possible with tubes ending on part of the conductor C, and tubes leaving the rest of the surface, since the lateral action does not depend on the direction in which the tubes run. If now the ball be brought nearer to and finally through the opening in G (see Figs. 49, 50), the tubes of induction will be drawn in with it, those which terminate on the conductor will shorten by the -motion of their negative extremities round the edge of the opening until they terminate on the inside. Other tubes which pass by the conductor will be pushed up to it, will part as already described ; one portion will run down to its shortest length within the conductor, the other will take up an equilibrium position outside, conditioned only by the arrangement of the tubes there. Finally, if the opening be closed up with the charged V ELEMENTARY PHEXOMENA OF ELECTROSTATICS 111 body inside, all the tubes will have been divided into two parts, and, ■clearly, just as many will leave the outside and terminate on remote ■conductors as start from the enclosed body, inasmuch as these are the parts of the original tubes which, with altered length, retain the termi- nations the system of tubes starting from the charged ball had before the closed ■conductor was brought into the field. 164. The modes of distribution outside and inside are independent, and we see how the contact of the ball with the interior of the closed conductor completely discharges it. At the place of contact or of spark between the conductors, the resistance to the lateral motion of the internal tubes is removed ; the ends of these run along the conductors towards the place of contact, and '^^°- ^^■ the tubes shorten and finally disappear into the connection between the conductors, giving up their energy there in producing a spark and in heating the substance of the conductors. Hypothesis of Incompressible Fluid 165. Another hypothesis, at first illustrative rather than a real way of accounting for the phenomena, may be mentioned here. It is that the dielectric-space outside a conductor is filled with an incompressible fluid, which also pervades the interior of the conductor. If then any additional quantity of incompressible fluid be forced into the conductor, an equal quantity must be forced out across the outer surface of the conductor, and across any closed surface which may be drawn in the dielectric so as to enclose the conductor. Thus if the space have an outer conducting boundary, an equal quantity of fluid must be forced inwards towards the conductor across that boundary. The fluid displaced outward across any element of the inner conductor may be regarded as the charge of electricity on that element, the equal and opposite negative charge which is its complement is the fluid displaced inward across the corresponding element of the outer conductor. Corresponding elements are those connected by a tube of displacement, and the displacement across any cross-section of a tube is proportional to the electric induction at that cross-section. This hypothesis is helpful in the description of electrical facts, but by itself is inferior to some others as regards the explanation which it affords of phenomena. A theory, however, has been recently put forward by Larmor, which aims at accounting for electrical phenomena by supposing that a fluid ether pervading space is endowed with an elasticity which resists rotational displacement, and explains a linear current as a vortex rino- in the ether, and magnetic force as velocity of flow of the medium. 112 MAGNETISM AND ELECTRICITY chap. Some account of this and other ether-theories will be given in Vol. II., and with this in view a fairly complete sketch of the theory of both irrotational and vortex motion of an incompressible fluid, is given in a later Chapter. We shall therefore not enter further into fluid theories at present. Specification of Electric Induction and Electric Intensity. Energy of Field 166. We now assume, according to the statement made above, that what we, by an analogy with the ordinary elasticity of matter, regard as the electric strain in the dielectric medium at any point is measured by a directed quantity which we shall call the electric induction and denote by D, and which we shall also assume to be connected in general by a linear relation with another quantity called the electric force or electric inteTisity and denoted by E. These two quantities are precisely analogous to the magnetic induction and magnetic intensity defined and discussed above in section 56, Chapter II. ; and the electric induction is 4nr times the quantity called by Clerk Maxwell in his treatise the electric displacement. In the case of an isotropic dielectric the electric induction is taken as in the same direction as, and in simple proportion to, the electric intensity. We can now express the energy of the system of electrification as the potential energy of the state of electric strain in the dielectric, which we suppose to have its seat where that strain exists. We take the amount of the energy per unit volume as measured by the work done by the electric intensity in producing the electric displacement, or for the case of an isotropic medium, as given by the equation E=SH = ^^^ W In the more general case, in which D is proportional to E, and inclined to it at the constant angle 6, as they grow up together, we have i^y ^cose.dH = —"EDcose . . . . (2) Surface Integral of Electric Induction 167. In the case of magnetism the surface integral of magnetic induction is equal to 47r times the quantity of magnetism within the sur- face, and in a similar way we have here the surface integral of the out- ward normal component of electric induction taken over a closed surface drawn in the electric field equal to 47r times the quantity of electricity within the surface. The theorems that have been proved above in pp. 35 to 43 with respect to magnetic intensity and magnetic induction all hold for the electric quantities, when E is put for H, D for B, with ^' ELEMENTARY PHENOMENA OP ELECTROSTATICS 113 the corresponding substitutions of components, and k, which we call the electric inductivity of the medium, is put for fjt,, so that D = h'E. Thus we identify in the same way as at p. 40 the electric intensity E, produced at. any point in an unlimited uniform dielectric by a point- charge of amount 2 of electricity distant r from the point, as the quantity qjlcr^, which measures the repulsive force exerted on unit quantity of electricity placed at the point in question. Also, as before, the field, in any given case, is the resultant of the fields thus produced by the individual point-charges of which the given distribution may be regarded as built up. This synthesis leads, it will be found, to correct results. Energy in Case of .3!olotropic Medium 168. Putting/, g, h for the components of electric induction (that is 477 times the corresponding components of electric displacement denoted by the same letters by Maxwell), and P, Q, B, for those of electric intensity, we have for the most general linear relation between in- duction and intensity / = k-^^P + \^Q + \^R \ g = h^^P + \^Q + k,^JR I (3) h = k,^P + k^^Q + \^rS The energy per unit volume has now the value dE=^ [{Pdf + Qdg + Rdh) . . • (4) that is dE = ^ {{k^^P + k^^Q + k^-^R)dP + {k^^P + k^^Q + k,,R)dQ + {\^P + k^^Q + k^^R)dR}. But we suppose the energy to depend only on the state of the medium : henoe dE must be a perfect differential in the variables P, Q, R. Thus we have ?| = i- {k,^P + k,,Q + h,R), with two other equations for dEjdQ, dEjdB. Hence since d^E/dPdQ ^d^EjdQdP, &c., we get the relations ^12 = Kv hs = hi' hi == "^13- The energy per unit volume can thus, from (4), be written E = -{k,,P' + h,Q' + k,,E' + 2k,,QE + M,,RP + 2k,,PQ) (5) I 114 MAGNETISM AND ELECTRICITY chap. which by a suitable choice of axes can be reduced to the form E = ^ ihP^ + hQ^ + k^R^) (6) where P, Q, R are now the components of electric intensity with reference to the new axes, and /Cj, h^, k^, the electric inductivities in these direc- tions, are called the principal electric inductivities of the substance. Charged Spherical Conductor in Uniform Dielectric. Energy of System 169. Now let us consider an infinitely extended uniform dielectric surrounding a conducting spherical surface, of radius a, charged with electricity. As we have seen above, the tubes of induction issue normally from the surface of the sphere and extend radially outwards, having their farther extremities on conductors, which for our present purpose may be regarded as infinitely distant from the sphere, so that the field round the latter may be taken as symmetrical about the centre. To find the energy of the field we have, taking the surface integral of D over a spherical surface of radius r concentric with the given one, and putting 4s'7rQ for the value of this integral, so that Q is the charge of electricity, 47rr^D = 46'kQ, or D = Qjr': The value of E is D/Zc, so that E = Qjkr^, and the directions of D and E are the same. The energy of •electric strain in the field is thus (if dvs be put for an element of volume) [Edrs = ^[ ED . i^rr^dr = ~ . . . . Vl) Spherical Condenser 170. If now a concentric spherical surface of radius 6 round the former have uniformly distributed over it a charge — Q, the tubes of induction will terminate there, and there will be no external electric field. This surface of course in practice would be the inner surface of a hoUow shell of conducting material. The negative charge upon it merely represents the external ends of the tubes of induction. The energy of this arrangement is given by 2k \a bJ' as may be seen either by substituting b for oo as the superior limit of the integral in (7), or by observing that a uniform distribution — Q on a spherical surface of radius b, would produce external to itself an in- duction everywhere equal and opposite to that produced hj+Q on a concentric surface of radius a, so that the resultant induction is every- where zero. Thus we have simply to subtract from Q^/2ka, the energy external to the outer surface due to—Q, that is Q^/2kb. This latter V ELEMENTARY PHENOMENA OF ELECTROSTATICS 115 way of regarding the matter is, however, a mathematical fiction : what we have physically is a conductor as internal and external boundary, and across that the electric strain is not propagated in either direction, inasmuch as the conductor will not sustain any such strain. Tubes of Electric Induction, Unit Tubes 171. Let us consider now a tube of electric induction, and take first a closed surface made up of a portion of the tube bounded by end surfaces everywhere at right angles to the line of induction. Since there is no ■component of induction at right angles to the sides, the sum of the fluxes of induction across the ends must be equal to ^tt times the quantity of electricity within the surface. Thus if Dj, Dg be the inductions at any point of the ends at which the lines of induction enter and leave respectively the closed surface, and q be the quantity of electricity within the tube, we have, integrating over the ends. |D,(i^2 - JDi^'S'i = 4,rg (8) Hence if the tube be thin, and there be no electric charge within the portion considered, we get, putting S^, S^ for the areas of the ends, now very approximately plane, DA = DA (9) Again, if the tube pass through a continuously electrified surface, we see by taking one end on one side of the surface, and the other end on the other side, but infinitely near the surface on the two sides, that Dj - Di = 4,ro- (10) where a is the electric density (or quantity of electricity per unit of area) at the part of the surface at which the tube is taken. If Dj = 0, that is if the induction be zero behind the surface, as in the case of a conductor in the substance of which one end of the tube is taken, this gives D2 = 4Tcr (11) and E=ip. . . (12) The direction of E is normal to the surface of the conductor. If the cross-section of the tube be so chosen that "D-^S-^ = 1, this relation will by (9) hold for every part of the tube which does not contain any electric charge. Such a tube is called a unit tube, and the surface integral of electric induction taken over any closed surface in the electric :field is equal to the number of unit tubes which cross the surface in the outward direction minus the number of those which cross the surface in the opposite direction. The quantity of electricity within the surface is I 2 116 MAGNETISM AND ELECTRICITY chap. thus equal to 4nr times this excess of the number of outward directed unit tubes over the number of inward directed unit tubes, or, as it is sometimes put, is 4-Tr times the excess of the outward over the inward drawn lines of induction. When the word line is thus used it. is to be understood as signifying a unit tube. Tension along Lines of Induction in an Electric Field. Tractioii on Surface of a Conductor 172. We can now see that the energy contained in a portion of a naiTOW tube, (if there be no electricity within the portion) is 8:rJ- 1 fD2 EBSds = ^\^Sds, Avhere S is the area of cross-section of the tube at any place, ds an element there of the length of the tube, and the integral is taken along the part of the tube considered. But D^Stt/c . Sds is the work done by a force 'D^jSirk . >S in a displacement ds, so that the force per unit area is -DySirk. In fact we see that we may regard a tube as formed by drawing out the negative charge on one end against an inward pull of amount D^/Stt^ per unit of area. This is equivalent to supposing that a tension of amount D^/Svlc exists at every point in the medium along the lines of induction. 173. As an example let the radius b of the outer spherical surface in the case considered above be increased by an amount db. The change of energy is 2k \b b + db) 2k 62 ■ The tension in this case overcome through the distance db at eacb element of the outer surface in carrying out the outer ends of the tubes is therefore Q^/SttW. But— o- being the density on the outer surface fT^ = Q^j\Qw%^, so that the inward tension along the lines of induction just inside the outer surface is 2'7ra^/k, The outward 'tension on the inner surface is obviously also 2ira-'^lk where a- is the electric density there. The traction on the outer surface of a conductor being taken as D^/Stt/c we have its equivalent 2'n-a^/k as the value of the outward pull exerted by the external medium per unit area on the surface of the conductor, and likewise of the equal and opposite pull exerted per unit area by the conductor on the medium. We have a good example in an electrified soap-bubble, which if the electric surface density be a, is pulled on by the external medium with a force per unit area 27r(r\ V.' ELEMENTARY PHENOMENA OF ELECTROSTATICS 117 causing an apparent diminution of amount Jtto-V/A; in the tension of each surface of the fihn where r is the radius of the bubble. It may be noticed here that, in the particular case considered above, by increasing the radius b of the outer spherical surface from a to infinity, we get for the work done and therefore for the energy of the system the equation = J 2k 62- ■2ka *■ ^■ Constancy of j'Eds along any Path from one Conductor to another. Potential." Difference of Potential. Equipotential Surfaces 174. Next consider a single conductor of any form on which tubes of induction originate, but none terminate, and which is surrounded by a single closed conducting surface. The tubes of induction for this con- ductor leave its surface normally and pass outwards until they terminate on the surrounding conductor. The energy contained in a narrow tube, of cross-section dS at any point, is J_fD2 Htt] k dS , ds, in which the integral is taken along the tube from one end to the other. Since J3dS. is constant qlong the tube and is equal to 4nradS', where d^ is the element of area intercepted on the surface of the conductor by the tube, and a- is the density there, we obtain 1 p2 ,„, crdS SttJ k The value of I Eds taken along a tube of induction must be the same for every induction tube passing from the inner to the outer bounding surface. Eemembering that £ is the electric intensity at a point of the field, that is the mechanical force on a very small conductor charged with unit quantity of electricity and placed at the point, we can prove this proposition in the following manner. Let such a point-charge be carried round a closed quadrilateral path consisting of two curves along lines of induction, and two connecting pieces at right angles everywhere to lines of induction. The work done in carrying the charged body round this path is zero; for this would clearly be the case in a field due to a single point-charge, and therefore also in one obtained by superimposing the fields of a system of individual point-charges (Art. 167). The work done in carrying it along each element of the parts of the path, which are at right angles to the lines of induction, is 118 MAGNETISM AND ELECTRICITY chap. also zero. Hence the work spent in carrying the charge along one of the two sides of the curved quadrilateral which lie along the lines of induction is equal to that gained in carrying it along the other. Since the lines of induction leave or enter the conductors at right angles tO' the surface, the value of lErfs taken along any line of induction from one surface to the other has the same value, and therefore has the same value along any path whatever, whether a line of induction or not, leading from one surface to the other. 175. E must therefore in cases of electrostatic equilibrium be a function only of the co-ordinates, and we may denote it hj —d V/ds, where F is a single-valued function of the co-ordinates only. 9 Vjdx.dx + 9 Vjdydy + 9 Vjdz.dz or dVis thus a perfect dififerential. The energy within a tube of induction has the value where Fj, V^, are the values of the function V at the inner and outer surfaces. The quantity F", — V^ is called the difference of potential between the two surfaces. Integrating now over the surface of the inner conductor, and putting Q for the charge of electricity upon it, as measured by 47r times the flux of induction across a closed surface in the dielectric surrounding it, we get for the energy E = f (Fi - n) . . . . (14) If the outer surface be everywhere at a very great distance from the inner conductor the value of E will vanish, that is F will cease to vary with displacement of the point considered along a line of intensity ; in fact, all lines of intensity will at a great distance have become untraceable in consequence of the smallness of magnitude of the electric intensity, and the impossibility of determining its direction. 176. We may if we please define Vdq for any point as the work done in carrying a small charge dq of electricity (so small that it does not appreciably affect the distribution on the conductors) from an infinite distance from the electrified conductor to the point in question, along any path against the electric intensity. The work thus done depends only on the initial and terminal points of the path, inasmuch as no work on the whole would be done in carrying the small charge dq round a, closed path on which those two points are situated. This makes Fzero for all points infinitely distant from every part of the electrification. The value of V at any conductor is in this reckoning called the potential of the conductor, and its value for any point is called the potential at that point. The meaning and use of this function will be further illustrated' in what follows, and especially in the chapter on V ELEMENTARY PHENOMENA OP ELECTEOSTATICS 119 Fluid Motion which is given below as a preliminary to the discussion of general theories of electrical action. A surface at every point of which the potential has the same value is called an equipotential surface. Such a surface can evidently be drawn through every point of the electric field. Any equipotential surface may be taken as the surface of zero potential, and the potential at any point is then the difference between the potential at that point and the potential at the surface. Lines of electric intensity obviously meet equipotential surfaces at right angles, that is, the resultant electric intensity at any point of such a surface is normal to the surface. It is clearly a property of the function V that it cannot have a maximum or minimum value in space void of electric charge. For if there could be a point or region of maximum or minimum of potential it would be possible to describe round it a surface at every point of which E would be directed outwards in the case of a maximum, and inwards in the contrary case. Thus the surface integral of electric induction woidd in the former case be positive and in the latter negative, that is, the surface would contain a charge of electricity, which is contrary to the supposition. General Problem of Electrostatics 177. The general direct problem of electrostatics is ordinarily the determination for a given system of conductors insulated with given charges in presence of certain other conductors maintained at zero potential, or at given potentials, of the value of V for every point of the field, and the surface density of the electric distribution at every point of the conductors. In this problem there may be given for some or all of the insulated conductors, not the charges, but, what is equivalent, the outward normal component of the electric induction for every point external to and infinitely near the surface. A still more general problem than this is obtained by replacing some or all of the conductors by surfaces, not generally equipotential, over which an arbitrary surface distribution of V or of the electric induction, or of V over some and of the electric induction over others, is made. The possibility of always finding a solution of this more general problem has been the subject of discussion ; but in any case in which we may have to deal with it the question of the existence of a solution will be answered by finding one. It can easily be shown that for this problem, as well as for the other, if there exist one solution for a given set of conditions, there exists no other. The problem first stated is sufficiently general for most electrical purposes, and will be treated as fully as is necessary in a later chapter. At present we shall consider only some general propositions regarding systems of conductors, and the properties of the potential function. 120 MAGNETISM AXD ELECTRICITY Section II. — Electrostatic Capacity and Electric Energy of Charged Conductors. Electric Condensers or Leydens 178. The simple arrangement of two parallel conducting plates of the same material (copper or brass for example), in which the induction tubes extend across a uniform dielectric filling the space between them, is of considerable importance in practice; If, as is here supposed to be the case, the plates be of considerable dimensions in every direction in their own plane, and be opposite to one another, the lines of induction anywhere at a distance from the edge great in comparison with the distance between the plates may be assumed to be straight and at right angles to the plates ; and being straight and parallel at every such place the tubes of induction will be uniformly distributed. Thus the values of D and E are constant at every place well under shelter of the plates. Taking an area A in the dielectric parallel to the plates and crossed by such tubes of induction, we see from what has been said, that if d be the thickness of the dielectric the energy corresponding to the induction across A is where Vj — V^ is the difierence of potential of the two plates and Q the charge on the area A of the plate from which the tubes start. We have Fj — F2 = Ei^ = Hd/k, and Q = AD/^tt. Hence we obtain This ratio is called the electrostatic capacity of the area A of the plate from which the lines of induction emanate, for the case of electrification here considered. 179. In general the electrostatic capacity of a conductor is defined as the ratio of the charge on the conductor to the potential of the con- ductor when aU other conductors in the field are maintained at potential zero. Thus if V„ = 0, the capacity of the area A in the case just con- sidered would still be Akj^nrd. In all cases the potential of a conductor, and therefore its capacity, is affected by the presence of neighbouring conductors, unless the conductor in question is surrounded by a closed conducting screen maintained at zero potential. We shall denote the capacity of a conductor by C. The electrostatic capacity of a conductor can easily be calculated in a number of simple cases. For example by (14) above the energy of the spherical distribution alone in its own field is Q^j^ha, so that taking the potential at an infinite distance from the centre of the sphere as zero we have for the potential V of the sphere Qjka. The capacity of the sphere is thus ka. y ELEMENTARY THEORY OF ELECTROSTATICS 121 Similarly we could show that the capacity of a sphere radius a enclosed in a concentric sphere of radius b, and at zero potential, is Jeabl{b — a), which for b infinite reduces to ka, as it ought. We shall return to the calculation of capacities in particular cases later. Energy in Terms of Electrostatic Capacity. Energy of a System of Charged Conductors 180. The energy of the electrification thus considered can be ex- pressed by either of the equations E = i(7(F, - F,)2 = i^ (16) From this and (15) it is clear that for a given induction between the plates, the energy Aaries directly as d, that ,is the energy of the medium varies directly as d, for a given charge of electricity on the plate from which the tubes start. The physical reason for the slight amount of •energy in this case is obvious : the extent of medium strained to a given intensity varies directly as d. The ordinary explanation by the proximity of the negative charge at the final extremities of the lines of induction refers to the same fact, but does so somewhat obscurely. On the other hand for a given value of Eds along a line of intensity froin one plate to another, that is, for a given value, U say, of Dd or, which is the same thing, a given difference of potential between the two plates, the energy is given by and is inversely proportional to d. The charge in this latter case is Ahi V-^ — V^j^sird or, as before, C{ V-^ — Fg). This illustrates the ■ so-called condensing action of the arrangement. The smaller d, the greater is the induction required to produce a given ya,]ue Dd, or of D^c^/Stt, that is of the energy con- tained in a unit tube. An arrangement of this kind is generally called a condenser, but there is, properly speaking, no condensation of any sort. Lord Kelvin and Lord Rayleigh have recommended the substitution of the name leyden. 181. We now consider the more complicated case of a number of charged conductors of any form insulated from one another. Let the system for definiteness be supposed enclosed within a single conductor S^, and let S^, S^, S^, denote the surfaces of the conductors, and let tubes of induction only proceed from S^, and terminate on S^, Sg, and on the surrounding conductor S^. Further let tubes of induction proceed also from /Sg, and terminate on S^, S^, and on S^, and so for the other 122 MAGNETISM AND ELECTRICITY chap. conductors. The tubes of resultant induction will fall into groups, one for which the tubes originate in 6^ another for which they originate in S.„ and so on. The energy of the system will be obtained by calculating first the energy E^ for the resultant tubes which start from iSfj, next the energy Ej for those which start from /Sg, and so on until all the tubes in the field have been taken into the account. We get by what has gone before El = h^ads.iv, - To) + h^>Tds\{]\ - r,) + ^^,.ds\{r, - v,) + (18) where dS^ is an element of the part of the surface S-^ from which tubes pass to Sq, dS\ an element of that part of the same surface from which tubes pass to S2, &c., and the surface integrals are taken over these parts only; while J^i — ^o> ^1 ~ ^2 ^^^ ^^^ differences of potential between >Sii and />'„, S-^ and S^, But this may be written El = h\^dS,{V, - To) + i^ and corresponding potentials V\ F'g the change in energy "is \t V'Q' - it VQ. But by (24) we have -HSFV - 2r§) = ^(src' - %rQ' + %v'q - %vq) --= mr ^V){Q' + Q) (26) = \t{V' + V) {Q' - Q) This result is graphically illustrated in Fig. 51, and shows that the vsrork done in changing the state of the system is numerically equal ■either to the area of the trapezium .^ ^ C i^, or to the area of a trapezium E G 1) F. This illustrates the fact that the potential in- F Q' D E ...Q.. ^^ C V' V A Fie. 51. creases ;paH passu with the charge of the conductor, or, in the case of a system of conductors, that equal proportionate changes in the charges of all the conductors are associated with equal proportionate changes in the potentials. ELEMENTAEY THEOKY OF ELECTEOSTATICS 125- Coefficients of Potential and Induction and Electrostatic Capacities of a System of Conductors. Reciprocal Theorem 184. The potential of any conductor is thus a linear function of the- charges of all. ; Jlence we obtain the series of equations '^1 = PuQl + P21Q2 + ..'..+ PmQn ^2 = ^12^1 + ^22^2 + ••••+ PniQn Vri = PinQi + P^nQ^ + ....+ PnnQn (27) ■with the condition expressed by the theorem stated in (25), that P2i=Pn' • • ■ ■Pkh=Phk The coefficients ^jp ^^j, .... are called coefficients of potential. Their physical meaning is clear : a coefficient of the form p^j, is the potential produced at the conductor distinguished by the suffix k, A^ say, by unit charge on' the conductor itself when all other conductors are without charge on the whole : a coefficient of thfr form Phi: is the potential produced at the conductor A/e by unit charge on Ah when all the other conductors are without charge ; and this, as we have seen, is eqUal to the potential produced at A^ by unit charge on An, when all the other conductors are without charge. The number of independent coefficients of the formj^^j; is of course n, and the number of the form p^k is i^ (n — l), so that there are ^n{n + l) in all. By solving equations (27) we obtain of course a set of equations Ql = "11^1 + '"12^2 + ■••• + CinVn V2 ~ ^21 '1 "^ ^22 '2 + • • • • ~t* ^inf^n Qn = CniVi + Cn2^2 + ••••-!- CtmVn ■ P») for which also there hold the conditions Cj7, = c/iit, &c. _ The coefficients of the form Cij are the electrostatic capacities of the conductors. Each denotes the charge pn the conductor indicated, by the suffix when that conductor is at potential unity, while all other con- ductors are at zero potential. The coefficient Ckk denotes the charge on the conductor A^ when the, conductor Aje is at unit potential and all others, are at zero potential, and, being equal to c^a is also the charge on Ai^ when A^ is at unit potential and all the others are at potential zero. Such coefficients are called coefficients of induction. 126 MAGNETISM AND ELECTRICITY Properties of the Co-Efficients of Potential and Charge of a System of Conductors 185. The following additional general results are easily proved for the two kinds of coefficients. Every coefficient of potential is positive, and any one of the form pjjc is intermediate in value between zero and p^ji or pici^. For if Ah be charged with a unit of positive electricity, and all the rest of the conductors be insulated without charge, tubes corresponding to total induction of amount 47r pass outwards from Ah. At any other conductor, Aj^ say, just as many unit tubes originate as end upon it, since the charge is zero. The potential, therefore, in certain directions must increase from Aje, in all other directions diminish. There must therefore be a conductor, the potential of which is the highest potential in the field, and this clearly must be A^, since, as we have seen, there cannot be a place of maximum potential in space void of electricity, and from any other conductor the potential increases outwards in certain •directions. The potential of Ajgis therefore less than that of Ah and greater than zero, that is Phh >Pkh{= Phk) > 0. Similai-ly we can show that Pkk >Phk{= Pkh) > 0. If an uncharged conductor be enclosed within another whether uncharged or not, for example Ai within Atc, no lines of induction pass from one of these conductors to the other, for as explained (Art. 141), the distribution on the outer conductor is unaffected by the presence of an internal but uncharged conductor. The potential of Ai is then the same as that of A^, and we have Phk = Phi- As regards the capacities it is clear from (28) that they are all positive. For let the potential of Ah say be unity, and all the other conductors be at potential zero, we then have Qh = Ghh, The potential diminishes in every direction outwards from Ah, and therefore Qh must be positive. The coefficients of induction, c^j, are however all negative. For let Ah be charged as just specified. Then since the potential of Aj^ is zero, and the only other conductor not at zero potential is Ah, there can be at no point near the surface of Ajc a diminution of potential in the direction outwards from the conductor. Hence tubes of induction can only terminate on A^ and the charge on it is negative, that is c^j is negative. Again, since all the tubes in the case supposed originate on Ah, the sum of the negative charges on the other conductors must be numeri- V ELEMENTARY THEOHY OF ELECTROSTATICS 127 cally less than the charge on Aj^ ; unless A^ be completely enclosed by the other conductors, when the positive charge on Aj^ is exactly equal and opposite to the sum of the charges on the other conductors. Hence where the expression on the right denotes the sum of all the coefficients of induction for the system. Energy of a System of Charged Conductors Expressed as a Quadratic Function of Potentials or Charges 186. By equations (27) and (28), the energy (Art. 161) of the electrified system can be expressed as a homogeneous quadratic function of the potentials or of the charges of the conductors. For physical reasons the energy of the system cannot be negative, and hence both quadratic functions must be positive whatever may be the values of the variables. Clearly if we form the functions we have by (22), (27), (28) E = ii^nei" + ^Pi^QiQ^ +•••■+ F22«2' + 2^^2362^3 + ••••} (29) and ■ E = ifciiFi^ + 2c^^vj^ + .... + c,,r/ + 2c,,r^r, + ....} (so) The condition which must hold in order that E may be positive whatever be the values of Qj, Q^, ■ ■ • V^, V^, . . . is simply that if each of the homogeneous quadratic functions be converted into a sum of squares each of these squares must be positive. We therefore write for (29) 2E = — {(jPnQi + Pvfii + Vm% + ....)' + (P-A + «">Si + • • ■ 0^ + (»3303 + «34«4 +•■••)' +••••}• • ■ • (31) SO that the first square contains all the Q'%, the second all except Q^, the third all except Q^ and Q^ and so on. Equating coefficients from (29) with those on the right of the expression just written for E, we get exactly \n{n—V) equations for the determination of |»i(m— 1) unknown co- efficients, so that the resolution (which in the general case can be effected in an infinite number of ways) is unique. It is clear that since E is positive whatever values of Q-^, Qj. • • • may be taken, we have jBll > 0, ffl2a^ > 0, »33^ > 0, Determining the coefficients we find easily that these conditions will be satisfied if i'li > 0. >0, PlV i'lS) Vl3 Pl2' Pw Pis Pia> Pii> Pas >0, 128 MAGNETISM AIJD ELECTEICITY chap. In the same way precisely similar conditions are found to be satisfied by the c coeflScients. It is to be observed that by partial differentiation of (29) and (30) we obtain by (27) and (28) Thus 9E/3&, 9E/9 Vj,, are what are called reciprocal functions in dynamics. The properties of such functions will be discussed in Chapter VII., which is devoted to general dynamical considerations. Reciprocal Relations. Exploration of an Electric Field 187. The reciprocal relation ^aj; = 2>kh asserts, as we have seen above, that if a charge exist on A^ and all the other conductors be insulated without charge, the potential at the conductor A^, is the same as that which would be produced at A^ if the same charge were on A^, and all the conductors except Aje were insulated without charge. We can show that the same result holds if any or all of the conductors other than Ak, Ale are maintained at zero potential. For let all the conductors which are at zero potential be regarded as constituting a single con- ductor Ag, of which the charge is Qs ; and let Q be the charge on Ak. Then Vk = phhQ + PksQs T's = VhsQ + PssQs = 0, and therefore Vk = iphk — -pksjQ- If the charge were transferred to Aje, and Au insulated without charge, we should get in the same way Yh = [pkh — -phsJQ, Ps, and since p^h = Phk, we get rn=V, (32) 188. The reciprocal relation just discussed enables the electric field due to a charged conductor to be conveniently explored. The conductor is insulated but is kept uncharged, and one electrode of a delicate electrometer is connected with the conductor, while the other electrode is connected with the earth, or with some conductor the potential of which is taken as zero. Then a small charged sphere carried by an insulating handle is placed with its cenjire at any point of the field. While this is in position the electrodes of the electrometer are connected for an instant, so that the conductor is reduced to zero potential. The sphere is then moved from point to point in the field, and the positions V ELEMENTARY THEORY OF ELECTROSTATICS 12» noted for which the electrometer shows no deflectioD. These lie on su surface which is an equipotential surface for the conductor when the latter is charged. For the potential at the conductor due to the electri- fication of the sphere is equal to the potential which would be produced at the sphere by a charge on the conductor equal to that on the sphere ; and this part of the potential is the same for all positions of the sphere for which there is zero deflection. By the principle of superposition, this must be an equipotential surface for J all charges of the conductor. The convenience of the method consists in the zero potential of the- conductor, which therefore does not lose or gain charge, while the exploring sphere, which can be insulated so that its charge remains practically constant, is changed in position in the field. Nature of Charges in Conductors in Different Cases 189. The full discussion of equilibrium distributions of electricity we defer until the subject of potential has been more fully treated, as it will' be in the chapter on Fluid Motion, which is given below. But fromi what has gone before there flow easily a number of useful propositions. It will be readily seen that of a system of conductors there is always- at least one, the electrification of which has at all points the same sign. If the conductors in the field have all positive potentials, or all negative- potentials, that one is the conductor whose potential has the gi-eatest numerical value apart from sign. If, however, the potentials of the- conductors are some of them positive, and some negative, the two conductors which are respectively at the algebraically greatest and the algebraically least potential have each electrification of the same sign at every point. The consideration of tubes of induction will show readily whether the electrification is all of the same sign or of opposite signs on one con- ductor. For example, if there be only two conductors in the field, and these have equal and opposite charges, all the tubes which originate on one terminate on the other, and, as may be seen by considering a surface distant from and enclosing both conductors, there can be other tubes in the field, hence the electrification is wholly positive over one and negative over the other. If an insulated conductor have zero charge and be placed in presence of another charged conductor, the electrification of the latter will have the same sign at every point, that of the latter will have the positive sign at some points, the negative at others. Energy Change due to Relative Displacement of Conductors under Different Conditions 190. Another theorem to which the reciprocal relation (24) at once leads is that if an electric system be displaced, subject to the condition that the charges are kept constant, the loss of energy entailed on K 130 MAGNETISM AND ELECTRICITY chap. the system mimis the gain of energy when the same displacement takes place, subject to the condition that the potentials remain constant has the value J t{(Q' — Q){V'—V)} where Q is the constant value of the ■charge of a conductor in the first case, and V— V is the change of potential, while V is the potential in the second case, and Q' — Q the ■change of charge. The loss of energy in the first case is ^'tQ{V— V) and the gain in the second is \% {Q' — Q) V. Hence the difference specified above is ^{2C(F - V) - %{Q' - Q)V}. But Qi, Q2 , V\, V\, .... are charges and corresponding potentials ■after the displacements in the first case, and Q\, Q'^ Fj, V^, . . . . the charges and potentials after the displacement in the second case. The configurations being the same this represents two states of the same system, and we have by the reciprocal relation 'ZQV='ZQ'V'. Hence t {Q-Q') Fbecomes tQ\ V- V). Thus h{%Q{V- V) - %{Q' - q)V\ = |}S<3(F- V) - W{V- V')} = mQ' - QKV -V) . . (33) which was to be proved. If the displacement be very small Q'— Q in the one case and V'—V in the other will be very small quantities for each conductor, and their product is a quantity of higher order of smallness. Hence to the degree ■of approximation involved in neglecting the quantity on the right of (33) the loss in the first case is equal to the gain in the second. 191. If now the conductors be allowed to alter their configuration mechanical work will be done, and the potential energy will be •diminished to the same extent. Then if the potentials be restored to their former values, and the charges be correspondingly altered to enable this result to be effected, the energy supplied to the system must be sufficient to make up the energy consumed in doing the mechanical work and to bring up the energy to its former value, and the further amount needed to supply the gain which the energy would have received if the potentials had been kept constant during the displace- ment. Thus if W be the work which must be spent on the system in the latter case, there must be supplied from without to the system in the latter case a quantity of energy 2W+^'t (Q' — Q) (V'—V) or approximately 2 W, which also must be the energy supplied when the •condition of constancy of potential is imposed throughout the change. Characteristic Equation of the Potential 192. The characteristic equation of the potential has been dis- cussed above (p. 46), and will be otherwise demonstrated later. It ■expresses simply the surface-integral of electric induction for a small rectangular parallelepiped of the medium. For the sake of clearness we give a proof in this connection also^ taking in the case of an seolotropic medium. V ELEMENTARY THEORY OF ELECTROSTATICS 131 Let the element be taken with its centre at 0, the origin of co- ordinates, and its edges parallel to the axis of co-ordinates x, y, z, and of lengths dx, dy, dz. Let / be the normal cojnponent of electric induction {D) at the centre of the face (of area dydz) to the left of 0, and/j that at the centre of the opposite face, and similarly let g^, g^, \, h^, be the normal components at the centres of the other pairs of faces. If the values of/jj/g, . . . vary continuously over the faces of the element, we have, neglecting infinitesimals of a higher order, for the part of the surface integrals due to the three pairs of faces, (fi -f^dydz + (gr^ - g^)dzdx + {h^ - h.^)dxdy. If p be the average amount of electricity per unit volume within the element, equating the sum of these expressions to 4iirpdxdydz and dividing by dxdyds gives the result 4^1 + £^1 + ^ii_:A = 4.p . . . (34) dx dy dz By the definition of the electric induction it is plain that if p be everywhere finite within the element the values of /, g, h will be continuous, and we shall have ..§/", 3ff , - , 3A - /■) =/i + r- aa;, fl-j = S'l + ^ dy, h., = h^ + —dz, dx dy dz and therefore instead of (34) If the medium be aeolotropic, that is if the relation between electric induction and electric intensity be that given by (3), we have 7\ Ti 7\ - 4tp = (36) with the conditions hy^ — T^^i' ' If the medium be isotropic, we have h-y^ = k^^ = h^^, ^2 — ^"'21 = 0> and therefore ^ (kP) + ^ {kQ) + ^ (kR) - 4^p = 0, dx dy ^ vz ' or since (Art. 155) P = — dV/d.v, ; s(4)-i('f)-l(4:)— • <-) ;' At every point of space at which p = the equation K 2 132 MAGNETISM AND ELECTRICITY chap. holds. For the case of h uniform throughout the field this is what is known as Laplace's equation, from its having been first discussed by Laplace in connection with the subject of gravitational attraction. It will be proved later (p. 139) that if the potential have an assigned system of values over the iDounding surface or surfaces of a simply con- nected space, then to a function F found to fulfil this equation as well as the surface conditions there will correspond a value of the electric energy less than that for any other value of V which fulfils the surface conditions but does not satisfy (37). It will be shown, moreover, that (as has already been stated) if there can be found one function V which satisfies definite surface conditions and (37) throughout the field, it is the only function that can be found to fulfil them, and is therefore the solution of the problem ; — ^given the surface distribution of potential (or its equivalent) find the corresponding value of V for all points of the field. From (38) is to be found of course equations (10), (11), (12) which have already been established for an electrified surface. It is only necessary to take the direction of the axis of x as the normal to the surface at the point considered, and to put a for pdx, supposed finite when dx, the thickness, so to speak, of the surface layer, is taken infinitely small. If we draw normals n^, n^ from the surface towards the spaces on the left and right of the surface respectively, and suppose Tc-^, \ to be the values of k on the two sides of the surface, we get instead of (10) the equation Ai^ + /t,^ + 4,ro-= ...... (39) or calling N^, N^ the components of electric intensity at right angles to the surface on the two sides yfejiVj - k^N^_^ - 4ff(r = (39') Equation (39) or (39') is usually called the surface characteristic equation. Different cases of it have already been exhibited above (p. 115). Electric Induction and Potential in Particular Cases 193. We shall now consider some simple particular cases of electric charges and the resulting electric induction and potential. Let first the electric system consist of a point-charge gj of electricity situated at a point the co-ordinates of which are a^, &j, Cp a quantity q^ at {a^, \, c^, , in a medium of uniform inductive capacity Tc, and consider the electric induction at {x, y, z). The induction at (x, y, z) due to q^ at (a^ \, Cj), is radially directed from {a^, b^, Cj), and is qjr^ where r^^ = {x — ajf + (y — &j)- -|- (z — Cj)^- Hence its components are V ELEMENTARY THEORY OF ELECTROSTATICS 133 The components of induction due to the other charges are obtained from this by simply substituting the suffixes corresponding to the co- ordinates of their positions. Calling /, g, h the components of the resultant induction, we obtain (/, 9, h) = S|^ {x-a,y -b,z- c)\ where % denotes summation for all values of a^, S^, c,, a^, &2' "2 '^^ a, b, c. If the point-charges form a continuous volume distribution we may ■denote the volume density at any element of the space by p, and replace the summations in the formulae above by integrations taken throughout the space occupied by the charges. Thus we get putting' d'us for the element of the space at which p is the density J r^ \dx Zy dz) If a- denote the electric surface density at a point a, b, c of an electrified surface, and c^^S'the area of an element including the point, these equations become The I'esultant induction is of course given in all cases by the equation 2)2 =/2 + g^ + h\ In the case of a uniform isotropic medium with induction and electric intensity both in the same direction, we have for the intensity the relation E = ?-, or {P, Q, E) =-^-L^ . . . . (43) by means of which the;value of E and its components are to be obtained from the formulae above. But since if V be the electric potential we obtain 134 MAGNETISM AND ELECTRICITY chap. Having regard to the definition of potential given above (Art. ] 56) we see that the potential at P due to a point-charge q of electricity situated at any point 0, the distance of which from P is r, is qjhr. For we have for the electric intensity at any point Q, the distance of whjgh from is x, the value qjkz^. The work spent by external forces in moving a unit charge nearer to § a small distance dx is qdx/kx^. Hence if V be the potential at P = \^'^-i • ■ • • (^^) The difference between the potential V at P, and the potential V at the point P' at a distance r' from 0, or the work spent in carrying a unit of positive electricity from P to P, is therefore F - F' = ^ - ^, (46) r r It is important to remark that this value is independent of the path pursued between P and P. It depends only on the distances of the points at which the charges are situated from 0. Further, the potential due to a series of point-charges ^'j, q^, q^ at distances r^^, r,, 1\, . . . from P is given by the equation \ii'^'\'-i <") F=2 For a continuous volume distribution we have and for a distribution partly volume and partly surface '' = \Fr''''^\r/''- ■ ■ ■ ^'') ■ In all these expressions the respective integrals are taken throughout the distribution, surface or volume as the case may be. In the particular case (already considered in Art. 158) of a sphere uniformly charged with a quantity q of electricity, and alone in the field, the potential is given by X kr V=^ (50) where r is the distance of the point from the centre of the sphere, and is thus the same as if the charge were collected at the centre of the sphere. The potential at the surface of the sphere is q/ka, if the radius s a, and the capacity is ka. V . ELEMENTAEY THEORY OF ELECTEOSTATICS 135- Approximate Values of Coefl5cients of Potential and Induction 194. We can now find an approximation to the coefficients of potential and induction in one or two particular cases of a system of conductors. For example, let the system consist of a sphere A of radius a, and a sphere B of radius 6 at a distance r from the centre of. the former. Let first the distance apart of the centres of the spheres be very great in comparison with the radius of either. Let the charge of A be 3i and of B q^. Then if we regard the actual field as due to the superposition of the field due to A upon that due to B, it is clear that on account of the great distance of any part of the surface of A from B, the field around A is to a correspondingly small extent influenced by the charge on B; and similarly the field round B is influenced to a like small extent by the charge on A. The potentials of A, B are therefore,, approximately, ^ ka kr 2 - kr ^ kb (51) These equations give the charges if the potentials are known.. Thus— kbr '>.... (52), ab If Tg = 0, (52) give or to a rougher approximation g'l = kaV^, 9'2 = - -9i (54:) which can be seen at once to be values which will nearly satisfy the conditions. The effect of the alteration of the distribution on each sphere due to the presence of the other is of course what is here neglected. 195. Regarding a condenser as an arrangement of two conductors, such and so close together that the coefficient of induction of either on the other is very large, we might find in a precisely similar manner the action of one condenser, A, on another. A', at a distance very great in comparison with that anywhere between the constituent conductors of either condenser. Denoting the capacities of the conductors of the condenser A by a, c, the mutual coefficient by &, and the corresponding 136 MAGNETISM AND ELECTRICITY CHAP. ■quantities for the condenser B by e, J\ g, we get for the coefficients of potential of the system of two conductors A b-^' 62 and for those of the system B 9 Pii = _ /2' eg -J Pii ey -r^ Pii ^ pil = f eg -P •values which will not be appi'eciably altered by bringing one system into presence of the other at a great distance r. Supposing the charges q^, gj, (i\, g'g of the two pairs of conductors A, B given, the coefficients just written down enable the corresponding potentials V^, V^, V\, V\ to be calculated by the introduction of !//«• as the mutual coefficient of potential of either conductor of A on either ■conductor of B. Thus four linear equations are obtained for the potentials, which can then, if the potentials be supposed known, be solved for the charges. It is easy to verify that this process gives the following coefficients of capacity and induction in which D is written for fcV _ (a + 26 + c; (e + 2/ + g), c„ = a + {a + hf{e + V + g)~^ C22 = c + {h + cf{e + 2f+g)jj c^i = 6 + (a + 6) (6 + c) (e + 2/ + g) B) (55) ■with similar coefficients for B obtained by interchanging a, b, c with ■e, f, g respectively. The coefficients of induction between a conductor of one condenser and a conductor of the other are (56) -^iB, = - kr{a + b) (e +/)—, ca,b^ = - kr{a + 6) (/ + g) —, \ ■CA.B, = - hr{b + c) (e +/)^. ca„_b^ = - kr{b + c) (/ + g)—- i If B consists of only one conductor f = g = Q, bo that if B be now H-r- — {a + 26 + c)e = a + e{a + hf—, c,., = c + e{b + cf D Ci2 = 6 + e(a + 6) (6 ^■ c) -=- D c.ijBi = - ker(a + 6) and the other coefficients vanish. D' C.J2B1 -lcer(h + c)—^ (57) V ELEMENTARY THEORY OF ELECTROSTATICS 137 If A also consist of only one conductor, then h = c=f=g= 0, and we have D = /fiV — ae. From these last equations the coefficients for the particular case, already discussed, of two spherical conductors placed at a great distance apart may be at once obtained. 196. It is interesting to note the effect on the potential of a con- ductor A produced by bringing an uncharged conductor B into the field. Let all conductors in the field, except that the effect on which is to be estimated, be without charge. The inductive action of the field on B is such as to assist the bringing on of B, and work is therefore done by the electric system. Thus the energy of A is diminished, that is, \Pv\9.i is diminished. But g^ remains unchanged, so that ^^ is diminished. It is clear in the same way that if an uncharged conductor B be connected to A,p-i^ will be diminished, and it is obvious that B pro- duces a greater effect than does any conductor which can be inscribed within it, and a less effect than does any conductor which can be described about it. It follows that the introduction of a body B without charge increases the capacity of A. For in this case the capacity Cjj of A, that is the ratio of its charge to its potential, is l/Pn- But it has been shown that jjj^^ has been diminished, consequently c^j has been increased. The addition of a conductor B without charge to A also increases the capacity, and the effect of B in this respect is greater than that of any conductor which can be inscribed in B, and less than that of any conductor which can be described about B. It is clear thus that the capacity of a conductor, or system of con- ductors, all parts of which are at the same potential, is less than that of the circumscribing spherical conductor. The capacities of a system of spherical conductors which circumscribe a given system of conductors form also a superior limit to the capacities of the conductors. As an example consider either of the condensers discussed above. If both of its conductors be at the same potential, unity say, the charges of the two conductors will amount to a + ^h + c, or e + '2,/ -\- g as the case may be. By what has just been stated this cannot exceed half the greatest linear dimension of the condenser. o' Determination of Field within and without a Conductor by Potential Method 197. We have seen that since a conductor cannot sustain or transmit dielectric strain, its substance forms an effective barrier against the penetration to the space within it of any effect due to external electri- fication. There is thus no electric field within a closed conductor 138 MAGNETISM AND ELECTEICITY chap. which contains no insulated electrified bodies. Now if we assutne that we may apply the theory of the potential (which is, as we have pointed out, an action at a distance method of procedure) to the space within a closed conductor, the distribution on the surface must be .such as to produce constancy of potential throughout the internal space. In so doing we consider the conductor as non-existent, so that the whole of the internal space is regarded as having the same property of transmit- ting electric strain as the external dielectric, and consider only the electric distribution and the external and internal fields so regarded as due to it. The distribution on the external surface of a closed con- ductor is thus always such as to produce constancy of the electric field within the whole space contained by it, inasmuch as it is independent of any internal charges. This will not in general be true for the distribution on the internal surface of a closed conductor, as this must be such as, with the distribution insulated within the internal space, to maintain the conductor at a constant zero potential. For it is clear that if the conductor be brought to zero potential, that is, produces no external field whatever, no tubes of induction terminate on its external surface, that is, there is upon its external surface no charge whatever,, and the internal charge is unaffected by this circumstance. Now it is found by experiment in all cases that have been examined that the equilibrium distributions on the external or internal surfaces are such as to fulfil these conditions ; and we further find by this method, and the conditions as to constancy of potential, solutions of problems which are consistent with the results of experience. Electrical distribution in general will be treated in a subsequent chapter, but we may here illus- trate the potential method by a few problems, although by doing so we anticipate to a slight extent the subsequent discussion. 198. The theorem of the surface integral of electric induction which we have seen holds for any system of point-charges, applied to a con- centric spherical surface of radius z described within the space, internal to a spherical conductor, shows that since there is supposed to be no electricity within the surface, the electric induction and intensity are there zero. For by symmetry of circumstances the induction D, if riot zero, must be at every point at right angles to the concentric spherical surface referred to. Hence we have, if z be the radius of this surface. that is or since D = — /i-3 Vfbx 4«2D = 0, D = 0, The potential is therefore constant. The value of the potential must. V ELEMENTARY THEORY OF ELECTROSTATICS 139 therefore, throughout the interior of the spherical surface, coincide with the vahie at the surface. It may be noticed that, if proof were needed, the same method might be applied to show that the electric intensity at any external point, due to the spherical distribution, is the same as if the whole charge were collected at the centre ; but this is quite sufficiently evident from the considerations adduced above. Surface Distributions consistent with Surface Values of Potential Green's Problem 199. The problem which will occupy us to a great extent in the discussion of distribution is the determination in certain soluble cases of the distribution over a closed surface produced by the presence of a* given external or internal distribution. An example of such a problem is the calculation of the induced distribution on a spherical surface connected with the earth produced by a single point-charge at an internal or external point. We have seen that if we have an electrified surface the normal components of induction on the two sides of it are connected (see (39) above) by a certain relation with the density of the distribution. Thus the equation '- = 1-/'- may be replaced for a medium everywhere of uniform ind activity h, by F =-—[-('— + —\lS 4;rjj' \dn^ dnj The disti'ibution over a surface, or system of surfaces, given in an electric field (of uniform inductivity li), which is consistent with an. arbitrarily chosen potential Fg at each point of the surface, and with the fulfilment of Laplace's equation wherever there is no electrification by the potential V elsewhere than on the surface, is that of which the density is given for each point of the surface by the equation For assuming that the strrangement of potential proposed can physically exist, let us suppose it made. There will be called into existence some, distribution on the given surfaces. (A particular case which would arise, under certain circumstances would of course be a zero surface distribution.) The condition in (58) holds for every case of a possible surface distribution : it must accordingly hold for this. It remains therefore only to show that there cannot be more 140 MAGNETISM AND ELECTRICITY chap. than one distribution fulfilling this condition, and the other conditions of the problem. If possible, let another potential V, at other points of the field than those at which the value of the potential has been assigned, be con- sistent with the given surface values of the potential, and the given distribution of electricity elsewhere in the field. Then V, at the given surfaces, must coincide with Vs. It is clear that a potential — Vy having the value — Vg at the surfaces, is a solution consistent with change of sign of the electrification everywhere from that which pro- duces -t- F^. Hence the potential V -V■^ is a solution consistent with zero electrification everywhere in the field, with zero potential at the given surfaces, and zero potential at an infinite distance. Hence the potential must be zero everywhere else, otherwise there would be a region of maximum or minimum of potential in space void of electri- fication, which we have seen above to be impossible. Thus if Fj coincides with V at the surfaces it coin- cides with V everywhere else, and the solution obtained is the only one. 200. We come now to an important method given by Green for the calculation of distributions. Let The function U— 1/hr is generally denoted by G, and the name Green's function has been applied to it by Maxwell and others. Thus G is the potential at P due to the induced distribution on the bounding surface. It follows from the reciprocal relation established above that the 142 MAQNETISJI AND ELECTRICITY chap. potential Gpp, produced at any point P' by the induced distribution on the bounding surface when there is unit charge at P, is equal to the potential G,^p at P due to the induced surface distribution when there is unit charge at P. This may be proved independently as follows : — Let a, approximately. This may be regarded as holding exactly when a (and therefore also/) is infinite. But when this is the case we fall on the distribution on an infinite plane maintained at zero potential under the influence of a positive unit point-charge situated at P. The image- charge — a// is here —1, and is situated at a point 7*, on the normal PA produced behind the plane, such that PA = AP', that is at the optical image of P in the plane regarded as a reflecting surface. Thus the density, at any element JS of the plane, of the negative induced distribution varies inversely as the cube of the distance PH of the element from P. A figure is unnecessary. That this arrangement of inducing charge and its image produces zero potential at every element of the plane is obvious since the distances PJ", P'H are equal. The electric intensity outward from M, into the region in which P is situated, given by the arrangement is clearly — 2h/i^, and hence since this is 47ra- we get the result expressed in (71) which is thus verified. Nothing is more easy than to verify by direct integration that the total charge on the plane is — 1. 203. If the sphere be now supposed uninsulated and charged to a uniform potential V, the distribution upon it will be the distribution just determined together with a uniform distribution of density a- = ^ Vj^sira. For the two distributions being separately possible may ELEMENTARY THEORY OP ELECTROSTATICS 145 be superimposed to give a possible distribution which -will make the potential at the sphere V, and afford an induced distribution due to the point-charge (q say) at P. It will therefore be the only possible solution. The potential at a point Q at distance B from the centre of the sphere will therefore be (1) when Pis external (/> a), '^-^i*^ for all external points, and for aU internal points ; (2) when P is internal (f<,a) PQ a q WpQ (72) for all external points, and F-<3= F + E a q Wpq (73) h . PQ for all internal points. The surface density at any element E will in case (1) be 1 isra hV- iP '■•)f.} (74) and in case (2) (if the two distributions be taken together, as they may be if they are supposed to be on an ideal single surface) -4^{^^-(«^-^%^} (75) Of course in the actual physical case of a hollow spherical conductor the uniform distribution, represented by the first term on the right of the last equation, is on the external surface, and that represented by the second term is on the inner surface, the two being physically independent in the manner already explained. The reader may prove for P external that if the total charge Q{ = kaV—qalf) of the sphere fulfil the inequality qa' >Q> - qa^ there is a circle of points on the sphere at which the density of the distribution is zero ; and may find its position and may verify that the densities at A and A' are given by the equations for A iovA' 4ira \ 4Tra \ /+ a . / + a \ -a)V '{f+a) f a \ (76) (77) L 146 .MAGNETISM AND ELECTRICITY chap It will be noticed that when the inducing charge is q at P, we have for Green's function, or the potential produced at a point Q by the induced charge, ^ = -^?l^Q ^''^ Avhether P be external or internal to the sphere. It is to be remembered that in the former case/>a, in the latter / with respect to 0. If any system of points P, 6, ... be given, a cor- responding system of inverse points F, §',... can be found, and if the ^ See ElectwstaUcs ami Magnetism, 2nd Edition, p. 144 etseq. . 148 MAGNETISM AND ELECTRICITY chaPj first form a definite locus, the latter will form a • corresponding derived locus. We shall call the first the direct system, the latter the inverse' system of points. Of course, if P', Q', . . . be regarded as the direct system of points, the corresponding inverse system is F, Q, . . . with regard to the same centre. Each point P' is the image of the point P in the sphere of radius a and centre 0. This is called the sphere of inversion and its radius the radius of inversion. The triangles OQP, OP' Q' in Fig. 54 are similar, and therefore the' angle OQP is equal to the angle OFQ'. Thus if P, Q be very near points, so that OP, OQ are nearly parallel to one another, the angle OQP is nearly equal to the angle P'Q'Q, that is, the line QP is inclined to Q<^ at the same angle as that at which Q'P' is inclined to Q'Q. Hence the inverses of any two lines or surfa.ces intersect at the same angle as do the original lines or surfaces. The inverse of a circle is another ; circle, and therefore that of a sphere is another sphere. For let PQ (Fig. 54) be the extremities of a diameter of the circle, and B any other point 'on the circle, then PPQ is a right angle. The inverse points are P', Q', B' and the a,ngle FB'Q' is equal to a right angle ± the angle POQ, according as OB does or does not intersect PQ. Hence, as B moves round the circle, B' moveS round another circle which is the inverse of the former. " ' i If h be the radius of the circle (or sphere) and C its centre, the circle (or sphere) inverts into itself when OC^ — h'^ = a^. The inverse of a straight line is a circle passing through the centre of inversion. For let P be a point on the straight line such that OP is at right angles to PQ, where Q is any other point on the line. Then if F, Q' be the corresponding inverse points, P' is fixed, and OQ'P' is a right angle for every position of Q', and Q' is the inverse of ail points on the line which are infinitely distant from P. Hence the locus of Q' is a circle of which OF is the diameter. It follows from this that the inverse of an infinite plane is a spherical surface passing through the centre of inversion. According as the centre of inversion is without or within the surface inverted, the space within the inverse is the inverse of the space within or without the original surface, and the space without the inverse is the inverse of the space without or within the original surface. Electrical luversion. BeriTation of Induced Distribution from Ec[uilibrium Distribution and Vice-versa 207. The point P* which We have called the electric image of P with regard to the spher€s AEA' is, it will be observed, the inverse of the V ELEMENTARY THEORY OF ELECTROSTATICS 149 point P with regai'd to the same sphere, taken as sphere of inversion. It has been shown that a charge qajf at P' will produce a potential at every point of the spherical surface equal to that produced by q at the point P, while, according as P is external or internal to the sphere, the potential due to the charge at P' will be a harmonic function for all external or all internal points. We shall call this the inverse of the charge q at P, reserving the term image-charge or image-distribution for the inverse distribution with sign reversed. If instead of a single point-charge at P there be a system of point- charges 2i, 22> • • ■ at points Pj, P^, . . . without or within the sphere, of inversion, a system of point-charges- q-fl\f-^, q.f'jf^, ■ ■ . situated at points P\, P\, . . . will be the inverse distributions, and will produce the same potential at the sphere as does the former system of charges. Thus to any distribution without or within the sphere of inversion corresponds another distribution which is within or without the sphere, and is the inverse of the former. The space or surface, as -the case may be, occupied by the inverse distribution is the inverse of that occupied by the given distribution. We easily see that if dxs, dzs', dS, dS' be elements of volume and elements of surface in the direct and inverse distributions, and p, p', 0-, 0-' denote the volume densities and the surface densities in the two cases, and d7S' dxs a6 ■ a^ p' dS' a^ /'* d8 /* a* y'2 P a' P a? 0- a? /'» (82) 208. If y be the potential at any point Q due to the point-charge q at P, we have V = qlQc.PQ). The potential produced at Q', the inverse of Q, by the corresponding image-charge at P', is qa/(fJc.P'Q'). Hence if r, r' be put for OQ, OQ' we easily find V r a V a r' Accordingly, since this is a ratio independent of the position of P, if V, V denote respectively the potentials at the points Q, Q' due to the whole direct and inverse distributions respectively, we obtain r^-V=-,7 (83) a r Thus if F is a constant over any surface S, V is not a constant over the inverse surface ;S' unless r is a constant, that is, unless the equipotential surfece is a sphere concentric with the sphere of inversion, when its inverse is concentric with it and is an equipotential surfacd of the inverse or image-distribution. 150 MAGNETISM AND ELECTRICITY chap. In the case in which F is a constant and r variable, we may reduce the potential of the inverse surface &' to zero by placing at the centre of inversion a charge — ha V. This will produce at any point at distance r' from the potential — a Vjr' which is equal and opposite to V or a Vjr'.. Thus if S be the surface of a conductor on which the direct charge is in equilibrium, the addition of this point-charge at to the inverse dis- tribution shows that the latter is the induced distribution on the inverse surface produced by a charge — aF at 0. The image-charge on S' corresponding to that on S is thus the induced distribution on S' due to^ a point-charge a V at 0. Thus by the process of inversion we obtain an induced distribu- tion on the inverse surface from a given equilibrium distribution all at one potential, or conversely obtain from a given induced distri- bution on a surface, a natural equilibrium distribution on the inverse surface. If S be not the surface of a conductor, but one of the equi-poten- tial surfaces of the given distribution, its inverse S' is maintained at zero potential by the inverse distribution and the charge —aV&tO. Generally the given distribution lies part within part without any chosen surface S. Let these parts be denoted by g^, g'g respectively, and their inverses by q\, q\. Then if be without >S', q\ is within and q'^ without S', and vice-versa if is within S. Now consider the distribution made up of the two parts of the inverse distribution q\, q\, and the charge — a, V at 0, and suppose all points of /S' to be at zero potential. The whole quantity of electricity ■within S" properly distributed over that surface will in each case produce the same potential at all points outside the surface, as is produced by the internal distribution. If the internal distribution be annulled and this surface distribution be substituted, the total potential at all external points will be unaltered, while that of each point on the surface and within it will be zero. The amount of the charge thus distributed will be q\ if be without S, and q'o — aV in the other case; while in each the density will be { — kdVjdnflA'n; where dVjdn is the rate of change of the total potential outwards from the surface >S". Inversion of Uniform Spherical Distribution. Problem of Two Parallel Infinite Conducting Planes with Point-Charge between them 209. We may illustrate the method of inversion by applying it to one or two simple examples, reserving more recondite ' cases for later discussion. First of all we shall invert a uniformly charged sphere. Let the potential of the sphere be denoted by V, its radius by /3, the radius of inversion OA (Fig. 55) by a, the distance of the centre of inversion from any point F of the image by r', the distance, of the same point 9 V ELEMENTARY THEORY OF ELECTROSTATICS nx from the centre of the inverse sphere by/, and the radius of the inverse sphere by a. We have /3 = ± "" ■/'- a^ according as is external or internal to the given sphere. But j-'3 r"6 iirjS' Hence, substituting the value of /3 just found we obtain /2 - a2 kVa a- = 4:Tra r"^ according as is external or internal. According as is external or internal to the given sphere, and is therefore external or internal to the inverse, the spaces external and iO Fig. 55. internal to the former are respectively the spaces external and internal or internal and external to the latter. Hence, according as is external or internal, the potential of the inverse distribution at every iuternal point or at every external point of the inverse surface is the same as that of a charge k Va at 0. Also, since the potential of the given sphere is the same for all external points as if its charge F/3 were concentrated at its centre C, the potential of the inverse distribution is the same at every point, external to the inverse sphere when is external, and internal when is internal, as that of a charge q' =k Vj3.a/0C concentrated at the image I (Fig. 55) of the centre of the given sphere. But OC = + a^Kp - a^) and /8 = + a^aKf^ — a?) according as C is external or internal. Hence q' = -kYa, that is, the charge is the image in the inverse sphere of the charge -kVa at 0. 152 MAGNETISM AND ELECTRICITY CHAP. For 0/we have OI.OC ^ a\ or 01 = ± P- f that is, / is the image of in the inverse sphere. These results are those which have already been obtained above (pp. 143 et seq.). 210. Next we shall find the induced distribution for the case of two infinite parallel planes with a point-charge between them. This will afford an example of the method of successive influences introduced first by Murphy ' for the solution of the problem of the mutual influence of conductors. We shall then invert this system and from it obtain the distribution on two mutually influencing spheres. Let AB, Fig. 56, be the traces of two parallel planes on a perpen- dicular plane through F, a point between them at which a charge of y. ^ A S Fig." 56. amount g is situated. Let a, j8 be the respective distances of P from G, D, the points in which a perpendicular through P meets the, planes, and E a point on the plane ^ at a distance 7 from G. The planes are to be supposed rnaintained at zero potential, and it is required to find the induced distribution upon them and the field at any point between them. If the plane B were removed the density at E due to g at P would l>y (71) be .— qa/2ir(a^ + 7^)?. But the induced electrification which q at P would induce if A were removed, produces the same field to the left of P as that due to a charge — g' at a point /, distant from I) on PP produced, and the corresponding electric density at P is therefore 7 a + 2/8 2,r {(a + 2/3)2 + /}?' These two electrifications of A produce respectively the effects on the electrification of P of charges —q, +q to the left of A on PC pro- 'Murphy's Electricity, Cambridge, 1833. V ELEMENTAKY THEORY OF ELECTROSTATICS 153 duced, the former at a point Jj, distant a, the latter at a point J^ distant « + 2/S from C. The electrifications of B thus produced have on A the effects of charges +g, — g' at points I^, /g distant 3a + 2/3, 3a + 4y8 to the right of A on GB. The corresponding densities at E are therefore 9 3a + 2;8 2 3a + 4^ 27r {(3a + 2(8)2 + y2}t 27r{(3a + 4^8)2 + y^)! In the same way another pair of densities at E could be found corres- ponding to point-charges -(-g, — g at the respective distances 5a -|-'4/8, 5a 4- 6/3 to the right of A, and so on. The electrification of A is that which would, if B were removed, be produced by -|- g at P and an infinite trail of images I^, I^, . . . of charges — 2, +3. — 2, . . . at points to the right of P on OB produced. The potential at every point of A or to the left of it is plainly the potential due to 4-2 at P, and the image-charges to the right of P; and this is equal and opposite to the potential at the same point produced by the electrification of A. Similar results hold for B and the images to the left. The potential at any point between the planes produced by the ■electrification of either is that due to the trail of images behind that plane, and the total actual potential at any such point is the sum of the potentials due to g at P and the two trails of images. To verify that the potential of each of the planes is zero let V be the potential at any point E of the plane A. Then k \PE J^e) ^ k i ^\hJ! ~ h^^l " ^y^ ~ J^:^ ) ' ^^^^ where n has every integral value froln to oo. Since J-^E = PE, the first term is zero ; further, each series is con- vergent, and the terms of the two series (which are arranged in the same order in both) are identically equal. Hence F" is zero, and the above process gives the required result. The charges and distances of the images Jj, I^, . . . are given by the table Images I-rn-x Iin Charges - q + g Distances from P 2{n - l)a + 2»i/3 2n{a + /3) where n has every positive integral value from 1 to oo. The charges and distances of the 'images J^, J^, . . . are given by the same table when a and jS are interchanged. 151 MAGNETISM AXD ELECTEICITV CHAP. Clearly the density o- at ^ is given by the equation (2to + l)o + 2(» + 1)13 {[{2n + l)a + 2{n + l)Pf + y^}' (2m + l)a + 2nl3 :S[: {[(2w + l)a + 2nfif + y2}»J (85) where n is any positive integer. The density at any point ^ on £ is given by this equation with a and /8 interchanged, and 7 taken as the distance from D to F. Distribution on Two Spheres in Contact Obtained by Inverting Induced Distribution on Two Parallel Planes 211. We now invert the sokition just discussed. Let the centre of inversion be F, the radius of inversion a, and let the planes and suc- cessive images be inverted, omitting the charge at F. The inverses of the planes are spheres touching at F, as shown in Fig. 55, and th& Fig. 57. distribution on either sphere is the inverse of the distribution on the corresponding plane. For the inverse charges corresponding to the trail of images Jj, I^, . . . , and their distances we have Images Charges Distances from P — qa 2(n - l)a 4- 2np a 2 2(w - l)a + 2w/3 2«(a + /3) where n has every positive integral value from 1 to 00. The table for the images J-^, J^, is formed frohi this by interchanging a and /?. V ELEMENTARY THEORY OF ELECTROSTATICS 155^ The diameters of the spheres A, B are respectively a^ja, a^j^, and therefore the inverse charges corresponding to Jg, I^, . . . are within the sphere B, and the other series within the sphere A. The potential at any point on the planes or behind them is zero, and therefore the potential at any such point due to the distribution on the planes is equal to that of a charge —q situated at P, that is —q/r, where r is the distance of the point from P. The potential at any point on or within the spheres is therefore —q/a, a constant quantity. Again, since the potential produced by the electrification of either plane at any point on the plane or in front of it is the potential due to the trail of images behind the plane, the potential at any point on or external to either sphere produced by the distribution on the sphere is the potential at that point of the trail of images within the sphere. The charge on each sphere is therefore equal to the sum of the image- charges whose positions fall within it ; and the distribution thus found is the equilibrium distribution when the spheres are freely electrified in contact. In carrying out the solution we shall in inverting reverse the signs of all the charges, so that J^ will be +qja. Denoting the charge on the sphere B by Qj,, and the radii of the spheres A, B by 7\, r^ respectively, summing the image-charges, and substituting in the result V for +q/a, a^j2r^ for a, a^l2.r^ for /3, we get I. 7l=» Qs^W^^^^S, ,,, -^ , ^, . (86), whete, as indicated, n has every integral value from to oo. A similar expression with j\, r^ interchanged holds for Qj^. The capacities of the spheres are of course equal to these expressions divided by V. Multiplying the expression for the density at any point of the plana A by a^//* [see (82) above] where r' is the distance from P of the corresponding point E on the sphere A, making the substitutions already specified above, and besides putting a*(l/'/^ — Ijh'^) for 7^, we get for the density at -^ . _ 2kVr^%^ S r (2w + l)?-^ + 2ttr^ "' ~ '^ -^ [[r'^{(2w + 1)9-2 + ^nr^V + 4»-iV - '"'V]'^ (2ra + l)ra + 2(w + ly^ - " [»-'^{(2m + \)r^ + 2(w + \)r^\' + A:r^^r^^ - r'\^]i_ (87) This expression is convergent (unless / = 0;, and from it the density at any point can be approximately calculated in terms of the potential V, the radii r^, r^, and the distance, /, of the point considered from P. But if / be very small the value of the density is known from other considerations to be very small also, hence the- calculation need not be- 156 MAGNETISi[ AND ELECTRICITy chap. •carried out except for moderately large values of /. That the density is zero when r' = 0, is evident from the fact that the spheres are there in contact, and since in the immediate neighbourhood of that point the surfaces are very close and very nearly parallel, a surface element there is practically within a closed conductor, and there can be no sensible charge upon it. The charge on each sphere may be regarded as the difference of the sums of two harmonic series. Each series is divergent, but the two sets ■of terms taken together as in (86) constitute a convergent series^ and hence the charge can be approximately calculated for given values of V, r^, r^. Equation (86) may be written Q. n=oo r,r„ ^ r.-. rj + r^ -^ {n + 1) {{n + 1) {r^ + r^) - r^} ^ ' n=0 and therefore may be further transformed to 1 -^^ as may be verified by taking as successive terms under the sign of inte- gration the product of the numerator into successive terms of the series \ + 6 + ^ + . . . , and then integrating term by term. This is the form in which the result was given by Poisson.^ Of ■course the corresponding value of Q_i is obtained by interchanging r^ and r^ in the exponential. Case of Two Equal Spheres. Electric Kaleidoscope 212., When r^ = r^, we have = iFj-i log, 2 = -693147 ;i;Frj (90) or the charge on each sphere is to the charge on the sphere when alone in the field and at potential V, as %e2 is to 1. If i'j be small in comparison with r^ n=a) <2^ = AF-^ + AF-?i^5; , ^ ,, = ^Vr, (91) ^l + ''"2 '"l + »'2 «(w +1) , since %{lln{n + 1)} = 1. The charge is therefore nearly the free charge \ M'emoires de I'JnstUut, 1" partie, p. 1. See also Plana, M^m. de I'Acad. des Sci. de Turin, Set. II., t. vii, p. 71, for a fuller deTelopment of Poisson's method and results. V ELEMENTARY THEORY OF ELECTROSTATICS 15T which the sphere would have if alone and at potential V. The mean density is Ic VJiirr^. On the same supposition we have 2 ^2 '"2 ^ (m + If r^ 6 n—O and the mean density is ^k Vir^j'^TTr^. Thus the mean density of the small sphere is to that of the large sphere as ir^jQ, or 1'645, is to 1. This result is important in its application to the interpretation of the result of the method, sometimes employed, of determining electric dis- tribution by bringing a small conducting ball into contact with the charged conductor at different points, so as to receive charges the amounts of which are compared. The electric density at the point before contact is to the mean density on the ball very approximately as 1 is to 1'645. This result holds whether the charged conductor be spherical or not, provided its curvature be continuous round the point of contact over a distance great in comparison with the radius of the ball, and be small compared with that of the ball. In the case of two infinite planes intersecting at an angle 2irjn, {n a whole number) and influenced by a point-charge at a point P between: them, the number of image-charges is »i— 1. They are placed like the images in a kaleidoscope, of which the planes are the mirrors, and P is the object, and consist of charges —q, +q, alternately, when taken in order round the circle on which they lie. The inversion of this gives the distribution on two spheres cutting at an angle ^irjn. We leave the theory of electrostatics for the present at this point, and proceed to a short discussion of steady flow of electricity as a pre- liminary to a chapter on electromagnetism. Other cases of distribution and questions regarding a field occupied in difiereiit regions by different dielectrics will be considered in a later chapter. CHAPTER VI STEADY FLOW OF ELECTRICITY IN LINEAR CONDUCTORS Process of Change from the State of Equilibrium to Another.' Electric Current 213. Consider a condenser so charged that while one plate is at zero potential the other has a charge Q and is at potential V. According to the theory given above, the energy E of the condenser is given by the equation l-\\^',S,s^iQV (1) Stt in which one integral is taken over the surface of the charged conductor, and the other along a line of induction from the charged conductor to the other. Now consider two condensers perfectly similar in size acd all other respects, and so situated relatively to one another that they are without mutual influence when independently charged. Let one of these be charged as just described, and let its charged plate be connected by a fine wire to the corresponding plate of the other, while the second plate of the latter is kept at zero potential. It is found by experiment that when this is done a new state of the condensers is reached, in which the field between the plates of the condenser becomes the same in both cases, and that the intensity, at any given point, of the field of the originally charged condenser is, as nearly as can be observed, half what it formerly was. Thus we have half the former field-intensity and therefore also half the induction ; but, as the area of the charged plate has been doubled, the value of the enregy given by the equation written above has only been halved. In other words the difference of potential between the plates has been halved, the charge has remained constant. If the condensers are not equal in capacity the potential resulting firom the operation stated has the ratio to the original potential of the capacity of the originally charged condenser to that of the .condenser CHAP. VI STEADY FLOW OF ELECTRICITY 159 made up of the two when placed in contact. In all sharing of charges between conductors the resulting distributions are consistent with con- stancy of the total charge, that is with constancy of the surface-integral of induction across a closed surface surrounding all the charged con- ductors, and described in their field. This is the result of the equalisation of potential between the charged and uncharged plates connected together : we have to inquire what has become of the difference of energy. That there is expenditure of energy is due to the fact that a spark may be produced when the contact is made, and in any case heat is produced in the connecting wires. The mode in which the transfer of energy is imagined to take place is described in general terms in Arts. 127, 128 above. The tubes of induction move outwards laterally (from a disturbance of the balancing lateral action set up by making the connection) with their ends on the wire, and the transference goes on until they have taken up the new •equilibrium arrangement, with the same number of tubes in each of the two equal condensers. As the tubes move their energy is in part absorbed by the connecting wires and conductors, along which the tubes are guided, and passes from the medium across the bounding surface of the conductor. The energy thus absorbed is dissipated in heat in the ■conductor, and its total amount is independent of the nature of the con- necting wire, provided the latter is not such as to make any addition to the united capacity of the two condensers. The rate, however, at which the energy is thus transformed and at which the redistribution is effected depends very much upon the wire and its arrangement, as we shall see later. The whole question of flow of energy in the dielectric medium will be fully considered in the chapter which follows on general electromagnetic theory. The amount of energy thus lost from the system may be calcu- lated from the considerations put forward in Section II. of the preceding •chapter (Art. 180). Let the two condensers, instead of being equal, have capacities C^, G^, and have initially charges Q^, Q^ before they are put in communication. Before contact the energy of the system was JGi'/C'i + IQilO^, after contact it is l{Qi + Qi^Wi + ^2)- The loss of energy is therefore UQi(^2 — Qi^ifl^^i^'lfii + ^2)}' which is essentially positive. If t be the time of transition the average rate of passage of energy from the conductors to whatever form or forms it may take, is K<2i^2 - «2Ci)7{- Vb), and Bab for the resistance of the wire between A, B, we express all these results, including Ohm's law, by the equation — r = ^^ W If a, h be other two points in the same conductor, and Bab be the resistance between them, we have Va- Vb Vg- V i ,,, that is, the slope of potential per unit of resistance along the wire is the same at every point. This gives also Va- Vb={VA- Vb)^ (4) ' R ab It is to be observed that equation (2) or (3) defines resistance of a conductor between the cross- sections at which Va — Vb is applied, and also unit of resistance. The former is that coefficient which multiphed into the measure of the current gives the measure of the difference of potential between A and B; the latter is the resistance existing between A and B when unit difference of potential exists between theni, and unit current flows in the wire. It also expresses the physical fact just stated above as Ohm's law, inasmuch as it asserts the propor- tionality of 7 to V^— Fg when the wire is constant, or the proportionaUty of 7 to the slope of potential along the wire. It is not unusual to apply the term electromotive force to the differ- ence of potential between two points or two equipotential surfaces in a homogeneous conductor, when thus considered with reference to flow of electricity from one to the other. We shall, however, generally use the VI STEADY FLOW OF ELECTRICITY 163 phrase in a somewhat different but not inconsistent sense. It is to be carefully observed that neither electromotive force nor resistance is a Jorce, in the ordinary dynamical sense. It is not generally necessary, though it may be often convenient, to regard electromotive force as the cause, of a current. The two things really exist together, and either, if it serves any purpose, may be regarded as producing the other. Flow in a Conductor containing an Electromotive Force. Heterogeneous Conductors and Circuits 217. Equation (2) cannot be taken as fulfilled by a conductor made up of different homogeneous portions put end to end, or by a conductor moving across the lines of force in a magnetic field. For such cases we have where Va, Vs denote as before the potentials between cross-sections A, B, and R is the sum of the resistances of the homogeneous portions of the conductor contained between these cross-sections in the former case, or the actual resistance of the conductor in the latter. In such cases the conductor is said to contain, or to be the seat of, an electro- motive force Bab, or, as we say frequently, an electromotive force B is said to be in the conductor. The total electromotive force producing a current in the conductor is now Va — Vb +Bjs, of which the part Vj, — Vb is frequently called the applied electromotive force. 218. Since in a heterogeneous conductor supposed at rest in a non- varying magnetic field (2) applies in the first case to every part, except any, however small, which includes a surface of discontinuity, the electromotive force is said to have its seat at the surface or surfaces of discontinuity. Its presence is manifested by the existence of a finite step of potential across the surface of contact. In the other case the electromotive force has its seat in every part of the conductor moving in the field, according to a law which we shall discuss in connection with electro-magnetic induction. Consider a closed circuit made up of different homogeneous linear conductors placed end to end, and let B be the sum of the electromotive forces which have their seat in the circuit. Let adjacent points be taken on opposite sides of each surface of discontinuity, so that two points in each homogeneous part close to its extremities are thus obtained. Let the difference of potential between each latter pair of points be measured, taking them in order round the circuit in the direction in which the current flows. The sum of these differences taken in order is equal to the sum of the parts of B contributed by the discontinuities. For going round in the direction of the current from M 2 164 MAGNETISM AND ELECTRICITY chap. a point in one of the homogeneous parts to tlie same point again we have V.i = Vb, and (5) gives V = | (6) But denoting the successive liomogeneous parts in their order round the circuit by the suffixes 1, 2, . . . , n and the differences of potential between the pairs of points in each near their extremities by Fj— F/, Fj— Fg', . . . , F„— Vn, and the corresponding resistances by Bj^, jR^, . . . , Bn, we have "^ R^ E, ■■■■ En R ^' where i2 = ii!i + ^2+ • • • +^'i- Hence 2(r- T") = ^ (8) H is called the electromotive force in the circuit. For Vj, — Vb the difference of potential between two points A, B in a homogeneous part of the circuit we get evidently 7a- 7b = E^ (9) The example of contact electromotive forces, as the electromotive forces at the surfaces of contact of the heterogeneous parts of the circuit are called, most usually given is that of a voltaic cell ; but as the question of the existence of the electromotive forces observed in this case is not without difficulty, we shall not discuss it at present, but pass on to some general statements and some results regarding certain aiTange- ments of homogeneous conductors which are of great importance in practice. Meaning of Resistance. Rate of Production of Heat in Condnctors. Joule's Laws 219. First, the meaning of the resistance of a conducting wire may be put in a somewhat different light. In a homogeneous conductor, to "which (2) applies, a quantity of electricity measured by 7 is transferred from potential V^ to potential Vb per unit of time. The loss of energy is thus 7 (F4 — Vb) per unit of time, and this in the case supposed is found to take wholly the form of heat in the wire. Thus if A be the activity in the wire expended in heat we have A = y(F^ - Vb) = ^^^^^^ = f^ ■ ■ ■ (10) Thus B may be regarded as the amount of energy transformed into heat per unit of time in the portion of the conductor of which B is the ■VI STEADY FLOW OF ELECTRICITY 165 resistance when unit current flows, or 1/^ is the activity spent in heat in the same portion of the conductor when unit difference of potential is maintained between its extremities. The fact that the heat developed per unit of time in different con- ductors is proportional to the resistances of the conductors and to the squares of the currents flowing in them was established by the experi- ments of Joule,^ and the law of development of heat is therefore generally referred to as Joule's law. In no circumstances is a portion of a homogeneous conductor, in which there is no gradient of tempera- ture, cooled by the passage of a current through it, though heat may be absorbed by the passage of a current across a junction of two dissimilar metals or along an unequally heated conductor. Thus B is always a positive quantity. In the more general case to which (5) applies we have for the rate of transformation of electrostatic energy as before 'y(Va — Vb). But y'^B = y(F^ - Vb) + yEjB ■ ■ • • (11) We interpret the second term on the right as the rate at which energy is evolved in consequence of the existence of the electromotive force Uab in the condlictor. The sum of this and the rate at which electrostatic energy is yielded by the system is the rate of evolution of heat. Either of the terms on the right may be negative but not both ; that is, the electromotive force ^ may enable work to be done against a difference of potential, thus increasing the electrostatic energy, or work may be done by the electrostatic difference of potential Va — Vb against the internal electromotive force if that opposes the current. But in all cases a positive value of 7^-B results, that is to say, work of this amount is always spent in the conductor in producing heat. 220. We may consider also a part of the circuit, between the terminals A, B, of which there exists an applied difference of potential Va— Vb, and in which electromotive forces, for example, those due to a cell or cells of a voltaic battery, aiding the current, as well as other electro- motive forces, for example, those due to voltameters or storage cells in which energy is spent in producing electrolytic decomposition, have their seat. This part of the circuit may or may not be heterogeneous. If the sum of the former or positive electromotive forces be 'ZB^b, and that of the latter, or negative electromotive forces, be 'ZB'jb) and B be the total resistance of the part of the circuit, the^ rate at which energy is spent in heat in it is y^B = y{Vj - Vb) + y%EAB - y%E'AB that is y'-B + y%E'AB = y(F^ - Vb) + y%EAB ■ ■ ■ (11') The right-hand side of (11') shows the rate at which electric energy is 1 Phil. Mag., (S. 3), vol. xix, 1841, p. 260, and vol. xxiii, 1843, pp. 263, 347, 435, or •Collected Papers, vol. i, pp. 60, 123. 166 MAGNETISM AND ELECTRICITY chap. evolved in the circuit, the left-hand side shows the rate at which energy- is spent in heat, and in working against the negative electromotive forces respectivel}'. Equation (11') applies of course also to a complete circuit, though in that case, since A and B are coincident, Vj — Vg is zero. Arrangements of Electric Generators in Series and in Parallel 221. Let us suppose that we have a number n of equal generator of electric currents, such as a number of equal voltaic cells, each of electro- motive force U. If these be joined in series, that is, so that the electromotive forces of all act in the same direction along ii single linear arrangement, the total electromotive force of the sj'stem is found to be n times that of one cell. Let the circuit be completed by an external conducting wire of resistance B. In general heat is generated within the cell as well as in the external joining conductor, that is to say, if the rate of generation of heat within the cell be y^, each cell has an internal resistance r. From what has been stated above the current 7 is given by r = ^ - (12) E + nr If nr be great in comparison with M, which will always be the case if B is fixed, and n is taken great enough, hnt little advantage is gained by increasing n further. For the current is then approximately B/r, or that produced by a single cell when it is short-circuited, that is, has its terminals joined by a short piece of thick wire. To join single cells in series is only advantageous when B is sO' large that the condition stated does not hold. But r may be virtually diminished by joining the cells in what is commonly called parallel. To fix the ideas let a Daniell's battery be supposed employed. Each cell consists of a plate of copper immersed in a solution of copper sulphate and a plate of zinc in a solution of zinc sulphate, in com- partments of the containing vessel separated by a partition of porous earthenware which permits conducting contact between the liquids and retards their mixing together. A number m of equal cells of such a battery are placed abreast, and all the copper plates are joined together to form one terminal, and all the zinc plates to form the other terminal plate. The electromotive force of such a compound cell is H simply, but its internal resistance is r/m. If then n of these compound cells be joined in series, and the circuit completed by a resistance B the current obtained will be nE mnE ,,„, 7 = = ... . ('■"} „ r niR + nr E + n — m In practice it is sufficient, if the cells are similar in all respects, to- join the zinc plates which form the terminal plates of each series of VI STEADY PLOW OF ELECTRICITY 167 n cells, and the copper plates which are the other terminal plates of these series, and to leave the intermediate zinc and copper plates of the different series unjoined. Arrangement of given Battery to produce Maximum Current through given External Resistance 222. If E be not too great and the total number of cells mm be suitable, we can arrange the battery so that for the given value of B that of 7 may be a maximum. The numerator of the above expression for 7 is constant, since the available number of cells is mm, and it can be shown that mi? + nr is least when m andm are so chosen that inB = m\ For we have identically mti + nr = { JtnE - Jnrf + 2 JmnRr. The second term on the right does not depend on the choice of m and n. Hence the right-hand side is least when the other term, which is essentially positive, is zero ; that is, when mB = nr, or the external resist- ance B, is equal to the internal resistance mr/m of the battery as arranged. If it is not possible to fulfi] this condition exactly with the given number of cells, the arrangement which most nearly fulfils it should be chosen. This theorem can only be applied when the resistance B and the battery which is to work through it are given. It is an entire fallacy to suppose, as is sometimes done, that of two batteries having equal electromotive forces, that which has the greater resistance is better adapted for working through a high resistance than the other, owing to its more nearly fulfilling the condition of external equal to internal resistance. The arrangement just arrived at gives not only the maximum current in the external part of the circuit, but produces there also the greatest activity which can possibly be obtained, when the current is used only for the production of heat in the external part of the circuit. For the activity in B is given by E E'^R A = mnEy—fr = m^n^ - — — r^ . . . (14) ' mB + nr (mU + nry ' which is a maximum under the same conditions as 7. Arrangement for Maximum Current not that of Greatest EfSciency 223. This, however, is not to be confused with the arrangement of maximum economy or efiiciency. In fact, in the arrangement under discussion, as much energy is spent per unit of time in heat in the battery itself as in the external resistance ; and therefore the ratio of the rate at which energy is usefully spent (in the external resistance) to tha 168 MAGNETISM AND ELECTRICITY CHAP. whole rate of expenditure is i, that is, half the energy expended is wasted. But the most economical arrangement is that in which this ratio most nearly approaches to 1, that is in which as little as possible of the energy given out by the battery is spent in the battery itself, and consequently as much as possible in the external part of the circuit. For economy of working, the internal resistance of the battery, and the resistance of the wires connecting the battery with the usefully working part of the circuit, must be made as small as possible. This subject will however be further dealt with in the discussion of electric motors. Theory of a Network of Conductors — Two Fundamental Principles : (1) Principle of Continuity 224. We shall now consider shortly a network of linear conductors in which steady currents are flowing, and in which are situated any internal electromotive forces that may be possible. Besides the considerations advanced above, two principles, first Stated explicitly and applied to this subject by Kirchhoff,^ are available for the discussion of problems regarding such a system. The first is the principle of con- tinuity already expressed, for a single wire, by the statement that the current has the same value at all cross-sections if the flow is steady. This expresses the fact that the rate of flowJ.nto any portion of the wire at any instant is precisely equal to the rate offloWA,out of the same portion. The same principle gives the result that, when steady currents are maintained in the various parts of a network oh conductors, the total rate of flow of electricity towards the point at which several wires meet is equal to the total rate of flow from that point at the same instant. Thus the current arriving at A, Fig. 58, by the main con- FiG. 58. ductor is equal to the sum of the currents flowing away from A by the three parallel conductors which connect A with £. (2) Sum of Electromotive Forces in any Circuit of Network equal to Sum of Products of Currents round Circuit into Resistances of Conductors 225. The other principle is contained in the following, which can be at once obtained by applying Ohm's law to any complete circuit which can be obtained in the network. In any closed circuit of con- ductors forming part of any linear system, the sum of the products 1 Pogg. Ann., hd. 72, 1847, and Ges. Abh., p. 22. VI STEADY FLOW OF ELECTRICITY 169 obtained by multiplying the current in each part, taken in order round the circuit by its resistance, is equal to the sum of the electromotive forces in the circuit. Thus let 7j, 72> • • ■> 7m be the currents in in wires forming a com- plete circuit, and having resistances rj, r^, . . ., r,„, and XU be the sum of the electromotive forces which have their seat in these conductors, we have ri*"! + r2''2 + ••■•+ ymrm = 2^ . • (15) To prove this, consider equation (5) above applied to each conductor. Let V^ be the potential at the first point of the circuit, which we shall suppose to be the initial point of the conductor of resistance r^, V^ the potential of the final point of this conductor, and the initial point of the conductor of resistance r^, and so on. We suppose here, and throughout what follows on this subject, for simplicity (though without losing generality by so doing), that there are no electromotive forces just at the junctions so that the potential at each has a perfectly definite value. The first conductor gives yi'i = Ti - Fa + ^12, the second, y-2^i = '^2 - ^3 + -^23' and so on. Adding these equations for the m conductors of the circuit we obtain (15), since the Vs disappear. Examples. (1) Two Points connected by Conductors in Parallel. Con- ductance. Besistance and Conductance of Parallel Conductors. Specific Resistance and Conductivity 226. Taking first one or two simple examples, we shall now apply these principles to obtain some useful results. Consider the arrangement shown in Fig. .58. Let the point A be at potential Vj,, the point B at potential Vb> and suppose that there is no electromotive force in the conductors to be taken into account. The currents from ^ to ^ by the wires of resistance, r^,r^, rg, respectively, are {Va — VB)l{r-^, r^, r^. Thus the total current is r = (F.-F.)(l + i + l). If B be the resistance of a wire which, with the same difference of potential between A and B, might be substituted for the triple arc between A and B without altering the total current, we have ry = (F^ — Vb)IR> and therefore It or R 1111 - + — + — Vi''i ^1^2 + "^fz + '"3»"l (16) 170 MAGNETISM AKD ELECTRICITV chap. The reciprocal of the resistance J2 of a wire, that is 1/^, is called its conductaiice. The last equation, therefore, affirms that the conductance of a wire which, in the sense above indicated, is equivalent to the three conductors of conductances 1/r^, l/'/'j, l/^'j, is equal to the sura of their conductances. The resistance of this wire is equal to the product of the three resistances divided by the sum of the different products which can be formed from the three resistances by taking them two at a time. These theorems are capable of obvious generalisation, with the following result : the conductance of a conductor, which is equivalent to any number n of distinct conductors joining two points. A, B, of a linear circuit, is the sum of the conductances of the separate conductors ; the resistance of the equivalent conductor is equal to the product of the n resistances of the arcs divided by the sum of all the different products which can be formed by taking these resistances w — 1 at a time. If B be the resistance of a wire I centimetres long and of cross-section (I square centimetres, the quantity Btajl is called its specific resistance. The reciprocal of this is called the conductivity. (2) Bridge Arrangement 227. As another example consider the arrangeitient shown in Fig. 59, and suppose that a single electromotive force, E, exists in the conductor of resistance, r^. By the principle of continuity we'get from the three points, A, C, D, the equations 76 = Ti + Jv 73 = Ti - 75' 74 = 72 + 75 • • • (17) where 7^ 73, . . . denote the currents in the wires of resistances r^, r^, . . . respectively, when the directions are as indicated in the diagram. Applying the second principle to the circuits BAGB,ACDA, OBBO, and noting that there is no electromotive force in any of these circuits except the first, we obtain by (1.5) and (17) the relations 7i(''i + J-a + '•o) + rfis - 75'-3 = M ■ 7i'-i - y.,r.i + y^r^ = ^ ^ • (18) 7i'-3 - y./Ti - y^{r^ + r^ + r^) = J "vi STEADY FLOW OF ELECTRICITY 17.1 These give for the current in CB or 7^, r. = ^^^^^V^^ (^,^, where D = r^rlr^ + r, + ,-3 + r,) + rlr^ + t^ {r., + r,) + ^ei^i + '-■>) (»-3 + r^) + r^r^{r^ + rj + r.f^{rj^ + r^) (20> The second two of (18) give ' '•sC'-i + r^ + j-3 + rj + (rj + r^) (5-3 + r,) " ^ ^ It is interesting to notice that if B be the single resistance equiva- lent to the five resistances r^, r^, r^, r^, r, connecting A, B as shown in the diagram yi. = U/(rQ + B). Hence by (lOj and (21) we obtain li = '•sC'"! + ^3) (^2 + ^'4) + ^•1^3(^2 + '•4) + '•2^4(^1 + ^3) (22) ^('■l + '■2 + '•3 + '■4) + ('■1 + '•2) ('■3 + n) The arrangement here discussed is the well-known " bridge " arrange- ment of conductors used in the comparison of resistances. We have many examples of its use in what follows. Analytical Treatment of a General Network 228. It is not difficult to deal with the problem of a network of linear conductors by an analytical method, but the main results are more instructively obtained by simple physical considerations. The chief steps of the analytical process are as follows, and may be fully worked out by the reader. Consider a set of n points, every one of which is directly connected with every other by a single conductor, the resistance and electromotive force in which are known. This will in- clude all cases, as if one point is in reality connected with another by more than one conductor, these can be reduced to a single conductor of equivalent resistance carrying a current equal to the sum of the currents in the separate conductors ; and if there be two points which are not directly connected a wire joining them can be imagined in which the current is zero. It is supposed, as before, that no electromotive forces exist at the points of meeting of the conductors. Since there are n points of meeting, and n—1 conductors radiating ftom each there are ^n(n—l) distinct conductors. The « points give, by the condition of continuity, n—1 independent equations connecting the currents in the n conductors, which suffice to determine the n — 1 differences of potential between them. For example, if we indicate the points by suffixes 1 , . . n applied to the symbols for the various quanti- ties, and denote by 77,^, the current along the wire connecting the point indicated by the suffix h with that indicated by the suffix k, we have as- a type of these n — 1 equations, yiii + yiii + ■■■■ + yhk + ■■■■ + yim = ^ ■ ■ (23) 172 MAGNETISM AND ELECTRICITY chap. But by (o)7m = /v'm-(Fa— F^. + ^m) where Khk is the conductance of the ■wire joining the two points, and therefore the last equation becomes ^kt(Fft - Vk + E„k) = (24) ■where the summation is taken for all integral values of h from 1 to n except A. If we write - Kwi = Kia + Khi +....+ Kh(h-V) + Kh(h + 1) + • • ■ + ^^hn •we shall have for the last equation KhiVi + Ku-iV. + ....+ KhhVh + ....+ KhrJn = Zto^Ai + ....+ KhnEhn = #„ . . (25) and there are n—\ such equations to be obtained by putting 1, . . n in succession for h. So far -we have considered a system complete in itself; but it is convenient sometimes to consider a system of conductors which receives current from without at definite points. We may therefore suppose that the system under consideration receives at the points 1, . . . n, electricity at the respective rates Q^, Q.-^, . . ., Qn- These rates must fulfil the relation Q-, + Q, + .... + Qn = a, since the state of the system is supposed to be steady. The introduc- tion of electricity at the rate Qh at the point A will modify equation (25) so that we shall have instead of (25) Kni\\ + EhiV-, + + ^AftFft + . . . . + KhnVn = KhiEhi + + KhnEhn - Qh = ^'h . . . . (26) If in (25) or (26) we put Vn = 0, the other quantities, V^, F„ . . ., Fi-j, will become the excesses of the potentials at the different points above that at the point w, and the solutions of the ?i—l equations of the form (25) or (26) will give these differences of potential in terms of the electromotive forces and the conductances. The values thus found for Fj, Fj, . . . being then substituted in the J?i(w — 1) equations of the form (5) will give the currents in the conductors. ■Solution of Equations. First Reciprocal Theorem. Conjugate Conductors. Second Reciprocal Theorem 229. One or two important results are easily obtained. First it is to be observed that, since Kjiic = Kkh, the determinant of the system of equations (25) or (26) is symmetrical, so that the relations Aftjt = Am hold for its minors. Also the potentials Fa, Fj of any two points are ^ven by the equations F. = ?i^^^) r, = ^i^^^. . . . (27) VI STEADY FLOW OF ELECTRICITY ITS obtained from (26). The summations are taken for all integral values oig from 1 to n-l. The quantities, Q, may be, of course, all or any of them zero, so that the equations just written down include also the- case of a self-contained system. We obtain by (15) yiik — -«/jil — + i^Air . • . (io) and similarly yim - li-lmA + J^lm t . . . (29) Now Elm does not appear except in '; and ',„, so that -^"//A A \ ^*'^ ^/A A x3*'ml = — (A;/, + Arnh - ^Ik - ^ml)KlikKim = ^7^ • (30) since Ai^ = A.u, &c. This result expressed in words is the theorem, that if a given increase- of the electromotive force ^m, existing in the conductor lik, produce a^ certain increase of current from I to m in the conductor Im, an equal increase in the electromotive force Eim in the conductor Im, will produce the same increase in the current from A to A in the conductor Kk. If the existence of an electromotive force in one of two of the con- ductors of the network does not affect the current in the other, this relation is also reciprocal, and the conductors are said to be conjugate. The analytical condition for conjugacy of the conductors is ^ih + AmA = Aiifc + Ami (31) Let us suppose that a quantity of electricity Q^ flows into the system^ per unit of time at the point h. Then the parts of Vi and F,„ which depend on Q^ are — Qk^ki/^ and — Q,Ak,n/^ respectively. These show by their form that the effect on the potential of the point I, for example, produced by the flow at rate Ca into the system at the point h is equal to the effect on the potential at h produced by an equal flow into the system at I. The part of Vi— Vm which depends on the entrance of electricity at rate Q^ at h is therefore Qki^hm—^M)/^- Similarly the part of F^— F"™, which depends on an outward flow Qh at k is — Qh(^km — ^m)/^- The total effect on Vj- V^ is therefore ^A(Aftm-Ato-Aw+Afc;)/A. Ob- viously, since Akm = ^mh, &c., this is equal to the part of Fa— V^ which would arise from an equal inward rate of flow at I and outward flow at m. aT4 MAGXETISM AXD ELECTRICITY chap. Cycle Method for a Network 230. The following method, due to Clerk Maxwell, may in some cases be conveniently adopted for a network of conductors. The net- work is considered as made up of a series of meshes or cells in which each individual conductor, except those forming the outer edge of the network, is common to two meshes. A current is supposed to circulate round each mesh in the same direction, so that the actual current in each conductor is the difference of the current round two adjoining meshes. Thus each mesh is a closed circuit with its own current in it. Let 7 be the current in any mesh, B the resistance, E the electromotive force, in its circuit, 7', 7", . . . currents in adjoining meshes which have conductors in common with the mesh-circuit under consideration, r', r", . . . the resistances of these conductors. Then we obtain at once for the mesh the equation yE - yV - y"r" - . . . =£ (32) The reader may, as an example, apply this method to the bridge arrangement. Activity in a Network of Conductors 231. Consider the sum SFft(7fti-t-7A2+ • •• +77m) taken for the w points of meeting. By the principle of continuity each term of this form is zero. But since 7/1*= —yieh it is clear that the same quantity may be written ^77,4(7/,— Fjt) where now the summation is taken for the 5^(w— 1) distinct conductors. This is the electrostatic energy spent in the network, and we have just seen that its value is zero. It follows by (24) that "^Rhkyiik^ = ^^hkyhk (33) that is, the rate at which the energy spent in heat in the conductors is ■equal to the rate at which energy is furnished by the internal electro- motive forces. By equation (28) the currents are expressed as linear functions of the electromotive forces. Hence by substituting for yjije on the right of (33) the corresponding linear expression, the rate at which work is spent in heat may be expressed as a homogeneous quadratic function of the electromotive forces. On the left it is already expressed as a homo- geneous quadratic function of the currents, in fact, as the sum of the products of the squares of the currents in the different conductors by the corresponding resistances, so that every term is essentially positive. It must be observed that while the quantity on the left of (33) always gives the rate of expenditure of energy in producing heat in any conductors to which the summation is applied, whether these form the whole system or not, the summation on the right embraces all the electromotive forces concerned, and that if the equation is applied every conductor in which heat is produced must be included on the left. VI STEADY FLOW OF ELECTRICITY 175 Currents fulfilling Ohm's Law give Minimum Dissipation of Energy in Heat 232. We can show that the rate at which energy is spent in heat in a network of conductors is a minimum, if the currents while fulfilling the law of continuity, and producing heat according to Joule's law, each satisfy the equation 7^* ={Vh— Vh+^KkflRhh- Let 7^ have this value, and let the actual current be jjiy^ = 7^ + ?/*, so that Vk-Vu Em , . yhk = ^ 1- ^ 1- ihk- ■lihk tChh Then we have ^y'hkRhk = ^y'hk{Vh - Vk) + %y'hkEhk + ^y'hkihk^hk, in which the sum is taken for all the conductors of the system. The first term on the right vanishes, since it may he written %Vh{y'k-^ + Yh2 + . . . +j'hn), in which the summation is taken with reference to all the points of meeting of the network, that is for all values of h from 1 to n, and the currents 7'^ fulfil the law of continuity. Again, the last term maybe written 't('yhk + ^hk)hk^hk- But tyuc^jik^hk may be written 2^Ai( ^h — y^k), which vanishes like the first term, since the currents ^7,;^ fulfil the law of continuity. Thus for the activity spent in heat we have •^y'hkEhk = 'ty'hk^nk + MhkRhk (34) that is, the activity thus spent exceeds the rate at which work is done by the electromotive forces by the positive quantity ti^SkRhk, which is the activity that would be spent in heat by the system of difference- currents fftj;. If ^iik be zero we fall back on the result already demonstrated. Elementary Discussion of Network of Conductors 233. Most of the results obtained above with respect to a network of conductors can be obtained by elementary physical considerations. It will be instructive to treat the subject shortly in this way. It is an easy inference from fundamental principles, and it can easily be verified by experiment, that the currents in the different parts of a system of conductors are not altered by connecting any two points, which are at different potentials, by a wire which contains an electromotive force equal and opposite to the difference of potential. The wire, before being put in position, will have the same difference of potential between its extremities as there is between the two points of the network under consideration. If then the end of the wire, which is at the lower potential, be joined to the point of lower potential the other extremity of the wire will have the potential of the other point, and may be made coincident with that point without changing the state 176 MAGNETISM AND ELECTRICITY chap. of the system. The new system obtained satisfies everywhere the principle of continuity, and equation (5) or, what comes to the same, the second relation formulated by Kirchhoff (Art. 224). Again, it is easy to see that if an electromotive force in one conductor ^ in a linear system produces no current in another conductor B of the system, either conductor may be removed without affecting the current in the other. For if for example A were removed, the potentials at the points of the system at which it was attached would be altered. Let then an electromotive force equal and opposite to that difference of potential be placed in ^ ; no current will flow in A, and the presence or removal of the conductor, after this has been done, will not affect the system. But it has been shown above (and it will be proved again presently in an elementary manner) that if an electromotive force in A produce no current in B, an electromotive force in B can produce no current in A. Hence, B can be removed also without affecting the current in A. Redaction of Network to Eridge Arrangement, First Reciprocal Theorem. Conjagate Conductors. Second Reciprocal Theorem 234. Let A, B, C, D be four points of meeting in a network of conductors, such that besides any other connection there may be between the points A, B, a distinct wire between these points exists, and similarly for G, B, and let AB be the only conductor in which there exists an electromotive force. The network can be reduced to a system of six conductors arranged as in Fig. 59 and such that the wires, AB, CD and the currents in them remain unchanged. For currents will enter any one mesh of the network at certain points and leave it at certain other points. One of the former will be the point of maximum potential, one of the latter the point of minimum potential. The circuit formed by the mesh consists of two parts joining these points, and to any point in one of these parts will correspond a point of the same potential in the other part. Every point in one may then be supposed in coincidence with points of the same potential in the other ; that is, the mesh may be replaced by a single wire joining the two points, and such that the currents entering or leaving it by wires joining it to the rest of the system are not affected by the change. Clearly the current will enter the system at one extremity A of the wire AB and leave it at the other extremity B. Thus A and B are the points of meeting of the network which are at the highest and lowest potential respectively. Thus the meshes of the system can be re- duced one after the other to two single wires, while CD is kept unaltered, until the network has been reduced to two meshes, one on each side of CD, connected by single wires to A, B respectively. Each mesh and connecting wire can be replaced by two wires joining A, or B as the case may be, with CD, and the whole system is reduced to an equivalent system of the form shown in Fig. 59. VI STEADY FLOW OF ELECTRICITY 177 Let now the electromotive force hitherto supposed acting in AB be transferred to CD while the resistances r^, r^ are maintained unchanged. The value of 7^ will be got from (21) by simply interchanging r^ and r^, r-^ + r^ and r-^ + r^, r^+T^ and r^-\-r^, in D. But these interchanges leave D unaltered, and thus the new value of 7g is the same as the old value of 7^. Hence an electromotive force which, placed in the conductor AB, produces a current in the conductor GD, will, if placed in AB, produce an equal current in AB. The distribution of currents and electromotive forces in the system which has been supposed above, in order that the reduction described might be effected, may be superimposed on any other distribution which is possible, and the conclusion which has been reached will not be affected by the latter. Thus we have the general proposition, stated in Art. 229 above, from which the definition of conjugacy of two conductors is obtained as before. 235. Again, the five conductors J.C, AD, BC, BD, GD in Fig. 59 may be regarded as the reduced equivalent of a network of conductors which a current 7g enters at A and leaves at B. The difference of potential between G and D Vc— Vo is 755*5. But by (21) ^ y£AT£3Li:I£A '^^ ^ °° '■5(''i + '•2 + '•s + '•4) + ('-1 + '•2) ('•s + '•4)' The resistance between the points C, D of the system of five conductors, is '•sl^'i + '•3) (^' s + ''4) '•6(»'l + '-2 + '•3 + '•4) + ('•1 + '"2) (''3 + '•4)' and if a current of amount 7^ were to enter at G and leave at D, the difference of potential between G and D would be the product of this expression by7g. The product multiplied ^y'rj{r^ + r^ gives the differ- ence of potential between G and A, and multiplied by rj(r^+r^ gives the difference of potential between G and B. Hence the difference of potential between A and B is the difference of these products, or '•si*'! + ''2 + ''3 + '•4) + (''l + '•2) ('•S + '•4) the value given in (35) for the difference of potential Vc — Vb. Hence, generalising as before, we have the following theorem, already proved in Art. 229 above. If a difference of potential Vc — Vb between two points G, D oi a. linear system arise from a current entering and leaving at two other points A, B respectively, a difference of potential Va — Vb = Vc — Vb between the points A, B will arise from an equal current entering at G and leaving at D. 178 MAGNETISM AND ELECTRICITY chap. Effect of Joining a Wire between Two Points of a Network of Conductors 236. The following result is easily proved, and is frequently useful. If the potentials at two points A, B, of a linear system of conductors containing any electromotive forces, be V, V respectively, and B be the equivalent resistance of the system between these two points, then if a wire of resistance, r, be added, joining AB, the cuiTent in the wire will be (V— V)j(R+r). In other words the linear system, so far as the production of a current in the added wire is concerned, may be regarded as a single conductor of resistance B connecting the points AB and containing an electromotive force of amount V— V'. For let the points A and B be connected by a wire of resistance r, containing an electromotive force of amount V— V opposed to the difference of potential between A and B, no current will be produced in the wire, and no change will take place in the system of conductors. Now imagine another state of this latter system of conductors in which an equal and opposite electromotive force acts in the wire between A and B, and there is no electromotive force in any other part of the system. A current of amount {V— V')l{B+r) will flow in the wire. Now let this state be superimposed on the former state, the two electromotive forces in the wire will annul one another, and the current will be unchanged. The potentials at different points, and the currents at different parts, of the system, will be the sum of the corresponding potentials and currents in the two states, and will therefore, in general, differ from those which existed before the addition of the wire. As an example consider a circuit between two points of which there is a difference of potential V, and let r, r' be the resistances of the two parts of the circuit between the two points. (These two parts may of course be any two networks of conductors joining the points.) Then the equivalent resistance is Tr'/{r + r'); and if another conductor of resistance B, and not containing any electromotive force, be connected between the two points, the new difference of potential V will be given by r = r — ~- (35) r + r since it has been shown above that V'/B is the current in the conductor. Hence in order that V may be approximately equal to V, B must be great in comparison with rr'/{r+r'). But rr'/{r+T^) can be written in either of the forms r/(l+r/r'), r'/{l+r'/r), which shows that r and / are each greater than rr'jir + r'). Hence if B be great in comparison with either of the two resistances r, r', V will be approximately equal to VI STEADY FLOW OF ELECTKICITY 179 v. no matter how the electromotive force may be situated in the circuit. This result is useful in connection with the measurement of the difference of potential between two points of a circuit by a galvanometer, as it is only necessary to mak^ the resistance in the circuit of the instrument great in comparison with that of either part of the circuit lying between the two points, to be sure that the difference of potential is practically unaltered by the application of the instrument. CHAPTER VII GENERAL DYNAMICAL THEORY Generalised Co-ordinates and Velocities. Kinetic Energy 237. The application of general dynamical principles to the discus- sion of the properties of a sj'stem of currents or magnets, or of hoth combined, is due mainly to Clerk Maxwell, and is one of the chief peculiarities of his treatise on Electricity and Magnetism. Further pro- gress has been made in this direction, and much light has been thrown on electromagnetic action by means of mechanical analogues which the djmamical method has suggested, and by the attempts which have been made to obtain some working dynamical hypothesis to explain the ether phenomena lately forced upon the closer attention of natural philosophers by the electromagnetic theory of light. As it would be impossible without this dynamical treatment to present the theories which it is one of the principal objects of this book to give some account of, and as the discussions of general dynamical theories in treatises specially devoted to abstract dynamics have not in general such applications as the present in view, and are therefore for the ordinary student a little difficult to translate into electrical language, we devote a chapter here to such general methods as we shall require in what follows. 238. It is to be noticed in the first place that the law of conservation of energy is not sufficient by itself to enable us to find the mutual actions or relations of the different parts of a system — that is, to find a dynamical explanation of the phenomena. We require in addition a general dynamical process, from which, under certain assymptions to be stated, and as far as possible justified, these relations can be deduced. Such processes are furnished by the principle of Least and Stationary Action due to Maupertuis and Hamilton, and Lagrange's dynamical method. The generality of the former method is so great as to render its application to every type of dynamical problem a matter of some difficulty ; but, as we shall see, it leads at once to the dynamical equa- tions of Lagrange, which in many cases, not easily attacked by ordinary dynamics, give a ready means of solution. CHAP. VII GENERAL DYNAMICAL THEORY 181 239. First let us consider a system of bodies which are subject to certain kinematical conditions expressed by equations connecting the co-ordinates of the different particles of the system. These equations limit the freedom of the particles of the system and enable just as many of the co-ordinates to be determined in terms of the remaining co- ordinates as there are independent equations of condition. Thus if there are 3» co-ordinates, and m independent equations connecting them, there are in all Sn — m independent co-ordinates, or parameters, from which the position at any instant of any part of the system can be deduced. The system is then said to have Sn — m degrees of freedom. As an example consider the motion of a rigid body. The kine- matical conditions imposed on it render impossible any alteration of the relative configuration of the particles composing it. But any point of it is free to move in any one of three rectangular directions, or the body may turn round any one of three rectangular axes through any point. Thus the body has left by the condition six degrees of freedom ; or, in other words, the position of the body is completely determined when there are given in position a point in the body, a line through the point, and a plane containing the line. First fix the point, this requires three co-ordinates ; next fix the line, this requires two co-ordinates ; last fix the plane, which requires one more co-ordinate. Thus there are six independent co-ordinates in all. If the point is constrained to remain in a fixed plane one freedom is removed, as two of the co-ordinates of the point, with the other three co-ordinates determine the position of the body ; and so on. 240. To the term " co-ordinate " then we give, following Lagrange, an enlarged or generalised meaning. The co-ordinates are the Sn—m independent parameters, the variations of which give the motion of the system. They may be either Sn — m of the ordinary position co-ordi- nates in terms of which the remainder may be found by the m equations of condition already referred to ; or they may be Sn—m other quantities ■\^, ,X'-- •' connected with the ordinary position co-ordinates by a set of 2n — m relations, making Sn known relations of condition in all. In the former case we may write the m kinematical equations in the form — ^i(«i. 2/1. »i' ) = ) F^{x^, y„z„... . ) = )- . . . (1) in the latter case the Sn equations are a^i =/i(>/'' ^' X ) ) h =/2("/'. -^> X. • • • ■ ) r • • • (2) Thus we have either as in (2) Qn — m unknown quantities «!, y-^,... ■f, (p, ... with Sn kinematical equations, or as in (1) Sn unknown quantities x^, 2/i, Si, , a'n, yn, 2») connected by wi kinematical equations ; in either 182 MAGNETISM AND ELECTRICITY chap. case the remaining Sn — m relations required for the determination of the unknown quantities are furnished by the equations of motion. These it is our main object now to establish. It is to be observed that if (1) or (2) involve the time t explicitly the kinematical relations vary with the time, and the results obtained in the following analysis will not in general hold. Where the contrary is the case will be stated. If the time does not enter explicitly in these equations the kinematical conditions are said to be invariable. 241. Denoting time-rates of change of co-ordinates, or velocities in the ordinary and in the generalised sense, by the fluxional notation, *i> Vv ^v ■■■> '^>'I>!X> ■■■ ^^ g6t from (2) the following — dx, ■ dx, ; d\l/^ dtf, \ i It is to be clearly understood that equations (2) connect x^, y^, z^, . . . yfr,^,... without involving ^,, ^)'P^ + ....+ 2(^, ^)U + ....} (4) in which (-i|r, i|r), ...(i^, (^)... denote co-efiBcients which are functions of the co-ordinates and masses only, since dxjdffr,... are such functions. Theory of Action 242. The action of a system for the part of the motion which takes place in any interval of time from t^ to tj is double the time-integral of the kinetic energy, or J to Tdt (5) Since 2Tdt = Smsds, this becomes A = ■S,\'m&ds (6) where s^, s^, are limits of s corresponding to t^, t^ Hence the action may be defined as the sum of the space-integrals of the momenta of the difierent particles of the system, or, which is the same thing, the sum of the products of the space average of the momentum of each particle by the length of the path described in the interval tj^ — t^^. According to the condition imposed on the system, there are two VII GENERAL DYNAMICAL THEOEY 183 theorems of stationary action : one, which is generally referred to as the Principle of Least Action/ for which the condition imposed is that the system should move from one specified configuration to another with constant sum T + V oi kinetic and potential energies ; the other an analogous theorem, subject to the condition that the time of motion, not the total energy, is given. These theorems, expressed analytically, are that (a) BA = 2B['Tdt = ... (7) with the condition ST+8r=0 and (b) s{*\t - V)dt = (8) with the condition 8*1 - 8 The integrated terms vanish, and if SA also vanishes we must have, in place of the variational equation of motion (9) above, since B'^, Scf), .... are independent, ^ Sr _ 35' dV i^_^ + ^=o \ ■ ■ ■ (20) dtd(j> 9<^ 9<)!> ' which are therefore the equations of motion of a conservative system in terms of generalised co-ordinates. These equations were first given by Lagrange (M^canique Analytique, Seconde Partie, Section IV.). "We shall see later that if the system is not conservative they are subject to certain modifications. If we put Z for r— F the equations may be written in the compact form d dL dL did^ ~ df ^ " I (20') "} In this form they may very easily be deduced from the theorem (&) by a process similar to that used above. Generalised Momenta. Eeciprocal Theorems 246. The quantities dTjd\p; dT/d^, . . . obtained by partial differen- tiation of the homogeneous function T of the velocities, are called the generalised components of momentum of the system, and we shall here denote them by the symbols ^, »?, f, . . . They are evidently connected by the relation ^^^'^^-^■••■ = '''- • -■ • • (^^) which we shall usually write in the less cumbrous form V + = 2T (22) 188 MAGNETISM AND ELECTRICITY chap. It is obvious that f ,»?,..• • are linear functions of the velocities as given in the equations (./r, xj,)^ + {ij,, 4,) + .... = i \ (^, ,^)f + (<^, .^)<^ + ....=,? V . . . . (23) which are independent and just as many^as there are co-ordinates. By means of these yp-. d$ ■■■■) + y + ^a-f + ^a| + ---- We thus have the reciprocal equations Also we have 3n _ ,. 3^« _ (26) or 32^ 3^3£ =,,32^, .3| Bi/r 3| 3;/f d^ '^ '^dij, ^ Ignoration of Co-ordinates. Modified Lagrangian Function 247. If the kinetic and potential energies be independent of certain co-ordinates x, x', ■ ■ it is now obvious that the components of momen- VII GENERAL DYNAMICAL THEORY 189 turn corresponding to these co-ordinates are constant. For dTjdx, ^^I^X> • ■ • , 9 ^/9x. • • • are zero, and hence, by Lagrange's equations we have dtdx ' dtdx = K, = K , . 8X 8x' where k, k , . . . . are constants. The last equations may be written in the form (x. x)x + (x. x')x' + = « - {("A. x)'A + (<^, x)i>+ } ) (x. x)x + (x' x')x' + = «: - {(v. x');^ + (<^, x)^ +••••} ^ (28) Our object is now to find what the theorem hS= hildt = 0, with time of transition given, becomes when ;)^, j^', . . . . are eliminated by means of equations (28), in order to deduce therefrom the corresponding modified equations of motion. We shall not at first suppose that the momenta k, k, . . . . are necessarily constant. Take the expression ^[idt = \(^-^^ + ^8/ +....+ ^8x + -8x + .. . )dt; J J ^d\p d\p d-^ dx ' write in it k for 'dLjdx, k for dZ/dx, ■ ■ ■ ■, and transpose, and the equation becomes di 8J(Z-Kx -k'x'-.. . . )dt + j(x8« + x8'c' + . . . . - g-^Sx - ^g^Sx' -•••■) ^f/S^-^Sx + ^Sf + -8<^ + . .)dt. . . (29) The expression on the right is the variation of >S when i/r, i/r, . . . are varied, while %, X' • ■ ■ • ^^® ^®^* unchanged. Since the co-ordinates are independent, their variations are arbitrary, and this variation of S must vanish. Hence (29) becomes 8 [{L - Kx - « x' -....)dt+ |(xS« - g-^Sx + x'8k' -....)dt = (30) For brevity we piit L' = L - Kx- kY - ■ ■ ■ (31) lyo MAGNETISM AND ELECTRICITY chap. so that L' = l{iy^ + r,4> + .-■■ ~ «X- k'x' -....)- V . (31') The first expression on the right is the difference between the kinetic energy depending on the momenta corresponding to the velocities y, , . . . . and that due to the momenta k, k, . . . . If now the co-ordinates x< X>- • ■ ■ ^^ ^^^ appear explicitly in T or V, K, K, . . . are constants, dL/d^,, .... are zero. We have then S\L'dt = (32) for the theorem expressed by (8) in which the time of transition is fixed. The equations of motion are therefore now obtained in the manner indicated above, but with L' substituted for Z. Hence they are d ZIJ dL' r - — = dt dip dip dJJJ_ IIJ_ > • • • (33) dt d4> dij} Lagrange's Equations with Gyrostatic Terms 248. It is to be observed in performing the operations indicated in equations (33) that k, k , . . . . are to be treated as constants, while X, x'' • • ■ • ^re explicit functions of i/r, ^, . . . . Now T = i(# + ^-^ + • + KX + k'x' + ....); and if («:, k), (k, k) . . . . denote the ratios of the consecutive first minors of the determinant of the system of equations (28) to that determinant and A' = |{(k, k)k2 + 2(k, kV + } . . . (34) the values for x, x'. • • ■ • derived from (28) are X = g- - (i^J/ +^N + 60 + ... .)\ \ = 37 - {iM' + N' + ea + ) (35) where 21 = {«. «) ('A. x) + «) ii', x) + «') (lA. x') + ■ • ■' . «') (-A. x') + ■ ■ ■ . ^ = («. «) ('^. x) + («> «') (^, x) + ■ ■ (36) VII GENERAL DYNAMICAL THEORY 191 Hence, substituting in the value of T above, we get T = Klf + v'l> + ----- i'^xM - 4>%kN ) + K = ^1 + /f, say (37) Hence L = T^Jr K - {2K - ^%k2I - ^%kN - ) - V = 2\ + ^P%kM + ^%kN + - K - V . . . (37') so that L' contains terms of the first degree in -kJ/-, (p, . . . . We have therefore dL' dT. ^ ,^ — T = -4 + ^kM dip diP Again d dL' d dT, . dM ;^ dM dt ■dxp dt dip ^ 9^ ar dT, dK ,^ dM ,„ dN dv + i>%K — • + <^2k vr + • • 3i/f dxp d{j/ d\j/ d\j/ dij/ and similar expressions are obtained in the same way for the variables , ^, 6, 9,. . . . Hence equations (33) may be written d dT, dT, j^ (dM dN\ .^ (dM dO\ dK dV .\ dt-^-^'''^^\'U'^^^^"^'^'w^""''^^^^^ ddT^_dT^ ; (dN^_ dM\ A^(^_£JO\ a^ dV^ If in addition to — 3 V/d^fr, — d Vld(f), forces '^', ', act on the system, these quantities must be used instead of the zeros on the right of (38). ^, $',.•• • are here applied forces which tend to alter the total energy I' + F" of the system. 249. The terms in these equations which have yp;^,d, as factors are called gyrostatic terms for a reason which will appear below from an example or two which we shall give. It will be seen that in each equation no g3rrostatic term with the velocity corresponding to that equation as a factor appears, and that in the -\|r-equation the multiplier of (j) is the multiplier of -yfr in the ^-equation with its sign changed, and so on. Equations (38) were given by Lord Kelvin in 1873, and applied by him to important problems in fluid motion.^ They are ref)roduced here for the sake of an important case of fluid motion to be considered later in connection with a vortex theory of the ether, and because the idea of a gyrostatically dominated medium will have to be discussed in its bearing on magneto-optic rotation. 1 " On the Motion of Rigid Solids in a Liquid circulating irrotationally through Perforations in them or in a Fixed Solid." Phil. Mag., May, 1873. 192 MAGXETISM AND ELECTRICITY chap. The equations may be established by transforming T into T' + K where T'is a homogeneous quadratic function of the velocities ■\jr, X, . the motion of the system will be precisely the same as if the system had a quantity of kinetic energy Tj, and an additional quantity of potential energy K. In fact, we see that in any given case of motion the potential energy which exists may be regarded as the kinetic energy corresponding to velocities which are thus ignored. As an example we shall see later that if we suppose a liquid to circulate round infinitely thin cores immersed in it, ^ is a quadratic function of the cyclic constants of the motion, and represents the kinetic energy of the fluid motion, while the kinetic energy of the cores them- selves is I'l, and does not involve any terms of the first degree in the velocities of the cores. Hence in this case the modified Lagrangian function is L' = T^- K -V, so that L' is the same as if the circulation were zero and the potential energy of the motion were increased by K. Application of Lagrange's Equation to Theory of Gyrostatic Pendulum 252. As an example of the process of forming the equations of motion of a dynamical system by Lagrange's method we consider here a case of some importance in the theory of magneto-optic rotation. Imagine a pendulum (Fig. 60) the bob of which is carried by a rod terminating at its upper end in one of the forks of a Hooke's universal joint and contains a gyrostat (Fig. 60) rotating about an axis coincident with the line joining the centre of inertia of the whole mass with the point of support — that is, the centre of the cross-link of the joint. We suppose the pendulum to be kinetically symmetrical about this line, and that the rod carrying the other fork of the joint is fixed in a vertical position. In order that Lagrange's equations of motion may be used the kinetic energy must be expressed in terms of quantities which com- pletely specify the position of every part of the system at any instant. Thus the expression of the kinetic energy in terms of the angular velocities with reference to axes fixed in the moving system — for example, the Eulerian velocities Wj, a>^, cog — is unsuitable. This is a point which is sometimes overlooked in the application of the Lagrangian method, and errors arise in consequence. 253. Let ^ be the angle which the vertical plane ZOB, Fig. 62, through the point of support, 0, and the centre of inertia, makes with a fixed vertical plane through the former point, and denote by (p the 194 MAGNETISM AND ELECTE.ICITY CHAP. angular velocity with which this angle is increasing. Let 6 be the inclination of the axis, OB, of kinetic symmetry to the vertical ; Q its rate of increase ; C the moment of inertia of the pendulum, without the gyrostat, round OB; 0' that of the gyrostat round the same axis; and A the moment of inertia of the pendulum, including the gyrostat, round Instead of a kinetically sym- metrical case which ought to be used iu practice for such a pen- dulum, a ring-shaped bob is substituted in Fig. 60 to display the gyrostat. A short piece of steel wire having the upper end fixed and ver- tical, the other end fixed to the rod of the pen- dulum and directed along its axis, may be substituted with almost perfect equivalence for the Hooke's joint. Fig. 60. This Figure shows a gyrostat resting on a thin edge on a glass plate. The case is represented as cut open to show the fly-wheel, which is pivoted on a spindle turning in bearings attached to ■ the case. As the section indi- cates, the fly-wheel is a thin disk with a massive rim. [This cut is reduced from Thomson and Tait's Natural Philosophy (Vol. I. Part 1, p. 397), to which the reader may refer for further information regarding gyrostatic action.] Fig. 61. any other principal axis through the point of support. Finally, to specify the position of the gyrostat at any instant, denote by yjr the angle which a plane fixed in the gyrostat and containing its axis makes with the plane ZOB. The motion of the pendulum can be found from the kinematical VII GENERAL DYNAMICAL THEORY 195 theory of the Hooke's joint ;^ but the following is perhaps simpler Consider the equivalent suspension : a perfectly flexible untwistable wire, of which one end is soldered or screwed to the upper end of the pendulum rod, the other fixed so that the wire cannot turn. First let the inclination 6 of the rod to the vertical remain constant, and the circle in which the centre of the bob moves be ^_5C.(Fig.63), and let A,B, be the two positions of the bob at the beginning and end of an interval of time U. Since the wire is untwistable for any position of the pendulum, it is simply bent. For the position A the bending is about an axis in the direction aa' , for B about an axis in the direction hV, and these two axes include an angle (pht. Now that line through the centre of the bob, and fi.xed in it, which was horizontal and a tangent to the circle at A makes with that which at B is horizontal and a tangent to the circle an angle U, and both these line are fixed in the bob and lie in a plane right angles to the rod. For if the wire b^ unbent when the bob is at A, and the rod^ brought to the vertical, the line through the centre of the bob which was horizontal re- mains so. Then if the wire be bent about W from the vertical position a line parallel to hV is brought up to be tangent to the circle at B. These two lines, therefore, include an angle ^U. But the direction of the axis has changed from OA to OB, so that the line which is now tangential to the circle makes an angle ^S< with the former direction of the tangent. Now, if the pendulum had been simply carried round so as to make the line which was tan- gent at A also tangent at B, the pendulum as a whole would have rotated about the vertical at through an angle U, and therefore through (p cos 9. St about OB. Conisequently the pendulum has rotated in the actual dis- placement through an angle ^ (1 — cos 6)St about OB apart from the angle turned through in consequence of the displacement of that axis. The angular velocity about the axis OB is thus ^ (1 — cos ^) or 20 sin^ J 6. Combined with this motion round the axis OB is a rotation about a perpendicular axis in the plane ZOB. This is clearly due to the C' 1 Thomson and Tait, Vol. I. Part I, p. 86. 2 196 MAGNETISM AND ELECTRICITY CHAP. motion of the pendulum which is carrying the bob in the direction of the tangent at £, and since the velocity of the bob in this direction is

sin^ ^6, round the axis of symmetry OB. 4> sin 6 cos (ji + 6 sin (ft, round an axis perpendicular to OB in an axial plane fixed in the body. <^ sin 6 sin , round the third rectangular axis. It remains to specify the motion of the gyrostat. Clearly it will have the same motion as the rest of the pendulum round the two rectangular axes last mentioned, together with a rotation round OB. The latter, by Fig. 64, in which is the angle ZOB makes with the fixed vertical plane, and ■\lr the angle which the plane fixed in the g3T:ostat makes with ZOB, is plainly ^ cos 6 + xfr. The kinetic energy is therefore given by the equation 2T = C(l - cos e)^^ + A(j>^ srn^ + ff^) ■ + c'(4' + cos ey (42) Fig. 64. in which the first term and the last are re- spectively twice the kinetic energy of rota- tion round OB of the pendulum without the gyrostat, and of the gyrostat alone, and the middle term is twice the kinetic energy of the whole system round the other two rectangular axes specified. If C = the equation gives the kinetic energy of a kinetically ' This direction, in the more general case when the pendulum is not kinetically symmetrical ahout 0£, must be so chosen that it is one of the principal axes of the hody, while 0£ and the axis at right angles to ^'O-B are the other principal axes. The com- ponent angular velocity about the former is ^ sin 9 cos ^ + C sin 6 . + C'cos (9. - C sin 6.ipe.= 0, ^(f + 4' cos 6) = 0. 255. The last equation shows that the angular velocity, \ir + of the gyrostat round its axis remains constant during the motion,'_a well- known but remarkable result. Putting dTjdxff or GX-yJr +

cos 6, and therefore (p. 190 above), since i/r is the velocity for the ignored co-ordinate, iV = cos ^, M = 0, &c., and „„ ,dT jdT .dT 2T= if— + 4>— + ^ — 3f 9.^ 9(9 = ^(« _ C'cose) + {C(l - cos 0)2 + A sinH}4,^ + {K - C' cos 6) cos e + C'^^cos^e + A6'^ = {C(l - cos0)2 + Asm^6}<}>^ + Ad^ + ^ . , . (44) as might, of course, have been obtained from (42). 198 MAGNETISM AND ELECTRICITY chap. The kinetic energy is thus reduced to the sum of two quadratic functions, one, Tj, of the velocities ^, 6, the other, K, of the momentum K corresponding to yjr. Thus 2^1 = {C(l - cos 6f + A sin 2(9}<^2 + AO'^, IK = ^,- The equations of motion by (38) are reduced to two, which have the form {C(l-cose)2 + ^sin2e}<^ + 2{C(l-cos0) + ^cos0}sin(9.(9<^-Ksine.(9 = O ) .^g. . .AB-{G^-{A-G)co&e]^\a.B^'^ + mgh%\-a.e = ^ ) ' The last term on the left of the first of these equations, is the gyrostatic term QKdNjdO, and there is no other. These equations can be reduced, when 6 is small throughout the motion, to x, y co-ordinates and take then a symmetrical form which exhibits better the gyrostatic terms. As they will be of use later, we give them here also. They are best obtained by transforming the kinetic energy to the new co-ordinates, viz. x, y, taken in a horizontal plane through the centre of inertia of the pendulum, with the projection of upon this plane as origin. We have approximately 1 — cos ^ = r^/2h^, where r ( = \/x^+y^) is the distance of the centre of inertia from the origin, 6 = (xx+yy)jhr tj, = {xy — yx)jr^, so that, neglecting terms of higher order than r'^/h^, we get 2T = ~(x^ + f) + ^-, (46) Hence 2T, = ^^{x^ + f), 2K = ^, For the calculation of the gyrostatic terms we have = ^ + J/x + iVy, where ^ = Ki-2i)' ^^=-Kr^-2i) The equations of motion are, since dTJdx, dTJdy, dKjdx, dKjdy, are all zero, dt dx '^\dy toj^^te ~ / dt ■by '^\bx dy)'"'^ dy ~ ) VII GENERAL DYNAMICAL THEORY 199 Now -r -^ ~ -a.x, — -~ = Ay, dt dob at dy ^ dV _ X dV _ y dx h dy h dy dx /'■ and therefore equations (47) become. Ax — Ky + mghx = Ay + KX + mghy I] w 256. These are the equations of motion, referred to horizontal x, y co-ordinates, of the centre of inertia of a gyrostatic pendulum, the inclination of which to the vertical is always small, or (with change of sign of h) of a gyrostat supported, with its axis nearly vertical, by a piece of flexible wire or on gimbals placed in the prolongation of the axis helow the centre of inertia. If the gimbal support is used both axes about which the gyrostat turns are supposed to be on the same level ; if they are not, equations (48) require modification. Taking in this case the axes parallel to those about which the gimbal rings turn, and putting \ for the distance of the centre of inertia from the axis parallel to y, Jtj for the moment of inertia round the axis parallel to y and Ag, A^ for the distance and moment of inertia with regard to the axis of X, we obtain the equations for this case by substituting for h and A in the first equation — Aj and A^, and in the second —h^ and A^. It will be found later that equations (48) are similar in form to those which enter into tlje dynamical explanation of the efl'ect, discovered by Zeemann, of a magnetic field on the spectrum of a gas. Generalised Eorces. Applied Forces, and Forces of Constraint. Lagrange's Equations deduced from Cartesian Equations of Motion of Set of Particles 257. We do not enter here on the solution of these equations ol motion : many examples will "present themselves later in electrical applications. For the sake of considering Lagrange's equations from different points of view we shall deduce them from the ordinary Cartesian equations of motion of a set of particles. Let F-^^, F^, denote forces in the ordinary sense which act on the particles m^, m\, ; then in any possible displacements Ssp Ssj, of these particles the work done is hW = F^Ss-^ + F^Ss^ + = %FSs. If Zj, Zj, Zp . . . , &i, Ey^, S^i, .... be the components of F^... ., Ssj parallel to the axes we have 8W = -^{Mx + Y8y + ZBz) (49) 200 MAGNETISM AND ELECTRICITY chap. We have supposed X, Y, Z, here to be the components of the force actually accelerating a given particle. This force is the resultant of the forces applied from without the system to the particle, and the forces of constraint due to the fulfilment of the internal conditions of the system. Thus for the components of applied force we may -write X' , Y', Z',... ., and for the components of constraint X", Y", Z", .... Hence, since X=X'+X',.... SW = 2(Z'8a; + Y'&y + Z'Sz) + S(Z"8k + 7"% + Z"Sz). But, the forces of constraint referred to are due to the mutual actions of the particles of the system, and the resultant of the forces of constraint on any one particle is accompanied according to the Third Law of Motion, by an equal and opposite force on the rest of the system. Hence for the whole system %{X"Sx + Y'By + Z"Sz) = .... (50) and we have SW = %{XSx + 7% + Z'Bz) .... (51) There are also forces of constraint applied from outside the system for which the work in any possible arbitrary displacement vanishes. Such for example are the forces applied to certain particles of the system by frictionless unyielding guide-pieces, and they act at right angles to the guiding surfaces along which the particles to which they are applied travel. Since there is no displacement at right angles to these guiding surfaces these forces are taken as included with forces X', which give (50). In general we shall omit the accents in using (51), and it will be understood, unless the contrary is stated, that X, Y, Z, denote the externally applied forces and not the actual forces on the particles. 258. Calculating Sx, Sy, Sz from (1), we have by (51) = *8i/r + 4>S<^ + (52) where The quantities '^, *,.... given by (50) are called the generalised components of external applied force corresponding to the co-ordinates 259. The Cartesian equations of motion of a set of particles m^^, m^,... may be written briefly in the form ™i(*i. Pv ^i) = (^1' ^1' -^i) I .... (54) ^" GENERAL DYNAMICAL THEORY 201 ■T-^ ^^1' ^i> ^v-- are the actual component forces on the particles indicated by the suffixes. If the particles are, as we suppose, subject to the conditions of constraint (2), these forces are due to the applied forces and the forces of constraint together. Equations (54) and (53) give 2,m (x- h V— +a — 1=* V 3./. ^3^^ 0^; V . . . . (55) But .. 3^ _ ^ / . 3fl3\ _ d dx _ d / dx\ . dsb by (3), and similar equations hold for the y, s, co-ordinates. Hence we have /..3a: , ..32/ .. 3a\ d dT dT V Si/' ^3./r dtlfj dtd^ dyj, with similar equations in (j), 0,.... Thus we again obtain Lagrange's equations ddT _dT _ \ dt djp 3i/' d dT dT = ^ dt d^ dff, (56) If the forces are conservative, so that '^ = — 9 V/d-^, these equations coincide with (20) or (20') above. One general remark we may make here to guard the reader against possible error in the use of Lagrange's equations. The co-ordinates yjr, (f>,.... used must not only be independent, but they must be such as can express the position and, with the kinematical equations, the configuration of the system at any instant during the motion ; that is, a?!, yj, Sj must be expressible as in (1). Work done by Forces of Constraint in Variation of Kinematical Conditions 260. It is also to be observed that there is nothing in the process which equations (56) have been established which would be aifected by the explicit appearance of t in equations (1) and (2), and that there- fore the equations hold in this case also. If t, however, does explicitly appear in (2) we have dx dx , dx ; dt dij;^ d,j>^ ^ instead of (3). Thus the kinetic energy is no longer a homogeneous quadratic function of the velocities xfr, r + ----)- hi^oi'o + Voh + ••••) = hWr + f o) + i*(<^r + <^o) + • • • • (66) since, identically, ^ri'o + Vr4>0 + = lo\^T + Vo'I't + (67) Thus we have the theorem that the work done by the^system of impulses is equal to the sum of the products of each time-integral of impulsive force, into the arithmetic mean of the velocities at the beginning and end of the interval t during which the change of motion is effected. If •v^q, 4>o,- ■ . be all zero, that is if the motion has taken place from rest, ii^f + V + ) = K*"A + H + ) . . (68) or the kinetic energy is equal to the sum of the products of the time- integrals of the forces into half the corresponding velocities acquired. It is to be observed that, provided the interval of change r is very small, the total work of the given system of impulses is independent of the order or manner of their application. The work done by any particular impulse, however, depends on this order. The reciprocal relation expressed by (68) is very important. A similar theorem holds for forces and corresponding displacements of a system from stable equilibrium, provided the potential energy is a homogeneous quadratic function of these displacements. VII GENERAL DYNAMICAL THEORY 205 Motional Forces. Dissipative Forces, Dissipation Function 262. In equations (38) are included what have been called gyrostatic terms, -which naay be regarded as forces depending on the first powers of the velocities ^p-, , .. ., that is, as such forces are called, motional forces. It will be observed that if we multiply (38) (with forces, '^', + (72) at 2F is therefore twice the rate at which energy is dissipated by the forces derived from F. F has been called by Lord Eayleigh the Dissi- pation Function. It is of great importance in the theory of mutually influencing currents, as we shall see below. Controllable and Uncontrollable Co-ordinates. Thermokinetio Principle 263. In certain applications of dynamics to electrical and magnetic problems, and to the discussion of the laws of thermodynamics, it is convenient to consider the co-ordinates of the system as divided into two sets, those which from their nature, we may call uncontrollable co-ordinates, that is, co-ordinates which cannot be directly and individually affected from without in any specified manner, and controllable co-ordi- 206 MAGNETISM AND ELECTRICITY chap. nates, which can be so changed by the action of an external system. We have examples of uncontrollable co-ordinates in those which deter- mine the positions and states of the individual molecules of a body; and of controllable co-ordinates, on the other hand, in the volume of a body, the position of its centre of inertia, the orientation of a plane fixed in the body, the state of the body as to electrical charge and the like, which are all under the influence of external systems. 264. Separating the independent co-ordinates of the system into two sets yjr, , X, X', ... we have for the equations of motion dt ^J, dij/ dtj/ d_dT _ dT 3f^_ „ ^3X ^X 3X (73) Multiplying these equations by \}r, X> i° order, and adding we get, as in (69) I (r + 7) = *;^ + *<^ + + Xx + X'x' + (74) or dt d(T + V) = ^dtj/ + M4> + + X^x + X't^x' + (75) where d'^, dcf), . . . . , dx, «^%', • • • are any variations of the co-ordinates compatible with the kinematic conditions of the system. The work done on the system by the forces acting on the con- trollable coordinates is '^dyjr + ^d(f> -|- . . . . If this be denoted by — dW, + c^JF will represent the work done by the system on external bodies. If with this we put dQ for Xd^ + X'd^ + (75) may be written dQ = dT+dV+dW (76) which simply expresses the conservation of energy. 265 . The equations of motion of a system which has a constant sum of energy are obtainable from the action-theorem (8) above, namely, {'{ST- W)dt= where the time of transition t^ — t^ from one configuration to another is supposed constant. It may here be remarked that those of the more vu GENERAL DYNAMICAL THEORY 207 general system, in which there is action between the system and external bodies, resulting in the passage of energy from one to the other with or without dissipation, can be obtained in like manner from the equation r {ST - SF - %®S6)dt = . . . . (77) in which denotes any coordinate, so that S@S^ denotes S^Si|r + SX8^ or —BW+ BQ, that is, the whole work done on the system fr-om without. This equation written in the form P (Sr - 8F - 5*8^ - BQ)df = . . . (78) J to with SQ regarded as a quantity of h,eat furnished, to the system from external bodies, has been called the Thermokinetic Principle} It may also be noticed that while (73) lead as already shown to (75) and (76), it is impossible to obtain from them (77), from which, on the other hand, they are derivable. In other words it is vain to attempt to deduce the principle of action from that of energy ; in fact, as already stated, the principle of energy by itself is powerless to yield a dynamical theory of phenomena of any kind. Dynamical Analogues of Thermodynamic Relations 266. Let us now suppose that %,%',•■•• are the uncontrollable co-ordinates, so that while X, X', . . . are unknown dQ remains the quan- tity of energy introduced into the system from without through these co-ordinates. Further, let no product of the form '\p-j(^, that is of the velocity corresponding to a controllable co-ordinate by a velocity of the other set, enter into the expression of the kinetic energy. If there were such terms the reversal of all the uncontrollable velocities would alter the energy of the system, a result which cannot hold in any of the applications we shall make of the method now under discussion. Calling the kinetic energy corresponding to the velocities of the controllable co-ordinates T^ and that corresponding to the velocities of the uncontrollable co-ordinates Tu, the equations of motion for yfr, ^, . . . will be, since dTuj^Y = 0> &c. ±'^T, _dT, ^oTu dV ^ ^\ dt d^' df ^<(' ^ V . . . . (79) Let us suppose also that if the coefficients of the terms in T^ 1 See a paper On the Laws of Irreversible Phenomena by Dr. Ladislas Natanson, Phil Mag , May, 1896. See also v. Helmholtz, Das Princip der Kleimten Wirhxmg, CrelU, Bd. 100, pp. l37,- 213, and Wied. Ammleti, Bd. 47 (1892), p. 1. 208 MAGNETISM AND ELECTEICITY chap. involve the controllable co-ordinates at all, it is only through a common factor /(i^, ,...) of each coefiScient, so that Tu =/(f, <^, . . . .){i[x. X]X' + [X. X']XX' + ■•••} • (80) Thus (x, x)' (x> x)> • • • denoting the whole coefficients as above (^, ■^) = f (^fr, ^, . .) i'x, xl> ^^^ so for the others. When this is the case r« 9^ fdi^' ^ ' Thus the equations of motion become dt df 3'/' fdf " dij, ^ , . . (82) From the first of these, if the only variable quantity be supposed to be T„, we obtain Again, if we multiply d dTc dT, d dTc _ dTc_ dt Zxj, dtj/' dt dcf, 30 ' ■ ■ ' lay yjr, and the equations of motion are dL dL For the uncontrollable co-ordinates we have in like manner (93) (94) 8^ ^ 3x 8x dSy dt - ^^ VII GENERAL DYNAMICAL THEORY- 211 Thus- according to the assumptions there are no forces depending on the cyclic co-ordinates. Thus changes of energy of the system cannot be introduced by change of the cyclic coordinates of the system, but change of the cyclic velocities may do so. A system fulfilling this ondition is sometimes said to have an adiabatic motion. The work communicated by the system to without or dW is — 't^i'd^jr, so that if dQ be the energy obtained through the unconstrainable co-ordinates dQ = d£ + dW = di; - :S,^d>l, .... (95) From the equations of motion in terms of s^, . . . just obtained for the unconstrainable co-ordinates dQ = S^s^xdt = Sxofs^ (96) If the system be monocyclic and there is only one cyclic co-ordinate ;)^ dQ = xds^. Dividing by 2T = x^x ^® 8®^ f = 2rf(log«,) (97) so that l/T is an integrating factor of dQ, and dQ/T is a perfect differential. The above brief discussion of cyclic systems contains its general principles. Further information on the subject will be found in the examples which occur below. We here close this dynamical chapter. Special forms of the Lagrangian function and their applications will be discussed in their proper connection, and we shall find in the dynamical analogues of electrical theorems further examples of the use of Lagrange's equations. The reader may, however, be referred to v. Helmholtz's papers in Grelle's Journal, Bd. 100, pp. 137,213, to Professor J. J. Thomson's Applications of Dynamics to Physics and Chemistry, and to Professor G. H. Bryan's Report on the Present State of Thermo-dynamics, Part I., B. A., Report, ■Cardifi", 1891. Thomson and Tait's Natural Philosophy (Vol. I., Part I.) ■contains, besides a full account of general dynamical theory, a very valuable discussion of the cycloidal motions of gyrostatically dominated systems. CHAPTER VIII MOTION OF A FLUID Section I. — General Kinematical Theory Fundamental Assumptions. Lagrangian and Eulerian Methods 270. The analogy between the theory of electroniagnetisin and the theory of the motion of an incompressible frictionless fluid is so close, and later researches having for their object the djrnamical explanation of electromagnetic action have emphasised so much more forcibly the fact that some vortex theory of electric currents is probably the true one, that it seems desirable to preface the discussion of electromagnetic theory by a chapter on fluid motion. Much space will thus be saved, sijice the proofs of general theorems here given need not be repeated ; and the analogies and theories put forward will be rendered intelligible, without its being necessary to have recourse to treatises on hydrodynamics for a sufficiently full discussion of what are as much theorems of electro- kinetics, as of vortex-motion of a fluid, if, in point of fact, the two things are not in some sense identical. 271. A fluid is a substance isotropic, and, if it is incompressible, perfectly uniform in quality, so far as it is here considered. The smallest part which in the course of mathematical analysis we have to deal with will be supposed to possess the properties of the fluid in bulk, so that no reference is necessary to the statistical treatment which becomes inevitable when the grained structure and relative motions of the molecules of the substance are taken into account. We suppose also that the mutual action between two portions of the fluid is at right angles to the interface separating them, so that tangential forces are regarded as non-existent, whether the fluid is at rest or in motion. This action taken per unit of area at any point of an interface is called the " pressure in the fluid at the point." The forces acting on a portion of a fluid thus consist of forces calculable from the pressures at the difierent points of the bounding surface, and the applied forces, which, from whatever cause, act on the matter contained in the portion considered. Such forces are the force of gravity, electric and magnetic 3<^ ,,, ^^ = -8^' " = -3^- " = -^ • • • • W The function ^ is called the velocity-potential. It may be single-valued or multiple-valued. In the latter case the motion is said to be cyclic, in the former acyclic. In all other cases of fluid motion the velocity is not thus derivable from a potential function. All such motion is said to be rotational, in contradistinction to the motion for which a function <^ exists, which is termed " irrotational motion." Rotational motion is also frequently called vortex-motion. The justification of these terms will appear in the discussions which follow. Equation of Continuity. Theorem of Divergence 278. To complete this short kinematical discussion we have to establish an equation which is fulfilled by the motion, whether rotational or not, expressing the fact that, whatever rate of increase or diminution there may be of the quantity of fluid in any portion of space within which the motion of the fluid is considered, it must be equal to the rate of flow in or of flow out, as the case may be, of fluid across the boundary of the space. Consider a rectangular parallelepiped of the fluid of edges dx, dy, dz, having its centre at x, y, z, where the velocity is u, v, w. If p be the mean density of the fluid within it at any time the mass of fluid contained by it is p dx dy dz. Consider the flow in the direction of x at the two ends of an infinitely thin filament of the volume, having its length parallel to x, and its left-hand and right-hand ends at the Tpoints x — ^dx,y + ^6^dy, z + ^B^dz, x+\dx, y+^Ojdy, z + ^d^.^'^' where ^j, ff^' ^re quantities numeri- cally less than unity. If o- be the cross- section of the filament, the rate of flow in at the left-hand end is to quantities of the first order of smallness Ipu - ^d«>-^ + ¥idy-^ + l^arfa-^jo-, and the rate of flow out at the other end is / , , dipu) , ^ , 3(pm) , „ , 3(pM)\ 216 MAGNETISM AND ELECTRICITY chap. where 9 {pu)jdx, &c., are the rates of variation of pw at the centre in the directions of x, y, z. The excess of the rate of flow in above the rate of flow out is there- fore d{pv) - dx -~— a-, ox and this is the same for each filament in the direction of x. Hence the total excess, for the parallelepiped, of rate of flow in, in the direction of X, above rate of flow out in the same direction is got by replacing cr by dy dz, and is ^ — dx dy dz. ox Similarly for the directions of y, z, we get the excesses d(pv) 3(p«') - ^; — -dxdyaz, - -^- — dxdydz; oy oz and the total excess is -f '!?!> + ?^ + ? . . • • • (11) 3w 3^ 3w _ du dw dv du clz ~ J dz dx ~ ' dx ~ dy '^ = ■ (12) ■*'"! MOTION OP A FLUID 219 If the quantities on the left in the last three equations be denoted by 2^, 2r], 2f the equations may be written f=0, ^ = 0, ^ = (12') 283. In the case in which the motion does not possess a velocity- potential the quantities ^, r), ^ are at least not all zero. It is easy to show that they are the angular velocities about axes parallel to those of X, y, z, of an element of the fluid. For consider a small sphere of rigid material having its centre at the point x, y, z. Let its motion be parallel to the plane of y, z, and 6 be its angular velocity round the diameter parallel to x ; let v, w denote the component velocities of its centre in the directions j/, z respectively, v-\-d,v, w + dw those of a point on its surface the co-ordinates of which relatively to the centre are dy, dz. Then we have '0 + dv=^v — 6dz, w + dio = vj-\-ddy, and therefore the relative velocities are dv= — Qdz, dw = Qdy. Now the velocities parallel to y, z, of the fluid at the point x, y + dy,z-\-dz, taken relatively to {x,y, z) are dv/dy .dy + dv/dz.dz, dw/dy . dy + dw/dz . dz. If s denote ^{dw/dy + dv/dz) these relative veloci- ties may be written dv/dy . dy + sdz — ^dz, sdy + dwjdz . dx + ^dy. The first two terms in each of these expressions are the velocities arising from a pure strain in the fluid ; the last term in each is, by what we have just seen for the small sphere, the velocity arising from a rigid body rotation of the element of fluid, of angular velocity ^, round an axis parallel to the axis of x. Similarly rj, f, are the angular velocities round the other two axes. It is to be observed that ^, rj, ^ are the angular velocities of rigid body rotation of an infinitesimal element of the fluid at x, y, z, but that, unlike a rigid body, the fluid has generally values of f, r], ^ which vary from point to point. It is also to be carefully noticed that an element of a fluid may move round an axis without having any rotational motion. As stated below, the quantities ^, i], f have been termed the components of the curl of the velocity. The curl and the divergence of a quantity are intimately connected, being in fact together the result of a single vector operation. The curl of electric and magnetic quan- tities, like their divergence, plays a great part in electrical theory, and especially in the general theories which are discussed later in this treatise. Permanence of Velocity Potential 284. We can show that under certain conditions, if a velocity- potential exist for any portion of fluid at any instant, the same portion of the fluid possesses a velocity-potential at any instant before or after. It is obvious that if we integrate between and t, and put «„ for the a;-component of the velocity when t = 0, the equation holds dx n .dx . . d r« da ° Jo 3a 3aJo 220 MAGNETIS]iI AND ELECTRICITY" chap. Here wis the same thing as'x, since the motion of a particle of the fluid is considered. Also dxjda = 1, when t = 0. We have likewise 3 = [%j/d, + .Ar,.^, ca Jo o» oflsJo W-- = 1 w^dt + \i^\ w^dt, 9a Jo 9a oa Jo since dylda = 9^/9« = 0, when if = 0. It is to be observed that in these equations t may have any value positive or negative. Adding these results, we obtain 9a; dy dz da da da Similarly two other equations can be written down by substituting in succession h and c for a, and at the same time v^^ and Wg for Wg- Multiplying these three equations respectively by da, db, dc, adding, and remembering that 3ic 9uC ooij dx = ^r- da + ^rr db + -— dc, (fee, aa 00 dc we find udx + vdy + wdz - {u^da + VffLh + w^dc) = I {udx + vdy + wdz)dt + ^d\ {v? + v^ + w^)dt . (14) Jo ' Jo Hence, if, when t = 0, u^da + v^db + w^dc be a complete differential of a function of the variables a, b, c, the quantity udx + vdy + wdz will also be a complete differentia] at time t (positive or negative) provided that icdx + vdy + ibdz is one ; for, u, v, w being functions of the co- ordinates and the time, the second part of the right-hand expression is a complete differential. We shall see later that this condition will be fulfilled if the applied forces acting on the fluid are conservative, that is are derivable from a potential function, and the density is a function only of the pressure. This proposition will be proved by another method later (see Art. 290 below). 285. It may be noticed that if the expression on the right of (14) be transposed to the left the expression obtained is (Art. 243) the change, dA, in the " action " involved in subjecting the terminal positions of the fluid particle to the variations dx, dy, dz, da, db, dc, in the given circum- "^'"^ MOTION OF A FLUID 221 Stances of the motion. By the principle of Least Action this should vanish ; hence the equation may be deduced from this principle. It the variations in the terminal positions be zero, this view of the equation shows that, if the fluid move from a given configuration, lor which the velocities are specified, to another given configuration, the motion for which the action is least is that for which the work done upon the fluid m the passage is equal to the change which is produced m the kinetic energy. Flow and Circulation. Circulation round Curve expressed as Surface Integral of Normal Spin 286. If ds be a line of elementary length drawn in the fluid, and q be the velocity of the fluid a" its centre, we define the flow along the line as q cos ds, where 6 is the angle between q and ds. Since „ u dm V dy w dz oos6 = - — + -/+ --r, q CIS q as q a.< the flow becomes dx dy dz\ , u- V 13 —- + w —- )ds, . ds ds ds/ or, as it is frequently written, udx + vdy + wdz. The value of q cos Q is the component of velocity along ds, that is, — 90/9s if the motion have a velocity-potential ^. Hence the flow along ds is —di^. Thus if we integrate along any curve s we get for the total flow along the curve \ qco&Bds = - {^^- ^^ (15) where 0j, ^^ are the values of at the final and initial ends of the curve. Thus, where there is a velocity-potential, 1. (udix + vdy + wdz) = <^q - ^j^ . . . . (16) in which the suffix attached to the sign of integration indicates that the integral is taken along the curve s. If the curve is closed, then, as we shall prove presently, whether the potential is single- or multiple-valued, if it exist at every point of the path of integration, and a surface can be drawn having the path for its bounding edge and situated wholly in fluid possessing the velocity-potential, the flow is zero. The flow round a closed curve has been called by Lord Kelvin the circulation in the curve. 287. In considering circulation we shall take the ordinary left-handed system of axes represented in Fig. 65; that is, the axes will be supposed 222 MAGNETISM AND ELECTRICITY chap. so drawn that Ox can be turned into coincidence with Oy by a turn through 90° round 0~, in the counter-clockwise direction to an observer looking towards the origin from a point in Oz. Integration round the curve is taken, as explained at p. 49, in the direction in which an Fio. 65. observer standing on that side of the area enclosed by the curve towards which a normal is considered drawn would have to pass round it so that he should have the area on his left hand. Thus in Fig. 65 the integration is taken in the direction of the arrows, and a normal, if drawn at any point of the triangular area would be drawn towards the side remote from 0. By precisely the same process as that adopted at p. 49 above we can show that =^ 2{U + mrj + nQdS (17) where dS is the area of the triangle ABC, and I, m, n are the direction cosines of the normal drawn to it in the direction indicated. Now we can divide in this way any area S, whether plane or not, enclosed by a curve s, into elementary triangles, take the flow round each triangle in the manner indicated, and add the results. The flow along each side common to two triangles, being taken in opposite directions for the two, has no influence on the result, and there remains only the flow round the given bounding curve s. Thus we have the theorem I {udx + vdy + wdz) = 21 (li + mrj + n^dS . (18) where the second integral is taken over the area S, and the first round its boundary s. The second integral is thus zero over any area within which ^, jj, f are each zero, that is if the motion be there irrotational. It is to be noticed also that the integral taken over the surface S depends VIII MOTION OF A FLUID 223 only on the bounding edge, and that different surfaces having the same bounding edge and wholly situated within the moving fluid will give the same value of the integral, each of them being equal to the circulation in the edge. It follows, therefore, that if a surface S over which the integral is zero can be drawn with the given curve as bounding edge, all other surfaces whatsoever having the same bounding edge will give also zero integrals. It is to be observed that the proof given above of the vanishing of the circulation only holds if the elementary triangles fill up the whole space within the circuit : in fact, the theorem only holds for a single closed circuit if the circuit is reduciile, that is can be diminished to a point without passing out of the space within which the motion is irrotational. Eeducible and Irreducible Circuits 288. If the circulation be taken round a circuit which is irreducible, that is cannot be contracted to a point without passing somewhere out of the region in which the motion is irrotational, we can deal with the problem as follows. Let a surface be drawn having the given curve as its outer bounding edge, and let the motion be irrotational only over the shaded portion of the enclosed surface as shown in Fig. 66. The theorem that the circulation is zero in the bounding curve will hold if the integral be taken round the outer curve and round the two regions Fig. -W. A and £ in the opposite, or negative, direction with reference to their areas. This can be seen at once by considering the path indicated in Fig. 67, wbich is the former diagram repeated for clearness without the 224 MAGNETISM AND ELECTRICITY CHAP. shading. This path is clearly reducible and is all described in the positive direction with reference to the shaded area which it bounds. The circulation in it is therefore zero. The straight parts leading from the outer boundary to the inner curve are described each in opposite directions and therefore have no effect on the result. The circulation round the outer circuit is thus equal to the sum of the circulations in the opposite directions round the two smaller circuits in the interior. The theorem of zero circulation in a fluid, in which there are any number of regions in which the motion is rotational, can thus be applied if the boundaries of these regions are taken into accoimt as shown above. Analysis of Motion at any Point in a Fluid 289. Having distinguished, as above, between rotational and irrota- tional motion, we consider now the nature of the motion at any point. If u, V, w be the velocities at time t at the point x, y, z, those at the same instant at a; + x, 3/ + y, s + z, infinitely near the former, are 8m " 3a; du dy M + x— + y— + z^^, &c., &c. du dz Thus the component velocities u, v, w, at the second point relatively to the first are du du du\ " = ^^3^ + ^; dv dv dv w dx dy dz I dw dw dw Xt;- + y:^ + Z (19) 9a! dy dz The first of these can be written du 1/&W du\ 1/du dw\ and similar expressions can be written down by symmetry for v, w, f, 77, f, having the values stated above, p. 219. Hence, if for brevity we use the notation du dv dw dx ' 3v' dz , /9m ^ = H9i" dw\ h = ,(dv du' ^\^^ dy du\ (20) dy) we have for the relative velocities the equations u - flsx + Ay + g'z + 17Z - ^y 1 V = Ax + Jy + /z + ^x - fz' '- w = srx + /y + cz + ^y - tjx f (21) VIII MOTION OF A FLUID 225 Thus the motion in the most general case consists of two parts : a motion in the direction of the normal to the quadric surface ax^ + Jy2 + cz2 + 2/yz + 2gzx + 2Axy = const. . (21') and a rotation (of which the component angular velocities are ^, »?, f)' round an axis having direction cosines (^, rj, ^)l{^ + rf^. + ^)i. The motion corresponds to the general strain of an elastic body. The former part may be called a motion of pure strain, and its axes are those of the quadric (21'). If we take the axes coincident with the principal axes of this quadric, we have for the velocities u', v', w' along them, due to the pure strain the expressions u' = a'x', v' = 6'y'' W = c'z' .... (22) so that aiV,c', are the time-rates of elongation, per unit length, of lines in the direction of x', y', z'. If we were to suppose that the motion consists of a pure strain and a rotation, both of them arbitrarily assumed, we should in en- deavouring to suit this to the state of relative motion get precisely the same analysis of the motion as has been given above. Thus there is only one possible analysis of the motion into a motion of pure strain and a rotation. Constancy of Flow along any Path moving with the Fluid 290. We shall now prove a theorem due to Lord Kelvin, which is of very great importance; as it embraces almost the whole theory of" fluid motion. The statement of the theorem in dynamical language will, however, be given later, when we have dealt with the dynamical equations of motion. Let us consider an element ds of a line moving with the fluid, and calculate the rate at which udx + vdy + wdz is altering for the element. We get at once — {udx + vdy + wdz) = udx + idy + wdz + udu + vdv + wdw (23) dt The time-rate of variation of the flow along a finite curve is obtained by integrating the expression on the right along the curve. Thus we obtain the theorem — I {udx + vdy + wdz) = (udx + vdy + wdz) + 1 {q^^ - q^'^) (24) where q^, q^ are the velocities at the final and initial ends of the curve. We shall see that under the same conditions as stated in Art. 284 the first part of the expression on the right of (28) is a perfect differ- ential, and hence that the integral taken round a closed curve vanishes^ that is the circulation remains constant. Hence if the circulation round the curve is once zero it is always zero. Hence (18) iff, ?;, f be zero in a portion of a fluid they remain zero in that portion. Q 226 MAGNETISM AND ELECTRICITY chap. Section II. — Dynamical Theory Equations of Motion of a Fluid 291. We now consider the dynamical equations, that is, the equations obtained by equating the value of the acceleration calculated above to its value obtained from a consideration of the density, pressure, and applied forces. Consider again the parallelepiped having its centre at x, y, z and its edges of lengths dx, dy, dz parallel to the axes. The mass of the element is pdx dy dz, if p be the density of the fluid at the element, and therefore, if X be the applied force per unit of mass in the direction of x, the total applied force in this direction on the element is pXdxdy dz. If p be the pressure at the centre, the difference of pressures on the two ends, reckoned positively when towards the right, is, by the process used in Art. 277, —dpjdx.dx. Hence the force to the right due to the pressure is— dp jdx . dxdydz. Thus we get the equation dp' I pX - —-\dxdydz = pu dx dy dz, or substituting the value of w from (2), and proceeding similarly for the other two pairs of forces, we obtain the three equations ,, 1 9» 3m 3t4 3m 3m ' p ox at ax oy oz 1 3p 3w 3m 3« 3u p dy dt dx dy dz 1 3/> dw dw dw dw I p dz dt dx dy dz > (25) If the forces are derivable from a potential function, il, the potential energy per unit of mass of the fluid, we can write these equations, using u v,ib for brevity, _ /3n 1 ^\ _ . _ /3fi l^\_- _ /?5 1 ^^ _ • /o5'\ \3a; pdx J ' \dy p dy) ' \dz pdzj Hence we see at once that, if p be a function of p, iidx+My+wdz is a perfect differential. For, multiplying the equations in order by dx, dy, dz, and adding, we get, putting dp/p =f'{p)dp, — d{Q +/(p)} = udx 4- vdy + wdz, which proves the statements made at the end of Art. 284 above. 292. The statement of Lord Kelvin's theorem given in (24) now becomes y I (udx + vdy + wdz) = -\ — - \u, - \c^\ . (26) '^"^ MOTION OF A FLUID 227 where the suffix s after the bracket on the right denotes that the value ot the enclosed quantity at the initial extremity is to be subtracted from that at the final extremity of s. Let us suppose that a velocity-potential exists ; then dx' dy' dz' dw _ dv du dw dv du dy ~ d^' di^ dx' dx^ dy' and the equations of motion become + l«2) }■• P dx dtdx ' ^dx^" ^ •" ^ -;"''>■ . . (26') &c. &c. ) Multiplying the first of these by dx, the second by dy, and the third by dz, adding and integrating along any curve s drawn in the fluid, we get j (Xdx +Ydy + Zdz)-{^=\-^^ + y^l . . (27) If X, T, Z have the potential O, this equation becomes ^b*mA-t*id- ■ ■ ■ <-> This result is generally put in the form of an indefinite integral, thus Cdp dfj) f- ^^ n-iq^ + F{t) (29) where F(t) is an arbitrary function of t, which is added to complete the indefinite integral, and which if not expressed may be regarded as in- cluded in d/dt. The pressure is thus indeterminate ; all that we can obtain by (27) or (29) is the difference of pressures between two points at the same time. If, however, the pressure be given throughout the fluid for any value of t, it is completely determinate for any other time. The Lagrangian equations of motion of the fluid are comparatively rarely employed. They may be constructed by putting £, y, z for the quantities on the right of the three equations of (25) respectively, multiplying then the three resulting equations by dxjda, dy/da, dz/da and adding for the first equation, then multiplying by the same quantities, but with the variable a, replaced first by b, then by c, for the remaining equations. The equations are thus , dx , „, 3w ... „. dz 1 dp „ \ ("-^)9^ + (^-^>^ + ^^-^)a^ + p£ = n (30) Q 2 228 MAGNETISM AND ELECTRICITY chap. Kinetic Energy. — Rate of Variation of Energy in Given Space 293. If T denote the kinetic energy of the fluid motion, and dts an element of volume, we have, integrating through the fluid, T =\ j"p(w2 + „2 + vfi)dvs (31) Multiplying the equations of motion (25'), in the case in which the forces have a potential, by m, v, w, and adding, we obtain / . . .\, fdQ . 30. 312 A, / ^p dp 8«\ , p(MiH-w + Mni')rfCT + n -— » + :— w + ^^K rfCT= -(u~- + v^ + w-^)drs. ' '^\8a; dy oz J \ dx oy dzj But it has been shown above (Art. 280) that |(pcit:T) = 0, and hence the last equation may be written, if di^jdt = 0, in the form where Fis the potential energy of the fluid within the space and at the instant under consideration. By integrating the expression on the right by parts, taking I, m, n as the direction cosines of the normal drawn inwards to the bounding surface at any element dS, we obtain |(^ + n =\pi^ + mv + nv,)dS +|^g + I + ^)d^ (33) If the fluid is incompressible the last term is zero and we get the theorem j(T+V)=\p{lu + mv + nw)dS . . . (34) which expresses the fact that then the time-rate of indtease of the whole energy of the -fluid within the space S is equal to the rate at which work is done by the pressure of the fluid at the bounding : surface on fluid crossing it, the inward direction being taken as positive. Stream-lines 294. A curve drawn in the fluid so that the tangent at every point is the direction of the flow there is called a stream-line. The equations of a stream-line are dx dy dz ds — = — = — = — (35) u V w q * ' "^in MOTION OF A FLUID 229 or, if the velocities are derivable from a potential, — = ^ = 2i = ^ t^v\ 3^ 9^ a^ 3^ ■ ^ '' dx dy dz ds ' ' ' whicH shows that, if the component velocities are So derivable, the stream- line at any point of a surface, for which ^ has a constant value, is in the direction of the normal to the surface at that point. ' ' In the case of steady motion (29) becomes i ^ + n + J?2 = C (36) r where C is a constant, and this may be applied of course to a stream- line. We may, however, integrate along a stream-line without assuming the existence of a velocity-potential. Since, the motion being steady, du/dt, .... are each zero, we have by the equations of motion du du du 1 dp 3n OX' oy tiz p ox Ox ) which by the equations of a stream-line may be written du 1 dp dx 30 dx ds p dx ds dx ds Adding these equations, we find for an element els of a stream-line J %2) ^ \dp da ^ ds p ds ds of which the integral along the stream-lme is |,2 + j^ + n = C/\ . . . . . (37) where C is a constant which has the same value along each parti6ular stream-line, but in the general case has different values for different strea,m-lines. Motion in Two Dimensions. — Conjugate Functions , 295. In the case of irrotational motion of an incompressible fluid which is independent of the values of one of the co-ordinates z, say, or, as it is called, motion in two dimensions, the equation of continuity is 230 MAGNETISM AND ELECTRICITY chap. The equation of a stream-line may be written in this case vdy. - vdx = 0, and the equation of continuity is the condition that M = - dij/jdy, V = + dij/jdx, .... (38') so that the last equation should be capable of being written in the form I'"'-!*-" w where i/r is a function of x, y, t in its most general form. It is called the- stream-function. If the motion be steady the integral equation of a stream-line is ./, - C (40) The whole system of stream-lines is given by taking C in the last equation as a parameter which varies from stream-line to stream-line.. If the motion be irrotational (38') gives dx dy' dy dx and the equatioQ of continuity is Equations (41) give also 9a;2 9y2 ^t + ^t^O (42) which expresses the fact that t^\_ = \(d'^->^ldx'^ + d^-\^jdy^)'\ is zero. Of course by (41) ^,1), are identically zero. The differential equation of an equipotential line is ^■'"= + |* = « w and by equations (41) that of a stream-line is 'i^'-'i-"'" <") so that the equipotential lines and the stream -lines are two systems of lines in the plane of x, y at right angles to one another. "*""! MOTION OF A FLUID 231 Along an element Ss of an equipotential line the variation Si|r of the stream function is ^^Sx + ^^8y^-vBx-uSy (45) that is, St/t measures the rate of flow of fluid across Ss in the direction from right to left to an observer looking along Ss in the positive direction. 296. Apart from the possibility of realising the motions, it may be noticed that it follows from what has just been shown that the curves ^ = const., 'v^ = const., form a conjugate system, in the sense that when either set is taken as the equipotential curves the other set is the corresponding stream-lines. [Fig. 79 below shows such a conjugate system of lines.] Hence (/> and '^ have been called conjugate functions. Their theory is very fully developed in modern treatises on the Theory of Functions of a Complex Variable,'^ and in works on Hydrodynamics and the Mathematical Theory of Electricity. Some account of their applications to electricity will be given later, but as excellent works on the Theory of Functions are now available no space will be devoted to proofs of their purely mathematical properties. General Theorems for Incompressible Fluid in Irrotational Motion. — Integral Equation of Continuity. — Tubes of Flow 297. Several theorems of great importance, which are all analogues of well-known theorems in electricity and magnetism, can now be proved for the irrotational motion of an incompressible fluid. The theorems will hold also for the electric and magnetic applications when the corresponding quantities are substituted in the equations. In the first place, for such a fluid the existence of the velocity potential (j) causes the equation of continuity (7) to take the forra 3^ SV^3!|^0 (46) 30:2 ^ 32/2 3*2 ^ ' which is identical with Laplace's equation, already considered at p. 46 above. Denoting the expression on the left, as usual, by y^^, we see that throughout any mass of incompressible fluid moving irrotationally we have v^<^ = 0. 298. It follows at once that if d^/dn denote the rate of variation of ^ per unit of distance inwards along a normal to a closed surface S drawn in the fluid the equation {'^dS=0 (47) J dn s 1 See Forsyth Theory of Futictions, Camb. Univ. Press, 1893 ; Haikness and Morley, Theorv of FunctioTis, Macmillan, 1893; Klein, Ueler Siemann's Theorie der algebraischen Funciionen und Hirer Integrale; Maxwell, EUctricUy and Magiutism, Vol. I. ; Lamb, Hydrodynamics, 1895. 232 MAGNETISM AND ELECTRICITY chap. holds. For by direct integration of the three terms of y^(j} with respect to X, y, z respectively [[p<^.c?a;/dn is zero. Hence, denoting by Sj^ and S2 the ends of the portion of tube, the equation can be written Ifj'^^lt'''-" w that is, the surface integral of normal flow outwards over one end face is equal to the surface integral of normal flow inwards over the other. 300. For any surface whatever the integral of normal flow across it may be regarded as the sum of integrals taken over portions o-, , o-g, 0-3, .... of the surface, so chosen that the value of each integral is unity. The tube of flow marked out by stream-lines drawn through the points of the boundary of any portion a- is called a unit tube. The flow across any surface is then equal to the number of unit tubes of flow which cross the surface. Equation (47) expresses the fact that the number of unit tubes which cross a closed surface withiji which v^jdx, without regard to sign, while it cannot have a maximum, may have a minimum value. The maximum numerical value is obviously precluded by the theorem that there is neither a maximum nor a minimum in the algebraic sense ; there is nothing, however, to prevent it from having a minimum numerical value. For example, this value may be zero at some point or points in the fluid, as we shall see later. Section III. — Green's Theorem Proof of Green's Theorem. Surfaces of Discontinuity 307. George Green of Nottingham gave in his famous Ussay on the Application of Mathematical Analysis to the Theories of Electricity and' Magnetism a theorem of pure analysis which is of the very greatest importance in all branches of physical mathematics. We give a proof here, with some examples of its application to particular problems. The extension of the theorem to multiply connected spaces will be given later, when fluid motion in such spaces is considered. Let U, V, denote two finite, continuous, and single-valued functions of the co-ordinates x, y, s of a point within a closed surface S^ (Fig. 68),. and k any other arbitrary finite, continuous, and single-valued function of X, y, z (or a constant), and let the derivatives of these functions be- finite and continuous also. Denote by E the integral w \ dx dx dij dij dz dz ) 236 MAGNETISM AXD ELECTRICITY CHAP. taken throughout the space enclosed by the surface S^. Integrating by parts, we obtain dV \—-dydz + ^^ dxdx + ^^ dxdy) \dx dv oz > and of course an exactly similar expression for E, which may be written down from this by interchanging U and V. The triple integrals in this equation and its companion are taken throughout the space within the surface ; and the elements of the Fig. 68. •double integral, which are furnished only by the enclosing surface, are taken as negative where a point moving in the positive direction along X, y, or s, as the case may be, enters the surface, and as positive where the point emerges. Take first the motion of a point parallel to the axis of X. Draw a normal inwards from the surface at each of the points of entrance or emergence, and let /j, Zg denote the cosines of the angles which the normal makes at A,B, with the positive direction of the axis of x at an entrance and an emergence respectively. Let a straight rectangular filament of the space, of cross section dydz, intercept elements dS-^, dS^, of the surface at the feet of these norma/ls. •Consider the positive sides of these elements to be those turned inwards . to the enclosed space, then we have dydz — l^dS^^ at an entrance, and ■dydz '= — l.^dS^ at an emergence. Each pair of elements therefore contributes to the integral the portion - (mi^^dSh - (UhH^-^dS),, and, since we can exhaust the whole surface by pairs of elements, we ■obtain for the first term of the double integral the expression ' UkH^-dS ox taken over the surface. '''"I MOTION OF A FLUID 23T Proceeding in the same manner for the directions y, z, and putting ■m,n, for the direction cosines of the inward-drawn normals at the corresponding elements of the surface, we find for the whole surface integral in (53) the value J„ V dx dy dz) In the companion equation to (53) we get, of course, a similar surface mtegral, except that U and V are interchanged. Denoting the expression between the brackets in the surface mtegral just found by d V/dn, since it is the rate of variation of V inwards along the normal at dS, and using dU/dn in the similar sense in the companion equation, we obtain finally which is Green's theorem. 308. The necessity for continuity and finiteness of the functions as specified above will be evident from the statement of the theorem in (54). If, for example, the value of 3 Ujdx is discontinuous within the limits of integration, that of 9(F9 Ujdx)/dx in the second expression for H becomes infinite, and the triple integral involving this term is indeterminate. Any region of such discontinuity must in the application of the theorem be excluded from the space considered. This may be done as follows : — Let P (Fig. 69) be a point within the space at or near which 9 U/dXj&c, one or ihore, are discontinuous. Describe a small closed surface S round F so as to include the region of discon- tinuity. We can apply the theorem to the space included between ^S" and /S^^, if that be simply connected, provided we add to the surface integral the value of - {nc%dV'/dn)dS taken over >S'. The normal is, of course, supposed to be drawn from dS towards the space throughout which the volume integral is taken. If the region of discontinuity is a mere point, F, then by supposing S shrunk down to infinitely small dimensions round F we can obtain as nearly as we please-the proper finite value of U for the given case. If there be a number of such points of discontinuity F, Q, ■ within the space considered, each must be dealt with in the manner 238 MAGNETISM AXD ELECTRICITY chap. just described, that is, the triple integrals must be taken throughout the space included between the outer bounding surface and the infinitely small closed surfaces described round P, Q, &c., and the proper values of the surface integral over these latter surfaces added to that taken over the outer surface. In the case of a cluster of such points or a finite region of discontinuity, a closed surface described round the cluster or region in question, and included in the surface integration, will enable the theorem to be applied to the remainder of the space. 309. It follows that if any discontinuity such as is here considered occur at points of a closed or unclosed surface, we may apply the theorem, provided we exclude the surface of discontinuity by a proper surface integration. Let, first, the surface of discontinuity be unclosed. Imagine a closed surface surrounding it described, and then shrunk •down until it forms an infinitely thin shell, >Si (represented by the dotted line. Fig. 70), having the surface of discontinuity between its Fig 70. -t'le. 71. faces. We find B then for the rest of the space within the external containing surface S^, by adding to the surface integral over S^ the value of -[vJMJJIdn.dS taken over the surface S, as already de- scribed. If the surface of discontinuity be closed, we have only to suppose a surface S (Fig. 71) described round it, infinitely near it, and add to the integral over S^ the value of — \Vk^.dV jdn.dS taken over S, to obtain the value of E for the space included between the outer closed surface S^ and the inner S. The space within the surface of dis- continuity may be treated separately by describing a closed surface within it, and infinitely near to it at every point, and using this as the outer bounding surface of the space now considered. Any other dis- continuities within either space must of course be dealt with in addition by the method stated above. vin MOTION OF A FLUID 239 Existence Theorem for Potential Function.— Motion of Minimum Kinetic Energy.— Uniqueness of Value of Potential Function for Given Space and Given Values at Surface 310. A question of considerable importance, though rather from the mathematical than from the physical point of view, may be shortly discussed here. Can a function be found which has a specified arbitrary, but continuous, system of values over the bounding surface or surfaces of a simply connected space, and satisfies the condition throughout the space 1 Practically the same mathematical proof of this proposition has been given by Lord Kelvin and by Lejeune Dirichlet, though it has been objected to on certain grounds by some writers. Mathematically the question resolves itself into whether or not it is possible to determine ^ so that it shall have an assigned value at eveiy point of the bounding surface and satisfy the condition (.55) in the interior. We shall give Lord Kelvin's proof here, as it will at the same time establish for us another very important theorem of fluid motion. 311. Let <}> denote any function whatever of x,y,z which has the assigned system of values at the bounding surface and v a quantity which is zero at every point of the surface, and has the same sign at every internal point as the expression on the left of (55) has there, and therefore vanishes wherever this expression is zero. If <^' = <^ + 6'c, where 6 is any constant multiplier, then, putting Q for the integral mm'- 0- m'\^^- taken throughout the portion of fluid considered, and Q' for the same integral with <^' used instead of ^, we have, by integration by parts, since v is zero at every point of the surface. Every term of the first integral on the right is positive by the condition imposed on v, and every term in the second integral is positive from its form. Hence we may write the result thus : Q' = Q - m${n -6) (56') where m and n are both positive. If therefore 6 be positive and less than n, Q' is less than Q ; that is to say, unless (55) be satisfied a 240 MAGNETIS-M AND ELECTRICITY chaf. function can be found which shall make Q < Q. If, however, (55) be satisfied, Q"^ Q; that is, ^ then gives the smallest possible value to the integral Q. But, since the integral is essentially positive (for every term is positive), there must exist for it a lower limit ; that is to say, there exists a value of ^ which makes it equal to this lower limit. This value satisfies (55). 312. One ground on which this existence theorem, as it is called, has been objected to is the assumption that a function v can be found to fulfil the condition stated. Whatever opinion may be held as to the validity of this and other objections, there is no doubt that if (f> fulfil (55) Q is greater than Q by the second integral on the right. This, stated in physical language, is the theorem that the kinetic energy of the motion given by the velocity-potential ^, which has the assigned values at the surface and satisfies (46), is smaller than that of any other motion fulfilling the surface condition by the kinetic energy corresponding to the difference between the velocity-potentials of the two motions. We here still call if> the velocity-potential, though the component velocities are -kd(j>/dx, .... 313. Further, if there exist one motion fulfilling the stated conditions, that motion is the only one that fulfils them. For, if possible, let ^j be the velocity-potential of another motion fulfilling the conditions, then — ^i also fulfils (55) and is zero at every point of the boundary. By a theorem which we shall prove immediately, the fluid under this potential is at rest ; that is, the motion corresponding to the potential — (^j is equal and opposite to that corresponding to ^ ; that is, ^ = <^i throughout. In proving the theorem on which this result depends, we suppose /j = l, which is the only case relevant to the fluid motion we are considering. But as the reader will see at once the theorem is true also in the more general case. Integrate the expression «-(sy-(i)"*sr throughout any region the bounding surface of which is S. Integrating by parts, we get - fjLvV-'^'K'^y'^^^ • • (57) If the motion is irrotational throughout the space, y^^ = 0, and the second integral on the right is zero. If then (1) .the fluid is at rest. The same conclusion follows also from the considerations- set fokth above as to lines and tubes of flow, . _ ;,.../- 314. Again, if instead of an arbitrary distribution of velocity- potential we have given over the surface a continuous distribution of normal velocity, then, if there exist a solution of (46) fulfilling the former condition, there must also exist one fulfilling the latter. For conceive a fluid contained within a flexible envelope, and impress upon the envelope from without the distribution of normal velocity specified. This must satisfy the equation I q„dS=Q, where q„ is the normal J s velocity. If the normal velocity can be expressed at every point as the rate of variation in that direction of some function of the position of the point, the surface condition corresponds to a certain arbitrary distribution of 0, and by the last proposition- there is one, and only one, solution fulfilling the prescribed condition. ' Similarly there is one, and only one, solution if ^ is given over one part of the surface and d^jdn over the remainder. With the proper changes, which will appear later, in the specification of the symbols, these theorems as well as those which follow, are of great importance in electricity. Deduotions from Green's Theorem. — Sources and Sinks 315. One or two important consequences of ' Green's theorem it is convenient to deduce here. Supposing k a constant, we get from (54) 1\\{UVW- yV^U)d.dyd.=l{r'^ - U^^)dS. (58) where the first integral is taken throughout the space considered, and the second is taken over its surface, including those parts of the surface which exclude regions of discontinuity from the first integral. This equation is of great service in many parts of physics. It is sometimes, though wrongly, asserted to express Green's theorem. If both U and V fulfil Laplace's equation the left-hand side of (58) is zero, and we have \u'^d^=\vfdS (58') }s dn is an 316. As an example of the use of this result we shall put U= 1/r, where r is the distance of any point F from the element dxdydz, and for V a function of x, y, z which fulfils Laplace's equation and is every- E 242 MAGNETISM AND ELECTRICITY chap. where finite and continuous within the region of integration. We have therefore y^ V= 0, throughout the space considered, and also y''' 17= for the same space except just at P. For P" becomes infinite when r=0, so that if F is within the space considered we must exclude it by describing an infinitely small sphere round it and extending the surface integral round that sphere. The value of dU/dn for this sphere is d U/dp = — 1/p^, where p is the radius ; and the surface is 4nrp^, so that the contribution to the surface integral is where ^p denotes the mean value of taken over the infinitely small sphere, centre F, that is, the value of <^ at P. For the rest of the space considered the left-hand side of (58) vanishes, and we have s or s s 317. To find a physical interpretation of this equation, imagine fluid to enter an indefinitely large space at a point, and to flow uniformly in all directions radially from the point. Let the amount of fluid introduced into the space per unit of time be m, then the same amount m crosses every concentric spherical surface in each unit of time. The rate of flow per unit of area at any place at distance r from the point is therefore — mjiirr^. The function mj^sirr thus fulfils the analytical conditions for being the velocity-potential of the motion, and we shall adopt it as the value of ^ for the motion. We call the point from which the fluid diverges a 'point-source of intensity m, and we see that the potential of such a source, at distance r from it, may be taken as mj^sirr. Had we considered a flow uniformly converging radially to a point we should have obtained the same numerical value of the velocity, but with opposite sign. The potential for the same amount of fluid m carried inward per unit of time across each concentric spherical surface would in this case be — mj4nrr. We call the point of convergence a sink of intensity m. Instead of fluid we may have sources and sinks of heat and electricity, and in the latter connection we shall have to consider the subject fully later. We shall then see that the electrodes by which a current enters and leaves a conducting body jplay the parts of electric source and sink for the flow of electricity, the theory of which is precisely that of the irrotational flow of an incompressible fluid. >'™ MOTION OF A FLUID 243. When ^ is taken as electric potential, and sources and sinks are distributions of positive and negative electricity, we have corresponding theorems on the distribution of potential and lines of force in the electric field. ^^^- ^o""^ consider two equal and opposite point-sources A, B (Fig. 72), at a distance dn apart, the positive direction of dn being taken from the negative source at A to the positive at B, and let A C B Fig. 72. r, r + dr be the (nearly equal) distances from these sources of any point Q in the moving fluid. Let also 6 be the angle which the bisector CQ of the angle AQB makes with -the positive direction of dn, that is with AB. Since the flow at any point Q is compounded of a flow of amount — m/4nrr^, towards A, and another of amount mj^'n-{r + drf, from B, and these have potentials — ml^irr, m/4iir{r + dr) at Q, the velocity-potential there is — m/47r .{!/?•— l/(r + c^r)}, that is — 'mdr/i'Trr^. But dr = —dn cos 0, and if we write m for mdnjiir the potential is m cos 6/r^. We call this a double point source, or doublet source, of intensity m. The potential of such a source at a point at distance r from the centre of the doublet can clearly be written also in the form d /I an \r Going back now to (59), we see that the physical interpretation of the result there stated is that the potential at F is the potential that would be produced by a distribution over the surface S of simple point sources, so that the intensity at dS is — ((^0/(^w)/47r per unit of area, and of doublet sources so that at dS the intensity is ^/47r, where is the potential at c^^S^. 319. The result here obtained is very important in many respects. The motion is shown to be producible by a surface distribution of sources which can be deduced as specified from the geometrical configuration E 2 244 MAGNETISM AND ELECTRICITY chap. of the bounding surface S^^ and the values at every element of this sur- face of the potential, and of its rate of variation along the normal there. The actual sources may be very different ; in fact, their nature may be quite unknown. The result is analogous to the replacement, in the theory of light, of the actual sources by what are called secondary sources, a notion due originally to Huyghens and developed very fully by Fresnel and his successors. Let us suppose, however, that the potential at any point in the space included by the inner bounding surfaces or situated without the outer bounding surface, that is, throughout the rest of space, is given by a single-valued function ^'. For this space, since P is external to it, we have by (59) i*'ie)--ijf^^=» in which the normal n' is supposed drawn inwards from S towards the space to which 0' applies. Adding this equation to (59) we obtain, since d/dn = —djdn', s s and a

' at the surface S *^-hm-t}^^ ■ ■ ■ ■ « s Thus the distributions, whatever they may be in the space separated from that in which P is situated by the surfaces S, may, so far as the motion of the fluid in the region in which P is situated be concerned, be annulled, and, if iS be a surface of discontinuity of ^, the surface distribution of simple and doublet sources specified by (59') substituted, or in the other case, that of continuity of potential, replaced by the arrangement of simple sources specified by (59"). If d(j)jdn = —d^'jdn', that is, in the case of continuity of normal flow across S, the second integral in (59') vanishes, and the distribution of sources is reduced to the doublets alone. If the surface S consist of two parts, (1) one or more surfaces at finite distances from P everywhere, and (2) an outer spherical enclosing surface every part of which is at an infinite distance, the surface integrals may be taken to vanish for the latter part, since for this outer surface the second integral vanishes and the first becomes the constant, C, of (50). Since we have in any case only to deal with differences of potential, this G may without affecting any result be taken as zero. For example take Green's problem (Art. 201 above). Let U be such a function that it is sensibly equal to Ijr in the immediate vicinity of P, is equal to zero at the internal bounding surface or surfaces S, and is VIII MOTION OF A FLUID 245 harmonic throughout the space for which the volume-integrals are taken. The existence of such a function is clear from the electric analogue. For let a unit charge be situated at P, and S be maintained at zero potential. The induced charge on S will be such as to give a potential exactly- counteracting the value, 1/r, of the potential at each element of S produced by the unit charge at P (k is here taken as unity). The value of IT at any point is, then, the potential there due to the induced distribution, together with that due to the induced charge at P. This is harmonic throughout the space external to S, except just at P, where it is 1/r. Thus let be the arbitrarily chosen potential at any point of the surface of S ; since the expression on the left of (58') (with the point P allowed for as in Art. 316) vanishes, the corresponding potential at P is ^^-il^fn"' ^''^ Kinetic Energy of Irrotational Motion. — Expression as a Surface Integral. — Cases in which Motion cannot exist 320. We now pass to a consideration of the energy of the motion. By Green's theorem, or indeed by direct integration, we can express the kinetic energy by a surface integral. For if the motion is irrotational we have, by integration by parts, since the non-integrated term vanishes, the normal to the element of surface, c^^S* being supposed drawn inwards to the space occupied by the fluid, and the surface integral being taken over the whole bounding surface of the space considered in the volume integral. Thus, if T denote the kinetic energy of the fluid. ^-ifl'^t'' ^'"'-^ Let the fluid extend to infinity, and the velocity tend to zero at every point very far from the inner bounding surface S-^, and let S^ be a surface' taken in the fluid so as to enclose S^ and be everywhere very distant from it. "We have fj'^ = ' («^) and i:>--f i4*?j'^*U>l Si S2 246 MAGNETISM AND ELECTRICITY chap. But if the velocity may be taken as zero at every point of S2 the second term on the right is l<^'i '^"^^^„ where C is the constant finite value of the potential at infinity, and by (62) this can be written -\^rJ''- Thus T==-y\{^-C)^dS, (63) If there is no flux on the whole across the outer boundary, this reduces to T=-y^^^dS, (64) which is the result that would be inferred, were it legitimate to do so by the preceding equation (61), from the fact that S^ is now the total boundary. 321. If the liquid fill the whole space so that S^ does not exist, T = 0, and u, v, w are by Green's theorem zero, at every point. Irro- tational motion is thus impossible in a liquid filling infinite space and at rest at infinity. It also follows from (61) — what has already been proved above, Art. 313, — that irrotational motion cannot exist within a finite, simply connected space with a boundary S-^^ fixed at every point ; for the velocity — d(f)/dii at right angles to the bounding surface must be zero at every point, and hence u, v, w must be everywhere zero. Multiple Connection of Spaces. — Reduction of Connectivity by Diaphragms. — Number of Irreconcilable Circuits 322. So far we have considered motion in simply connected spaces only ; we have now to consider spaces of multiple connectivity. Such a space is represented in two dimensions in Fig. 73. A space of multiple connectivity admits of the drawing of at least one section or diaphragm so as to give a section having a closed curve as boundary, without dividing the space into disconnected parts. A surface for which one such diaphragm can be drawn is said to be 2ply connected, or to be of connectivity 2. If «, — 1 such diaphragms can be drawn it is said to be wply connected, or to be of connectivity n. The space enclosed by an anchor ring, or the space external to an anchor ring, is an exaitiple of a space of connectivity 2. If we suppose ■^™ MOTION OF A tl/UID :; 247. an anchor ring to have a region such as'^, Fig. 73, attached to it, it is changed into a space of connectiyity 3 ; if two such regions are attached, as in Fig. 74, it becomes a space of connectivity 4 ; and so on. The diaphragms which in the different cases reduce the connectivity to 1 are shown by dotted lines in the figures. 328. We shall now prove that in a space of connectivity n it IS possible to draw n — l independent and irreconcilable irreducible. Fis. 73. circuits. This can be proved in various ways. First it is to be observed that if a complete circuit cannot be drawn so as to cross one particular diaphragm without at the .same time meeting another diaphragm an odd number of times, these two diaphragms divide the space into unconnected parts. This is clear from the fact that the circuit, having been carried across one diaphragm, cannot be completed without passing at least once through the other; that is, beyond the Fig. 74. first diaphragm is a portion of the space from which the path cannot •emerge into the remainder without piercing the second diaphragm. It follows that, if the diaphragms are so drawn as to preserve connectivity of at least 1 for all the space, it is possible to draw a ■circuit so as to cross any particular diaphragm once, and any other an even 'number of times, of which as many are crossings in one direction ss in the other. But this latter condition comes to the same thing 248 MAGNETISM AND ELECTRICITY chap. as prohibiting the passage of ' any other diaphragm at all. It i» therefore possible, since there are n—1 diaphragms, to draw n — 1 such circuits. Also these circuits are iEreconcilable. For if two of them were reconcilable — that is, if one could be changed into the other without somewhere passing out of the space! considered — then in the process either each would have to be altered so as to pass through both diaphragms, or one would be withdrawn from one diaphragm and made to pass through the other. But, since each passes through its diaphragm only once, either alteration is impossible without taking the circuit across the closed curve which, as stated above, forms the boundary of the section made by a diaphragm. 324 . The same thing may be proved more shortly thus. For every region added to a multiply connected space necessitating an additional diaphragm it is clear that a new circuit not reconcilable with any existing circuit can be drawn so as to pass through that diaphragm. But when there can be drawn only one diaphragm — that is, when the connectivity is 2, as in the anchor ring — only one independent circuit can be drawn ; hence the number of independent irreconcilable circuits is equal to the number of diaphragms — that is, one less than the con- nectivity. Cyclic Uotiou in Multiply Connected Space. — Cyclic Constants 325. The circulation in a circuit crossing only one of the diaphragms and crossing it once only, is the same for all such circuits. This is easily seen from Fig. 75, which represents part of a multiply Fig. 75. connected space. The circulation in the path AEBOFB indi- cated by the arrows is zero, since the circuit is reducible ; and, since the parts BG, DA are infinitely nearly equal and opposite, the circula- tions in the remaining parts taken in the same direction are equal. Let the circulation in either be k ; then k is what is called the cyclic constant of the circuit and by (16) is equal to the change of the velocity- potential along the part of the closed path from B to A, or along th& other part of the closed path from G to D, ""« MOTION OF A FLUID 249 826. Let now k^, k^, .. . . . k^_^ be the cyclic constants of the w— 1 independent circuits, and consider any compound circuit in the space drawn so as to pass through the /", P", &c., diaphragms, and let the excess of positive crossings of the/" diaphragm above negative crossings be^^, the corresponding excess for the /c"" diaphragm^* and so on. Then the circulation in the circuit is PjKj + PkK^ + . . . . To see this we have only to notice that the compound circuit is reconcilable into the independent circuits of which one passes Pj times through the y* diaphragm, another crosses p^ times the A*"" diaphragm, and so on. Green's Theorem in a Multiply Connected Space 327. In a multiply connected space Green's theorem must be modified by the reduction of the space to simple connectivity by barriers. Then the surface-integrals over the barriers must be intro- duced, and the theorem applied to all the compartments each simply con- nected, of which the space now consists. If the value of U on the positive side of a barrier exceeds that on the negative side by k we have for the surface integral due to the two sides of the diaphragm AdV/dn .da: Similarly if k' be the difference of the values of V on the two sides of the same barrier we get tc'\dU/dn . da for the corresponding part of the surface integral in the companion expression of (58). The theorem is then Qi being taken as a constant) = _ 11" V^^dS + 2k' [^<^^} - ff fFV2i7cZa!(^y6?« . (58') where the first surface integral in each expression on the right is extended over the whole original bounding surface ;S^, and the others are taken over the barriers. This extension of Green's theorem is due to Lord Kelvin. Kinetic Energy of Cyclic Motion of Fluid 328. The kinetic energy of the fluid motion in a multiply connected space must now be expressed. It is clear that the energy will not be affected if we suppose each of the w — 1 diaphragms we have imagined drawn in the fluid to move at each point with the motion which the fluid has at that point. The diaphragms will convert the spaice in which the motion takes place into a simply connected space ; and to get the whole energy it is only necessary, since the motion is supposed irrotational. 250 MAGNETISM AND ELECTRICITY chap. to include the two sides of each diaphragm- in the surface integral. Thus, drawing normals from the two surfaces of one of the diaphragms into the moving fluid, we have for that diaphragm the contribution to the surface integral h'^/'-K''- where the suffixes 0, 1 refer to the two sides of the surface. But, since the motion is the same on both sides of the surface, d(f> d(j> and the integral is (.)'^dS. Now, as has been seen above, ^o~0i i^ ^'^^ same for every element dS of the surface, and is equal to what we have called the cyclic con- stant of the diaphragm for the state of motion considered. The kinetic energy of the whole motion is therefore given by the equation T= P + «"|g<..,.) . . (65) 2 where the first integral is taken over the bounding surface, properly so called, and the remaining integrals over the diaphragms. Uniqueness of Motion in Multiply Connected Space 329. If the cyclic constants /Cj, k^, . . . . are given we can show that the irrotational motion in a multiply connected space is quite deter- minate. For let the space be rendered simply connected by means of diaphragms, and let there be two values of the velocity potential 0', if)", which give for the difference of potential on the two sides of the successive diaphragms the values ^0 ~ <^'i = "i' 4'"o - 4'"i = "i Then we have that is, the function (f>' — 0" has the same value on both sides of each diaphragm. Hence, if the velocity-potential be made ' — (f>" at each point, the value of ^™ MOTION OP A FLUID 251 will be zero for any closed curve, whether passing through a diaphragm or not. We can therefore apply Green's theorem and so get for each point of the space 2^ _ M.' = dx dx • • ■ ■ ; that is, the two solutions are identical. Perforated Solids Moving in a Liquid. Ignoration of Coordinates 330. "When there exist moving solids in the liquid and some or all of these have perforations through which cyclical motion takes place, we have an excellent example of the principle, of ignoration of co-or- dinates discussed in Arts. 247, 248, above. , Equations of motion are applicable to this case which are precisely analogous to the equations there established for a gyrostatic system. Let s, ^„ denote the parts of the velocity-potential depending on the motion of the solids and the cyclical motion respectively, then the kinetic energy of the fluid becomes, since in (65) we must write ^j -f- (j)^ for 0, ^ = - |lj(<^. + s + i>c)dS + KlJ^^^^^ d., + ....} (65') But since ^g, c are both velocity-potentials, we have by (58) the first integral on each side being extended over the surfaces of all the solids, and the remaining integrals over both faces of each dia- phragm. It is clear that all the integrals on the left are identically zero, since the cyclic constants of a + Mbiib + . . . . where the suffixes do not relate to the summations but mark the different solids, and each contains as many terms as there are velocities Qa, or 5t, &c., necessary to fix the motion of the solid referred to (in ordinary coordinates six, namely, the linear velocities of the centre of inertia of the solid, and the angular velocities of the solid round three rectangular axes through that point), and the quantities ^a, ^j are functions of the ordinary coordinates, from which the components of velocity normal to the surface at any point are to be calculated. Again, we can write (fie = K(0 + KIO + . . . . where a, «',.... are functions of the coordinates to be determined from the conditions : — that w is a cyclic function which diminishes in value by unity for each time its variation is taken from point to point in the positive direction round a closed curve passing once through an ideal barrier across the first channel, and returns to its former value when a circuit is completed not cutting this barrier, that y^o) = at all points, dco/dx = 0, . . . . , at infinity, and doajdn = at all points of the surfaces of the solids. The functions «', w", .... fulfil similar con- ditions for their respective channels. The justification of this specification of the potential lies in the fact that each constituent of the motion is in itself possible, and corresponds to an independent part of the motion which exists at the boundary of the system, that is the surfaces of the solids and the barriers of the channels, and that the combination of these partial motions forms a possible motion. If there were two possible motions of the fluid corre^ spending to generalised velocities j^, %, .... of the solids, and cyclic constants k, k it could be shown by reversal of one of them that the fluid would be at rest, since the kinetic energy would be zero {see Art. 328). 332. We thus have for the kinetic energy of the fluid T = la^lil + «12?1?2 + ••■•+ i«2292^ + + i(K> f)"^ + (k, k)kK + where a„ = - pj^j -^ dS, a,., = - pj .^^ -^ dS = - p|<^i ^dS,.... (by (58) since V^^i = 0, V^02 — ^)' ^^^ ^° °^' *^® quantities (^j, (f)^, . . . . being the functions of the coordinates associated with velocities q^ ^^, .... respectively. To include the kinetic energy of the solids it is only necessary to add another homogeneous function of the velocities q.^, q^, . . . . This gives for the total kinetic energy depending on the solids a single homogeneous quadratic function but with different coefficients from, VIII MOTION OF A FLUID 253 those specified above. I)enoting the quadratic function of the velocities by Tp that of the ks by K we have T= T^ + K as at p. 191 above. 333. Now regarding pK, px, .... as components of momentum, let X'X> • ■ • • ^^ t^6 corresponding generalised velocities. These will be given by equations of the same form as (35) p. 190 above, ^j, ^2' ■ ■ • • taking the places of the velocities xfr, (p, . . . . and (for uniformity of notation) M^, M^, . . . . those oi M, N, . . . . Thus we get 2 «X +«'X' + = -K - q{%KM^ - ?25«-3^2 P and therefore for the modified Lagrangiau function r = T^ + pq{2,KM^ + pq^^KM^ + K -V . (66) from which the differential equations of motion are to be derived as already explained at p. 190 above. Applications will be found in the discussion of General Electromagnetic Theory given below. The part of the kinetic energy which consists of terms involving «s is equal to the quadratic function K together with terms involving products of KS and velocities of coordinates. Now all the terms in- volving «s and no others, arise from integration over the barriers. Thus -we have by (37) Chap. VII. Hence by the condition by which (65") is derived from (65') above, and the value of Z", namely, —IpXKd^d'^n.da; Thus S(^S«i/) = S«0c^. = -J^.fi^.. . . (67) 334. We shall show that the velocity x associated -vfith any mo- TOentum px is the rate of flow of liquid across the barrier. For taking the expression for the kinetic energy and remembering that 254 MAGNETISM AND ELECTRICITY chap. and that v^w = 0, we have for the value of x associated with a par- ticular K p dK JJJVSa; ox dy dy dz dzj J an J an jdn for by (58) since kw, k, , ^r- = - l-r-(KCl) + KO) +....) OtcT = - \-^d(r Ok jdn^ J dn the rate of flow of liquid across the barrier. It is easy to show that lep is the impulsive pressure that would have to be applied to generate from rest this part of the cyclical motion. It is clear from (67) that if the solids be merely infinitely thin cores round which the fluid circulates, the terms of Xi^Zx-M) vanish on account of the smaUness of the surface of the solids, and L' is the same as if there were no cyclic motion of the fluid, and the potential energy V were increased by K. This result leads to an important theorem as to the mutual forces between the cores which will be proved below (see Art. 359). 335. The process just explained of specifying the energy of the motion by one homogeneous quadratic function of the velocities specifying the motions of the solids, and another of the cyclic constants of the circulatory motion and thence forming the equations of motion of the solids, thus ignoring the coordinates of the fluid particles them- selves, precisely corresponds to the case of motion of a rigid system explained in Arts. 247, 248, above. The only possible motion of the fluid corresponding to the motion of the solids thus defined and the cyclic motion specified by the constants «,«',.... is the actual motion. For consider two motions consistent with the motions of the solids and the motion of the fluid at the diaphragms. One of these motions reversed would reduce the solids and the fluid at their surfaces to rest, and the' yiii • MOTION OF A FLUID 255 flow across the diaphragms to zero. But by (65) above this would reduce the kinetic energy of the motion to zero, which is inconsistent with the motion of any part of the fluid. The two motions must thus be identical, that is there are not two possible motions fulfilling the conditions stated. Section IV. — Vortex Motion Vortex Lines and Vortex Tubes 336. We have seen above (Art. 288) that the quantities ^, »?, fare the components at time t of the angular velocity of an element of fluid round axes parallel to the axes of co-ordinates, and it has been proved that if any closed circuit s be drawn in the fluid the circulation round it is equal to twice the surface integral of (^, r}, f ) taken over any surface S bounded by the circuit. This theorem is expressed by (18), namely, I {udx + vdy + wdz) = 2 (Z^ + mij + n^)dS. Js Js 337. A vortex-line in the fluid is a line the direction of which at every point is that of the axis of resultant rotation at the point. If dx, dy, dz be the projections of an element ds of any vortex-line on the axes, the differential equations of the line are ^ = ^ = ^ (68) ^ V i The region bounded by the vortex-lines passing through any closed curve drawn in the fluid is called a vortex-tube. If I, m, n be the direction cosines of the normal to the surface of a vortex-tube at any point they evidently fulfil the relation l^ + m-q + nt, = Q (68') The position and direction of the vortex-lines vary with the time in consequence of the motion of the fluid, and we shall see presently that we are entitled to regard the vortex-tubes as moving in the fluid from one place to another, preserving their identity, inasmuch as they are always made up of the same portions, of the fluid. Permanence of a Vortex Filament 338. If the closed circuit referred to above be drawn so as to em- brace any system of vortex-tubes, and, while the circuit is kept in a fixed position, the surface be drawn so as to cut across the system at different places, the value of the surface integral on the right of(18) is the same for every such position of the surface. Consider now an infinitely thin vortex-tube, or vortex-filament, and let the circuit fit 256 MAGNETISM AND ELECTRICITY chap. close round it, while the surface is so drawn as to cut across the tube at right angles at any desired place, while the portion of the surface connecting this cross-section with the circuit lies everywhere close to the sides of the tube. The latter portion of the surface, by (67), con- tributes nothing to the surface integral, and we see that the integral over the cross-section is the same wherever the section is taken. But if ft) be the resultant angular velocity at any cross-section of area dS the surface integral is mdS. If w' and dS' be the angular velocity and area for any other normal section of the tube, we have iodS = m'dS' (69) that is, the angular velocities at different cross-sections are inversely as the cross-sectional areas. This theorem holds even when the values, of ^, Tj, f change abruptly so that there is a sudden bend in the vortex- filament, provided u, v, w are continuous. It follows evidently from the theorem just proved that a vortex- filament cannot terminate within the fluid and must therefore either form a closed ring or have its ends on the surface of the fluid. The constant product asdS of the angular velocity into the cross Section of the filament is called the strength of the vortex. Vortex Surface 339. Any surface drawn in the vertically moving fluid, so that no vortex-linep cross it is called a vortex-surface. At every point of such a surface the condition (68') is fulfilled. Now, for any circuit drawn in the fluid and moving with it, the circulation, by Lord Kelvin's theorem, remains constant, and therefore if the circuit is drawn on a vortex- surface the circulation remains zero, as the circuit moves with the fluid. Hence, as the fluid moves, vortex-surfaces remain vortex-surfaces and contain the same particles of the fluid. Further, as the intersection of two such surfaces is a vortex-line, vottex-lines move with the. fluid and contain the same fluid particles. 340. Consider now any case of steady motion of a system of vortices. Draw vortex-lines through any stream-line whose equations are dx dy. dz u V w ' these lines define a vortex-surface. All stream-lines drawn through points on such a vortex also lie on the surface. For the particles on any vortex-line at any instant are there travelling along the stream-lines, and as vortex-lines move with the fluid they occupy in succession the positions of the series of vortex-lines which at any instant lie on the surface. By (29), since the motion is steady, 9^/9^ = 0, and F(t) reduces to a constant of integration invariable along each individual stream4ine, but "^"i MOTION OF A FLUID 25 7 in the general case variable from one stream-line to another. Hence on such a surface — + n + ig' = constant (70) along any chosen stream-line. We shall show that this equation holds also along a vortex-line, and therefore over the whole surface. Since the motion is steady, we have three equations of the form du du du dW 10 := u— + v— + w— = — ox &y dz ax in which W is put for - (jdp/p + il). Multiplying these by f, rj, ^, respectively, and adding, we obtain ./ du dv dw\ ( du dv clw\ / du dv dw\ _ dW dW dW dx dy dz But, since ^, rj, f are proportional to dx, dy, dz, the components of the element ds of a vortex-line, the expressions on the left and right of this equation are proportional respectively to the space-rates of variation of \c^ and W along such a line. Hence ^5'^— W is constant along a vortex-line, and the proposition is proved. Determination of Velocities for Given Spin and Expansion of Fluid ^ Sil. We shall now consider how from the equations dw dv ^ _ dv, dm ^y _ ^ ^ dy dz' " dz dx' dx dy' du dv dw dx dy dz where f , »?, f, 6 axe known quantities, the \e\oci\,iesu,%w are to be determined. First we shall show that if the problem has a solution it has only one. For let there be two solutions u',v',w' \ u",v",w", which fulfil the given equations; then clearly u' — u",v' — v",io' — w" will be velocities of another possible state of motion of the fluid. Let these be denoted by «ij,t?j,Wi. But for this motion 3wj _ dv^ ^ Q :dy dz ■'•••• du, dv, dw,. - — i -I- ^ -f — ^ = 0-, ' 9a! dy dz 258 MAGNETISM AND ELECTRICITV chap. so that the velocities u^, t\, ^i\, are derivable from a potential, and the motion is without divergence. Let the space in which the fluid is contained be simply connected, then the velocity at the bounding sur- face and at right angles to it must be zero ; that is, if \, fi, v be the direction cosines of the normal to the surface, we must have for each possible motion that is, Xu + ij.v' + vw = Xm" + fiv" + vw'' = , dd, „ if Auxiliary Function called Vector Potential.—" Curling " To find %(,^, v^, w^, write dH dG dF dH dG dF "^=3^ -Tz' ''■^ = Yz-Yx' "^ = 9^^8^ • (^2> so that the solenoidal condition is satisfied. Again, by (c), if J denote the divergence (dFldx + dGjdy + dHldz) of the quantity {F, G, H), \ i ^ I y^ I i so that, if F, G, IT can be found to satisfy the condition .7=0 at every point, we shall have V^F + 2f = 0, v^ dF dy dz dH dx 10 = Wj + lOg = d^ dG dz dx dF dy , 260 MAGNETISM AND ELECTRICITY chap. 344. The quantities F, G, H are the components of a directed quan- tity A {cf. p. 49 above), and, since they are derived from equations (74), which are identical with Poisson's characteristic equation for electric or magnetic potential, by the same process of integration as that used in the latter case, they are frequently called " components of vector-poten- tial." The electromagnetic vector-potential will be further used and discussed in the chapters on electromagnetism which follow. 345. Clerk Maxwell suggested the term curling for the operation by which u^, v^, w^ are derived from F, G, H, and that for shortness the con- nection (72) between (Mj. v^, w^) resultant q^, and (F, G, H) resultant A, might be indicated thus : ^2 = curl A (79) According to this notation we have also (^, r), f), resultant a, and {u, V, w), resultant q, connected by the equation 2o) = curl q (80) The condition for the existence of a velocity potential, that is, the vanishing of ^, t), f may thus be expressed by the equation (in which q ■denotes the resultant of %, v, w) curl q = Q. We shall frequently use such expressions in what follows, in order to abbreviate the equations. Velocity due to Long Straight Filament. — Velocity due to Element of any Filament 346. We may now work out two examples. First, let the vortex be a straight filament of strength m, lying along the axis of x, and ex- tending from a; = — CO to x = -|- oo in an unlimited fluid. Let us find for any point distant a from the filament the velocity due to the vortex. From the conditions of the problem G = IT = 0, and 27ri^ = m\dx'/r, where dx' is an element of the length of the filament. Hence, by (72), dF dF Again, let s = 0, so that the point lies in the plane of x,y, and we have r = \/ x^ + a^. Hence v^ = 0, and _ m d C dx' _ m \ . — CO The velocity is thus inversely as the distance a of the point from the axis, and at right angles to the plane determined by the axis and the point. 347. This result could have been obtained at once from the simplest MOTION OF A FLUID 261 considerations. In whatever direction the velocity at the point in question may be, it is clear from , the symmetry of the case that the direction of the velocity at any point d in the circle, radius a, drawn round the filament as axis, will be got from that at any other point c in the same circle by simply turning the whole system of fluid and vortex round the axis through the angle doo (Fig. 76). It is easy to show Irom this that there can be no component at any point in the plane through that point and the axis. For, take a closed path abed (Fig. 76), consisting of two equal circular axes, having the filament as axis, and two equal straight lines he, ^a,both parallel to the axis. The circulation cp A yh Fig. 76. round this circuit is zero. The circulation along ah is clearly equal and opposite to that along cd. The circulation, if any, along he must, be equal and opposite to that along ^a. But this is impossible, since 6cr can be turned into the position ad by turning the whole system round the axis through the angle, doc. This circulation is therefore zero. That there is no circulation at right angles to the axis in the plane containing it and the point in question is clear also from the fact that, the circulation in the circuit efgh is zero. The velocity is therefore per- pendicular to this plane. Taking thie velocity, then, as q, we have for the circulation round the circle of radius a the value 'iiraq. Hence 2Traq the same result as before. m, or q = 27r a ' 262 MAGNETISM AND ELECTRICITY chap. 348. From this example, also, we get with great ease an expression for the velocity at any point due to an element of a vortex-filament, namely : ma dx' that is, if P (Fig. 77) be the point at which the velocity is required, A B P Fig. 77. ds the length of the element AB, m its strength, and d the angle between the line drawn from P to the centre of the element and the element itself, '^-Z^"- » The direction is at right angles to the plane ABP, and is that in which the point would be carried if it were situated on a rigid body rotating with the vortex element. This is precisely the expression given by Ampere for the magnetic force due to an element of a conductor carrying an electric current of amount m/iir. It must be observed that other expressions might be obtained for the velocity due to an element, which would, when integrated for a complete vortex-filament, give the same result as that of (82). Any term of proper dimensions which, integrated round a closed curve, would give a zero result might be added to the expression in (82) without affecting the velocity distribution due to the complete vortex. ^Velocity due to Closed Vortex Filament 349. As another example, consider the velocity due to an endless vortex-filament of any form and of strength m. If a> be the resultant angular velocity, and a the section at any point, so that (oa — m, and ■ds' be an element of length of the filament there, an element dvs of volume of the filament will be ads'. Thus ^drs = dzs . wdx'jds' = "^'i" MOTION OF A FLUID 263 a-wdx' = mdx'. Similarly r^dvs = mclij', ^dzs = mdz. Thus, we get by (72) and (75), dropping the suffixes. m '' = % where the integrals are taken round the filament. In the case of the most general system of vortex-tubes, putting doo'dy'dz' for dvs, -\\\{^'l^'l-^h'^'^''y''' ■ ■ ■ ^'''^ with the symmetrical equations for v and iv. The volume integrals are taken throughout all space where |', rj', f ' are not zero. Velocity Potential of System of Vortices 350. The velocity-potential due to a system of vortices can be found as follows. Take, first, a single endless vortex-filament. We have m f/, , 3 1 , , 3 1\ ,„,> a line integral taken round the filament. Writing the equation in the form ^ {{Xdx' + Yd}/ + Zdz'), we see that it is equivalent to m a,/dZ dY\ /3X dZ\ /dY dX\)^^, "^ = 2;J\^37' - 3?) + '"fe - 3^') + Hs^' - W)^''^' in which the integral is taken over any surface S' having the filament as bounding edge, and I, in, n are the direction cosines of the normal to any ■element dS' of that surface. Now we have oz r 01/ r Hence so that dy' ~ d^ ~ \dy'^ "*" 3«'V r ~ dxdx' r ' dX dZ 3^ 1 ^ _ ^' _ _ ^ 1 dz' dx' dxdy' r' dx' 3y' dxds' r' OT f 3 /, 3 3 9\1 ,„, 2irJ dx \ dx dy dx / r 264 MAGNETISM AND ELECTRICITY chak and similar formulas hold for v, w, with the respective substitutions of djdy, djdz for djdx after the integral sign. Hence the potential from which u, v, to are derived is given by the equation mf/, 3 3 3\ 1 ,„, m f ~dS' (85) r- where S is there the angle between the line r and the normal {I, m, n). The integral is the solid angle subtended by the filament at the point at which M, V, w are to be found. Clearly the potential is cyclic, since the solid angle changes by 4 tt as the point considered is carried round from being close to the surface on one side to an infinitely near position on the other side — that is, passes round a closed circuit threaded through the filament. The potential in the most general case is the sum of the values given by (84) for the potentials of the individual voirtex-filaments into which the system can be regarded as divided. The velocities and velocity-potential above found will, of course, be the actual velocities and potential in the case in which 6, the expansion, is zero. Of. (71). 351. By the result obtained in Art. 317 above, the value obtained for dG dF\\ , , , which, integrated by parts, gives ^ = I ||- <^g^ + F{mw - nv) + G{nu - Iw) + II{lv - mu)\dS + p\\\{Fi + Gr, + HC)dxdydz . . . . (88) in which the first integral is taken over the surface of the enclosing vessel. The first pai-t of the surface-integral is zero if the fluid is enclosed by fixed walls ; the whole surface-integral is zero if the vortices are all within a finite distance of the origin of co-ordinates and the fluid fills all space and is at rest at infinity. For, when quantities in the surface-integral which remain finite when the surface is taken at an infinite distance from the vortices are omitted, F varies as l/(x^ + y^ + 2^) and «, v, w as ll{x^ -f- ^^ -l- s^)^. Hence the surface integral vanishes. Inserting the values of F, G, H in the volume integral, we get for an infinite liquid of density p i^ + VV + i^ dxdydzdx'dy'dz' . (89) -miw It is worthy of notice that the volume integral in (88) is equal to the sum of the products obtained by multiplying the total rate of flow •of matter through the circuit of each vortex-filament into the strength of the filament. This theorem the reader may easily prove for himself. If the fluid is of finite extent the surface integral in (88) must be retained and taken over the whole bounding surface. It is to be noticed that the volume integral contributes nothing to the value of T except for those parts of the fluid where there is vorticity. Thus in the case of an infinite fluid the value of T in (87), which is an integral taken throughout the whole mass, is reduced by (88) to an integral which extends only to those parts of the fluid where vortex motion exists. This mode of calculating the energy is that given by v. Helmholtz (Crelle, Bd. Iv., 1858, s. 45) in his famous memoir on vortex motion, from 268 MAGNETISM AND ELECTRICITY chap. which, and Lord Kelvin's paper on the same subject (Trans. E.S.K, vol. XXV., 1S69), most of the results given in this section have been taken. As we shall see later, the expression given in (89) is, to a constant factor, precisely that for the energy of a system of electric currents replacing the vortex-sj'stem, and having components of current at any point x, y,z proportional to the values of f, -r], J for the same point. 357. If, for example, the vortex-system reduce to two distinct endless filaments of strengths m ( = a)a), m' ( = a)'o-') respectively, the corre- sponding distribution of electric current is that of distinct currents proportional to m, m! flowing in two linear circuits coincident with the filaments. In this case (89) takes a very simple form. We may replace, for any point x, y, s on either filament, dx, dy, dz by ads where ds is an element of length of the filament, and similarly dx'dy'dz by a'ds. By Art. 349 ^dxdydz = mdx,. . .sxii (89) becomes T= £{m^ jl^ ds,ds, + 2mm' jj^-^^ dsds + m'^ jj"-^^ ds'.ds','^ (89') where 6^2> ^' ^'12 ^^e the angles between the positive directions of two elements, both on the first filament, one on the first filament and the other on the second, and both on the second filament respectively, and ^1?' ''• *'i2 *^® corresponding distances between the elements of each pair. The two integrations in the first term are both taken round the first filament, the two in the last term are both taken round the second filament, and those of the second term are taken one round the first filament, and the other round the second. In the general case to which (89) applies, the whole system of vortices might be divided up into filaments, and dealt with by a process similar to that used to establish (89'). Thus we should obtain r = A|2(^2f f«_^2 ^,^^g^) + 2%{mm' fl^ dsds')\ . (89") where the first term denotes the sum of the values of the expression in the brackets taken for each filament, and that of the values of the second expression in brackets taken for every distinct pair of filaments. The value of T is thus a homogeneous quadratic function of the strengths of the vortex filaments. We shall see that the electrokinetic energy of a system of linear currents exactly coinciding with the vortex filaments is a homogeneous quadratic function of the magnitudes of the currents, with precisely the double integrals as coefficients which are displayed in (89"). 358. Again, integrating (87) by parts, we get for a fluid filling all space ">^iii MOTION OF A iFLUID 269 since the surface integral vanishes. By the equation of continuity for an incompressible fluid this may be written But, by integration by parts, also I w (yr z —)dxdydz = -\\\{ifi - u^)dxdydz = 0, with two similar results for v, w. These give by subtraction from the expression on the right of (91) T = 2pJ|j{M(2/^ - zrj) + v{z^- x^) + w{xr} - y^)}dxdydz (92) an integral, again, confined to the vortex region of the fluid. 359. The fact that the elements of the volume integrals in the above expressions for the kinetic energy are zero except where there is spin of the fluid involves some important consequences. For example, take the case of a fluid in irrotational motion in a multiply connected space^-that is, circulating round cores or through apertures in fixed solids. The motion normal to the bounding surfaces of the solids or vessel is everjrwhere zero, and we may suppose the fluid to extend throughout the rest of space if the velocity there is everywhere zero. Then we have simply a case of motion tangentially discontinuous at certain surfaces, and it has been shown that such a motion can be regarded as produced by a vortex-sheet extended over those surfaces. Hence the kinetic energy in all such cases can be calculated by (89) or (92) properly modified to suit the very great spin which must be regarded as existing within a very thin sheet of the fluid at the surface. It is clear from the expressions for the relative velocity^ in Art. 352, that the case in which the solids immersed in the fluid are infinitely thin cores, round which the fluid circulates in irrotational motion, the strength of the vortex sheet directed along the surface of any core in the direction at right angles to that of the relative motion of the fluid, is equal to pK where « is the cyclic constant for the core. For the line integral of the relative velocity round the core is -the strength of the vortex, and this is also «. The electric analogue is a corresponding distribution of currents along the cores, and the force-systems between the cores are the same in amount as those between the conductors replacing the cores in that distribution. The forces are however opposite in sign in the two cases, as will be explained in the dynamical theory of currents given below. 360. We shall now consider briefly the action of vortices on one another taking only the case of parallel rectilineal vortices in an infinite incompressible fluid. The motion of the fluid due to such a system will be the same for all values! of z, and we may therefore treat the motion as two-dimensional. Consider then any such system of vortices ■210 MAGNETISM AND ELECTRICITY chap. having their axes all parallel to the axis of z. Let at any point (x, y) the angular velocity in the vortex-motion be f, and the components of velocity there be u, v. These must be component velocities with which the vortex-filament is changing in position there, inasmuch as vortex-filaments, as we have seen, move with the fluid. It is clear that as any vortex-tube moves its cross-section remains unchanged. For the tube remains always composed of the same particles of fluid, the height parallel to z of any portion of it remains constant, and, the fluid being incompressible, the volume must remain constant. Hence the cross-section remains constant. But, by Lord Kelvin's theorem. Art. 290 above, I f dx dy must remain constant for any tube whatever. We may write this as ^f„„ where f,„ is the mean value of ^ over the cross-section of area A. Since A remains constant, f,„ must also remain constant. This holds for a tube, however small in dimensions of cross-section, taken in the system ; hence ^, the angular velocity of any point in the system, remains constant. We can now prove that for the whole system the equations [utdS = 0, {v^dS =0 (93) hold, when the integrals are extended over all parts of the plane of x, y where there are vortices. The first integral may be written in the form Integrating by parts we obtain The line integral is to be taken round a curve encircling the whole system of vortices. If the vortices be all within a finite distance from the origin, and the curve of integration is taken everywhere at an infinite distance, the integral, as will be seen by considerations similar to those stated in Art. 355 above, must vanish. But by the equation of continuity the remaining part becomes a line integral which also clearly vanishes when taken round a curve at an infinite distance from the vortex system. Hence we have (93). Since any element ^dS does not vary with the time, we have, inte- grating with respect to the time, ^.xifiLS = C, [yt,dS = C (94) i vni MOTION OF A FLUID 271 / where C, C" are constants. Take now mean values of oj^, y^ for the system, such that x,\^t,dS =^xt,dS, ySt/'= - \i\ogrdS' (96) we have ** = - V "^^ ^ ^ Of course it is needless to introduce into_i/r the value of ^belonging to any part of a vortex-tube for the calculation of the motion of the axis of which ■^ is to be used. The function i/r is the stream-function and fulfils the differential equation 3V + SV = 2^ (98) 272 MAGNETISM AND ELECTRICITY chap. from which, according to the theory given above, the solution might have been derived. Of this equation (94) is the solution appropriate to this case. When the fluid does not extend to infinity in all directions, but is bounded by a surface parallel to the axis of z we must add to this solution a complementary function -f^^, so that >/' = ^jriogrc?^' + ^o ■ • • . (98') This function must be so chosen as to enable the boundary conditions to be satisfied, and to satisfy the equation It represents the stream-function due to the vortex-sheet which, according to Art. 352 above, may be supposed to exist on the surface, and is such as to reduce the velocity to zero for every point external to the surface. Equation (98) is thus, by (98'), satisfied at every point inside and outside the boundary. 362. We shall now consider one or two particular cases. Suppose that there are two infinitely thin rectilineal vortices A, B, of strengths m^, m^, at a distance r apart in an unlimited fluid. Taking the axis of x along, r, and the origin at A\ we have, for the velocity of B, ■Vj = mjirr, and, for the velocity of A, v^ — —^nj-n-r. The velocities are thus inversely as the " R s) ? rf) Fig. 78. strengths of the vortex-filaments, and the pair of filaments (not the fluid around them) move as if they were rigidly connected with an axis A parallel to s, and in the plane of the vortices, at a distance frorn the ■origin m^rj{m^ + m^. If the vortices are of the same sign this axis lies between the vortices at G^ ; if the vortices are of opposite signs it lies on the line AB produced beyond the stronger vortex, and at a distance given by the same formula, account being taken of the signs of the values of m^, m^. If m^ = — m^, G is at infinity, and the pair of vortex- filaments move with constant velocity »ij/(7r . AB) at right angles to the line AB joining them. The stream-lines due to a pair of vortex-filaments are given by the equation »»i log »-i + ^2 log rj = C, where i\, r-g are the distances of the filaments from any point the motion at which is under consideration. Diflferent stream-lines are obtained by giving different values to the constant C. VIII MOTION OF A FLUID 273 363. The stream-lines of a pair of equal and opposite vortex-filaments are given by the equation log r^-log r^ = constant, or (100) and are two sets of circles surrounding A, B respectively, as shown in Fig. 79. o ' 1' J> The straight line running up the centre of the diagram may be regarded as the limiting circle common to the two sets. Since there is no motion across the plane of which this straight line is the trace on the plane of x, y, no change in the motion of the fluid would be pro - duced on either side by replacing it, after the motion has been set up, by an infinitely thin fixed sheet of matter cutting off communication between the portions of the fluid on the two sides. Thus the motion of a single rectilineal vortex which is parallel to and at a distance d from a fixed infinite plane wall is m/2'7rd. Again, since there is no flow across the cylinder bounded by any circle of either set, we may suppose the surface of that cylinder replaced by a material wall cutting off all communication between the fluid on one side and the fluid on the other side of the surface. Then clearly, if we have a filament of strength m at A, or one of strength — m at £, that- filament must move always at right angles to the line drawn from it normal to the cylinder, and each point of it will therefore describe a circle round the axis of the cylinder. It will be observed, however, that this solution, in the case in which the external filament is, say. A, presupposes a certain circulation, that due to the filament £, in a circuit round the outside of the cylinder but not embracing A. The amount of this is — 2m. If a vortex-filament of strength -|- m be placed along the axis of the cylinder, and we add its circulation everywhere to that of —m at B, the circulation outside the T 274 MAGNETISM AND ELECTRiCITV chap. cylinder will be zero, and the section of the cylinder by the plane of x, y will remain a stream-line. The velocity of the filament at A will be -mlir.(llAB - l/CA) at right angles to AB. But by the equation of the circular section of the cylinder GB .CA = c\ where c is the radius, and therefore we have AB =CA ~ CB = CB CA The velocity of the filament A becomes therefore m/ CA 1 \ " X V(7l2-Z72 " ca)' If the solution is to provide for a specified circulation « in the circuit referred to, the term k . GAj2ir must be added to the velocity of A just found. Since the radius c is the geometric mean of the distances CA, CB of the filaments from the axis, the filament B is called the image of the filament A in the cylinder. 364. It is an easy deduction from the preceding theory (for example from the theorem stated in the third paragraph of Art. 356), and it is of great importance in the electromagnetic analogue, that the kinetic energy T (taken for unit length parallel to j) of a system of rectilineal vortices is given by r=y|^i41ogri2rfcriC^ monly employed for its repe- tition in illustrated courses of lectures. A wire is stretched horizontally in the mag- netic meridian, and above it or below it is placed a mag- netic needle which rests parallel to the wire, provided no current flows in the latter. When, however,, a current is made to flow the needle is deflected through an angle round its axis of suspension, and takes up a position of equilibrium intermediate between its original position and that at right angles to the wire. It is in fact acted on by a deflecting ■couple due to the current, and finally rests in stable equilibrium when Fig. 80. CHAP. IS FACTS AND THEOEY OF ELECTROMAGNETISM aVt the restoring couple due to terrestrial magnetism balances the disturbing couple. If without alteration of the current the conductor be turned " end for end," the needle is turned round in the opposite direction. This clearly shows that the so-called current has directional quality, what- ever the real nature of the phenomenon may be. ■ Again, if the wire without reversal of direction relatively to th& current is transferred from above the needle to below it, the direction of turning is also reversed. Thus if the wire run, say, from south to north above the needle and back from north to south below it, and a current be made to flow in the wire, the two parts produce effects on the needle which conspire to deflect it in the same direction. Hence by winding the wire a large number of times in the plane of the magnetic meridian, so as to make a coil surrounding the needle, it is possible to -obtain a greatly enhanced effect, and a feeble current flowing in the circuit may be made to produce a large deflection of ^'"*- *^' a "magnetic needle properly suspended. Each turn of wire exerts a couple on the needle, and the resultant couple is the sum of all the couples exerted by the simple turns of wire. Fig. 81 shows such an arrangement of wire, and is in fact a picture of a now obsolete form of" galvanic multiplier" or galvanometer. To specify the direction in which the needle is turned we have first to specify that in which the current is considered to flow in the, wire- Imagine the wire stretched north and south above the surface of a table, and let a magnet resting on the table, with its length east and west, and its north-pointipg end turned towards the vertical plane; through the wire, be brought up towai-ds it from the west side of that plane. In consequence of the alteration of the magnetic field in the vicinity of the wire a current will be produced which is taken as flow- ing from south to north in the wire. If the wire be below the table th& current will flow in the opposite direction, and this may be verified by- noting that the deflections of a needle produced by the currents in the two cases are in opposite directions. The direction here assumed for the current agrees with the con- vention based on the use of a voltaic cell, and, for the reason indicated in Art. 216, generally adopted. A simple form of voltaic cell consists of a, plate of zinc amalgamated with mercury and a plate of copper placed side by side, but not directly in contact, in a vessel containing dilute sulphuric acid. When the plates are connected externally by a wire, a cun-ent flows in the circuit thus made up, the direction of which is assumed as being from the copper plate to the zinc plate along the wire. If then the current in the wire stretched in the south and north direction above the needle were produced by connecting the copper plate of such a cell to the south end of the wire, and the zinc 278 MAGNETISM AND ELECTRICITY chap. plate to the north end the deflection would be in the same direction as when the current is produced by moving a magnet along the table towards the wire (supposed above the table) in the manner already described. The direction in which the current is supposed to flow being thus fixed, we can specify that in which the magnet turns round under the action of the current. Supposing the wire to run north and south above the needle and the current to flow from south to north, the ndrth- pointing end of the needle will move towards the west, the other end towards the east. This rule may very easily be remembered by the following mnemonic device : Hold the right hand with fingers pointing along the wire in the direction in which the current flows, and with the palm turned towards the needle. The north seeking end of the needla will turn towards the outstretched thumb. Or, remembering that the northern regions of the earth have magnetism of the opposite kind to that of the north-pointing end of a needle, we may keep the rule in mind by remembering that the terrestrial magnet may be regarded as turned into position across the plane of the equator by currents circulating round the equator in the direction of the sun's apparent motion. Ampere's Theorem of Equivalence of a Current and a Magnetic Shell 367 The explanation of Orsted's experiment is an important application of the general electrodynamic theory of Ampere, contained in the famous memoir^ which is justly regarded as the Principia of ■electromagnetism. The fundamental theorem of Ampere's memoir, so far as this part of the subject is concerned, is contained in the following statement ; Uvery linear conductor carrying a current is equivalent to a simple magnetic shell, the hounding edge of which coincides imth the con- ductor, and the moment of which per unit of area, that is the strength of the shell, is proportional to the strength of the current. Thus in Ampere's view a current, whatever the form of the circuit in which it flows, is equivalent to a certain distribution of magnetism, that is to say, it produces a magnetic field affecting magnets placed in it just as the field of a certain actual system of magnets would. Indeed, as we shall see, he put forward the theory that an actual magnet is nothing else than a congeries of electric circuits of molecular dimensions cairying currents of electricity, and that the difference between a magnetized and a nonmagnetized body consisted in a similar orientation of these molecular circuits conferred on them in the case of the former body by the act of magnetization. The direction of magnetization of Ampere's equivalent shell may be specified as follows. Let an observer stand on the shell near its edge, and face so that the surface of the shell is on his left hand, the edge on his right. Then, if he is looking in the direction in which the current is flowing the face on which he stands will be that covered with ' ThioHe dcs pheuomines iledrodynamiques, Memoires de I'lnstitut, IV., 1823, IX FACTS AND THEORY OF ELECTROMAGNETISil 279 northern magnetism, that is magnetism of the opposite kind to that of the earth's northern regions. If he is looking in the opposite direction to that in which the current is flowing, the face of the shell on which he stands is that covered with southern magnetism. This result may also be remembered by the rule already given by means of the magnetization of the earth regarded as produced by currents flowing from east to west round the equator. 368. The theorem of Ampere stated above is founded on experi- ments proving the truth of the following more elementary theorem, which we shall now consider : The magnetic field produced by the current in a plane closed circuit is the same at all points, the distances of which from every part of the conductor are great in comparison with every dimension of the circuit, as thai produced hy a small magnet placed anywhere within the eircuAt, luitJi its centre in, and its .axis at right angles to the plane of the circuit, and having a magnetic moment piroportional to the current flowing, and to the area of the circuit. The following is an experiment by which this theorem may be verified. A circular circuit is mounted in a vertical plane on a sliding piece which can be moved along a horizontal slide. The circuit is arranged so as to be in the magnetic meridian, and the slide is there- fore in the east and west magnetic direction. A needle is now set up Avith its centre on the east and west (magnetic) line through the centre of the circuit, and is provided with a pointer moving round a circular scale so as to show angles of deflection directly, or has rigidly attached to it a mirror by which a ray of light from a lamp is reflected to a scale, and which thereby measures the deflection of the needle. When a constant current is sent round the circuit (by means of wires from a battery at some distance and twisted together to prevent this part of the circuit from producing any direct effect on the needle) and the position of the circuit is changed along the slide, deflections of the needle are produced which show that the magnetic forces at the centre of the needle are very nearly in the inverse ratio of the cubes of the distances of the centre of the needle from the centre of the coil, when these distances are great in comparison with the dimensions of the circular conductor. Now we have seen above, p. 28, that this is exactly the result that would have been produced by placing a small magnet with its centre at the centre of the circuit, and its length at right angles to its plane. If the circuit mounted is not circular the same result will be found to hold whatever be the form, provided the east and west line through the centre of the needle passes through the plane of the conductor within or near the circuit, and the distances are measured from the plane of the circuit to the centre of the needle. A small magnet placed along this line with its centre in the plane of the circuit would produce deflections of the needle following the same law of variation with distance. By properly choosing the moment of the magnet the deflec- tions oif the magnet and circuit may be made identical. By reversing 280 MAGXETISMAND ELECTRICITY chap. the magnet and using it along with the circuit, the two being displaced together, it can be shown that if there is an exact balance of effects at any distance, there is balance at all distances, provided of course the current is not altered from one experiment to another. To complete the demonstration it is only necessary to notice that if the area of the circuit is altered in any given ratio the force at the centre of the needle is changed in the same ratio, and, further, to test whether a magnet and a current which produce the same magnetic force at distant points upon an east and west line passing through the circuit, as described above, also produce the same magnetic effect at all other distant points. Experiments to prove these facts can obviously be arranged with great ease, and it is not necessary here to enter into details regarding them. Definition of Unit Current, Proof of Ampere's Theorem 369. It is now possible to define the measure of a current by means of its magnetic action. We take the numerical measure of a current as proportional to the intensity of the magnetic field produced by it at a given point. This mode of numerically reckoning current, as we shall find later, gives results consistent with those obtained from other methods which are sometimes used. We define unit current as that current which flowing in a circuit of unit area can be replaced by a magnet of unit magnetic moment without altering the magnetic field produced at a distance from the circuit. Unit magnetic moment has already been defined, and in the C.G.S. system is the moment of a doublet composed of two opposite f)oint-charges of magnetism, each 1 C.G.S. unit, placed at a distance of one centimetre apart. Thus, when the area of the circuit is one square centimetre, and it is replaceable as regards magnetic action by such a doublet, the current flowing is 1 C.G.S. unit. It is important to observe that the magnet equivalent at distant points to the plane circuit may be supposed replaced by a very large number of equal small magnets uniformly distributed over the area en- closed by the circuit, with their centres in and tbeir lengths at right angles to its plane. The same field as before will be produced if the total magnetic moment is the same as before : for, as has been seen, the force which a magnet produces at distant points is not affected by the position of the magnet within the circuit provided its direction is always the same> But the process of distribution of a large number of small magnets converts the equivalent magnet into an approximation to a uniform magnetic shell, the strength of which (that is its magnetic moment per unit area) is simply the measure of the current, and the larger the number of small magnets the closer is this approximation. 370. We can now prove Ampere's proposition stated above, that any linear circuit carrying a current is equivalent in external action tO' a magnetic shell, the edge of which coincides with the circuit. Let FACTS AND THEORY OF ELECTROMAGNETISM 281- a current of 7 units flow in the circuit, represented by BAC in Fig. 82. As shown in Fig. 82, we may suppose the circuit converted by cross conductors into a network without any displacement of the boundary,, and we may suppose that round each mesh a current y also flows in the same direction as that flowing in the original conductor. These mesh currents give equal and opposite currents, that is zero current, in each conductor common to two meshes, and the system reduces at once to the current supposed to flow in the boundary and that alone. Thus we may suppose the action of the latter Fig. 82. current the same as that of the system of mesh currents imagined. But each of the meshes may be taken so- small that it may be regarded, with as little error as we please, as a small plane circuit : and each of these small circuits is by the preliminary theorem replaceable by a small magnet, or by a magnetic shell of strength equal to the cun-ent. But this replacing of each of the meshes by a shell would give a shell of strength 7 bounded by the circuit. Hence the theorem. It is to be observed that any point at which the action of the finite shell is considered need only be at a distance from any part of the equivalent shell great in comparison with the dimensions of a mesh, hence the limitation as to distance imposed in the preliminary theorem does not here apply. It is only necessary to take into account the finite thickness of the wire, and therefore consider magnetic action at points at a distance of several diameters of the wire from the boundary. It is also to be carefully observed that provided the boundary of the shell, that is the circuit, be undisturbed, the meshes may be supposed to have any position we please, in other words the shell is defined only by its boundary. Theorem of Work done in carrying a Unit Pole round Closed Path in Field of Current 371. From the theorem of the magnetic shell we can at once derive a theorem of the greatest interest and importance in electrodynamics. Let a unit magnetic pole be carried in a closed path from any point P in the field of a circuit back again to the same point. The work done against 6i by magnetic forces is zero if the path does not pass through the circuit, and 4777 if it does. To prove these statements let the equivalent magnetic shell be constructed in such a position as not to intersect the closed path. Then it is clear that if work is done in carrying the pole from the point P to another Q on the path, just as much work will be gained in carrying the "pole from Q to P along the remainder of the path. Thus the work is zero. 282 MAGNETISM AXD ELECTRICITY CHAP. Again let the shell be imagined constructed so as to be infinitely near to the starting point P, and let the pole be carried round from P to another point Q infinitely near to P, but on the opposite side of the shell. Now we have seen at p. 32 that the work gainedj or spent in so ■doing is 47r times the strength of the shell, or ^ir^. But although the shell was supposed fixed when the pole was carried round from 7' to Q, and the forces at each point were the same as those given by the current, it is obviously not necessary to suppose the shell in the same position when the pole is carried through the remaining short distance QP required to complete the closed path. We therefore now suppose the shell in a position clear of the element OP of the path : and it is plain that since the forces are finite and the element QP is infinitely short, the work done over QP is zero. Hence the work done in carrying the pole round the circuit is 477-7. 372. If the path be laced round the circuit any number of times, n, the work done in carrjang the pole round the circuit is ^irny. For let the full line in Fig. 83 represent the path, and let the different spires of the path Q, Sj TJ, W, be connected by the dotted ^^^^ path P, iJ, T, V. Let the pole be sup- posed first carried round the closed path PQBP, next round the path BSTB, then round TUVT, and so on, the final path traversed being PBT . . . P. By this process 47r7 work is done (or gained) in every closed path PQRP,BSTB, &c., and the paths PB, i?7',&c.,added are each traversed twice in opposite direc- tions, so that the work done in them is zero. Hence the work done on the whole is simply that done in the path PQBS ...P which was given. In the same way it can be shown that if the circuit pass any number of times n through the closed path the work done in carrying a unit pole round the path is again 477^7. For, take the simple case of Fig. 84, in which the •circuit passes twice through the path. Let the path be converted into two separate paths, each enclosing the circuit once, by drawing the line SQ. The work done in carrying the pole round SBQ is 4 be the angle of deflection, and M the magnetic moment of the needle, then the deflecting couple is 2'7TyMrhos^l{a^+r^y^ and the u 2 292 MAGNETISM AND ELECTRICITY chap. directive couple due to the earth is ^ilf sin (f>. Hence for equilibrium we must have equality of these couples, which gives , = (^^i.tan, (7) The arrangement just described is that of an ordinary tangenlj galvanometer with eccentric needle. If there are n turns of wire arranged in a thin coil so that the dimensions of cross-section may be neglected, the equation becomes (a^ + r^VJ V=-li/^*-^ («) If a = 0, so that the centre of the needle is at the centre of the coil, the last equation becomes ^ = £^*"'^^ (9) The multiplier of HtaxKJ) in these equations for y is called the constant of the galvanometer. The value of this multiplier requires correction in general for the cross-section of the coil, and sometimes also for the length of the needle. These corrections, as well as the mode of using such instruments, will be fully discussed when we deal with the subject of, galvanometry and the measurement of currents generally. Energy of Current-Carrying Circuit 384. We shall now calculate the energy of a circuit carrying a current in a magnetic field. This will consist of two parts, one independent of the previously existing field, the other represented by the work which has been done in establishing the current in presence of the field. Let the components of magnetic force of the previously existing field at any point be a, /8, 7, and its inductivity /i (supposed at present independent of the magnetic state of the field), and let the circuit produce a field intensity a, /8', y, where the intensity previously exist- ing was a,^,y. By the specification of magnetic energy given in Art. 56 above, the energy in the medium is given by the equation ^ = ^^« + «T + (/8 + /3T + (y + 7') VJ7 • (10) where dns is an element of volume, and the integral is taken IX FACTS AND THEORY OF ELECTROMAGNETISM 293 throughout the whole field. This breaks up into three integrals, so + ^JM(a'^ + ;3'2 + y'2)cZOT + j;;:|M(aa' + j8;8' + yyO-^cr .... (11) Mutual Energy of Two Current-Carrying Circuits _ 385. The first of these is the magnetic energy of the field previously existing, the second that of the circuit introduced supposed existing alone, and the third that due to the introduction of the circuit into the previously existing field. It is this last expression we wish here to consider, and to change into another form which will be useful in discussions regarding the actions of magnetic fields on conductors. The integral can be written in the form — LiHH'cGsedCT where H, H' are the previously existing and the induced force respectively, and B is the angle between their directions. Now we have 4 ?ry ■■ [vLdi (12) where y is the current in the circuit, ds an element of a line of force where thej intensity is H', and the integral is taken round a line of force due to the current, which, as we have seen, threads through the circuit. Further, the total induction through the circuit due to the previously existing field is the integral l/t(?a + myS + ny)dS, in which /, m, n denote direction-cosines of the normal to the surface element dS, and which has a constant value for all surfaces having the circuit for bounding edge, since the magnetic induction fulfils the solenoidal condition. Hence, multiplying both sides of (12) by this integral, and dividing by 47r, we obtain y\lJ.{la + m^ + ny)dS = 2~ U(^"' + m^ + ny)dS x lH'(fs. Now let surfaces be so drawn as to be successive equipotential surfaces for the intensity due to the circuit, that is, for the intensity H'. 294 MAGNETISM AND ELECTRICITY chap. Each of these has the circuit for bounding edge, and the first integral on the right of the last equation has the same value for each, that is, the value which it has for any surface having the circuit for bounding edge. Hence we can evaluate the right-hand side of the last equation by multiplying the (constant) value of the first integral for each equi- potential surface by the value of J3!ds for the step from one equi- potential surface to the next (since for a given pair of successive equipotential sui-faces Hds has the same value), and so on throughout the whole field. But plainly, since la + ra^+ny = 13.008 0, this gives simpl}' the integral LhH' cos 6 doS where dzs, as here considered, is an element of volume, bounded by a tube of force between two successive equipotential surfaces ; but may, of course, be any element, since a volume integral cannot depend on the manner in which the elements are taken. Thus we get for the mutual energy Tc/ of the circuit and field Tef = — LhH' cos edzs = y L(^a + m^ + ny)dS . (13) that is, the mutual energy is equal to the product of the current into the surface integral of magnetic induction through any surface which has the circuit for bounding edge. This theorem is of enormous import- ance in electrodynamics. The proof here given is not that usually adopted, which treats the circuit as a distribution of magnetism acted on by the previously existing distribution : but it seems preferable to arrive at the theorem from the expression for the energy of the whole medium occupying the field. Electrokinetic Energy 386. The jnagnetic energy we are here considering is energy de- pending on the state of the field at any instant ; and it is not necessarily equal to the energy which has, up to that time, been thrown into the field from a battery or other source. That there is energy depending on the state of the field as distinguished from the total amount which has been furnished by the source is clear from the dissipation of energy in hysteresis ; but we shall return to this subject later. According to the conclusion adopted above as to the nature of magnetic energy we regard it as kinetic or, as we call it, electrokinetic energy, and hence in a system not subject to dissipative forces we must so choose the sign of the energy that the mutual forces of the system will tend to cause the amount of energy to increase. Thus a circuit being brought into a field must tend in virtue of the forces exerted upon it by the field to move IX FACTS AND THEORY OF ELEOTROMAGNETISM 295 so as to increase its electrokinetic energy, that is, we must so choose the sign of the surface integral of magnetic induction that in any actual case the mutual forces may tend to its increase. Thus if T^f denote the mutual energy of the circuit and field, and d-^ any small change of position or configuration of the circuit, and '^ the force producing it, the work done by this force is ■^(^i|r. Thus or ^JIl^J^l (14) if N denote the magnetic induction through the circuit and 7 be main- tained constant in the change. Electromagnetic Force on Element of Circuit. Most General Specification of Current 387. The force '^ which thus tends to increase T^f by increasing the magnetic induction through the circuit is called the electromagnetic force on the circuit. The circuit, if free to move as a rigid whole, will change its position in obedience to this force so as to increase N, and whether fixed in position as a whole or not, will tend to increase its area so as to include a larger total induction. The resultant of electromagnetic force on each element of a circuit can only be in the direction at right angles at once to the magnetic force and to the element, because the element, if free to move in that direction, would increase the magnetic induction through the circuit at the greatest rate. Thus there is no electromagnetic force in the direction of the magnetic force on an element, since a displacement in that direction would not alter the electrokinetic energy of the circuit. The direction of the force, ■*■, on an element of the circuit and the corresponding directions of the current and the mag- netic induction at the element are shown in Fig. 91, in which ds is supposed perpendicular to the plane of B and '^. The value of N is to be taken positive or negative according as the direction of the magnetic induction through the circuit agrees Avith (as here), or is opposite to that in which a right-handed screw moves when the handle is turned round in the direction in which the current flows. In general, however, the elements of the circuit are inclined to the direction of the magnetic induction. Fig. 91. Let the angle between the direction of the current in an element of the circuit and the positive direction of the induc- tion be e, and let the element be displaced through a distance d-^ in a 296 MAGNETISM AND ELECTRICITY chap. direction at once normal to itself and to the direction of the magnetic induction. The element may be supposed moved out along guiding wires placed in this direction at its extremities. Let ds be its length. The change in N is the product of the induction B at the element into the component of the length of the element in a plane at right angles to B into the dis- placement ; that is, dN = SsinOds . dyfr. Hence dTcf = yB sin Bds . dxj, . . . (15) and ^ = yB sin dds .... (16) This formula is applicable also in certain cases, when apparently no progressive change takes place in the in- duction through the circuit, as, for example, in the case of the Barlow wheel described below (Art. 402). The Fig. 92. action on the element of the circuit in every case is due to the magnetic induction there existing, and the ■element experiences a force causing it to cut across the lines of in- •duction, and of the amount given by equation (16). If the direction cosines of ds be I, m, n, we have, using the com- ponents a, b, c of B, the equation . „ Umc - iihY + (na - Ic)^ + (lb - maY\i sm 6 = -^^ ^ ^ '■ —^j and therefore for (16) the alternative form * = y{{mc - nhf + {na - Icf + {lb - maf]Hs . (17) Substituting in this for the values lylajmyja-jny/a u,v,w, the com- ponents of the current in the directions of the axes taken per unit of the area a- of the cross-section of the conductor we find * = {{vc - wb)^ + {wa - ucY + {uh - va)'^}ids . a. From this, supposing ds in the direction of y, so that lo = w = 0, and B in the plane of yz so that a = 0, we get ^ = {vc - wb)cr.ds (16') that is vc — wb is the electromagnetic force per unit of volume on the conductor in the direction of x. Denoting the components per unit volume in the directions of the axes by X, Y, Z, we find X= vc - wh Y = wa - uc Z =ub - va (18) which are called the equations of electromagnetic force. It is to be particularly observed that u,v,w are here the components of the total current flowing from whatever cause, and include the IX FACTS AND THEORY OF ELECTROMAGNETISM 297 currents due to variation of electric displacement, and also the so-called convection currents produced by the motion of charged bodies. Cur- rents due to the two last-mentioned causes have not yet been con- sidered, but they are of great importance in the general theory of the electromagnetic field, and will be fully discussed in that connection. Mutual Energy of a Current and Magnetic Distribution 388. An apparent difficulty in connection with the subject of the mutual energy of a circuit and a magnetic distribution arises here. It is not the case that when a circuit carrying a current in a magnetic field is interrupted so that the current is annulled, there is any work done in the circuit or in any other way which it is possible for us to detect, corresponding to an annulment of the mutual electrokinetic energy T^f. The mechanical value of the current itself as thus tested is found to be quite independent of the existence of permanent magnets in its vicinity. The expression for the electrokinetic energy, however, enables us to calculate the forces on the circuit : and it must not be concluded that the energy, though not rendered available when the circuit is broken, does not exist. This subject will be further discussed in a later chapter. 389. The expression for the force on an element of a current-carrying conductor in a magnetic field may be applied to find the turning moment of a thin uniformly magnetized bar magnet on a conductor. Such a magnet, we have seen, may be regarded as made up of two equal and opposite point-charges of magnetism at its extremities. Choose an element ds of the conductor and draw lines to it from the extremities of the magnet. Let these lines make with the positive direction of the axis of the magnet angles 6-^, 0^, and with the element angles c^^, ^j. First we shall suppose that all these angles are in one plane, and that the positive pole corresponds to the angle ^-^. The force exerted on the element by the field of the positive pole is by the ■expression found above /lymds sinJr.^^ (where m is the pole-strength and r the distance of the pole from the element), and is in the direction of the normal to the plane through the element and the magnet. The pro- duct of this into the perpendicular distance of the element from the axis of the magnet, that is, into j-j sin 0.^, is the moment turning the element round the magnet. Hence the total turning moment due to the two poles is ^sin 6^ sin r- ,. ' r we shall show that these are the x and y components of vector-potential at the point P. In order to prove this we have to show that a = & = 0, and. that _3^ _ 3^ 8a! dy 302 MAGNETISM AXD ELECTRICITY chap. The first statement is obviously true, since if=0 and ;• does not contain z. To prove the second, we perform the differentiations, and find 3(? 3^* ixtxds dx iy r^ ■ {uy - vx). But if 1^ be the angle which the line r makes with x we have yjr = sin 0, xjr = cos , also «6 = 7 cos {6 + ^)l} dx oy r^ cr = -'^i^sine. But this by the conventions as to the axes with respect to which F, G, H are taken, states that /lyds sin O/r^ is the magnetic induction in the negative direction of the axis of s. This agrees with the result as to the direction of magnetic force due to a current stated in the mnemonic rule in Art 391. Of course any quantities ■x^, yjr, say, may be added to F and G respectively if they fulfil the condition ^ -^A = dx dii Fig. 95. "^ and are of proper dimensions. For example, if Z7 be a function of the co-ordinates y,x, the values yjr^dUjdy, ^ — dV/dx would fulfil the condition. 396. More generally, when the current is distributed through any space vs, the three components of vector-potential of the current are given by integrals taken throughout the whole space xs, namely F=iA.\-da, G = f>.\-d^, H=A-dzs . . (21) if fi has the same value throughout the medium. Should, however, the medium be different in different parts, the values of F, G, H will require modification by the addition of certain integrals taken over the sepa- rating surfaces of the different parts of the field. It will be noticed that the expressions here given for the vector- potential are identical in. form with those which we should write down for the ordinary scalar-potential of matter of density u, v, or w. The vector-potential is, however, a directed quantity (hence its name), which has X, y, z components, produced respectively by the x, y, z components lij, V, w of the current. The expressions for these components will be modified in Chapter XI. for the case of propagation in the electro- magnetic field. IX FACTS AND THEORY OF ELECTEOMAGNETISM 303 397. In order to calculate the magnetic intensity we have to properly ■differentiate the integrals in (21). This may be done as follows : The integrals, it is to be noticed, are here regarded as taken throughout all space, since elements of space where there is zero current of course ■contribute nothing to the sum. Hence to find, for example, the value of i?" at a point (x,y + dy,z) we may suppose the whole distribution of currents displaced through a distance dy in the direction of y negative. That will bring the current system into the same position relatively to (x, y, z) that it really occupies with respect to {x, y + dy, z). But by this process F becomes F+dF/dy . dy, and n changes at all points to %c + dtt/dy . dy. Thus we get dF Cldu , = '^\r?7,'^'' (22) dy dy the integral being taken throughout all space as before. Thus we obtain b = dH dy dF dz ' dG dx dG dz dH d_F dy (dw ^]r\dy dz) n /du _ 3«>' ]r \dz dx ri (dv du ]r \dx dy dv\,_\ dvj (23) As an example we apply these equations to find the magnetic in- duction produced by the system of radial currents imagined in Art. 392. If r be the distance of the point P, at which the magnetic induction is to be found, from the element dzs of the current system, we have 47rJ r \fix r j'' cy r-^/ 47rJ»- \ r^ r-^ I and similarly a and & are zero. Thus the result already obtained is verified. It is well worth noticing that equations (23) give a, h, c as the com- ponents of vector-potential of the quantity of which dw dv du dw dv dw dy dz dz dx dx dy are the x,y,z components respectively. We shall sometimes denote these for convenience by ^, i], ^, and write (23) in the form {a, h, c) = 4j (^. V> 0^-^ (24) 304 MAGNETISM AND ELECTRICITY chap. Thus, as expressed shortly by the use of the term curl, the magnetic induction at any point is the vector-potential of the curl of the current. Magnetic Intensity for (1) Case of Straight Vertical Conductor with One End on Surface of Earth, (2) for Plane Current Sheet with Lines of Flow Circles Round Point in it, and Current Inversely as Square of Badius of Circle 398. The fact noticed at the end of the last article enables us to solve the problem of finding the magnetic intensity at any point when a straight conductor, carrying a current 7, has one end placed in contact with the indefinitely extended plane boundary of an otherwise infinite mass of conducting material. Here the current is again radial and at any point at distance r^ from the point of contact is r^j^irr^ across unit of area of the hemi- spherical surface of radius o\ drawn in the conducting material with the point of contact as centre. The components are therefore given by («> 1'' vi) = ^^^ {x,y,z) (25) and are each double of the components in the former case, and the currents within the mass give zero vector-potential as before. The dis- continuity at the plane surface, where the components of current change from the value here given to zero, must however be taken into account. Thus if we take the axis of x along any radius on the surface and that of y, also in the surface, at right angles to that radius, we have ■ndvs = :~,dS lira'- if a be the distance of the point for which t] is calculated from the point of contact, and dS is the area of an element of the surface at that point. We should obtain the same value for any radius, and therefore r\ is directed at every point tangentially to a circle having its centre at the point of contact, and has the same value at every point of that circle. It only remains to find by integration over the plane or otherwise, the potential of the quantity thus calculated at the point where the magnetic force is to be found. 399. Without direct integration the magnetic intensity can be found by the following ingenious method due to Oliver Heaviside.^ Suppose two radial currents to flow, one in all directions from the point of contact B, and of density 'yjAs-nr^, at all points distant r■^^ from the radiant point, and another of the same density everywhere as the first, but flowing towards the radiant point outside, and from the radiant point within the conducting body. The current outside is thus zero, that inside the conducting body is radial and of density ll^irr^, which is the case supposed. The first current gives no magnetic intensity anywhere. The ' Electrical Pwpcrs, vol. i. p. 225. IX FACTS AND THEORY OF ELECTROMAGNETISM 305 magnetic force due to the second (which may be regarded as circuital) can be calculated by the ordinary theorem, thus : Let the point P at which the force is to be found be anywhere within the conducting body at a distance \ from the normal- drawn to the plane surface through the point of contact. Describe a circle round this normal as axis, and calculate the current flowing through it. The area within this circle on the sphere with centre at the radiant point, on which the circle lies, is 'iirr-^il — cos ), where is the angle between the axis of this circle and the line BP (see Fig. 96). The current through the circle is therefore 7(1 — cos ^)/2. If /S be the magnetic intensity its line integral, since it is tangential to the circle here considered, is 27rA/3, and this must be equal to 4^7(1 — cos ^)/2. Hence ^ = l^' cos ^) (26) If the point be outside the conducting mass and ^ be measured from the former direction of the normal, the equation is -iP ^=i<^ + cos<^) (26') Fig. 96. To this must be properly added, of course, the magnetic intensity due to the straight wire as given m (20), and that due to the currents radial or otherwise flowing to the further ex- tremity of the straight wire. If the wire is not normal to the plane surface the resultant force must be found in the ordinary way. The problem just discussed is that of a wire communicating with the surface of the earth, and the solution gives the magnetic intensity at any point in this case, on the supposition of uniform conductivity. It is of importance to notice that if we regard the distribution of 1? above specified as a current-sheet, we see that the magnetic intensity produced by it is the curl of the expression in (26) or (26'). This a very little consideration will show, is radial from B (Fig. 96) below the surface and towards B above, and of amount Y/r^ at distance r from K Experimental lUustrations of Electromagnetic Action 400 The actions of currents on magnets and of magnets on currents can be illustrated by a variety of simple apparatus. A common form is that shown in Fig. 97. On a stand is mounted a horizontal circular coil Along the vertical axis of the coil passes a metal stem on the upper end of which is a mercury cup. In this is placed a point attached to 306 MAGNETISM AND ELECTRICITY CHAP. the middle of a horizontal wire uniting two vertical pieces which dip into a horizontal circular trough just above the coil, and containing mercury. A current is passed up the central stem to the mercury cup where it divides and passes by the two side rectangles to the circular trough. A current sent through the horizontal coil produces a magnetic field equivalent to that which would exist if a complex magnetic shell replaced the coil. The lines of force curve round from one face of the coil to the other, and the sides of the pivoted rectangle tend to cut across them. Since the currents are both outwards from the centre in the two horizontal upper parts of the frame, and both downwards in the side pieces, the wires to an observer looking to either side must appear to move towards the same hand, the right or the left. Thus the frame is set into rotation. With a sufiSciently strong cun-ent in the wire rectangle the current in the coil may be dispensed with, and the wire will rotate under the influence of the earth's force alone. The current in the two horizontal upper parts of the wire frame is in opposite directions, and these parts therefore cut across the earth's vertical lines of force also in opposite directions. The vertical sides of the wire rectangle tend to move both in the same direction. The earth's force may be assisted by placing a bar magnet along the stem with its north pole upwards. If the magnet have its south pole up and be powerful enough to more than counteract the earth's field the rota- tion will be in the opposite direction. If a shorter stem with mercury cup be substituted for that used with the wire frame, and the pair of magnets, united by a cross bar and provided with a vertical axle by which they can be pivoted in the mercury cup, is put in the place of the wire frame, as shown in Fig. 98, and a current is then sent along the central stem to the cup and thence to the cir- cular trough by a connecting wire, the pair of magnets will rotate on the vertical axle. The vertical current in the central stem gives a circular line of force which moves the similarly directed poles, S, S', of the magnets both round in the same direction. The opposite poles at the upper ends tend to go round the other way; but as the field of the current at the lower ends is relatively stronger, Fis. 98. Fig. 99. IX FACTS AND THEORY OF ELEOTROMAGNBTISM 307 Fio. 100. the pair of magnets rotates in the direction in which the S poles are moved. A single magnet pivoted as indicated in Fig. 99 will behave in the same way. Obviously it may be regarded as a bundle of thin magnets, or rather as a combination of many sets of pairs like that last con- sidered. 401. Fig. 100 shows the rotation of a magnet round a current and of a current round a magnet. The magnet NS on the left is attached at the bottom of a jar of mercury to a piece of copper by which a current is carried to the mercury. The current passes to the vertical wire shown dipping into the mercury and thence round to the other side, where by a wire attached at its upper end by a flexible joint it passes to a wide-mouthed vessel con- taining mercury, in the middle of which stands a vertical magnet. The action requires no explanation. The flexible joint at the lower end of the magnet swimming in the mercury on the left allows it to play round the current, while in a similar way the wire on the right plays round the magnet, both, to an observer looking from above, in the clockwire direction. This apparatus is due to Faraday. 402. The apparatus shown in Fig. 101 is what is known as Barlow's wheel. A wheel of sheet copper is pivoted on a horizontal axis so that its plane is vertical, and its lower edge is made to dip in the mercury contained in a trough below as shown in the diagram. A horseshoe magnet is placed so that the lower part of the wheel is between its poles, and the lines of force pass across the wheel just above the part dipping into the mercury. A current is sent as shown from the rim of the wheel up the lower part to its centre, and if the magnet have the poles placed as marked the rotation is in the direction of the arrow. Fig. 101. Expression of the Eleetrokinetic Energy of any System by means of the Vector-Potential 403. The expression for the eleetrokinetic energy can be put into the form of a line integral by means of the vector-potential. Let at any point F, G, H be the components of this quantity due to the 308 MAGNETISM AND ELECTRICITY chap. currents in other circuits, and any magnetic distribution which may exist in the iield. Then substituting the values of a,b,c in terms of these components, we obtain instead of (13) But by the process given in Art. 79 above, the surface integral on the right can, in the case of a linear circuit, be shown to be equivalent to the line integral. j\ as as as/ taken round the circuit. Hence If the circuit is non-linear the equation is Tcf = {{Fu + Gv + Hw)dxs (29) where dxs is an element of volume, and the integral is taken through- out all the space in which the current (components u, v, w) flows. If F, Q, H are due to the currents in other circuits only, their values are given by (21). In particular, if the magnetic intensities are due to the currents in one other circuit, this equation can be put into a simple and easily remembered form. Let -r C sin^cosede (a + r)3 a + r (a - rY 3aV2 a - r 2 r J Ja^ + r^ - 2ar cos 3aV a/r ar 3a2 In integrating with respect to a we must use the first expression for the space within the sphere of radius r, and the second for all space external to that sphere. But fa, 2f"r J 1 2 Jr^'^'' + 3ja^'^" = 3 + 3 = 2f a IX FACTS AND THEOEY OP ELECTROMAGNETISM 313 The vector-potential at B in the case supposed is therefore simply l7i, in the direction A B. The component in the direction of ds^ is thus ^Jidr/ds^, since cos 02 = dr/ds^ This is due to a radial current starting from the centre of the element at A. But what we really have is an outward radial system at the positive end of ds^^, and an inward radial system at the other end. The vector-potential due to the former is therefore and that due to the latter is /dr d^r -^^rf^r^.^'^'O' so that the total vector-potential is ^f'-^^d^,^'''^''- The part (c) of the energy is therefore ^''y^'^'^ds^k,'^'''^''' and the total mutual electrokinetic energy of the current elements supposed made circuital by radial displacement currents is given by „ /cose , csV \ , , ,„_. yc/ = A.y,y.(— + i^<^^xrf^. . . . (35) This result was first stated by Oliver Heaviside in the Electrician for December 28, 1888, where it is given without proofs 408. It can be shown by geometry that dV • /I ■ /I r , — ;— = - sin 6, sin S, cos )/, ds-^ ds^ when the differentiations are taken as here at opposite ends of the line AB. Hence (35) becomes T^^ ^'^LUih^cosi - \sm6-ySm0^eos,yj)d8^ds2 . . (36) where »; is the angle between the plane of r and ds^, and the plane of r and ds^ If k, m^, n^, k, m^, n.^, denote the direction cosines of dsj^, ds^, and we put cos^i = Zi,cos^2 = Z2. the equation takes the form F,f=qf^{2l,l, + m,m, + n,n,). . . . (37) 2r Since cos £ = cos e^ cos 02 + sin 0^ sin 6^ cos rj = IJ^ + %»»2 -I- rift^- 1 See also Heaviside's Electrical Papers, vol. ii. articles xlviii. and xlix. pp. 501, 502, also article 1., p. 506. 314 MAGNETISM AND ELECTRICITY ohap. It is to be noticed that the formula of (35) for the mutual energy of two elements reduces to the expression given by the equation W= niyiyr^'^hds-i ■ ■ ■ ■ (38) if the circuits are closed, as in this case the second term for the action of the elements in the summation from element to element obviously gives a zero result. In the case of two conductors of finite length AA', £B' (Fig. 103), closed by radial currents, we can easily calculate the contribution of t ^ y^ / x)/ 7\" /|^ A Fig. 103. B the second term in the expression of (35) for Tcf. For integrating ^d^r/dSj^ds2,dSjds2 along the two conductors in the directions of the arrows we easily find dsj ds^ = i{A'£' - A'B + AB - AB') . (39) *M rf«2 Hence the energy in this case is ^c/ = Wiy^f j^cfei ds^ + ^' {A'B' - A'B + AB - AB') (40) The expression ^ , , /cose 1 - k d^r \ ,.,, was given by von Helmholtz ^ for the mutual energy of two current elements. His expression, therefore, passes into that given by the theory of Maxwell, in which all currents are supposed closed by dis- placement currents, if h be put equal to 0. This has been noticed by Helmholtz himself, and by other writers, but in quite another connection. ^ "Ueber die Theorie der Electrodynamik," Crelle's Journ. Ixxviii. p. 309; Oes. Werke ii. p. 745. IX PACTS AND THEORY OF ELEGTEOMAGNETISM 315 It will be observed that the proper expression for the mutual inductance of two current elements is given by the equation ^=,..,..,(2^%.^),. . . . (42) which leads to the equation (34) already given for two closed circuits M= J{^^ds-^ds2 (43) Elements of the kind here considered of course can be combined to give a conductor of finite length, since the radial currents at two extremities in contact cancel one another. Forces between Two Current Elements 409. The formulas found above enable us to calculate the force between two current elements which have been rendered circuital by radial displacement currents, or rational current elements, as they have been termed by Heaviside. For example, to find the force in the direc- tion of the line joining the elements it is only necessary to dififerentiate (35), (,36), or (37) with respect to r. The differential coefficient with minus sign prefixed, or —dTcf/dr, is the applied force necessary to increase r,and therefore dT^f/dr is the internal force between the elements. Thus if we call this force X, we have X = - ^2 {2l^l,_ + m^m^ + ni^a) .... (44) This shows that if X be regarded as a force on, say, the element at £, and the elements be arranged as shown in Fig. 102, the force is in the direction to diminish r, that is, it is an attraction. Similarly, if we suppose the value of r to be subjected to increase in the direction from ^ to ^, the applied force will be positive, and the internal force is an attraction of the element at A towards £. A different law of force which was given by Ampere in his famous paper is expressed in the symbols here used by the equation 1 3 X = - 2y,y2 "5 (cose - - COS 6j COS ^2) cZSjt^Sg • ■ ('^^') and was deduced by him on certain suppositions, and without taking into account the current in the dielectric. This law is easily deducible from Neumann's formula for the mutual inductance of two closed circuits, by an application of the calculus of variations.^ The result stated in (44) is in accordance with the observed fact that if currents be made to flow in two conductors, crossing one another at an acute angle so that the currents flow both towards the acute angle on one side and away from it on the other, the acute angle in conse- quence of the electromagnetic forces between the conductors tends to 1 For further information see Absolute Measurements, vol. ii. part i. p. 125. 316 MAGNETISM AND ELECTRICITY CHAP. diminish. Forces act as shown in Fig. 104 so as to draw the conductors nearer to parallelism. On the contrary, if one of the currents in Fig. 104 were reversed, as shown in Fig. 105, the forces would be changed to repulsions and the conductors would tend to set at right angles to one another. Fig. 104. Fig. 105. The particular case of two straight parallel conductors is illustrated in Fig. 106. The lines of force round the conductor AB are circles (of which one is indicated by the dotted line) of which AB is the common axis. The other conductor CD, carrying a current in the field of AB, tends to move so as to cut across the lines of magnetic induction most rapidly, that is, it tends to move along the radius, and in the direction shown by the arrow. The force on AB is equal and in the opposite direction. If the conductors be long and their circuits be each completed by a long wire at a great distance, the magnetic induction through either circuit will be increased by the approach of one wire towards the other if the currents are in the same direc- tion, and diminished if they are in opposite directions. Thus there is attraction in the former case, repulsion in the other. The same way of viewing the matter may obviously be applied to the case, which we have already referred to, of two conductors inclined to one another at an acute angle. PACTS AND THEORY OF ELECTROMAGNETISM 317 Explanation of Actions on Conductors by Stresses in the Field. 410. Later we shall see that, according to a theory of stress in the magnetic field, where magnetic force H exists in an isotropic non- magnetic medium, there probably exists also a stress of the nature of hydrostatic pressure combined with a tension along the lines of force. No completely satisfactory theory of stress has yet been elaborated, but according to that given by Maxwell a stress of the nature of hydrostatic pressure of amount /MH^/Sir exists at every point, combined with a tension of amount /iH^/47r along the lines of force. Thus, according to Maxwell, there exists at every point a pressure across the lines of force and an equal tension along them. Such a theory as this gives at once an explanation of the apparent attraction of one conductor by the other. Consider for simplicity two long parallel straight conductors, the cir- cuits of which are completed by wires at a great distance. The two sets of lines of force, which are in opposite directions round the circuits,, conspire between the conductors, and are opposed in the rest of space :: but if the currents are in the same directions the lines of force are opposed between the conductors, and conspire elsewhere. Thus in the former case the hydrostatic pressure is greater between the conductors than elsewhere, and the conductors are driven apart, in the latter case the pressure is less between the conductors than elsewhere and the conductors are pushed nearer to one another. Ampere's Experiment. 411. But besides these qualitative results those of many careful quantitative experiments confirm the theory given above, and of a few of these we shall here give a short account. Some of the apparatus with which these experiments were made can be readily used to illus- trate in a number of ways the attractions and repulsions referred to above. The general result demonstrated by the experiments is the truth, on which we have already insisted at some length, that a current system is equivalent to a distribution of magnetism. The first experiments made of any importance were the famous four electrodynamic experiments of Ampere. These were made with the apparatus shown in Figs. 107 — 111. The principal piece of apparatus is a stand arranged so as to allow a ♦ part of a circuit, generally itself very nearly a closed circuit, to turn freely round a vertical axis. The arrangement is clearly shown in Fig 107. Two conductors, one a tube, the other a stout wire within it, are attached to binding screws in a heavy sole plate and carried verti- cally to some distance, then bent at right angles as shown in the drawing. The end of the wire projects beyond the mouth of the tube, which carries a projecting lip, so that two mercury cups at their ex- tremities are in the same vertical. These cups bring the turned down 318 MAGNETISM AND ELECTRICITy CHAP. Fig. 107. ends of a wire frame such as that shown in the figure into contact with the tube and wire, and therefore with the binding screws, and leave it free to turn round a vertical axis passing through the cups. A current can therefore be sent (by attaching the terminals of a battery to the screws or otherwise) round the circuit. The frame shown is a double rectangle of wire, the parts of which are insulated from one another at the points of crossing. The current flows as shown by the arrows, and two nearly complete circuits are ob- tained, the areas of which are approxi- mately equal. The arrangement of tube and enclosed coaxial conductor prevents the communi- cating wires from having any appreciable effect, and limits any electromagnetic action experienced to the frame. The modification of Ampere's stand shown in Fig. 108 is due to M. Nodot. A vertical platinum wire is hung by a silk thread as shown, passes down through a mercury cup, the bottom of which is a plate of mica perforated by a hole just large enough to give the wire enough of clearance. The frame is attached below, and a point at its lower end dips into a mercury cup vertically under the platinum wire above. The current is let in and out at the cups. The frame turns with great freedom, under even small forces. 412. When the circuit as here arranged carries a current it shows no tendency to set itself in any par- ticular position in the earth's magnetic field. If, however, the frame suspended consists of a simple turn of wire, it sets itself so that its plane is at right angles to the horizontal magnetic force of the earth, and so that the direction of the current in the circuit, if looked at from the north side, is in the opposite direction to the hands of a watch. This illustrates the equival- ence of a current and a magnetic shell. Ampere's first experiment. — A wire was doubled on itself as shown in Fig. 109, and attached to binding screws at its extremities. The two Fig. 108. FACTS AND THEORY OF ELECTROMAGNETISM 319 parts of the wire, although close to one another along their whole length, did not touch. When the wire carried a current and was brought near but not too close to one side of the suspended frame, there was no deflection of the latter. Thus the effect of the current in one part of the doubled wire neutralized the effect of the current in the other part. |l The neutralization is not in this case complete, but can be rendered quite exact by replacing one wire by a tube enclosing the other. Ampere's second experiment. — In this one of the two parts of the doubled wire of the last experiment was Fjg. io9. made a zigzag close to the other, as shown in Fig. 110. There was still neutralization of the effect of one portion of the doubled wire by the other. It follows that the effect of an element of a straight conductor can be replaced by that of a small crooked or sharply bent conductor, having the same beginning and end as the straight element, and the same current flow in both. This is in other words the important theorem that the electromagnetic force on a straight element may be considered as the resultant in the ordinary dynamical sense of those on any number of com- ponent elements at the same place. (See Art. 387.) Am/plre's third experiment. — A conductor was arranged so as to be free to move in the direction of its length. Fig. 110. It consisted of an arc of wire supported on the convex surface of mercury in two troughs and attached to a light arm of wood pivoted at the other end, so that the wire was only free to move as specified. A current was sent along the wire from one trough to the other. No arrangement of magnets or of circuits carrying currents brought near to the arc of wire was found to pro- duce any displacement of the conductor. Ampere therefore inferred that the electro- magnetic force on the arc of wire had no component in the direction of the length of the conductor. This agrees, it will be noticed, with the law of force given in Art. 391 for an element of a conductor in a magnetic field. Amp&re's fourth experiment. — A nearly closed conductor, B, Fig. Ill, was hung on the stand, so as to be free to move round a vertical axis, and two others, A and G Fig. ill. similar to the first, were arranged on the two sides of B, The dimensions were so chosen that each linear dimension of B was n times the corresponding dimension of A, and 320 MAGNETISM AND ELECTRICITY chap. 1/wth of the corresponding dimension of C. Further, the positions of the conductors were made similar, that is the position of A with respect to B was similar to that of B with respect to C, and hence the distance of any element of C from any element of B was n times the distance apart of corresponding elements in B and A. Currents of equal strength were sent in the same direction through A and C, and a current of any convenient amount through B in either direction. The actions of A and C on B were found to be opposed and exactly to balance one another. 413. From the result of the fourth experiment Ampere inferred that if the action on the movable conductor be made up of the actions, on each of its elements, of forces due to each element of the other con- ductors, the action of each pair of elements varies inversely as the square of the distance between them. To prove this in his manner we call ?-j the distance between an element &i in B and an element a^in A, and 7-2 the distance between two similarly situated elements Cj and h^ in C and B, and let /(i\), /(rg) denote the forces between the elements of the respective pairs per unit of length and per unit of current in each case. If (is be the length of each of the two elements of B chosen, those of the elements a^ and c^ of A and C are respectively ds/n, nds. Since B is in equilibrium the forces for corresponding pairs of elements are to be taken as equal, and therefore if 7 be the current in A and C, and 7' the current in B, — vy/(''i) = nds'^yy'fi^-i), or that is the law of the assumed force between a pair of elements is that it varies as the inverse square of the distance. This is entirely an action at a distance way of looking at the matter. The truth no doubt is that there is no direct action of one element on the other. Each circuit produces a magnetic field, and the conclusion to be arrived at from the experiment, if B remains at rest and shows no tendency to deformation, is that the magnetic fields produced by A and C neutralize one another at every point of B. From these four experiments Ampere deduced his law of force given above. We do not give the discussion here, as it depends on assumptions which it is impossible to justify, and further involves the fundamentally erroneous notion of an unclosed current element. Of course the law of force found gives the total action experienced by a circuit as a whole (in fact it can be found by supposing a circuit to be slightly distorted so that its elements are placed in slightly different relative positions). From it can be deduced various results which are known to be true, such as that a solenoid is equivalent to a uniformly magnetized magnet ; but it is also the case that the same results are IX FACTS AND THEOEY OF ELECTROMAGNETISM 321 obtained by using any one of several other laws of force between current elements which have been put forward by various mathematicians. In tact, an mfinite number of laws of force between a pair of current elements are possible, and if their circuits are not supposed closed in sonae proper physical manner the search for the true law of force must be fruitless. Weber's Experiments 414 An exceedingly valuable series of experiments was made by W. Weber i by means of his electrodynamometer. This is shown in the chapter below on Measurement of Currents. The instrument consisted of two circular coils suspended by bifilar wires so as to be free to turn round a vertical axis, and a fixed coil, which could by levelling have the planes of its windings made vertical. The current was conveyed to the suspended coil by the bifilar suspension. Two forms of the apparatus were used : (1) One which had the movable coil system suspended within the fixed coil, so that their centres were as nearly as possible coincident. (2) One in which the fixed and movable coils were distinct, so that they could be placed at any required distance from one another, and in any relative positions. The deflections of the movable coil were measured by the ordinary mirror and telescope method. (See the chapter on Measurements of Currents, Vol. II.) In his first experiment Weber proved that the electromagnetic action on the suspended coil varied as the square of the strength of the current. The fixed coil was set up with its axis at right angles to the magnetic meridian, while the suspended coil had its plane in the meridian. The centres were coincident according to arrangement (1). Currents were sent through the coils, and to prevent too great a deflec- tion, the current through the suspended coil was kept down to 1/246-26 of the current in the fixed coil, by carrying the rest of the current through a thick wire joining the terminals of the movable coil. A magnetic needle, consisting of a magnetized steel mirror hung within a vessel of sheet copper, was placed 58-3 centimetres due magnetic north of the centre of the fixed coil, and the tangents of the deflections of this needle gave a comparative measure of the different currents used. The results shown in the following table were obtained : — No. of cells used. Comparative values of force between coils = A. Force on magnetometer needle in arbitrary units = B. Force on needle found by formula 5-15634V^. Difference B -p-ismWA. 3 2 1 440-038 198-255 50-915 108-426 72-398 36-332 108-144 72-589 36-786 -282 --191 --454 EUktrodyn. Maashest. I. (1846). 322 MAGNETISM AND ELECTRICITY CHAP. It will be seen that the mutual action between the systems wa& proportional to the square of the current, that is to the product of the strengths of the two magnetic shells. In another series of experiments Weber used the second form of apparatus; the movable coil was hung with its axis horizontal and parallel to the magnetic meridian, while the fixed coil was placed with its axis at right angles to the magnetic meridian and its centre, (a) in the (magnetic) north and south horizontal line, (6) in the (magnetic) east and west line through that of the suspended coil. Experiments were made in each case with distances between the centres of respectively 0, 30, 40, 50, 60 centimetres. A current was sent through both coils, and also through a coil set up about 8 metres from the fixed coil (so as to form a galvanometer with the magnetic needle mentioned in the other set of experiments), and through a reversing key, so arranged that the current in the suspended coil could be made to flow first in one and then in the opposite direction, without changing it in the rest of the circuit. The object of thus reversing the current was to determine and allow for the turning moment of the earth's magnetic field, when the axis of the suspended coil was deflected from the magnetic meridian. The corrected results of the experiments are shown in the table below, in which the second column for each series of positions gives the corre- sponding numerical values calculated by Ampere's formula for the force between two current elements. Positions of centres of coils. In magnetic east and west line. In magnetic north and south line. Distance of centres of coils apart. Couple observed. Couple calculated. Couple observed. Couple calculated. 30 40 50 60 22960 189-93 77-45 39-27 22-46 22680 189-03 77-79 39-37 22-64 22960 -77-11 -34-77 -18-24 22680 -77-17 -34-74 -18-31 The results for the greater distances agree very fairly with calculation from Ampere's formula, and it has been shown that for a closed circuit (which each coil was very nearly) Ampere's formula and the magnetic shell theory give identical results. It is to be remarked that in these experiments the two coils are not independent circuits ; but that they may be so regarded is plain fi-om the fact that the remaining portion of the circuit, if the wires are closed or twisted together, is of no effect, since it can be altered at pleasure IX FACTS AND THEORY OF ELECTEOMAGNETISM 32S without affecting the action between the coils, provided the current be maintained constant. The deflections 6, &', in the two cases agree closely for the greater- distances with the approximate equations ^ . 2MM' / d\ ^ ^, MM' (^ l\ which give the deflections of a magnet of moment M! , by another of moment M in the so-called " end-on " and " side-on " positions and at distances D apart, great in comparison with the dimensions of the magnets. This verified the proposition that the coils are replaceable by magnets. Investigations of the mutual actions of circuits have been made with a great variety of experimental arrangements by Cazin, Boltzmann, and others. Full accounts of these are given in Wiedemann's Elehtricitdt,. Vol. III. Those of Weber, which we have described, are of special interest, as his electrodynamometer is in a modified form still a standard instrument for the absolute measurement of currents. But the surest experimental basis for the theory of electromagnetism is to be found in the accurate consistency of the results of electrical measurements made by means of instruments graduated by methods derived directly^ from it. Y 2: CHAPTER X INDUCTION OF CURRENTS Section I. — Experimental Basis of Theoi-y of Current Induction Faraday's Experiments 415. The chief experimental results on which the theory given below is based, were first obtained by Faraday, though it is also now certain that some of the most important of them were afterwards independently arrived at by Joseph Henry of Washington. The general nature of the phenomena observed may be understood by reference to a few simple experi- ments with easily obtainable ap- paratus. Two flat spirals or coils of wire are arranged parallel to one another as shown in Fig. 112. One, ^, is in circuit with a gal- vanometer, the other, B, can be connected at pleasure to a battery, and a simple commutator enables the terminals of the battery to be reversed relatively to the coil B at pleasure, so that a current can be sent in either direction round the circuit of the latter. When the key is put down so as to send a current round the coil connected to the battery, a deflection of the galvanometer needle takes place showing that a current has been produced in the other coil as well. This current is transient ; for the needle returns to zero, and so long as the battery current is not varied there is no disturbance of the needle. When however the key is released so as to stop the battery current a transient deflection of the needle again takes place, but in the opposite direction to the former. It will be found that, if the current stopped is the same in amount as the current started, the Fig. 112. CHAP. X INDUCTION OF CURRENTS 325 transient deflection at the completion of the circuit, or " make ", as we shall call it, is equal to that at the stoppage of the current, or •" break." This equality has been elaborately verified by v. Helmholtz and others. It may be roughly shown by releasing the key as soon as it is tapped down ; when the transient current at make will be followed by that in the opposite direction at break, and the needle will scarcely move from the position of equilibrium, as the impulse received from one current is compensated by that received from the other. 416. Again let the circuit B after a current has been started in it be quickly placed nearer to and parallel to A. A transient current will again be produced in A. If the coil B after having been brought up be main- tained in the same position relatively to A, no deflection of the needle will take place unless the current in B is quickly varied or stopped. If however B be quickly drawn farther away from A, a transient current will again be produced, but in the op- posite direction to that shown by the needle when the coil was brought up. Further Fio. 113. the transient currents pro- duced by the approach of the coil and its withdrawal are m the same directions as those caused by the starting and stopping of the current respectively. • -cr- no If instead of the coil B a magnet be used, as shown m J^lg. 116^ similar results are obtained. The efifect of bringing a magnet from a. distance up to or within a coil is to produce a transient current in one direction, and of its withdrawal from the coil to produce a current m the opposite direction. Feliei's Experiments 417 The laws of current induction are well illustrated by a method of experimenting used by Felici. Two coils^, £ are arranged as shown in Fig 114 in circuit with a battery, and so as to act inductively upon two coils X, Y, which are in series m a secondary circuit con- taining also a galvanometer. The coil A acts upon Xand^ upon 1 „ and it is possible so to adjust that these two actions exactly balance, and the galvanometer indicates no current when one is started or stopped in the primary coils A, B. ■ ,. , j- • i Bv means of such an arrangement, consisting however of a single turn of wire, serving as secondary to two primary turns, one on each side of the secondary, Felici proved a number of propositions which we 326 MAGNETISM AND ELECTRICITY Battery rllllllh Galvr. Fig. 114. shall see follow immediately from the magnetic induction theory of Faraday. Instead of following in detail Felici's actual investigations, we shall shortly describe the ex- periments which can be per- formed with the apparatus re- presented in Fig. 114. Balance is first obtained; then for one of the coils is sub- stituted another differing only in area of cross-section of the con- ductor or in the material of the conductor. No disturbance of the balance is produced. The contrary result however follows if the form or dimensions of the coil are changed, unless the change is made up of different parts which compensate one another. Thus : a. The induction of one circuit on 'another depends on the form and dimensions of the conductors, and not on their area of section or on their material. By interchanging A and X, say, after balance has been attained it is found that the balance is not disturbed. Hence : &. The inductive effect of A on X is equal to that of X on A. The inductive effect of ^ on X is balanced by that of B upon Y ; and again by that of a third primary coil on a third secondary Z. Then the inducing current after passing through A is divided in any «hosen ratio between B and G, while the secondary coils Y and Z are joined in series so that the currents induced in the latter are in the same direction, but are opposed to the induced current in X. They are then found to balance this last induced current. It follows that however the inducing current is divided between B and each part produces an effect proportional to its strength, and such that the two effects added together precisely equal that which is produced by the whole current when flowing in one of the coils. Thus : c. The inductive effect is proportional to the inducing current. Further, the arrangement just described is so constructed that the sum of corresponding dimensions in B and Y, and C and Z, is equal to the corresponding dimension in A and X; when X is opposed to Y and V, thus arranged, there is no current through the galvanometer when the primary circuit is made or broken. This proves that : d. The inductions between geometrically similar pairs of circuits are ^proportional to the linear dimensions of the different pairs. 418. In another series of experiments made with only one pair of coils, a primary and a secondary, Felici obtained further important X INDUCTION OF CURRENTS 327 results. If the relative positions of a primary and secondary coil be changed, a series of positions will be found in which the starting or stopping of a current in the primary produces no effect on the secondary. The two coils are said to be then in conjugate positions. Felici found that if the secondary was quickly moved from any conjugate position with respect to the primary to another such position, without changing that of the primary, no induced current was shown by the galvanometer in the secondary, and this no matter how the transition was effected, or whether or not the current in the primary was altered during the change. Again if the secondary was moved from a conjugate to a non- conjugate position, an induced current was produced exactly equal and opposite to that set up in the secondary by interrupting the primary circuit while in the latter position. This was shown by moving the secondary quickly from the first to the second position and breaking the circuit immediately after, so that both induced currents had given their impulses to the needle before it had sensibly moved from its equilibrium position. The effect of changing the position of the coils so as to allow induc- tive action to ensue, was equal and opposite to that due to stopping the current at the second position, and therefore precisely the same as that which would have been produced by starting the current in the coils after they had been placed in the final position. Precisely similar results were found to hold when the primary was moved instead of the secondary. Induced Currents are due to Change of Magnetic Induction 419. All these results are confirmatory of the view, put forward by Faraday, that the induced currents are the effect of the alteration of the total magnetic induction through the secondary circuit. For, (1) the mutual action of two circuits in producing currents in one another by induction depends only on the geometrical form and the dimensions of the circuits (unless the material be magnetic), (2) the magnetic induction at any point produced by the primary is proportional to the current strength, for different currents in the same circuit are compared by the magnetic intensities they produce at a given point, (3) the total magnetic induction through a circuit A produced by a given current in another circuit B, is equal to the total magnetic induction through B produced by the same current in .4, or in other words the inductance of A through B is equal to the inductance oiB through A (Arts. 403,404), {4} the mutual inductance of two similar pairs of circuits is, as we shall show later, proportional to their corresponding linear dimensions. _ Again when a primary and secondary circuit are in conjugate positions the magnetic induction through either due to a current in the other is zero, that is their mutual inductance is zero. Hence, chang- ing the coils to a non-conjugate position when a current is flowing in 328 MAGNETISM AND ELECTRICITY chap. one, produces the same induction through the other as would be pro- duced by first placing the coils in the non-conjugate positions and then causing the currents to flow in the primary. Faraday's Theory of Lines of Force Induction 420. Faraday's idea that change of total magnetic induction through a circuit is the cause of induced cuirents is at the bottom of the whole mathematical theory of current-induction ; but there can be no doubt that it relates to a real physical change which takes place in the medium when, owing to changes in currents or displacements of circuits or magnets, what we call the magnetic induction is altered, and it is the main object of this book to discuss consequences of such changes. Faraday began by attributing to a circuit in a magnetic field an " electrotonic state," and afterwards interpreted this view by means of his notion of " lines of force " belonging to or moving with magnets or current-carrying conductors. According to this idea lines of induction fill the space surrounding a magnet or a circuit in which a current is flowing, come into existence when the current is started in the con- ductor and disappear when the current is annulled, expand out from or contract down upon the circuit or magnet when the current is increased or diminished or the magnet is strengthened or the contrary, move with the magnet or circuit to which they belong when that is displaced, and by their passage across a conductor produce inductive electromotive force in it and set electricity in motion. 421. The whole modern theory of lines or tubes of induction, though not in its mathematical details, seems to have been clearly present in the mind of Faraday. By their relative concentration or sparseness of distribution he represented the intensity of magnetic induction from point to point of the field ; and by the physical existence he rightly attributed to them, he obtained a much more vivid idea of the con- ditions and relations of magnetic and electric phenomena than was possible to men of far greater technical mathematical power, but inclined, perhaps for that very reason, to regard physical phenomena in too abstract a manner. The following extracts from his Experimental Researches will show how fully he stated the theory of induction which has only since the rise of practical electricity become part of the general knowledge of ordinary electrical students. " The moving wire can be made to sum up or give the resultant at once of the magnetic action at many different places, i.e., the action due to an area or section of the lines of force, and so supply experimental comparisons which the needle could not give except with very great labour, and then imperfectly. Whether the wire moves directly or obliquely across the lines of force in one direction or another, it sums up with the same accuracy in principle the amount of the forces represented by the lines it has crossed." {Exp. Res., Series XXVIII, 3082.) INDUCTION OF CURRENTS 329. (In this statement is contained the method of exploration of a magnetic field by means of the induced current, which is so important theoretically, and which has been used with such good effect in practice- for the determinations of the intensities of powerful fields such as those- of dynamos, and for the investigation of fictive magnetic distributions.) " If a continuous circuit of conducting matter be traced out or con- ceived of either in a solid or fluid mass of metal or conducting matter,. or in bars or wires of metal arranged in non-conducting matter or space,, which being moved crosses lines of magnetic force, or being still, is by the translation of a magnet crossed by such lines of force, and further, if, by inequality of angular motion, or by contrary motion at different parts of the circuit, or by inequality of the motion in the same direction one part crosses either more or fewer lines than the other; then a current will exist round it due to the differential relation of the two or more intersecting parts during the time of the motion : the direction of which current will be determined (with lines having a given direction of polarity) by the direction of the intersection combined with the- relative amount of the intersection in the two or more efficient and determining or intersecting parts of the circuit." (Series XXVIII,. 3088.) Faraday's Experiments on " Unipolar Induction" 422. In further elucidation of this subject Faraday made a number of experiments with a bar-magnet and a loop of wire variously disposed relatively to one another. He found that when the magnet was rotated about its axis while the wire was held in a closed circuit (as in Fig. 115, on the left) no current was produced in the wire. When the wire was made to touch one end of the magnet with one extremity, and the centre of the magnet with the other (as in the other diagram of Fig. 115) so that the circuit of the wire was completed through the magnet, and the magnet was turned round its axis while the wire re- mained at rest, a current was produced, while none was observed when the circuit was rigidly connected with the magnet and turned with it. The direction of turn- ing and the poles of the magnet being as represented in Fig. 115, on the right, the current is m the direction of the arrowhead on the wire. 423. This is the so-called "unipolar induction, though there is nothing unipolar about it. A vast amount of discussion has been devoted to it, yet the phenomena are simple consequences of Faraday's principle. When the magnet moves its field of force moves with it. Hence when the loop turned with the magnet, there was no cutting[of Fig. 115. 330 i[AGNETISM AND ELECTRICITY chap. tubes of induction and therefore no current. On the other hand when the loop was a detached circuit, there was cutting of lines of induction ; but every line of induction which cuir the wire once, cut it again at another point, so that the sum total of lines cutting the circuit was zero at every instant. Hence again there was no current. When the loop made rubbing contact at its ends with the magnet, the lines moving with the magnet cut the wire as the magnet rotated, and the effect of this was not fully balanced by cutting elsewhere, since the lines passed from one pole of the magnet to the other in external space, and completed their circuit within the magnet. For the con- ductor through which the circuit was completed was the substance of the magnet, and this moved with the field. When a wire was led along close to the magnet from the end to the middle to complete the circuit, and this portion of wire turned with the magnet the result was the same, as clearly it ought to be. Among his conclusions from these experiments Faraday says : " It is evident by the results of the rotation of the wire and magnet, that when a wire is moving among equal lines (or in a field of equal magnetic force) and with an uniform motion, then the current of ■electricity produced is proportionate to the ti7ne, and also to the velocity •of motion." " They also prove, generally, that the quantity of electricity thrown into a circuit is directly as the amount of curves intersected." {JExr. Res., Ser. XXVIII, 3114, 3115.) 424. The last quotation but one is equivalent to a conclusion which may be drawn from the dynamical theory above, and which we shall make much use of: the electromotive force in a moving conductor or portion of a circuit is proportional to the rate at which it is cutting across lines of induction. From this by proper summation we get the proposition practically stated in a former quotation, and used now in calculating total electro- motive forces, viz. : each portion of the system of tubes cut across or moving across the conductor produces its own effect, which is added to that of the others, so that the effects of the whole system of tubes which have jDassed across the conductor are integrated by the circuit. The last quotation states the rule followed for the estimation of the total quantity of electricity which flows in a transient current produced by the motion of a conductor in a magnetic field. This is simply the time integral of the current at any instant during the induction, a quantity which we can measure in an analogous way to that in which we measure an ordinary dynamical impulse, although we cannot estimate the impulsive force of which such an impulse is the time integral. We shall see later how the time integrals of transient currents can be measured, and discuss the construction of instruments for that purpose. At present it is sufficient to assume that such measurements can be made. ^ INDUCTION OF CURRENTS 331 Numerical Estimation of Inductive Electromotive Force 425. Faraday showed that by rotating the Barlow's disk placed in a magnetic field, as described in Art. 402 above, an electromotive force was produced in the circuit completed by the wire touching the disk at the centre and the mercury in the cup at the lower edge. We may now estimate, on Faraday's principle of the cutting of lines of magnetic induction, the electromotive force in the circuit. This will show how the electromagnetic unit of electromotive force is defined. (See also Art. 450.) Let the induction be supposed of uniform amount B, and to be directed at right angles to the disk, a be the angular velocity, and r the radius of the disk. The impressed electric force e at a point distant q: from the centre is axS, since the rate at which an element dx of the radius there- is cutting across tubes is coxSdx. Hence the whole impressed electromotive force in the circuit is (oBI xdx = ^diBr'^ = ^vrB if V be the speed of the edge of the disk. If the circuit be not closed there will be a difference of potential of this amount between the centre and the edge, and no work beyond that consumed in the friction of the bearings and the air will be required to drive it. If, however, B have not the same value at every point of the disk, internal currents will be set up in the metal, the mutual action between which and the magnetic field will oppose the motion (Art. 427). Thus energy will have to be expended in driving the disk across the lines of magnetic induction in the field, and will be dissipated in heat in the metal. Very marked heating effects can be produced in a copper disk by rptating it rapidly, so arranged that a portion of it only, between the centre and one part of the rim, is in an intense magnetic field. If, however, an electromotive force equal and opposite to |»rB be placed in the wire the current will be reduced to zero, and the latter electromotive force will be obtainable from the rotation and other circumstances for the disk. This process has been applied to the determination of resistances in absolute units. (See Abs. Meas. Vol. II.) The apparent constancy of the number of tubes of induction passing through the circuit may be explained by supposing the outer end of each radial portion as it leaves the mercury cup to remain connected with it round the edge. On this view -n-r^B unit tubes are added in each turn to the total magnetic induction through the circuit. Another arrangement, equivalent to that just described, is shown in Fig. 116. Two parallel rails in a uniform magnetic field directed at right angles to the plane of the rails, are connected by a slide which is moving with velocity v. The electromotive force is here vlB where I is the length of the slider between the contacts, and this is the difference of potential between the rails when the circuit is not closed by a second 332 MAGNETISM AND ELECTRICITY CHAP. connection AC. Thte directions of the motion, the induction, and the current are as shown by the arrows. The general specification of the impressed electrical intensity at an element of a conductor, moving with velocity s in a direction inclined at angle 6 to the magnetic induction B, is that it is the vector product of B and s, that is, Bs sin Q. Its direction may be inferred from Fig. 116. (See also Art. 498. 426. Faraday also investigated induced currents of higher orders than the first. The notion of mag- netic induction shows at once that an induced current in a secondary will produce a magnetic field, tran- sient like itself, but setting up induced currents in a conductor (which we may call tertiary) pro- case of a transient currenb in the the tertiary is first in one direction, to a maximum, A V = /r ^ ^ . / D Fig. 116. perly placed to receive it. In the secondary, the induced current in then in the other, as the secondary induction rises and again falls off to zero. From tertiary induced currents we can pass, of course, to induced currents of higher order. Electromagnetic Forces due to Induced Currents. Law of Lenz 427. Induced currents produce electromagnetic forces between the inducing and the secondary circuits. These forces are always such as to oppose the motion of the circuit or magnet setting up the induced current. For example, the current excited by bringing a magnet or circuit nearer to a coil, is in the direction to produce a magnetic field which tends to oppose the magnet or circuit's motion of approach. If the magnet or circuit is being withdrawn, the magnetic field of the induced current tends to cause approach of the primary system. Or if the inducing field is set up by starting a current in the primary circuit, the magnetic field of the secondary tends to produce withdrawal of the circuit to a greater distance. This is the well-known law of Lenz.^ It should be noticed that if this were not the case there would not be stability of equilibrium of any arrangement of circuits. The motion would start a current which would set up a force on the circuits aiding the motion, and so the disturbance would go on increasing of itself On this law F. E. Neumann founded a theory of induction which may be regarded as a sequel to Ampere's theory of electromagnetic action. One of the most important results was the formula [Art. 403 (34)] for the mutual inductance of two circuits, or, as it used to be called, the electromagnetic potential of one circuit on the other. Any discussion of Neumann's theory would be foreign to the aim of the present work. Pogg. Ann. 31, p. 483 (1834). 3^ INDUCTION OF CURRENTS 333 Self Induction T4. ^^f ■ 1^^® subject of self induction was also investigated by Faraday. it had been observed by Mr. Jenkin, who communicated the result to J^araday, that if a powerful electromagnet is included in the circuit of a battery, shocks can be obtained by making and breaking the circuit when the human body forms a shunt on the electromagnet coils. Thus let the body be mcluded in the cross contact piece AB o{ Fig- 117 and let the key Z^ be kept down while JT^ is de- pressed and raised. If the battery be powerful enough and the coil C have many convolutions the person making contact in ^ ^ will experience smart shocks. If an iron core be included in the coils, the I shock will be very decidedly more severe than if the ^ coils are used alone. A phenomenon due to the same cause is per- ceived every time the circuit of an electromagnet is broken. When the circuit is broken a spark is \ seen, which, by the use of a sufficiently strong current in the circuit, may be made a bright flash, fusing the surfaces of the metals at the contact. This spark does not occur on making the circuit, na. m. nor is it perceptible unless the circuit includes a coil of many turns of wire, or else, if of comparatively few turns, contains a core of iron. Faraday's Experiments on Self Induction 429. By means of the arrangement shown in Fig. 117, with a galvanometer substituted for the human body in the cross connection A B, but with the coil at G, Faraday made many interesting experiments on such currents. The galvanometer needle was prevented by proper stops from moving in the direction in which it was urged by the steady current due to the battery, that is, a current from A to B. When the circuit ABC was complete, and the battery circuit was broken at K^ it was found that the needle showed a current in the direction from B to A. The stops were then set so as to prevent the needle from returning to zero after deflection by the steady current from A to B, but at the same time leave it free to move still further in the direction of the deflection. When the circuit was completed it was found that the needle sustained a transient deflection. A platinum wire substituted for the galvanometer glowed when the battery circuit was broken at K.^, although it did not glow under the steady current. Also by placing an electrolytic cell in A B, chemical decomposition was produced. 334 MAGNETISM AND ELECTRICITY chap. Faraday's Theory of Self Induction 430. It was thus shown that the induced currents caused in the circuit by putting in and throwing out the battery produced all the effects of ordinary currents. Faraday explained them by the magnetic induction through the circuit itself, produced by its own current in- duction. When the battery is thrown in, lines of magnetic induction are passed through the circuit and the number of these is greater the greater the number of turn of wire in the circuit, and is also increased bj^ placing within the coil an iron core which becomes magnetized by the steady current. When the circuit is broken these lines of induction disappear from the circuit. Thus an electromotive force in one direction is produced by the creation of the field, and an opposite one by the withdrawal of the field. In the former case the induced current caused is opposite to the steady current which is being set up, in the latter case it is in the same direction as the steady current which is beiag annulled. The directions of these cuxTents follow the rule given for currents of mutual induction by the law of Lenz. When induction is produced through a circuit the current produced is such as to oppose the action, approach of magnets or circuits (or creation of magnets by the starting of a current in the inducing circuit), and is therefore in the direction to diminish by its own induction that which is being produced by the external action. Again when induction is being withdrawn the process may be re- garded if we please as consisting in the insertion of opposite induction to the first so as to annul it. This is diminished by the induction due to the induced current. This is precisely what took place in Faraday's experiments. At make a current opposite to the steady current beginning to flow was set up and caused a current to flow round the coil in the opposite direction, and therefore in the cross connection from A to B, the same direction as that in which the steady current would have flowed. Henry's Experiments 431. Faraday's experiments were made in the year 1831, and as we have seen fully established the fundamental principles of the whole subject of current induction. Henry's investigations were made independently about ten years later, and his discoveries practically confirmed the conclusions of Faraday. Some curious points of apparent difference, however, existed between the results of these two pioneers of electrical discovery which have only been explained comparatively recently. These will be referred to in connection with some of Henry's more important experiments in the chapter on The Experimental Verification of the Theory of Induction in Vol. II. We shall here only X INDUCTION OF CURKENTS 335. notice shortly ^ the points established by Henry in his earlier experi- ments. These experiments were performed with coils of copper ribbon and helices of wire of various sizes and numbers of turns. For example to show effects of self induction, a circuit was made of a small battery and a flat coil of ribbon, by connecting the ends of the coil to mercury cups which formed the terminals of the battery. Then the experimenter, touching one of the battery terminals with one hand, broke the circuit by lifting one of the coil terminals out of the cup. Each time he did this he experienced a shock due to the induced current, and a flash took place at the mercury cup. It was found that up to a certain limit when the length of the coil was increased the flash diminished in brilliance while the shock increased in intensity. When this limit was passed both flash and shock diminished. 432. These results were due to self-induction, and illustrated the combined effect of increase both of resistance and of inductive action. Lines of magnetic induction were thrown out of the coil by the altera- tion in the circuit, and an induced current flowed round the derived circuit formed by the human body and the coil as in Fig. 117. The diminution of the flash was due to the increased resistance of the circuit caused by the lengthening of the coil. The body, however, being of moderately great resistance, the gradual increase of resistance of the coil at first brought up the resistance of the circuit by an amount small in comparison with that by which the electromotive force was increased. 433. Experiments on mutual induction were also made by Henry. One of the most remarkable of these consisted in combining all his coils of ribbon into one large primary coil of about 5 J feet in diameter, which was suspended vertically. A secondary four feet in diameter, made of a mile of copper wire yV inch in diameter of cross-section, was placed co-axially with the primary at a distance between them of a few feet. With a distance of three or four feet between the coils, and with a large-surface battery of eight elements,' severe shocks were obtained by an experimenter placing his tongue between the terminals of the secondary, and breaking the primary. The shocks were quite perceptible when the coils were placed at much greater distances, 434. In connection with his mutual induction experiments, Henry made an important observation. By using a secondary of a few turns, and a primary of many turns in which was a battery of great electromotive force, he found that he could obtain an induced current of considerable amount but of low electromotive force. This result is that now achieved on a commercial scale in electric lighting operations by means of what is called a " step-down transformer." Besides using physiological effects in order to detect induced currents Henry employed a horse-shoe of soft iron surrounded by a magnetizing coil,and tested the direction of the induced current by observ- ing the nature of the magnetization produced. He found generally 1 A more detailed acooiiiit will be found in Fleming's Alternate Cwrmt Transformer^ vol. i. 336 MAGNETISM AND ELECTEICITY. chap. speaking that induced currents which gave slight shocks magnetized the soft iron and produced bright flashes, while those that gave severe shocks gave only slight flashes and feeble magnetization. 4-3.5. By means of the soft iron horse-shoe and experience of shocks, Henry experimented with currents produced in a tertiary coil, and even with induced currents of a higher order. The shocks experienced were true indications of the existence of such currents ; the inference obtained from the direction of magnetization was apt to mislead. For one :so-called tertiary current consisting, as we have seen, of two currents in •opposite directions, it is obvious that the apparent direction as shown by the soft iron will depend on which of the currents is effective in producing the magnetization, and this depends, as we shall see later, on a variety of circumstances. Application of Principle of Energy 436. A great step was taken in advance by v. Helmholtz, Lord Kelvin, and Joule in the study they made of the energetics of the voltaic circuit, and of electromagnetic action generally. Thus von Helmholtz and Lord Kelvin independently accounted for an induced ■electromotive force due to the motion of magnets or circuits by a reference to the theory of conservation of energy. Von Helmholtz's earliest expression of his views is contained in his famous essay on the "" Conservation of Energy."^ His discussion of the case of two circuits must however be regarded as imperfect, owing to neglect of the electro- kinetic energy of the system. The correct solution was given by Lord Kelvin in the paper referred to in p. 342 below. The foregoing sketch of the experimental basis of electromagnetic induction must sufiice for the present. Many other investigations of great importance have been carried out; but most of these will be ■described in connection with their various subjects as these arise in the further discussion of electromagnetic theory. 437. It must now always be remembered that, according to Maxwell's theory and Hertz's experiments, all the phenomena of electric and magnetic induction are the results of the propagation of electric and magnetic induction at a finite speed through the medium filling the field. Among the investigations referred to above and to be described later, are those which have established this great generalisation. We shall see that under certain conditions the phenomena which present themselves are quite different from those we should expect to find if inductive effects were transmitted instantaneously, as it was long the habit to tacitly suppose. In the discussion of many ordinary phenomena however this supposition can generally be made with safety, and of course in many questions the element of time is without influence at all. It was this fact, no doubt, that prevented the earlier experimental verification of Maxwell's remarkable theory. 1 Die Erhaltung der Kraft. Translated by Dr. Tyndall in Taylor's Scientific Memoirs, Part II., p. 114. Republished in Ostwald's ClassiJcerder exakten Wissenschafien. X INDUCTION OP QUERENTS 337 Section II. — Dynamical Them-y of Current Induction. 438. We have found an expression [(33) of Art. 403 above] for the electrokinetic energy of a system of currents which fulfils the condition such an expression ought to fulfil, of giving as its rate of variation in a given direction, the observed force in that direction on a circuit or part of a circuit carrying a current. If this expression be really and truly the electrokinetic energy of the currents or that part of the energy on which the various phenomena of mutual action depend, it ought to fulfil the ordinary conditions of a material system, and be subject to the dynamical equations which hold in every case in which mutual action takes place between the bodies of such a system, that is, when the distribution of energy among the different bodies undergoes variation. These dynamical equations come to the aid of the principle of conservation of energy, which (see Art. 237 et seq. above) is generally insufficient to enable us to account for the phenomena of a material system. They are the outcome of dynamical laws which themselves are the results of certain axioms, or postulated propositions the truth of which, or applicability of which for the formation of a system of dynamics depends on experience. 439. A well known and striking example of the inadequacy of the law of conservation of energy for the explanation of physical phenomena, and the necessity for having recourse to the results of experience to supplement it is to be found in the dynamical theory of heat. The first law of thermodynamics is simply the law of conserva- tion of energy applied to a substance taking in and giving out heat, and doing work against external forces. But by itself this law does not give any properly so-called thermodynamic result, and we have to use in addition the famous second law, which has for its foundation a certain postulate the truth of which appears from experience. So in electrical dynamics we have recourse to a dynamical theory established by Lagrange for the motions of molar matter, and by its power and exquisite flexibility admirably adapted for the subjugation to dynamical law of a new department of science. This application was first made fully and consistently by Clerk Maxwell, and there can be no doubt that this was one of the most important steps in advance ever made in electrical theory. Accordingly we have given in Chapter VII. above a sketch of Lagrange's dynamical method, and of some important investigations, by Lord Kelvin, Routh, Helmholtz, and others, connected with the application of the method to particular classes of problems which are analogous to those which are met with in general electro- magnetic theory. Electrodynamics. Electrical Co-ordinates 440. The application of the dynamical theory of Lagrange to the solution of electrical problems and the interpretation of electrical phenomena is only possible in so far as we are able to follow the energy 338 MAGNETISM AND ELECTEICITV chap. changes of our system, for on this depends our power to build up an expression for the kinetic energy of the system which will give the proper applied or internal forces observed in various actual cases. The process consists in settling from the observed forces in different cases what terms must exist in T, and then deducing from the expression found not only the forces observed, which would not increase our information, but other forces which have not been observed. These can then be looked for, and if found increase our stock of knowledge of phenomena, and confirm the theory by which they were discovered. 441. In this application we have first- to consider what terms enter into the expression for the total kinetic energy of a system of con- ductors in which currents are flowing. We shall have two sets of co-ordinates, one of electrical co-ordinates, the other independent entirely of the electrical or magnetic state of the system. These we shall denote by ^, i|r , p, q, . . . . respectively. Then T will be made up of three parts, one 1\ depending on the electrical co-ordinates alone, another T^ depending on the co-ordinates 'jp, q, . . . alone, and a third Tg depending on these co-ordinates conjointlj'. Thus ^1 = i{(<^. <^)'^' + + 2(<^, >/')# + ••■•}) ^\ = h{{P' vW +■••■ + 2(p, q)n +■■■■]}■ ■ (1) T, = {, p)p + j There is no reason so far as we know to believe that T^ exists, and in considering electrical phenomena we may disregard T^, and confine our attention to T^ Also T-^ is found to depend on only the velocities of co-ordinates, not the co-ordinates themselves. The question now arises, what are to be considered electrical co-ordinates of a system of currents ? In answering this question we are guided by the fact that the state of the system remains unchanged when the currents are all kept constant, and the arrangement and positions of the conductors in the field are unchanged. Thus we are led to take the currents in the several conductors as 0, ■>//•, and to the conclusion that the co-efficients (0, <^), (0, i/r), . . . depend only on the co-ordinates which fix the positions of the system. 442. In what follows we regard the magnetic energy as electro- kinetic energy, and we have already seen how it is measured. In some of our discussions we shall find it necessary to introduce the electric energy of the system if of sensible effect, and we may consider that as potential energy. (Whether we consider a quantity of energy as kinetic or potential altogether depends on our point of view and if a proper regard is paid to the signs of the expressions used, the same result will be reached on either hypothesis.) The tendency of scientific progress is to explain phenomena by the motion of matter, and the ordinary division of energy into potential and kinetic is likely sooner or later to be replaced by a more accurate classification. At present, potential energj' is energy we are able to define by the position of X INDUCTION OF CUKRENTS 33» co-ordinates of the molar matter of our system : if we knew all about it we might be able to express it in terms of the velocities of particles of our system the co-ordinates of which. are beyond our control or observation, the uncontrollable co-ordinates of Art. 263 above. From this point of view the ordinary transformation of potential into kinetic energy and vice versa, is only a process of re-distribution of kinetic energy between the dififerent parts of the system. So far as electrical phenomena are concerned we are quite unable to refer any portion of energy to the motions of particles of the system, that is to say we are ignorant of the connection between the general- ised co-ordinates and those of the particles composing the system. Hence we may regard both kinds of energy, magnetic and electric, as kinetic if it suits our convenience. It is very remarkable, however, that the generalised co-ordinates, should, as the process of derivation above shows, be capable of connect- ing any properly known physical state with the motions of the particles of the system. To understand the nature of these connections w& must first ascertain what are the equations (2) of Art. 240 above, by which the electrical co-ordinates are given in terms of the independent co-ordinates of the particles of matter composing the system. This in general we cannot do, and the problem may not be solved for a long time to come. Happily its solution is not necessary in order that we may intelligently use the dynamical method, by which to attain to a clearer understanding of the interrelations of observed phenomena. Electrokinetic Energy. Current- or Electrokinetic Momentum 443. We have already seen that the electrokinetic energy is capable of being expressed in the form ^ = ULiYi' + Silfi^yiy, + .... + Z,y/ + 2M,,y,y, + ....} (2) where 7j, 72> • ■ ■ ^^^ ^^^ currents in the different circuits, and Zj, Zg, . . . M^2' -^23' ■ • • ^^'^ their self and mutual inductances. Differentiating with respect to the different currents we obtain the inductions through the different circuits. Denoting these hyJS\, N^, . . .^ N]c, ... we obtain (3) ^2 = ^ = ^^aiVi + hy-i + • • dy. ■ ■ M^kyk + • ■ M.jkyk + . . . . 'dT iVi = _- = il/,jyj + Mk,y, + . . oyk . . Lkyk + . . . . where Mkj = Mjk- z 2 340 MAGNETISM AND ELECTRICITY chap. It is plain that N^, y^, . . . ^k, ■ ■ ■ are the generalised components ■of momentum of the system, when current is regsirded as a velocity. We shall call them the components of electrokinetic momentum for the sake of distinction from another quantity which perhaps is more properly called the magnetic momentum (see Art. 56 above). 444. To understand more fully the meanings of the quantities Xj, Zj, . . . Jl/j2, ^w, ... let all the currents be zero except say 7^. Then iVj becomes L-^ji, N^ becomes M^jy^, and so on. Thus Zj is the magnetic induction produced through the first circuit by unit current in it and similarly for the others. On the other hand M^^ is the magnetic induction through the second circuit (indicated by the suffix ^) produced by unit current in the first circuit, and in the same way Tlf^; is the induction through the /i;"' circuit produced by unit current in they*. The expression for the kinetic energy shows that the induction through the /'' circuit due to the current in the /c"' is equal to the induction through the ¥'' circuit produced by an equal current in the '', that is, that Mjk = Mjcj. For the energy term which yields the nduction is the same in both cases. 445. Applying now Lagrange's equations in the form (71) of Art. 262, taking E for the electric energy (energy of charged conductors) regarded as potential energy, and y-y, y^, . . . as the electrical co-ordinates ■corresponding to 7^, 72, . . ■ •, so that 71 = y^, It — y^ , remembering that T Art, 441 does not involve the electrical co-ordinates but only their velocities, we obtain for the equation of the ¥^ circuit :^>^ + ^ = ^, (4) M '^Vk 87*; where F denotes the dissipation function and Eh the proper impressed electromotive force. Now it is known from the experiments of Joule that the time-rate of dissipation of energy in any system of circuits has the value R^y^ + -^272' + ••.• + Bm^ + Hence the dissipation function is given by the equation F = \{R,y,^ + R^y^^ + .... + i2iyfc2 + ....). . (5) and (14) becomes '#-i=*.-«'r. (6) The quantities B^, B^, . . . Bk, ... are called the resistances of the ■circuits. They are in fact the rates of dissipation of energy in the ■different circuits per unit of the current flowing in each case. The quantities y^, y^, . . ., y^, . . . are the charges of the conductors, and we have seen, Art. 186 above, that E is a homogeneous quadratic function of these quantities. X INDUCTION OF CURRENTS 341 The force ^j,. is called the " electromotive force " in the circuit to which It belongs, and (6) asserts that if E be zero the rate of increase of electrokinetic momentum is equal to the excess of the electromotive force over that required to overcome the dissipative resisting force. In other words, if the electrokinetic momentum Nk is undergoing change an electromotive force acts in the circuit opposing the resisting force which causes dissipation of the electrokinetic energy. It is to be observed that while we speak of the energy of a circuit, the energy referred to is part of the energy of the field. When the energy is dissipated in the circuit it must travel from the iield into the conductor across its lateral surface. We shall see later how this flow takes place. Case of two mutually influencing Circuits 446. Consider now in particular a system made up of two circuits. Let the currents in them be ^p .^f (9) found by calculating the kinetic energies of the different parts of the system and adding them together. This may be put in the form T = llL^o,^^ + 2Jfa,iO), + Z^wa^) .... (10) which is precisely that of the electrokinetic energy of a pair of mutually influencing circuits. The kinetic energy does not depend on the angles through which the wheels have turned, but only on the angular velocities ; and thus the machine forms a good example of a cyclic system (Art. 269 above) with three independent cyclic velocities to^, w^, oig. Here the wheels A and B correspond to the two circuits, while Z^,Z2,M, represent their self and mutual inductances. The rotating arms and attached masses (as well as the wheels A and B themselves), in which the energy T is stored, represent the medium in which the circuits are situated, through which their mutual action is propagated, and which is the vehicle and store of electrokinetic energy. Let resisting forces be applied by the brakes to the wheels A and B proportional to the angular velocities to^, co^ respective!}', and let the external couples applied to the wheels be ©j, @2- Then Lagrange's equations for the two wheels, obtained by differentiating T, are d r ■ ■ ■ ^^^^ since the kinetic energy does not involve the positions of the wheels- A, B. These equations are precisely the same as those of current induction for two circuits (7) above. 453. The forces in these equations have a simple interpretation.. For example, Mdajdt is the applied couple or generalised force on J, rendered necessary by the acceleration dwjdt in B, that is, if B moves- with this acceleration a force of this amount must be applied to A to keep it from moving. The model may be made to illustrate the transient induced currents; at make and break of the circuits. Take the case of two circuits which we may call a primary and a secondary, and let there be no applied electromotive force B.^ in the latter. Then, to correspond, ©^ must be made zero in the equations of the model. Let now a couple ©j be 346 MAGNETISM AXD ELECTRICITY ch.u'. applied to ^ so as to start the system from rest. At the beginning w., is zero, and the equation of the wheel B gives That is, the angular acceleration of the wheel B is opposite to that of A. Thus <»2 acquires a negative value and — R^to^ is positive. There- fore &)2 increases in numerical value so long as Mdmjdt is greater than -B2«2 ^^ numerical value, and increases fastest at first, since then Wj = 0, and dcojdt has its greatest value. 454. The further changes can best be studied by integrating the equations, but this we shall do later for the electrical equations. We .shall see that if 0j be kept constant, tOg will rise to a negative maximum, then die away to zero, while a>^ approaches a constant value. The wheel A will then rotate steadily, while B does not move. All this is paralleled by the rise of the current in the primary, when an electromotive force is applied to that circuit, while none exists in the secondary. At the instant of making the primary an inverse •current begins to flow in the secondary, rises to a maximum, and dies away again to zero, as the current on the primary approaches its steady value. During the variable period the cross-bar and the wheel A are getting into rotation, and acquiring a store of electrokinetic energy. Similarly in the case of the circuits energy is being thrown out into the medium, and a store of electrokinetic energy is there accumulated, which can ■only be recovered in part or in whole by varying or stopping the current. 455. What takes place when the primary is broken can also be traced from the machine. After A has attained its steady state let it be retarded. The equations show that then dajdt being negative, and 4:{L^L,, - M^)R^R^ that is if (Ziffj - L.^Rj)^ > - ^M^RJi,^ which is obviously true, since the quantity on the left is positive, while that on the right is essentially negative. It is necessary further, in order that it may be impossible for either current to become infinite that both a and yS be negative. This involves the inequality (Zi/?2 + L.R^f > {{L^R.2 + L^R-,f - 4(ZiZ, - M^)R^R^] which is true if L-Jj^ >• M^, that is, if the mutual inductance of two circuits is less than the geometric mean of their self inductances. This is obvious from the energy equation, or from the lines of induction of a unit current flowing in either circuit. These lines all pass through their own circuit, but clearly do not all pass through the other. The other circuit however may consist of n turns, and hence M%^ ■ C aj " ^ / ■^/' / ... 1 Seconds , 1 ly V •005 •01 •015 Secondary Fig. 120. INDUCTION OF CURRENTS 351 It is to be noticed that while the current in the primary gradually increases towards its steady value the current in the secondary rises to its maximum and then falls off towards zero as the current in the primary becomes constant. The dotted curve in Fig. 120 is the curve of rise of current for this case on the supposition that there is no mutual inductance. It appears (and the reader may easily satisfy himself that it is so from the equations above) that the effect of mutual inductance is to make the rise of the current in the primary more rapid at first, and afterwards to retard the rise, as the dotted curve if continued would cross the full curve. 464. The time in which the current rises to its maximum can be calculated by finding the value of t for which d'y^jdt is zero. Differenti- ating the second of (19) and putting d^^jclt = 0, we find u. — p a From the values of a,^ given in equation (17) it is clear that t has its least value when M^ = L-Jj^. This is approximately the case when the primary and secondary coils are equal, and as nearly as possible coincident. Then Zj = Zg = M. When the condition if ^ = LJ^^ is fulfilled the value of d-y^ldt when ^ = 0, that is — EMI{L-^L^ — M^'), is infinite, and the current takes at once its maximum value. It is needless to say that this condition is never really fulfilled, as M^ must always be sensibly less than L-^L^. 465. The march of the secondary current at break will be discussed presently. We shall first find the whole quantity of electricity which flows through the secondary at make. By equation (19) above we have R^ 00 CO T y^dt = aS (e»« - e^i)dt = A.^^-^ . . (22) Using the value of A^ given in (20) and noticing that, by the quadratic of which a, /3 are the roots, a/3 = R^B^KL-^L^ — M^), we get j»'"-^,l « The current flows, theoretically, for an infinite time before it has become steady, but in any actual case the whole variable stage due to induction does not last sensibly beyond a fraction of a second. This result has been obtained, by the process adopted here, as an example of the use of general integrals obtained for the case specified in which currents are started in both circuits at the same instant. But o2 MAGNETISM AND ELECTRICITY chap. it can be much more easily deduced from the second differential equation of (24). Thus integrating over the variable period we find CO jr-yi + L.S ^dt + E.S y^dt = 0, and the current y.^ being zero both when t = and when t = the first of these integrals is zero, and we have CO the result already given above. The same process gives at once for the total quantity of electricity which passes in the secondary circuit when the primary is broken the same result with opposite sign, viz. : For the primary current is initially 7j, and finally zero, while 73 is both initially and finally zero. Thus exactly as much electricity passes through the secondary at break of the primary as passes at make, but the quantities are opposite in sign. This has been verified by direct experiment, and affords strong «vidence of the correctness of the theory from which the result has been shown to flow. 466. It is interesting to study the march of the current in the secondary circuit (see Fig. 120). First suppose the secondary circuit to be kept closed, while the primary is broken. Let the variable stage of the primary current extend over a time t, then this is called the dura- tion of the break. Then integrating over this period the differential equation of the secondary circuit we get n - My^ + Zo-yj + a A y^dt = or IT 72 = ^71 -5|')'2<^« (26) If the time t be very short the integral on the right will have a very small value, and may be neglected. Strictly speaking the break is INDUCTION OF CURRENTS 353 never instantaneous, but in a sharp break, we may say that the current in the secondary rises very quickly to the value My.jL^ nearly, and then gradually dies away. The mode of variation is illustrated by Fig. 121 Fig. 121. The duration t of the break is OM, and the ordinate MP is approxi- mately MyJL^. ; The energy initially is ^L^M^y^^/Z^^ = iM^y^/Z^, and at any time when the current is y^ is ^Z^y/. The rate at which the energy is dissipated in heat in the circuit is 22272^- Thus we have dt (IL^^ = R,y,^ or ^2^+^272 = 0. Integrating this and remembering that when ^ = 0, y^ = y-^MjL^ we find ME _^6 72 = L„R e iz (27) which shows how the l^current dies away in the secondary. It is to be understood that if the primary or secondary circuit or both consist of coils surrounding iron cores, the march of the induced currents is very different from that studied and illustrated here. The inductances are no longer constant, but vary with the current. For results in Such cases see a paper by T. Gray, On the Measurement of the Magnetic Properties of Iron, Phil. Trans. R.S. 184 (1893), A. Single Circuit with Self-Inductance 467. We can easily investigate the theory of a single circuit with self-inductance. We have only to take one of equations (13), putting Jf = 0. Dropping suffixes we get for the equation of currents 4-^^ E (28) A A 354 MAGNETISM AND ELECTRICITY Integrating we find " ^ R CHAP. When ^ = 0, 7 = 0, and therefore we must put A = — EjB. Thus the equation of current is (29) The rise of the current with time is shown in Fig. 122. The curve ter- minates in the Fig. at the value of the current after the lapse from the M Time Fig. 122. closing of the circuit of about three and a half times the time-constant L/JR, which is represented by OM. The part He'^'/^/B is the extra or induced current, and dies away to zero, as the total current attains its steady value U/M. The whole quantity of electricity which passes in the induced current at make can be found at once by integration. Let q be this quantity, then B f .f, . EL R^ . . (30) The quantity of electricity passing in the break is also easily ound. To render the problem definite let the battery be thrown out at a given instant, and an equivalent resistance be introduced. The current circulating is yo{ — EjB) at the beginning, and at any subsequent stage in the variable state has a value 7. The rate of loss of electrokinetic energy is —Zydy/dt, and the rate of dissipation is By\ Equating these two rates we get Ldyldt+By = 0, which of INDUCTION OF CURRENTS 355 course might have been obtained by putting ^=0 in equation (28). Thus we have J ^ Hjdt E Ri (30') that is, the quantity passing at break in the induced current is the same in amount but opposite in sign to that which passes at make. The equa,lity of these quantities of electricity is independent of any want of uniformity of distribution of the current over the cross-section owing to rapidity of variation of the current. (See Chap. XI.) The current at any instant after the removal of the electromotive force is given by the equation R which is illustrated by Fig. 123, the ordinates in which show values of 7 with successive values of t. OM is the value of LjB, the so-called time- M Time Fig. 123. constant. This is the interval in which the current falls to 1/e of its initial value. From equations (29) and (31) it is evident that the curves in Figs. 122, 123 are the same, but differently placed with r.espect to the axes. The difference between OA and any ordinate of Fig. 122 is the corresponding ordinate of Fig. 123. Theory of a Network of Conductors 468. As another example of the theory of current induction we may take a set of conductors, whether or not containing electromotive forces, joined so as to form a network. The dynamical equations are at once applicable to such a system, just in the same way as to a system of complete circuits, provided we use instead of resistances, inductances, and electromotive forces in circuits, the resistances, inductances, self and mutual, of the conductors, and the impressed differences of potential between their terminals. The two fundamental principles stated in Art. 224 above, and A A 2 356 MAGNETISM AND ELECTRICITY chap. applied to the case of steady flow, are applicable also to this more general problem. The first, the principle of continuity, requires no modification, the statement of the second principle requires to be changed in the manner indicated below. There is a diflSculty, however, in deciding just what is the self- inductance of a conductor joining two points in a circuit, or the mutual inductance of two such conductors in the same or in different circuits. Happily, however, there is no real practical difficulty, as in most cases the conductors to be considered are coils, which may be regarded as each so many complete circuits given in position and dimensions by the turns of wire. The total magnetic induction through each such turn of wire is quite definite and can be calculated, and different methods lead to the same result. 469. The difficulty here alluded to is apparently avoided by the use of the cycle method of dealing with a network described in Art. 230 above. Any network of conductors is regarded as made up of a series of meshes or cells, as in the arrangement shown in Fig. 124, which con- sists of three distinct meshes ABC A, ABBA, OBBO. Each individual conductor is common to two meshes, except those conductors which form the outer edge of the network. Maxwell supposed a current to circulate round each mesh in the same direction, so that the actual current in each conductor was made up of the Fig 124. difference of the currents in two adjoining meshes. Each mesh is from this point of view regarded as a complete circuit with its own current flowing round it and the self and mutual inductances of the system are quite definite, being those of the distinct circuits formed by the meshes. 470. There is no difficulty in writing down the electrokinetic energy and finding the equations of motion from either point of view. If in Maxwell's method we denote by L^, L^,. . . the self-inductances of the different meshes, each regarded as a separate circuit, in which flow currents 7,^', 72', ... by M-y^, M^^, . . the mutual inductances of the pairs of meshes indicated by the suffices, the electrokinetic energy has the expression r = j(Av? + 2^i27'iy'2 + • ■ • •) • • • (32) If Bj]c denote the resistance of a conductor common to two meshes j, k, one-half the rate of dissipation of energy in heat in the whole system, that is the dissipation function, is i^ = iS%(7',- - y*)'^ (33) where the summation is extended to all the conductors of the system. We can now write down the equations for the different meshes. They are of the type X INDUCTION OF CURRENTS 357 where Ej is the electromoti^?e force in the circuit indicated by the suffix j. There are advantages in this method of procedure, but it cannot be said to be of any practical service in inquiries concerning in- ductances. In such applications it is usual to write the electrokinetic energy in the form T = i%{Lji,{y'j - y\f + 2if„-i„!m,(y,- - y'ic){y'i - y'm)} (35) where Zj^ is the self-inductance of the conductor common to the two meshes y,^, and lf(,i)(im) is the mutual inductance of the two conductors common one to the meshes/, k, the other to the meshes I, tn. But this is after all simply to return to the other method, to which at present we shall adhere. There is no difficulty really in writing down the equations for all the different conductors by this method, applying the principle of continuity at each meeting point. Further only one symbol is required for the current in each conductor, so that the formulae are briefer. 471. If then we denote by Z^, L^ . . . M.^^, M^^, . . . the self in- ductances of the conductors 1, 2 . . . and the mutual inductances of the pairs of conductors 12,23, . . . the electrokinetic energy has the value T = MZiri^ + sjfi^yiy^ + •••• + hy2^ + '^M^^y^y^^ ....) (36) The dissipation function is F = i{B,y,^ + B.y,^ + . . . .) .... (37) If there is electric energy E such as that of charged condensers situated in the conductors, the equations of the circuits are of the type l^ + 3JE + a^ = ^,_r, .... (38) dt dyjc dyic dyk where Ek is the internal electromotive force in the conductor, and Vjc is the difference of potential between its terminals, taken with the negative sign, since we suppose Ujc to act with the current, and Vje to oppose it. Adding these equations for all the conductors forming a circuit, we get lf^^i^^....)..^ + i^+....'+f-.^ + .... = ^ (39) dtKdyj dyj+-^ / dyj dyj+i 3yj 3yj+i where JE is the total internal electromotive force in the circuit. The sum of the differences of potential between the terminals of the conductors is of course zero for every circuit By Art. 181 above if C,-, G.j+^, .... be the capacities in the succes- 358 MAGNETISM AND ELECTRICITY CHAP. sive conductors of the circuit and yj, yj+-^, charges, we have denote the corresponding Hence (39) may be written in the form S { ^ {Ljyj + J/,,y.) +Iifyj+^}=B . . (40) in which we shall generally use it. This may be taken as the most general form of the so-called " second law," given by Kirchhoff (Art. 230 above) for a system of linear conductors. We shall find many examples of the use of this equation when we come to the measurement of inductances, though in most of these we shall have to use only the less general form of equation, which does not include terms depending on electrostatic energy. At present we shall consider a few problems of practical importance, taking first a case investigated experimentally by v. Helmholtz, whose method will be described in Vol. II., in the chapter on the Experimental Verification of the Theory of Induction. Primary with Secondary as Derived Circuit H'I'I Ci- A'ej' -^D 472. Let the same arrangement as that described in Art. 427 above be made, namely, a battery and coil in circuit with a cross-connection between them as shown in Fig. 125. If 71,72 ^^ ^^^ currents in the coil and the cross-connection, and 7 the total current, r^, r^, r the resistances of the coil, cross- connection, and battery with connecting wires to AB, then we have by the principle of continuity Coil Fig. pose no is in the 125. 7 = 71 + 72 (41) By equation (7) we have, since there is we sup- mutual induction to be considered, and the only self-induction coil '"272 + ry = E L^ + r^y^ +.ry = E (42) for the two circuits EABE, EAGBBE. These by (41) may be written '•272 + '•(71 + 72) = ^ j ^fj + ('■ + ■'■1)^1 + '")'2 = ^i' (43) ^ INDUCTION OF CURRENTS 359 The value of 7^ derived from the first of these substituted in the second gives for 7 the equation dt r + r^ ■ '1 r + r^ ^ ' Writing trr^ for rr^ + r^r^ + o-^r we get by integration and hence for 73 V2 = 7^{l-^(l-«-^(^/)} . . (46) The quantity of electricity which flows through the coil during any interval t reckoned from the closing of the circuit is thus If after the lapse of the interval t after the make the circuit be broken by raising the key, the quantity of electricity which flows through the coil is found as follows. The current flowing through the coil in the break satisfies the equation. X^+(ri + r,)y = (47) from which, remembering that the current at the beginning of the break has the value given in (45), we find y-^r '—y--e iCr + r^r]e L ' . . . (48) The quantity which flows through the coil in the break is therefore 1 — e L(r + r. Oscillatory and Non-Oscillatory Discharge of a Condenser 473. Later we shall discuss fully cases of primary and secondary circuits in which the electromotive forces acting are simple harmonic functions of the time ; and the very important arrangement of primary and secondary circuits which we have in an ordinary make and break induction coil will be dealt with when its construction and mode of action are being considered. We shall next consider a very curious and important problem, which has played a great part in modern electrical discovery. 360 SIAGNETISM AND ELECTRICITY chap. Let a condenser be charged to any given difference of potential, and let its plates be then connected by a wire of self-inductance T and re- sistance B. The condenser will discharge along the wire from one plate to the other. Let, at any time t, the difference of potential between the plates be V, the current 7, and the capacity G. Then the energy of the condenser is \GV'^ in virtue of its charge, and the current has electro- kinetic energy \L'f: The total electrical energy is \(GV'^-\-lyf'). The total time rate of charge of this energy must be equal to the rate at which energy is being transformed into heat in the circuit, plus that at which energy is being radiated from the varying current system. If we neglect the latter part we shall have ji^(CF2 + Z/) + 7?/ = .... (48') But 7= — CdVjdt so that the equation just found is Solving we get, putting (B'^-4sL/C)y2L = a, or as it may be written F = e"2i'(^e«« + i?e-««) (50) V = e 2l' D cosh {at - where A and £ or J) and ^ are constants to be determined from the initial circumstances for any particular case. If a is real this represents an ordinary discharge gradually approach- ing complete equalisation of potentials between the plates, and, theo- retically, only reaching it in an infinite time. If a is imaginary, which will be the case if B^<^'^LIG, the solution is V=e'^' Acos^^i^- R^yt - 6\ . . (51) where A and are constants to be determined from the given initial circumstances. This represents an oscillatory discharge with gradually diminishing range of potential. The period T is given by T- - /'-^ (52) and the logarithmic decrement of the potential is BTj^L. X INDUCTION OF CURRENTS 361 The discharging current is -{CdV/dt')/B, and is obtained from (50> for non-oscillatory and by (51) for oscillatory subsidence of the potential. The values of the current at different times for the two extreme case& of non-oscillatory subsidence are plotted in curves in Fig. 127. 474. The existence of an oscillatory discharge depends, as we have shown, on the relation of the resistance to the inductance of the dis- charging coil, and the capacity of the condenser. If the inductance is great enough in comparison with the resistance of the coil, electrical oscillations will take place, and there is no manner of doubt that many electrical discharges which appear mere single sparks are each a succession of backward and forward discharges caused by successive oscillations of the potential of the condenser. The possibility of this form of discharge was suggested first apparently by v. Helmholtz, in his famous essay Die Erhcdhmg der Kraft, from certain unexplained phenomena of magnetization produced by passing Leyden jar discharges through a coil surrounding a bar of steel. The theory given here is practically that given by Lord Kelvin in a remarkable paper published in the Phil. Mag. for June, 1853. 475. The discharge of a condenser is thus similar to the motion of a deflected spring when resisted by a force proportional to the velocity of displacement. For the equation of motion (49) can be written -f-f-J— (-) which shows that L corresponds to the inertia of the spring, V to its displacement, \jG to the return force per unit displacement (that is, C may be regarded as the modulus of yielding, or permittance as Heaviside calls it), and B to the resisting force per unit of the velocity of displacement. In such a case we know that if the inertia is very small and the resisting force has a large enough value, the spring vfiW simply slip slowly back to its equilibrium position without oscillation about it, just as does a pendulum bob, of small inertia, deflected in a. highly viscous fluid like treacle, and then left to itself If, however,, the spring has a certain amount of inertia it will get into motion, and as it nears the equilibrium position will move more quickly than before,, overshooting that position and oscillating about it, if the inertia is. sufiiciently great. When the inertia is such that the spring is just brought to rest without passing the equilibrium position, and the slightest addition of mass would cause the spring to pass beyond that position before coming to rest, the motion of the spring is dead beat,. and the condition B^G=4 A I00_ so" M 40°' ■ 20 0" 20"' — 1 ^■'^ \ — ■ — •-— " J \ \ -^ ^ .^ 80\ innf* 1-6 2 2-4 /tiduciance (henrys) Fig. 131. {This illustration is taken from a paper by Professor Perry and Mr. H. Bayly in the Electrician for July 21, 1893.) In Fig. 132 curves are plotted as in Fig. 131 for each of a number of different values of R for a circuit in which the capacity of the con- denser included and the frequency are kept constant while the self- inductance is varied. The curves are drawn as follows. First R is laid down as an ordinate OA, say, along the line OY, and a length OM to the left is laid off along the axis of abscissae equal to IjCn. A value of nL — IjGnis then laid off as an abscissa ON. It is clear that MN is equal to nZ. The line AN represents evidently what we have called the effective impedance ^R^ + {nL — IjCnf. An ordinate Na is drawn from N of length equal to AN and gives a point a on a curve passing through A. Thus a curve concave upwards and symmetrical about OY is obtained, the ordinates of which by (62) express the ratio to E^^, for the different values of nL, of the amplitudes of the electromotive forces required to produce a given current. Similarly curves are drawn for other values OB, 00,... o{ R. Next, to obtain the curve of current from these curves of amplitude INDUCTION OF CURRENTS 369 of electromotive force 00' is taken equal to MN, and an equilateral hyperbola, having the lines O'Y', O'JTas asymptotes, is laid down on the diagram, such that the product of any ordinate measured from O'X, and any abscissa measured from O'Y' is ^q. Taking, then, any ordinate Na corresponding to nL of the iirst curve of amplitude of electromotive force, find the point on HE, which has the same ordinate. The corre- sponding abscissa from 0' is the current-amplitude, which, laid down as an ordinate Na, gives a point on the current-amplitude curve, and so on. Thus the various curves of Fig. 132 are drawn in which A'a, B'^, C'y, . . . are the curves corresponding to the resistances OA, OB, 00, . . .. respectively. In a similar way curves of amplitude of electromotive force for a given current, and of current-amplitude, can be drawn for each of different s a' / 'h\ ^ ^r^ ^^^"■^ ~~~— k ^^ y^ N ^T- ° ^ 7 ^^/\p //b' ^ >^ \/^ h\ v"^^ ^ L^^ L- ^^ •^^ > ^■e' — Fig. o 132. N Self- Inductance values of B, for L and n constant and variable, and for L and C con- stant and 11 variable. (See a paper by W. E. Sumpner in the Electrician for July 28, 1893, from which Fig. 132 and the mode of constructing the curves are taken.) It can be shown that if the resistance and self-inductance be wholly in a coil joined in parallel with the condenser instead of as here in series the same diagrams will be available, but the current curves will become those of electromotive force, and vice versa, and the lengths OA OB OG, . . . instead of representing the values of B, will now repre- sent those of IjB. Effect of Inductance on Signalling through a Cable or in Telephony 481. It is interesting to observe that (61) shows that when the- circuit contains a given condenser, and has a given resistance, the B B 370 MAGNETISM AND ELECTRICITY chap. addition of self-inductance up to a certain point increases the current, and that the maximum is obtained when CL'n?=\. This result is of importance in the theory of signalling through a cable. In that case the capacity is distributed along the cable, and the mathematical theory {which will be given in Part II.) is somewhat complicated, but the general result is the same as that just obtained. It will be noticed from (61) that the effect of inductance is only serious when n is considerable, that is when the frequency of alternation is great. For slow signalling through a telegraph cable the inductance may be neglected ; but it is a mistake to suppose, as is sometimes done, that it is necessarily deleterious, and that it should always be made as -small as possible. In very rapid ordinary working, and especially in telephony, the presence of a certain amount of self-inductance improves the clearness of the signals. From (59) it will be seen that for zero inductance the retardation of phase would have a value tan"^(?iC^) — 7r/2 depending on the fre- quency of the vibrations. In telephony the corresponding effect is a retardation of phase of the signal at the receiving end behind that at the sending end, and this produces confusion of the signals, inasmuch as in a composite sound vibrations of one pitch have a different retarda- tion from those of another pitch. Since in this case all the values of n are great, the value of nOIi/{l — n^CZ) is approximately — GBjnGL, or zero, and the existence of self-inductance L enables the value of 6 to be nearly — 7r/2 for all the actual values of n, and so removes distortion. Of course, on the other hand, excessive self-inductance may, by (61), produce too great attenuation of the signals. Difference of Potential between Terminals of Condenser. Resonance 482. By (57) the difference of potential V between the plates of the condenser is given by the equation V = E^sinnt - L-^ - Ry, or by (61) V = -^0 si" ('^^ - ^) /fio'N {(1 - n^CLf + n^G-^£^]h ■ ■ ■ ■ y°'> ) If the denominator of this expression be less than unity, the value of V will be greater than the numerator of (63'), and the amplitude of V will be greater than that of the impressed electromotive force. This curious result has been verified in practice by observations on the electric light mains carrying alternating currents between London and the Ferranti Company's generating station at Deptford. It was there found by Mr. Ferranti that the so-called " electrical pressure " (which may be taken as the square root of the mean of W) on the terminals X INDUCTION OF CURRENTS 371 of the alternator, working at its normal speed with a certain exciting current, was increased by connecting the machine to the mains. This ■was no doubt due to a partial fulfilment of the conditions necessary for a small value of the denominator of the expression in (63'), that is, it was a case more or less of what has been called resonance} Maxwell's Dynamical Analogies of Inductance and Capacity. Resonance 483. The above explanation of the effect thus observed was pointed out at the time by Glazebrook,^ Lodge,^ and others ; but the theory of the whole matter was really given first by Maxwell in the paper cited in Art. 479.* In that paper Maxwell points out analogies to inductance, capacity and resistance. The inductance of the primary coil acts like inertia, resisting the starting and stopping of the current in the same way as the inertia of a body resists a push or pull tending to set it into motion, or to bring it to rest. The capacity of the condenser acts like a spring, which must be bent or compressed if the body is accelerated or retarded, as, for example, when railway-buffers resist the motion of a carriage towards an obstacle. He then imagines as an example a boat floating in a viscous liquid, and kept in place by buffers at the bow and stern abutting against fixed obstacles. If there were no obstacles a long continued pull would move the boat, however great its mass. But on the other hand alternate puMs and pushes would produce very little motion. With the buffers in position, however, alternate pulls and pushes in, or nearly in, the period of the springs would produce in a short time considerable backward and forward motion of the boat. The work spent in each impulse is in great measure stored up in the springs in consequence of their resilience, and a very much greater motion results than could have been obtained with the boat free. This agreement of the period of the alternating impulses with the free period of the springs is called resonance. When a circuit is closed without a condenser the current in it is like the motion of the boat when free in the viscous liquid, and under an alternating electromotive force the amplitude of the current produced is small. But when a condenser is applied, and the period of the alternating electromotive force' is that of the discharge of the condenser, the relation between the inductance and the capacity is such as, for that value of the period, to give a very much greater alternating current. 484. The greatest amplitude of V is attained when n is so chosen as to make the quantity (1—n^CLf + n^C^R^ as small as possible with the given values of G, L, R. For this purpose w^ should have the value 1 EUdrieian, Dec. 19, 1890. =* Mledrieian, Deo. 26, 1890. 3 Ibid, April 24, 1891. ' Also Collected Papers, Vol. II. p. 12. 372 MAGNETISM AND ELECTEICITY chap. (2Z - Cm)l2GL". If R be so small that CR^ is negligible in com- parison with 2Z this gives iv^ = 1/GL, that is, n is then 27r times the natural frequency of vibration of the condenser and coil as arranged. We have then amplitude of V 1 /L ,.,, Y, = Wc- ■ ■ ■ (^*) which will be much greater than unity, since CE^ has been supposed negligibly small as compared with 2L. Primary and Secondary. Inductance and Capacity in Primary with Harmonic Electromotive Force 485. We now consider the problem of a primary circuit arranged as in the case just considered, with the addition of a secondary circuit containing no electromotive force. The equation of the primary is ^1^ + ^^ + ^lyi + Z.\yi is the lag in phase of the total current entering or leaving the parallel system at any instant behind that of the impressed difference of potential. The effective capacity of the system of parallels is in general in- determinate, and is of no practical importance. 488. Now let the circuit of AB be completed by a single conductor of resistance B, inductance L, and containing an electromotive force ^0 sin (lit + ^), so that f is the lag of the difference of potential ¥„ sin nt behind the electromotive force. The solution for the case in which this, conductor also contains a condenser G will be obtained by putting L — ljC'n? for L. The current in this conductor is V, so that we have for the differential equation dv L— + BT + To sin nt = E^ sin {nt + Q. (It Substituting the value of F already found, and remembering that the identity thereby obtained must hold for all values of t, we easily se& that it gives the two equations £o {(R + ^f + n\L + Lf}i I Instead of the first of (77) we therefore have (78) The first of these shows that the amplitude of the impressed difference of potential bears to that of the whole electromotive force in the circuit the ratio of the effective impedance of the system of parallels to the effective impedance of the whole circuit. The second equation shows that the lag in phase of the current behind the electro- motive force is given by substituting, in the expression (75) already found for one of the parallels and the impressed difference of potential, the whole effective inductance and resistance of the circuit for the corresponding quantities relating to the single conductor. These results were to be anticipated. As an example take the arrangement shown in Fig. 130. Let the inductance of the branch containing the coil p be zero, and iJj be its resistance, Z,E the inductance and resistance of the branch containing^ the magneto-electric machine. The inductance and resistance of the branch containing the condenser we shall suppose to be zero, and denote the capacity by C. We easily find n A l_ _ - O^i' " - 1 -f- n^C^Jt{^' 1 4- n'-C^Ji/ from which the current can be found by (79) and the second of (78) 37(3 MAGNETISM AND ELECTRICITY CHAP. Action of Induction Coil, or Inductorium, with Condenser across Break in Primary. Case I. Secondary Closed 489. A vltv imijortant practical case (if a circuit containing a con- denser is found in the ordinary induction coil, or inductorium, Fig. 13-1-, for producing high electromotive forces in a secondar}' coil by breaking the circuit of a primar}'. The primary is a coil of comparatively few turns of thick co]iper wire wound round a cylinder enclosing a core (if Sdft iron, preferably a bundle of thin iron wires, sufficiently insulated fnim one another to prevent sensible currents from being induced and Fig. 134. circulating in the iron. The secondary is a coil composed of a great length of highly insulated thin wire and is wound in a large number of turns on a cylinder outside the primary. To avoid damage to the insula- tion by internal sparks, the coil is woun(J in sections, so as to avoid placing close together pjarts which may, when the coil is in action, be at a great difference of potential. Owing to the large number of turns in the secondary, the change of total magnetic induction through it produced by stopping or starting the cuiTent is very great, and if the change is effected suddenlj- the electromotive force reaches a high value, though ofcour.se its amount varies during the passage of the secondaiy current. Coils have been constructed of such power as to give sparks of over 3 feet in length between the terminals of the secondary. Further practical details, and calculations regarding the mutual induction in actual cases are reserved for a later chapter. The circuit of the primary is made and broken by a vibrator of <_)ne ■or other of the two fonns shown, one in Fig. 134, the other in Fig. 135. In the former a brass ami mounted on a spring support carries at one •end a piece of iron, at the other a platinum wire which the .spring tends to make dip into a cup of mercury as indicated. The iron piece at tli(i other end is placed ju.st above the projecting end of the core or of an INDUCTION OF CURRENTS 377 iron armature attached to the coil. To hinder sparking the surface of the mercury is covered with an insulating layer of alcohol, from which the wire is not withdrawn when the break is in action. A battery of cells of low internal resistance is placed in one direction or the other in series with the primary by means of a commutator, and the circuit is completed by the contact of the platinum point with the mercury. When a current flows in the primary and the core is magnetized, the piece of iron, being magnetized inductively, is pulled down and raises the platinum wire at the other end from the mercury, thus leaking the circuit. When the circuit is broken the iron core loses its magnetism, the spring is allowed to act, and the contact with the mercury is restored. As sparks do not take place so readily in alcohol as in air, on account of its greater dielectric strength or power to resist rupture by electric stress, the formation ■of a spark is nearly if not wholly prevented. In another form of contact breaker more usually employed, the contact is made and broken by the action of an upright spring D '(Fig. 186) and the attraction of the magnetized ■core on an iron piece, H, carried by the spring at its upper end. One face, P, of this contact piece, shod with platinum, is held by the spring against another piece of platinum, P', mounted ■on one end of an adjusting screw, and so completes the circuit. Soon after the circuit is completed, however, it is interrupted by the magnetization of the core, and the current •quickly ceases. The secondary coil is for the most part used to give discharges across a spark-gap of adjustable width between points or knobs -attached to its terminals ; and these discharges are rendered uni- ■directional by occurring at the break, not at the make, of the primary, owing to the greater suddenness with which the break can be effected. The electromotive force, depending as it does on the rate ■of decay of the primary current, is enhanced by any arrangement which increases that rate. One of these is the immersion of the break in alcohol, another is the attachment of a small condenser, usually made of sheets of tinfoil separated by paraffined paper or silk, and contained in the wooden base of the instrument. One set of plates of this are connected to one side of the break, the other to the other side : for ■example one to the lower end of the metal piece C (Fig. 135), the other to the metal block supporting the spring D, and similarly in any other make and break arrangement. 490. The action of the condenser has been the subject of some dis- cussion. The following explanation is due to Lord Rayleigh,^ but it assumes a closed secondary in which currents are induced in the manner described above. In most cases of use of an induction-coil the 1 Phil. Uag. Juno, 1870. Fig. 135. 378 MAGNETISM AND ELECTRICITY chap. secondary cannot be regarded as closed in the ordinary sense except when a spark is passing, and the riyime of the current is therefore very different from that here supposed. We shall return presently to this point. We suppose then that the secondary is closed, that the discharge of the condenser which takes place at break of the circuit is oscillatory, and that the diminution of current due to resistance may be neglected, as only the first one or two oscillations are concerned. The equations of the primary and secondary are, if the variation of the inductances,, produced by the iron core, be neglected. ^'dt ^^ dt ^ C. dt ' dt (79) These give by elimination of 7, {L,L,-M^)^^ + ^y,dt = E. . . . (80) Also the second of (79) integrated is ^Ti + ^272 = ^ro (81) where 7^ is the current in the primary just before the break begins. Equation (80) shows that the oscillation in the primary is the same- as if the secondary did not exist and the self-inductance of the primary were changed from Z^ to L^ — M^jL^. Since the period of oscillation is liTiJ LO (Art. 473), where L is the effective inductance, the secondary diminishes the period of oscillation in the primary. It may be remarked here that if two coils be made up of wires coiled side by side so as to be nearly identical both in geometrical arrangement and in jjosition L^L^ will be onl slightly greater than M'^ and the period will be exceedingly short, y Equation (81) indicates that the two currents oscillate together, the cun-ent in the secondary being at its maximum when that in the primary is at its minimum and vice, versa. After half a period has elapsed from the beginning 7^ has changed to — 7^ and we have 72 = 2^-70' and this is the largest value that 72 can have. It is double the maximum the secondary current would have in the case of a simple break in the primary however suddenly brought about. It is thus the oscillatory discharge of the condenser in the primary that according to this theory gives rise to the enhanced value of the secondary current, and it is the first maximum of current, of approximate amount IMlLcj.'^^, which produces such effects as the- '"^ INDUCTION OF CURRENTS 379. magnetization of needles when surrounded by the secondary coil. It. was found by Lord Kayleigh {he. cit.) that the magnetizing effect of the induced current m a secondary coil produced at break of the primary was greater the smaller the number of turns in the secondary, that is to say the smaller i,, which is in accordance with the foregoing theoretical result. It may easily be verified in practical cases whether the assumption made above that no serious damping has been brought about during the first few oscillations is well founded. It is only necessary that the period 2its/LU should be small in comparison with the time-constant Ljlt. Action of Condenser in Induction Coil. Case II. Secondary Open 491. If the secondary be unclosed and have no sensible capacity very little current will be set up in it, and d|rj is applied to the system. Let the system have zero potential energy, but be acted on by dissipative forces given by a function F (Art. 262 above), which is a homogeneous quadratic function of the velocities -v/y-i, ■v/r^ . . . of the system. Let also the functions T and F contain no products of velocities except those in which yjj-.^ appears. We have P = K^i/i' + hi^i + .••■ + '^hAh + 26i3fiv^3 + ••••): Hence by (71), Art. 262, the equations of motion are (82) 11". (■hS'l + '»22^2 + ^12^1 + ^22'/'2 = ^ «13'/'l + «33'/'3 + \A + ^33'/'3 = ^ (83) In response to the force '^j the motion of the system will be simple harmonic in the same period, 27r/ra say. Thus representing any -«|r by gin« -where i = ^ —\ we get i/y-= in-^ and therefore (83) becomes (w«n -f- 5n)'Ai + (w»i2 + ^i2)'A2 + (''^«i3 + ^is)'A3 + = *r (mai2 -F 6i2)v('i + (*'»^»22 + ^^t"- ^^ f> 84) (wajg + \^i>^ + (magg -I- 633)i/'3 = 1 put Mag. (Ser.4) 49, March, 1875, and Ser. 5, 21, May, 1886; Theory of Sound, Vol. I. (2nd edition) p. 433. 382 MAGNETISM AND ELECTRICITY CHAP. hi + *11 -....}fi = * (85) Substituting in the first of (84) the values of yjr^, i^g, .... obtained in terms of \/rj from the remaining equations, we get The factor in brackets on the left is a complex quantity, and may therefore be written in the form R' + inZ', where B' and Z' are real. We may call B' the resistance of the system, and Z' its self inductance. Separating the real and imaginary parts of the factor in brackets in (85) Ave easily find, if there be m co-ordinates h~m R' = K -2 £-'2 h = 2 h L' = «n ^ ahh ^ h=i 7i = 2 ^ hh (aihhh - ahhh-ihf hh{i>\h + n^ahh) (ahh{b-hh + n^ahh) (86) It is clear that as the frequency nj2-ir increases B' increases and Z' •diminishes towards the limits R L = a h = in he. 11 • -sS-s 7i=2 h=- /i = m '11 ^ all, ^ ahh (aihbhh - ahhbihY bhhahf? (87) Thus when n is great L' is independent of the J-coefficients, that is the resultant inductance is not affected by the dissipation. It is to be observed also that if there is no kinetic energy of the system the third term in (87) does not appear and we have the same minimum of resistance as when the frequency is very small. When n is very small h=m R = L' = a, bhh A=2 A=m y ■ {aih^hh - (thhb^hY h-2 ahh tthjfihh (88) that is B' is independent of the inertia coefficients, and Z' has its maximum value. ^ INDUCTION OF QUERENTS 383 495. These results find immediate illustration in electrical problems, i'or example take the case of a primary and secondary circuit already fully treated m Art. 485 above. We have with the notation there adopted a,, = L^, a,, = M, a,, = Z„ \^ = B„ \, = 0, 6,, = i?,, and therefore tne ettective resistance and self-inductance are, for the primary, R\ = Ji, + (89) Thus if the frequency be very small the secondary has no effect on the primary. If the frequency be very great B', approaches the limit H^ + MmjL^^ a,nd Z\ to Z^-M^/L^. For the secondary we get as before R', = R, + nWR^ R^^ +n2/vi2[ R^liAL{' (90) the limiting values of which are B^, Z^ for very small frequency, and B^+UmjZ^^, Z^-M^jL^ for great frequency. 496. The reader may work out for himself the case of a series of conductors forming primary, secondary, tertiary, &c., circuits, but such that no mutual induction exists except between the primary and secondary, the secondary and the tertiary, and so on. For example consider four circuits. The current in the fourth is due to the inductive action of the third. The reaction of the fourth in the third causes the latter to have an effective resistance B'^ and self- inductance X'g at once calculable from (89) by substituting for i2j, Z-^, B^, Zg, M the quantities B^, L^, B^, Z^, M^. Then if iJ'g, Z'^ be used as the resistance and self-inductance of the third circuit the fourth may be ignored. The effective resistance and self-inductance of the second circuit due to the reaction of the third can be found in the same way, and the third then ignored. Finally the effective resistance and self-inductance of the primary can be found by another application of the same formula. When the alternation is very rapid it will be found that the phases of the currents in the different circuits of the series depend in the case of very rapid alternation on the induction coefficients only, and differ successively by half a period. 497. Another of Lord Rayleigh's examples {loc. cit.) is the case of two parallel conductors, of resistances B^, B^, self-inductances Zj, Z^, and mutual inductance M, joining two parts of a circuit in which an alternating current is flowing. If 7 be the total current the portion 7^, 384 MAGNETISM AND ELECTRICITY chap. 7.3 into which it would divide itself between the parallel conductors were there no inductance would be B,^l{B^+B^,B^r^l{R^+B^. We may take 7 as the velocity for one co-ordinate, and for the other velocity a quantity 7' such that the current in the first conductor is •^17/(^1 + -^2) + 7' and in the second B^yKR^ + B^y — y'. We obtain easily ^-* {B,+ii,r y + B^Tif, ^ + i{L,-2M+L,)y'-^ ^=^i^/-^^(^^ + W (91) (92) It is supposed that there is no energy of electric charge to be taken into account. The literal coefficient of the first term in the value of T is ajj, that of the second term is a^^, and that of the third is a^^. Similarly the coefficients of the terms of F are \^, h^ respectively, and \^ = 0. Thus we get at once for the effective resistance and self-inductance of the parallel connection K = _1_ {re+ ,,. KA-^ )A- -iL,-M)R,Y ■\ R^ + R^\ 1 - {R^ + R.^)^ + n\L^-2M+L,f^ L'= — ^ ^LL MU {h-M)R,-{L.^-M)R,Y' ] Z; - 2M+ L., \ 1^ "^ {R^ + -R.,f + n\L^ - 2M+L.,f f j As noticed above L^L^ — M^ is necessarily positive, though it maybe made very nearly zero by winding the two parallel conductors together. Also L^ — 1M-{-L^ is essentially positive as it is twice the kinetic energy of the system when the current in the first conductor is + 1 and that in the other —1. The reader will notice that when n is very small B' = B^R^I(B^+B^^ while that of L' = {L^B^^ ^2MBJt^+L^R-,^)l{B^+B^^, and that when n is very great B' = {B^{L^-Mf + RlL^-Mf}l{Li-2M+L^f,L' = {L^L^ — M^)/(Z.^ — 2M+L2). He may also find from (91) the currents in the two conductors at any instant, and their phases with reference to the total current. Another very interesting example is obtained in the Wheatstone Bridge arrangement of conductors, when inductances as well as resistances have to be taken into account. This will be fully dealt with, however, in the discussion in Vol. II. of methods of comparing inductances. (See also Absolute Measurements, Vol. II., Part II., Chapter VIII.). CHAPTER XI GENERAL ELECTROMAGNETIC THEORY Section I. — Medromagnetic Theory of Light Electromotive Intensity at Element of Moving Conductor. Total Electro- motive Force 498. We have seen above (Arts. 420 — 425) that if an element of conducting material move with velocity s[ = («^-|-2/^+i^)*] in a field in which the magnetic induction at the element is B (components a, b, c), an impressed electric intensity e is produced which is equal to the- vector-product of s and B. For the components P, Q, B o{ e we obtain by the process of Art. 387 above (P, Q, E) = {cy - hi, az - ex, hi, - ay) . . (1) The direction of e is found to be that in which a right-handed screw would advance if the handle were turned round from the direction of £ to that of s. (See Fig. 136.) But at any point of the field the value of B may be changing with the time. The line-in- tegral of the electromotive intensity just found taken round a linear conductor forming a circuit, gives what is sometimes called the total electromotive force, so far as that is due to motion of the circuit. To this there must be added the total electromotive force in the circuit due to that part of the time-rate of Fig. 136. change of magnetic induction through the circuit which is independent of motion of the circuit. The 'total electromotive force from all sources in this circuit will be obtained by taking the line-integral of E round the circuit, where E is the resultant electric intensity and has the components (P,e,i?) = («2/-6«-gJ-3^. «^-''*-3^-3^' *«'-«2'- ST-B^j (2> c c 386 JIAGNETISM AND ELECTRICITY chap. Here F, G, H are the components of vector-potential, and x, y, i are the components of the velocity of the element of conductor, considered as cutting across the tubes of induction. This velocity, it must be noticed, is the velocity of the conductor relatively to the tubes, and not with reference to any system of axes connected with moving matter. The function "^ includes the electrostatic potential, and also any other terms giving components of electric force which, integrated round a closed circuit, give a zero result. For example, when the total electromotive force for a moving conductor is deduced from the time- rate of change of the line-integral of vector-potential round the circuit considered, there appear, besides the terms of the components in (2) depending on the vector potential, terms which are respectively _ /I, I, 1\ (F^ + Gy + Hi) and which vanish when integrated round a closed circuit. 499. We have here considered electromotive force due to variation of the magnetic induction through a closed circuit, but we are now led by phenomena which have to be explained to take the view that when- ever a portion of matter, whether conducting in any degree or not, is in motion in a magnetic field so" that it cuts across tubes of magnetic induction, inductive electromotive intensity given by (1) is produced in it, and further that electromotive intensity is set up wherever magnetic induction is varying in value. In an insulator, however, no ordinary conduction current producing dissipation of energy can exist, but the substance is the seat of an electric field intensity, the value of which may be constant or may vary with the time. If the latter is the case, the time-rate of variation of the corresponding electric induction divided by 47r is taken as an electric current which with the con- duction current constitutes a circuital system of currents producing the magnetic field, and is subject to all the laws of currents. The justifi- cation of this assumption lies in the agreement, so far as investigation has yet proceeded, of its results with observation. By its aid we regard all currents as closed ; for example, when a conductor is being charged from an electric machine to which it is connected by a wire, the electric field' surrounding the conductor is undergoing change, and a current exists in the dielectric which renders the charging current in the wire circuitall. The hjrpothesis is perfectly reasonable from the point of view of flow of energy already referred to in Arts. 145-148 and elsewhere above. Into the wire there flows across its lateral surface a certain amount of energy which is dissipated in the conducting substance, producing heat ; into a portion of the dielectric in which the electric field intensity is being increased there flows an amount of energy which is not dissipated, if the medium is a perfect insulator, but remains stored as energy of electric strain. XI GENERAL ELECTROMAGNETIC THEORY 387 Flow of Energy in the Insulating Medium, and Dissipation in a Conductor. Displacement Currents 500. If the wire or other conductor receiving energy from the medium were a perfect conductor, there would be no dissipation in it, but energy would enter it. Into an ordinary conductor energy can be conveyed in consequence of the fact that it has to some extent the properties of an insulator as regards supporting electric strain, tem- porarily at least, so that electrical strain penetrates its outer strata, and energy is dissipated in the matter in the interior in consequence of the ultimate breaking down of the strain which is there set up. Thus the inward flow of electric energy upon any portion of the dielectric medium occasions a time-rate of variation of the electric induction, which, to a numerical factor, is related to the rate at which energy is received per unit of volume, as is the conduction current to the rate of dissipation of energy per unit of volume in the conductor. It is the flow of energy in the medium of which the magnetic field is an accompaniment, wherever it occurs: the ultimate disposal of the energy as increase of electrical strain and corresponding stress, and as increase of electrokinetic energy, or in the production of heat, does not itself have any effect except in so far as it indirectly reacts on the inarch of the phenomena. The closing of the circuits of all currents in conductors, by means of •displacement currents in the dielectric, constitutes the most remarkable feature of Maxwell's theory, and leads at once by a natural development to his great generalisation, the electromagnetic theory of light. This we shall now endeavour to explain, referring the reader for a most instructive discussion of electric displacement, and displacement to Heaviside's articles on Electromagnetic Induction and its Propagation?- Impressed Electric Force. Illustration by Ideal Magneto-Electric Machine. Energy is received by System at Seat of Impressed Forces 501. Returning now for a little to a conductor moving in a magnetic £eld, take the case, described in Art. 425 above, in which a conductor, or .a straight wire at right angles to the tubes of magnetic induction, is moved at a uniform speed in a direction at right angles at once to the -wire and to the induction. The electric state of the wire remains unchanged while the motion continues. A difference of electric potential between its extremities is maintained which it is consistent with experience to take as the line-integral of e along the conductor. This difference of potential of course tends to produce a backward current in the conductor, and thus to annul itself, and is prevented from -doing so by the inductive electromotive force. An electric field is thus -produced, surrounding the moving conductor and maintained by the ■inductive action. 1 Electrical Papers, Vol. I. C 2 388 MAGNETISM AND ELECTRICITY chap. 502. This is an example of impressed electric force at each point balancing an electric force which otherwise would set up a current, and involve redistribution and dissipation of energy. For although as the wire moves carrying its electrification with it, the electric displacement is being continually changed in position in the medium, once the steady state has been set up there is neither expenditure nor gain of energy in continuing the motion. The energy stored in the electric state is dissipated when the motion is stopped. The impressed electric intensity is the electric intensity at every point, which balances that due to the electrification produced. Thus we have an illustration of the important fact, insisted on with great force by Heaviside, that in considering electric and magnetic intensity and the consequent flow of energy in the field it is necessary to take account of the impressed intensities, and make a sharp distinction between them and the intensities set up in consequence of changes of the medium produced by them. 503. If the wire we are considering touch at each end one of two parallel rails of conducting material, which are connected elsewhere, a current of amount 7 will flow round the circuit as shown in Fig 116. The electromagnetic force on the sliding wire will tend to stop the motion which produces the current, and if I be the length of the slider its amount will be BZ7. To keep the slider in motion therefore, work will have to be expended on it at rate B^7«. But B» is the im- pressed electric intensity, e say, along the slider at each point, and the time-rate at which energy enters the system, from without, per unit length of the slider is 67. When dissipation in the slider is taken into account, the actual rate of flow of energy to the medium per unit of length of the slider is - E7, where E here denotes the actually existing electric field-intensity parallel to the slider, and the rate of dissipation in the slider itself is (e + Ey7. It is to be remembered that E, which is t\\e field-intensity , is opposed to e. 604. Thus in considering the delivery of energy to any system from without we have to bring the impressed forces properly into the account. So far we have dealt only with impressed electric intensity, but the same thing holds for magnetic intensity. If we had such a thing anywhere as a magnetic current G, and h denoted the impressed magnetic force at the place, the energy delivered there per unit time would be )iG. We shall consider the flow of energy later in the present chapter, and again in Vol II., when we shall have further to consider impressed forces, and to classify them. The reader should however refer to Heaviside's Electromagnetic Theory, Vol. I. Chapter II. Complete Specification of Current in Imperfect Conductor or Imperfect Insulator. Fundamental Circuital Equations of Field. 505. The total current in a not perfectly conducting material is the sum of the ordinary conduction current, the displacement current, and a ^i GENERAL ELECTROMAGNETIC THEORY 389 current which we shall consider in a later section of this chapter due to moving niatter carrying with it an electric charge. The total current in the direction of x at any point is thus u = kP + ~P + ex (3) ■where k denotes the conductivity of the medium, h the dielectric mductivity, and e the electric charge per unit of volume of the moving matter at the point in question. The product ex may be taken as the definition of the measure of the convection current at any point. Similar equations in Q, Q, y, B, R, z, hold for the components v, w of the total currents. Applying the circuital theorem obtained in Art. 379 above, that is, assuming that the experimental results which have been found to hold for circuital conductional currents hold also for currents made circuital by displacement currents, and for convection currents with circuits however completed, we obtain for matter at rest relatively to the medium {hp + ^-kk) E = curl H, where p stands for djdt. 506. Again we have another circuital theorem, namely, that the total electromotive force round a circuit is equal to the time-rate of diminution of magnetic induction through it. This for an infinitesimal circuit may be expressed hj — p'B = curl E. To complete the analogy between this equation and the one stated above Heaviside introduces a non-existent quantity ^'H, which may be regarded as 47r times the magnetic current. Thus we obtain the two corresponding circuital equations {hp + 4tk)E = curl H"l (/yj + 4ir5')H = - curl Ej ^ ' From these two relations Heaviside has derived his solutions of pro- blems of wave propagation, and they are equivalent to a form into which Hertz has thrown Maxwell's equations, and which he has used in his theoretical researches on electromagnetic radiation. Since there is no divergence of the current anywhere, and the magnetic induction. is also without divergence, we hav the additional two equations div. E = 0, div. B = (5) the first of which holds at all points except where there is electrification, the second at every point, whether within or without magnets. Here we suppose that there is no impressed force, magnetic or elec- tric, to be taken into account, and that the state considered is one set up by an electromagnetic wave propagating itself. It may however be remarked that no polar electric or magnetic intensities, that is intensities derivable from potential functions, can enter these equations, inasmuch as the curl of every such intensity is zero. 390 MAGXETISM AND ELECTRICITY chap. Propagation of a Plane Electromagnetic Wave 507. Let the electromagnetic wave be a plane wave, propagated along the axis of s with wave front parallel to the axes of x, y. That the equations are consistent with such a disturbance it is easy to show by eliminating E, say, from (4). Thus we obtain Ijxp + 4,r^) {hp + 47ric)H = v^H . . . . (6> the well-known equation of wave propagation. Precisely the same equation could have been found for E by eliminating H. It will be noticed that H and E are directed quantities at right angles to one another and both by the restriction imposed in the disturbance lying in the wave front. Taking ^r = 0, we have for the periodic solution of (6) according as E or H is the quantity sought, (E, H) = (Eo, H(,) exp. i{mz - nt) . . . . (7) where z = ^ — 1, and 2Tr/n is the period of vibration. Substituting from this in (6) we find that the condition m^ - Ti'-k/j. - iTrKfini = (8) must bo satisfied. Thus since we suppose n to be real, m^ is essentially complex, and therefore so also is m. Writing in = q — ri, where q and r are real ; squaring and equating real and imaginary parts we obtain j2 _ ^2 _ jj2^^^ qr = - 27rKfji.n .... (9) Since r- must be a positive quantity since r is real, these equations give r = - ^^{ V/fcS + 16,rV/n='' - k\>^ J2 n \J u. I Jk^ + 167rW?i2 + k\i (10) in which the radicals in each case have the positive sign, and r is macje negative for the reason that the disturbance is not to increase in amplitude as it travels out from the source. Thus 5 is a positive real quantity. By (9) the solution becomes (E, H) = (Efl, Ho) exp. rz exp. i{qz - nt) . . (11) cerned with the real part of the quantity of the ri enomenon observed. Hence we obtain finally (E, H) = (Eq, Hq) exp, rz . cos (qz - nt) . ... (12) XI GENERAL ELECTROMAGNETIC THEORY 391 where the zero of time is so chosen that B^, H^ are the maximum values of E, H in the plane s = 0. This result gives for the wave length and velocity of propagation V the values = — = 2^/j 1^ ( Jk^ + IQ^^Kyn' + k)i [ V = J2 /j ^( v/^2 + lQ„2^2jn2 + ^)i| (13) and for the ratio, at any time, of the amplitude in any plane z = \, to> that in the plane 2 = 0. exp. r\ = exp. 27r( Jk^ + Uir'^KV - k)i (14) Thus the wave length and velocity of propagation are less, and the rate of diminution of amplitude with distance from the source is greater the greater the conductivity, other things remaining the same. If « = 0, that is if the medium is an insulator, r = 0, and q = n Jkfi, V = ijkfi. (15) Plaue Wave in iEolotropic Dielectric, Principal Wave-Velocities 508. We shall now, following Maxwell, suppose the medium to be seolotropic as regards electric inductivity, but at the same time isotropic as regards magnetic inductivitj^. Taking/, g, h as before as components of electric induction, we have as in Art. 168 above g — k^^r + A22V + «23" } (16) If we suppose the induction taken in a direction in which it is parallel to the electric intensity, then, denoting the electric induc- tivity for that direction by Ic, putting hP, kQ, JcB, for/, g, h in (16), and eliminating P, Q, B we get the determinantal equation ^11 - ^I "^12' ^13 Ajo, ^■22 ~ "■' *23 ^13 "^23 Agg — K = (17) which is a cubic in k, with three real roots /(Jj, Jc^, \, say, which apply to three directions at right angles to one another. For if I, rrbjin 392 MAGNETISM AND ELECTRICITY chap. be the direction cosines of one of these we may write (16) in the form h-J, + AjoW + ijgW = hi three equations from which by substitution of \, \, Tc^ the three corre- sponding sets of values of I, m, n, namely l^, m^, n^, l^, m^, n^, l^, m^, n^, can be found. It can be verified easily that ^^ + m^mj + Mi»!'2 = 0, &c. These directions we call principal directions. 509. We now find the corresponding equations of wave propagation, taking components of induction in the principal directions. In so doing we shall suppose the conductivity k of the medium to be zero. The ■circuital equations (4) above for the determination of the three com- ponents F, Q, E of electric intensity in the three principal directions yield three equations of the form vf = v>.-^(£.f.g)...as, since dP/dx+dQ/dy + dH/dz is not in this case zero. Consider now a plane wave travelling at right angles to the plane Ix + my + nz = 0, with velocity V. The disturbance may be expressed by the equation {P, Q, R) = {P„ Q^, E,) exp. ^ {Ix + my + nz - Vt) (19) A Substituting from (19) in (18) and writing v^, v^, v^- respectively for l/k^fi, l//«2/^, 1/k^ we obtain the conditions expressed by :} (F2 - v^^)P + Vj^l(lP + mQ + nR) (V^ - v.2^)Q + v^m{lP ^ mQ -v nR) = (j\ . . (20) (F2 - v^^)R + vin{lP + mq + nR) = These by elimination of P, Q, B yield which since P + m^ + n^^l may be written l^ m v^ - v^ v^ - 1-2 "^ r2 - + 172— r2 = o ■ • • (21) '2 ' - 'Oz This is Fresnel's equation connecting the velocity of light in an seolotropic body with the direction of propagation, and v^, v^, v^ are the squares of the principal velocities of propagation. These are the velocities of an ordinary wave in the three principal directions, that is the velocities in three different isotropic dielectrics characterised by the constants \, \, \. ^^ GENERAL ELECTROMAGNETIC THEORY 393 ^ ^^0. Let I', m', n' be the direction cosines of the electric induction, ihe components are proportional to k^P, \q, kJR, and these by (20), since v^^^l/k^fi, &c., are proportional to l/(V^-v^^), mHV-v^), n/(r^ — v/). Hence U'+mm' + nn' = 0, or the displacement is in the wave front. Take the direction of propagation as that of s, and the direction of the electric induction as that of y, then n=l,l = m = 0, in' = 1 , Z' = «,' = 0. But since m' {x,y,z,t) be a solution of (26). Since each component of electric or magnetic force involves differentiation with respect to x or y, they will be given also by the solution 11 = ^ {x, y, z, t)- + y^{z,t), where i|r {z,t), is any function of z and t. This would give the differential equation (26) with the addition of a term ;^(s, t) on the right arising from y}r(z,t). We may choose %(s, so that f(z,t) + x(^>i)=^^> ^^^ therefore putting /(«; = ^^ C^^) ^i^ ^°^ affect the electric or magnetic intensity at any point. Thus we have finally for the equation of propagation ^^ = .> <-) If r denote i>Jj?-\-y'^-\-z^ or V'/j^+s^, the general solution of this- equation is ji = \{F^{rr-vt)^Flr + vt)\. . . . (29) where v^ljs/kiM, and F-^, I\ are arbitrary functions. 398 MAGNETISM AND ELECTRICITY chap. A solution adapted to the vibrator imagined is n = — sin {mr - nl) (30) where 7ii,= 27r/\ and n/m = v—ll\/k/ji. This satisfies the differential •equation and is of the form (28). Moreover, if we take $ as the maximum moment of the doublet, we see that in the immediate neighbourhood of the origin, that is, at any point whose distance from the doublet is a small fraction of the wave-length of the disturbance, the electric field should at each instant precisely correspond to that •of a small magnetic doublet of the same moment numerically as that •of the electric doublet at the instant in question. The lines of intensity for a magnetic doublet are shown in Arts. 29, 30 above, and the field is there derived from a potential V= —md{l/r)/dx. Writing ■<1> sin nt/Jc 'for m, we see that the electric intensity to correspond ought ito be given by a potential V= — - ^ sin nf -;— I - k oz \r, But since for the region considered mr is a very small angle, we may write approximately for all points very near the doublet _ sin nt n = - * r and by (25) the electric potential in the field, near the vibrator is 1 pn 1 . 3 /I' K = - --- = -<|i sm nt — - K oz k 'dz\y, which agrees exactly with the magnetic analogue. So far the solution agrees with what we should expect to be the ■case. Again it gives electric and magnetic intensity everywhere zero at an infinite distance, which also must be the case on account of the ■divergence of the waves. Electric and Magnetic Intensities in Field of Doublet 519. For the sake of a generalisation of this solution to come as a succeeding a,rticle the three components P, Q, R of electric intensity have so far been retained in the analysis. But in order to calculate the •electric and magnetic fields of the doublet we now take account of the fact that the electric intensity is everywhere directed along a meridian plane, and is symmetrically distributed about the axis. It is sufficient therefore to use cylindrical co-ordinates z along the axis from the origin, •and p at right angles to the axis. (See Fig. 137). Identifying p with y, we have ft B for the components of E in the meridian plane at the GENERAL ELECTROMAGNETIC THEORY 399 point considered, while the magnetic intensity there reduces to a. We can find these components from the solution (30) by the second and third of (-25), and the first of (24), by putting p for y, -, = sj'^^^, and writing the third equation of (25) in the form proper to symmetry round the axis of s, namely. R _ 1 _a_/ an p 9p \ 9p an\ p 9p \ 9p / Thus we find ■«y = — g J ( 1 — j sin {m,r - nt) - mr cos (pir - nt) I sin 6 cos 6 * r • ^■K = —A 2 {sin {mr - nt) - mr cos (nir - ni)\ - {3 sin (mr - nt) - Zmr cos (rnr - nt) - mh'^ sin (mr - nt)] sin ^ « = — [mr sin {mr - nt) + cos {mr - nt)} sin 6 For points very near the vibrator these equations become (31) (32) 3'l> "i —ir sin {mr - nt) sin 6 cos 6 hR = - —- sin (mil" — ni) sin ^^ '** / s . ^ a = — - cos (nir - nt) sin e* (33) 520. The expression here given for a is that of the magnetic intensity which according to Art. 391 above would be produced by a current 7 in an element of length ds such that 'yds = n^ cos (mr — nt). But n^ cos {mr — nt) is the actual current in the doublet at time ^ multiplied by the length of the element. For points very near the doublet therefore the theory leads to the expression stated in Art. 391, and hence to that expression for the magnetic intensity produced at any point r, 6 by an element of a steady current. When r is very great equations (32) become kQ = kR = $ - — m^ sin {mr - nt) sin 6 cos 6 r $ . — m^ sin (mr - nt) sin '■^O r n^ . , \ ■ n — m sm vmr - nt) sm 6 r ^ (34) 400 MAGNETISM AND ELECTEICITY chap. Direction of Vibration] " longitudinal light." Bate of Propagation in Different Directions and at Different Distances 521. From (34) we obtain some important conclusions. Since Q, R, a all involve sin (vir — nt) as one factor, with others which are numerical or dependent on and r, the electric and magnetic intensities are, at a great distance from the origin, propagated together with velocity n/m, and are in the same phase. Again (34) gives Q sin 6 + E cos ^ = 0, that is, the electric intensity E has no component along the radius vector r. The direction of electric vibration is therefore at a great distance from the origin . perpendicular to the radius vector from the centre of disturbance to the point ; that is, it is transverse to the direction of the ray. The mag- netic intensity is in the same plane and perpendicular to the ray, and of course being perpendicular to the meridian plane is at right angles to E. For points very near the origin, however, the direction of the resultant electric intensity is not at right angles to the ray, but has a longitudinal component. To obtain so-called " longitudinal light " therefore it is not necessary to go outside of Maxwell's theory of the electromagnetic field. The longitudinal component of the electric induction along the axis is given hy kB in (32) with 6 put equal to zero. Thus we have 2$ kR = — {sin (rnr - nt) — inr cos [mr — nt) r 2$ n/1 + mV^ sin (jwr — nt - t) (35) r where tan e = mr. The transverse component kQ of the induction is here zero, so hat the vibration along the axis is wholly longitudinal. It will be observed that at points on the axis the value of a is zero. On the axis of the vibrator therefore there is a very special state of things. There is vibration of the electric induction, but that is wholly longitudinal, and there is no magnetic induction whatever. If there are special phenomena which can be produced by light of longitudinal vibrations, and consisting of only one kind of vibration, they should be looked for along the axis of a vibrator. It will be noticed that as the amplitude of the longitudinal component varies as the inverse cube of r, its value will be very small in comparison with the induction elsewhere, as given by (32), except near the origin. The velocity of propagation of electric intensity along the axis of the vibrator is to be found by calculating drjdt from mr — nt — t&n-^mr^O, and is therefore n(l+mV)/m^r\ Jt is very great when r is small and apjDroaches the value n/m, or 1/s/kfi, as r increases. 522. In the equatorial plane on the other hand the value of kQ, which is here the longitudinal component, is zero, and kB is the icsultant electric induction. The directions of kB and a are at right XI GENERAL ELECTROMAGNETIC THEORY 401 angles to one another, the former in, the latter at right angles to the meridian plane through the point. The values of these quantities are given by (32) with sin ^ = 1. Hence (J) , kR = -J ijl - mV + TO"'r* sin {mr — nt - e)\ ■ (36) a = — T- n/1 + mV sin {mr - nt - e) ' where tan e = mr/{l — mh^), tan e' = — l/inr. The velocity of propagation of electric induction in the equatorial plane is thus w(mV — inh-^ + l)/m^r^(mh-^ — 2). This is greater than the velocity along the axis, except of course for great values of r, where it is n/m as in the other case. Moreover, it is infinite when r = 0, and when r^ = 2/wi^, or r = X/('n-v 2), and is negative at intermediate points. We obtain therefore the very remarkable result that the electric induction is propagated outwards and inwards in the equatorial plane from a point outside the vibrator. In the representation of the lines of electric intensity (and induction) in the electric field given below in Figs. 138 — 141, this point is the centre of the small circle seen on each side of the vibrator in Fig. 141. At this point the electric force attains any value which it there takes about 12 of a period before the corre- sponding value is attained at the origin. The magnetic induction and intensity are propagated in the equatorial plane with a velocity m(l + m^r^)/mV, which is also infinite when r is zero, but diminishes as r is increased towards the limiting value. 523. The reader may verify that the interval in which a zero or maximum value of the magnetic intensity travels out in the equatorial plane from the origin to a great distance r is rm/n — Tji, and that a zero value of the electric intensity travelling out in the same plane from a point in the circle of radius X/{-jr>s/2), round the vibrator, reaches a point, distant r from the origin, in the interval rm/n — r/2 after the instant at which the zero value reached the origin. When, however, this zero value reaches the origin, the current has its maximum value, and so has the magnetic intensity. In the succeeding interval, rm/n — T/4i, this maximum travels out a distance r, but the zero value of the electric intensity has arrived earlier by ^/4, so that at a great distance the electric intensity is a maximum at the same instant as the magnetic intensity, in accordance with equations (34). The foregoing discussion of velocities in the equatorial plane is in its essential particulars taken from a paper by Trouton in the Fhil. Mag. for March, 1890, to which the reader may refer for further information. ■ D D 402 MAGNETISM AND ELECTRICITY chap. Electromagnetic Theory of the Blue Sky 524. The action of Hertzian vibrators, it may here be remarked, is very instructive in connection with the dynamical explanation given by Lord Rayleigh ^ of the blue colour of the sky. Let a beam of plane polarized light of different wave-lengths, as in white light, be propa- gated across a space in which there are a number of particles all very small in comparison with any wave-length of the light. On the suppo- sition that there is a difference in the density of the ether in the two media, and not in its rigidity, it can be shown on the elastic solid theory of the ether that these particles will act as centres of disturbance from which light will be radiated, the amplitude of which will vary as the inverse square, and the intensity as the inverse fourth power, of the wave-length. The forces acting on the ether within the particle will be in the wave-front and in the direction of the vibration in the exciting ray, and the theory shows that the direction of the vibration in the scattered light lies in a plane through the particle containing the direction of the force, and is transverse to the direction of pro- pagation of the scattered ray. The light is therefore polarized in a plane through the ray at right angles to the plane just specified. Further, there is no light scattered in the direction of the force on the particle, that is in the direction of the axis of symmetry, while the- maximum of intensity is found for rays in a plane through the particle perpendicular to the axis of symmetry. When the exciting beam is not polarized, the light scattered by the particle in any direction parallel to the wave-front is wholly due to- the component of the exciting vibration which is perpendicular to that direction, and is therefore completely polarized in a plane through the. exciting ray and the scattered ray, which in this case are at right angles; to one another. These results are found to be in accordance with experimjents om the light scattered from a space in which small particles are suspended and through which a beam of plane polarized light is passed, as in. those of Tyndall on light passed along a glass tube filled with carbon, disulphide vapour, and viewed from the sides of the tube. Also it is. found that the light received from a part of the sky, distant 90° fromi the sun, is always polarized in a plane through the sun. 525. Returning to the Hertzian vibrations, we have found [(34) abbve] that at a great distance from the vibrator the amplitude of the- electric vibration is inversely as the square of the wave-length (and the intensity therefore as the inverse fourth power), is a maximum in the- equatorial plane and is there transverse to the ray. Each ray is, moreoser, polarized in a plane through the ray at right angles to th& meridian plane in which it lies. On the other hand, along the axis of the vibrator there is no- magnetic intensity, and the electric vibration there is comparatively ' Sue Phil. Mag. Feb. 1871, or Wave Theory of Light, Encyc. Brit. 9tli edition. XI GENERAL ELECTROMAGNETIC THEORY 40» small and is longitudinal. We may therefore account for the blue of the sky on the electromagnetic theory by supposing the particles to be Hertzian vibrators set into forced electrical oscillation by the electromagnetic waves passing across the space where they are situated. It is to be observed that this result will only hold at a distance of many wave-lengths from the particles ; in the vicinity of the vibrator the radiation is much more complex. We see that here again, the vibrators being supposed to be electric, the direction of the electric induction is at right angles to the plane of polarization of the radiation. The real proof of the relation of the direction of the electric vibration to the plane of polarization will be given below. To show that the present result is not conclusive, it is only necessary to recall that, if the vibrators were magnetic, the mag- netic and the electric intensities in the present investigation would be interchanged without alteration. Propagation of Electric Potential 526. The solution here given is equivalent to one which may be obtained by the following process. If the medium is at rest the equa- tions of electric intensity are by (2) (P,Q,R) = (^- 3* dx' dG dt dff 'dt 3*\ dz) (37) and the condition of zero electrification at any point of the field,. dP/dx + dQ/dy + dBjdz = 0, gives where dF dG dH dx dy dz , (38) The value of ^ cannot generally be taken as independent of th& time, in the case in which there are varying electric charges. Let us then assume that J= —Jcfid'ir/dt, and inquire what this assumption involves. In the first place we obtain by means of it the equation 3% and kfi. 32/ vV (39) which are equivalent. The solution of either of these equations gives ^ and /. ' D D 2 404 MAGNETISM AND ELECTRICITY chap. 527. If a system of electric currents, of components u, v, u; exist in the field, it is easy to prove from the values of the vector potential, ^iven in (21) Chap. IX. above, or the same modified for the case of propagation, that where there is zero divergence of the current (and this must be the case where there is no electrification), that the vector- potential fulfils the condition J=0. This condition is fulfilled throughout the dielectric by the induction currents there existing. It does not hold, however, where there is varying electric charge, and this there always is at the origin of the disturbance if the waves are due to the oscillations of electric charges on ■conductors. Hence we may consider only the vector-potential due to the currents at the conductors at which the changes of electrification ■are proceeding. Modifying the expressions for the vector-potential of the system of currents in the field to provide for propagation, with velocity l/y/k/ji( = njm), of values of the vector-potential, in the case of simple harmonic time-variation of the currents, we obtain {F, G, H) = ^ f (M)lJf^) cos (mr - nt)drs where (Mq, v^, w^) cos nt are the components of current at the element dm of volume, and r is the distance of the element dvj from the point at which (F, G, H) is to be found. The integral is taken throughout the whole space, including both conductors and field. We suppose now that e^ sin nt is the electric charge per unit volume •at the element dvi at time t. This of course is confined to the con- ductors at which the disturbance originates. We write then 1 f e •^ = - -r\- sin {mr - nt)d7S (40) where the integral is taken throughout the space containing electric ■charge. It can easily be proved that the differential equation (39) is satisfied by the solution (40). The quantity '^ may be regarded as in a sense an electric potential due to the harmonically varying charges. To this potential each element of charge, as that at drs, makes a contribution, which is propagated outwards in every direction from the element, with speed w/m. From this and F, G, E, as shown in (37), the values of P, Q, B are to be found. Further, it can be verified that this value of '^ with those of F, G, S satisfies the equation 3* ^u ^r- + -^ = 0. at From the values of F, G, If the components of- magnetic induction -and intensity can be calculated and the field completely determined. ^i GENERAL ELECTROMAGNETIC THEORY 405 This mode of solution is applicalDle to any space distribution of (^o> ''^o' '^'q)> or of the corresponding amplitude electrification density e^. Attention was directed to it by Professor FitzGerald in- 1890.i 528. This solution agrees with that of Hertz for the electric vibrator. Remembering that here the only place where J is not zero is in the vibrator, and that there the current is along the axis of z, we have ^'=0, (?= 0, and U = n —^ — COS (mr - nt) where ds is the distance between the two point-charges of the doublet.. Also we have * = - ^ — -i!— sm(TOr - nt) .... (41) kdz r ' ^ ' Now by (24) above /udU/dt = H, since in this case /xa = dHjdy. But in this case also J = dJSjdz, and hence we have which give '*3«3.+ 3«-^' ^''U ^ oz- 7T sn dz write by (41) n = -^ sin {mr - nt) (42) which is the value of 11 used by Hertz. Graphical Representation of Pield of Vibrator 529. Very instructive diagrams were given by Hertz in his paper- already referred to, illustrating the successive changes which take place- in the electric field of a vibrator in the course of a period. The lines of electric intensity (and induction) can be drawn from their equation,, which is easily found to be 3n , ,,..„ p^r- = const (4o) op For the differential equation of a line of intensity is easily found to be a / 3n\ , a / an\ , . d-ApYpr'-d-p[pi^r = ^ 1 B. A. Eep, 1890, also Phil. Mag., Sept. 1896. 406 MAGNETISM AND ELECTRICITY CHAP. which integrated yields (43), in which substitution from (30) gives {sin {mr — nt) - inr cos (mr - nt)} sin^6 = c (44) where c is a constant for any particular line. 530. These lines are given in Figs. 138 — 141 as drawn by Hertz. Fig. 138 shows the electric field as it exists at the beginning of an oscillation when the vibrator is in the neutral state, the remaining figures of the series give it as it appears after the lapse of successive eighths of a complete period. [N.B. The \ marked on the curves is one-half of the wave length.] In the immediate vicinity of the vibration the lines are not drawn, and those drawn are stopped at a circle surrounding the vibrator. The Fig. 138. vibrator is of a dumb-bell shape, and its field in its own immediate vicinity must be very different from that of a doublet, though they will agree at some distance from the origin. Fig. 139 shows the field after one-eighth of a period from the instant ■of zero electrification. The lines drawn are enclosed within the circle given by (44) for i; = |2', and c = 0. This circle travels outwards with the velocity of propagation of the magnetic intensity. In Fig. 140 another ^T has elapsed, and the lines and enclosing circle have spread out farther. After still another ^T (Fig. 141) a remarkable change has been set up in the vicinity of the vibrator ; the lines of electric intensity have begun to contract inward on the source, while in the outer parts contmuing their progress , outwards. XI GENERAL ELECTROMAGNETIC THEORY 407 The lines thus are throttled, so to speak, and break off at the neck into closed curves, which spring up first in the interior of the system as Shown by the small circles inside the outer loop of the dotted curve. Just at these points the electric intensity takes any possible value before the corresponding value is reached at the origin. Thus in Fig. 141 the electric intensity has just become zero at the small circles. As t increases from f .7 to ^T the curves, break off successively from 408 MAGNETISM AND ELECTRICITY CHAP. within until they have all broken off into two groups of closed curves seen to right and left of the origin in the first of this series oi figures. These are the cross-sections of what we may call an electric vortex which is produced at the points shown in Fig. 141 and remains sym- metrically situated round the axis of the vibrator. The circular axis of this vortex travels out at first infinitely quickly, but ultimately slows down to the velocity of light. As time now goes on from i{ = |y to i( = |y the lines spread out from^ the source as in the first eighth of a period, except that they are now reversed in direction, and as they move force outwards the closed tubes which have broken off, rendering them more concave and more elongated,, Fia. 141. so that they approximate more and more at all points to lines transverse to the radius vector drawn from the origin. These successive groups of closed curves in which the direction of the electric intensity is alter- nately right and left handed are cross sections by a plane through the axis of the successive half- waves thrown off by the vibrator. The waves are thus each made up of two successive so-called electric vortex-rings, each consisting of a system of tubes of induction surrounding its circular axis. The closed tubes of intensity and induction here considered travel out, carrying their energy with them, and this constitutes the radiation of energy which is continually going on. The energy radiated is at the cost of the energy supplied to the vibrator, just as the energy carried out by waves formed by a steamer is supplied by the fuel burnt in the furnaces. The vibrations are thus damped out at a rate much greater than that due to the dissipation of enfergy by the conduction current in ""^ GENERAL ELECTROMAGNETIC THEORY 409' the rod connecting the two spheres and the spheres themselves. This rapid dissipation is the chief obstacle in the way of obtaining maintained electrical vibrations of great .power. We shall return presently to the calculation of the rate of radiation of energy by electric waves, and to- ettects produced by the damping out of the vibrations. Experimental Verification of Theory of Vibrator 531. Hertz verified the foregoing theory experimentally, by making durect observations on electromagnetic waves by means of a vibrator a,nd receiver as described above. The arrangement of apparatus is shown m Fig. 142, except that the vibrator was modified by the substitution for the spheres of plates coplanar with the axis. The Induction Coil a Q o Fig. 142. vibrator was charged initially by an induction coil, the terminals of which were connected to the two sides of the spark-gap. As soon as the difference of potential between the spheres A, A' had become great enough, a spark passed and electrical oscillations were set up, which depended for their period on the dimensions of the apparatus, but were enormously more rapid than the action of the coil. The oscillations had therefore time to be damped out by radiation and dissipation of energy long before they were renewed by the coil. There was therefore a succession of oscillatory discharges separated by intervals of inaction. It was found that the receiver acted best when it was chosen of a particular size, though to get it to respond fairly well no very exact timing was required. Thus it acted to some extent as a resonator, though as will be seen later, in consequence of the rapid damping out,, an impulse is given to the resonator; which starts it at first in forced 410 MAGNETISM AXD ELECTRICIXy chap. -oscillation, but quickly dies away leaving the resonator finally vibrating in its o\vn proper period with a much slower rate of subsidence. The receiver was made of wire 2 mms. thick, and the diameter of the circle was 35 cms. The spark was produced between two small knobs, the distance between which was regulated by a fine screw, which moved one end of the wire. In some of the experiments a receiver in the form of a square 60 cms. inside, made of similar wire, was used. In this the spark-gap was at the middle of one of the sides. Approximate Theory of Hertzian Receiver 532. The following sketch of a rough theory of the resonator is all we have here space for, but the action of both receiver and vibrator will be considered more fully irl Vol. II. in the account there to be given of later work on Hertzian vibrations. Denoting by P the electric intensity, parallel to an element of the resonating circle, produced by the action and supposing that the electric intensity is oscillatory with period 27r/w, and is a function of the distance s of the element from some point of the circle, say the centre of the spark-gap, taken as origin, we have P = <^(s) cos nt (45) If (/) (s) be a periodic function of S, we get by Fourier's series 4.(s) = A + B eos-^ s + .... + B' siTa — s + ... . (46) o o where S is the whole circumference of the circle. The terms here •exhibited are a constant term and the two gravest simple harmonic •components. The second of the two last must disappear also for the origin at the spark-gap. Hence at this origin and diametrically opposite we have respectively <^(8) = A + B, ^{s) = A - B. 533. Hertz took the view that the action of the vibrator was most •effective on the portion of the resonator opposite the spark-gap, that is, that the vibration set up depended in the main onA — B. The vibration no doubt, consists in a backward and forward flow of electricity in the connecting wire from one knob to the other (which is of course only the manifestation at the conductor of a surging to and fro of the induction tubes in the ether), which gradually increases in amplitude, if the period of the receiver is nearly equal to that of the exciter, until the maximum difference of potential between the knobs reaches that required to produce a spark. Let V denote the difiference of potential between the knobs at any XI GENERAL ELECTROMAGNETIC THEORY 411 time, then, on the supposition that the exciting vibrations are damped, those in the resonator not, we have ~ + p'^V = A' e-"* sin nt (47) if 27r/p be the natural period of free vibration. The solution for forced vibration is A'e-"'' F= ,_ sin (nt - t) . . (48) where tan e = 2nKJ{n^ —p^ — k^). If /c be very small, F will be very great if p = w, and will have the same sign as A' or the opposite, according as ^>- or <^n. But if p=^n tan 6 is very great and, e = Tr/2, approximately. Thus where resonance, is just attained there is a difference of phase of half a period. When k is not small the forced vibration is a maximum when n^=p^ + ic-j'l, and this maximum is smaller the greater k. The constant term in ^{s) [46] gives a term A cos nt in P. This has the same value at any instant all round the circle. It may be inter- preted as the electric intensity set up by the exciter at each element of the conductor, due to variation of magnetic induction of the same value at every point, or if the magnetic induction is not uniform, it is that part of the electric force which is the same at each point of the circle. If we denote by E the part of the electric intensity impressed on the resonator which does not depend on variation of the uniform part of the magnetic induction, and is due in part to other causes, for example the potential '^ of Art. 526 above, by -i/r the angle it makes with the plane of the circle, and by 6 the inclination of its projection on the plance of the circle to the radius through the spark-gap, the tangential component at the spark-gap is — ^ cos i/r sin ^ cos mi Thus the amplitude of the electromotive intensity producing a spark is of the form A + Cs,m 6, where G represents —U cos yfr. Experiments with Different Positions of the Receiver 534. In the experiments it was sufficient of course to observe for positions of the centre of the receiver at different points of one of the four quadrants of the horizontal plane made by two rectangular horizontal lines, one the axis of the vibrator, the other through the centre of the spark-gap. The receiver was used with its plane (1) vertical, (2) horizontal. , . , , , . , . In case (1) no sparks passed when the circle was placed with its plane vertical, and the diameter through the spark-gap horizontal. Sparks however passed with increasing intensity as the receiver was turned round in its own plane so as to bring this diameter nearer the vertical. Whatever the position of the plane of the circle was the 412 MAGNETISM AND ELECTRICITV CHAP. X^ Fig. 143. sparks passed most freely when the gap was at the highest or lowest point of the circle. According to the theory given aho\e it is clear that for any vertical position of the circle A = 0. For a vertical position of the gap the action of C is equal and opposite in the two halves of the circle, for a horizontal position its — «- action is most effective unless- i|r = 7r/2. When the gap was at the top or bottom of the circle, if the receiver was turned round a vertical axis there were found two positions in which the sparks passed with maximum intensit}', and two in which there was almost extinction of the .spark. The positions of maximum were 180° apart, and the two- positions of minimum were at the two points midway between these. For the former i|r = 0, and = 7r/2, for the element opposite the gap, for the zero positions ■\lr = 7r/2. Fig. 143 shows a number of these positions. The longer lines are the positions of the spark-gap when the sparking was a mimimum ;. the short arrow-pointed lines are the directions of the electric intensity and indicate very clearly lines of electric intensity, which near the- vibrator resemble the lines in Fig. 139. On the other hand, Fig. 144 shows the positions of the gap for experi- ments made with the plane of the circle horizontal. For position I and the gap at &i, or b\, no spark was observed ; with the spark-gap at a^, a\, however, equal maxima of intensity of spark were found to exist. It is clear that in position I, the circle received a zero number on the whole of tubes of magnetic indu ction. For position II the magnetic induction through the circle was no longer zero. Two positions of mimimum sparking were found at i^ and b'^, and two maxima of unequal intensity at «2 and ti'g. The line a„ ft'2 was at right angles to the electric intensity, and thus the action was represented hy A+B at one position and hy A—B at the other. The two effects, the electric and Fig. 144. sp: ired at a^ and were opposed at a'g. the magnetic inductive, con- The spark lengths at a^ and were 3"5 mms. and 2'5 mms. respectively. When the spark-gap was at h^ or h'^, the electric intensity, which was at right angles to a^ a'^, was equally inclined to the circle at those ^^ GENERAL ELECTROMAGNETIC THEORY 413 points, and gave components along the circle which neutralised the electric intensity due to magnetic induction. In position III the two null points were found closed up nearer to a'g, the smaller maximum, while the greater was at a^ diametrically opposite. Over a considerable region opposite a^ only a small effect was observed. The spark length at a^ was 4 mms. In the positions IV and V, no positions of extinction were found for the gap, but only a maximum and a minimum, at a^, a\ in IV, and a^, a\ in V. The line a a', it will be noticed, has turned round through nearly 7r/2 in the passage from III to V, in order to keep always perpendicular to^the electric intensity. The spark lengths were 5-5 mms. at a^, 1"5 mm. at a\, 6 mms. at a^, and 2-5 mms. at a\. Other positions of the resonator gave results in accordance with theory. The circle was placed in position V of Fig. 144 with the gap at ftg, and turned round the diameter parallel to the vibrator so as to raise the gap. During the changes 6 remained 7r/2, so that C remained constant, but A changed with the inclination of the plane of the circle to the horizontal. Thus the inclination of the plane of the circle to the horizontal being ^, and A^ being the value of ^ for = 0, the value of A+B changed from -4g+jB through successive values oi A^cos = 7r/2, and (^ = 7r, A^cos^+B took the value 0. As was changed from tt to 37r/2, Ag cos estimate the activity necessary to maintain the action of the exciter. The vibrator, the dimensions of which have just been given, had its spheres charged to a maximum difference of potential of 1 cm. The difference of potential between the spheres was thus about 60 C.G.S. electrostatic units, and the charge of each sphere was 60 x 15 C.G.S. units. Thus $ for the vibrator was 60 x 15 X 100 and the energy radiated in half a period was (see Art. 549 below) 60^ x 15^ x 100^ x 87r*/3X*, X being taken in cms. If the velocity was that of light the wave length was about 550 cms., and hence in each half-period about 12000 ergs, passed from the vibrator to the surrounding medium. The whole energy of the vibrator when charged to the potential stated above 1 If Si, i'a be two areas in the same plane and having any boundaries, and r^^ be the- distance between an element dS^ in one and an element dS.2 in the other, then Gf.M.D.= • //log r^^ dS;idS^USiS2 where the integrals are taken over ;S„ /S'j. Of course iwhen the areas coincide we have the ff. M. D. of an area from itself. The knowledge of geometric mean distances is of great importance in the calculation of inductances of parallel conductors ancl close parallel coils of great radius. 416 MAGNETISM AND ELECTRICITY CHAP. ■was hx2x 60- X 15 ( = 54000) ergs. Thus about f of the whole energy was radiated in the first half-period, that is the amplitude of vibration suffered from radiation a diminution of roughly ^. The period of vibration being l'85xl0~^ second, the average rate of radiation was therefore about 1'34 x 10^^ ergs, per second, or, approximately, 179 horsepower. Very considerable power would therefore be necessary to maintain the vibrations of even such a small vibrator as that of Hertz. Reflection of Waves in Air from Metallic Surfaces. Standing Waves 538. Hertz carried out a series of experiments in his lecture theatre, a room 15 metres long, 14 metres wide, and 6 metres high, on the re- ilection of electric waves from a large plate of zinc, 4 metres high by 2 metres broad, which covered the middle part of one end wall. The clear breadth of the room free from obstacles was only 8'5 metres, on account of a row of iron columns along each of the two sides. The Fig. 146. ■exciter (which was an instrument with plates instead of balls which had been used for experiments in propagation along wires) was placed with its axis vertical at the middle of the end remote from that covered with the zinc plate. The waves generated were thus incident nearly normally on the conducting plate, with the electric vibration in the vertical plane through the axis of the vibrator. The same receiver, as before described, the circle of 35 cms. radius, was carried along the normal through the centre of the vibrator, and the positions of maximum and minimum sparking in the neighbourhood of the zinc plate observed. The positions I, II, III, IV of Fig. 146 were those of most intense sparking, and V, VI, VII those of least intense sparking. In the former set the spark gap was turned alternately in opposite directions, in the second set the sparking was the same for both right and left positions of the spark gap. ^i GENERAL ELECTS OMAGNETIO THEORY 417 When the spark gap was at the top of the circle, so that the electric intensity could have but little effect, feeble sparking only was produced m position V, a maximum at VI, and a minimum again at VII. Thus the magnetic induction was evidently a minimum at V and VII, and a maximum at VI. Clearly these results point to a standing wave of electric and mag- netic induction produced as represented by the full and dotted curves in Fig. 146. The diagram shows that the electric intensity has its phase changed by half a period relatively to the magnetic intensity, so that in the standing vibration the nodes of one correspond to the loops of the other. Apparently the node for the electric intensity was behind the wall-surface about -68 metre, and the next loop but one, about 6-52 metres in front of it, so that the wave-length was about 9-6 metres. With the period 3 x 10"* second, which was nearly that of the vibrator, this would give 3'2 x 10^" cms. per second as the velocity of propagation of the waves in air. This is very approximately the velocity of light. Multiple Resonance 639. Hertz experimented on the rate of propagation of electric waves along wires, and found a velocity differing considerably from that obtained for waves in free space. A vast amount of work has since been done on this subject, and we defer its discussion to Vol. II. But the general nature of the explanation is as follows : It has been suggested by Messrs. Sarasin and De la Rive that the wave-length observed in free space may depend to a great extent on the dimensions of the resonator, and may be connected with what has been called multiple resonance. It has been noticed by these experimenters, as well as by FitzGerald and Trouton,^ that the exciter apparently gives rise neither to a single vibration of distinct period nor to a limited number of distinct vibrations, but rather to such a complex of vibrations as would give a wide band of continuous spectrum. Thus all vibrations, a.greeing with possible modes of vibration of the resonator, would be reinforced. That this is not contained in the theory is true, but the theory is very incomplete. It is hard to believe that the vibrations can be perfectly simple. The following explanation of multiple resonance has been proposed by Poincar^.^ The logarithmic decrement of the vibrations of the exciter is probably much greater than that of the resonator, and so the vibrations of the exciter diminish in amplitude more quickly than those set up in the resonator. This is confirmed by experiments on the damping of the vibrations in the exciter and receiver made by V. Bjerknes.^ Thus the resonator, being started by the exciter, ^ Nature, vol. xxxix. (1889-9). p. 391. ^ Electricity etOptique, 2*0 Partie. 3 med. Ann. 44 (1891), p. 74. E E 418 MAGNETISM AND ELECTRICITY chap, continues its own vibrations after those of the exciter have become insensible, but then vibrates in its own proper period, giving vibrations of longer period and of greater wave-length than those which excited it. The wave-length being determined by interference, and used with the too short period of the exciter, gives too great a velocity of propagation. With this explanation Hertz has expressed himself as practically in accord.^ As he remarks, the oscillations of the exciter, represented graphically, do not give a curve of sines pure and simple, but a curve of sines the amplitude of which gradually diminishes. Such an oscillation causes all the resonators receiving it to vibrate, but those in tune with the exciter more violently than the others. This agrees with the theory given in Art. 533 ; and the fact that the apparent spectrum seems more extended when wires are connected to the vibrator than when the propagation takes place freely in air may be due to a greater damping effect in the former case. Effect of Size of Reflecting Surface 540. It may be noted here that it has been found by Mr. Trouton ^ that the size of the reflecting sheet has a great deal to do with the distance of the nodes from the surface. Using long narrow strips held (1) so that the length was in the direction of the magnetic component, (2) in the direction at right angles to that component, he found that the node was in the former case shifted outwards from the reflecting surface very markedly. For example with waves 68 cms. long the distance of the magnetic node varied from 24'2 cms. for a strip 16 cms. wide to 17 cms. (^ wave-length) for a large sheet. This effect was due no doubt, as stated by Mr. Trouton, to the action of the charge periodically accumulated at the edges of the sheet. Smallness of size in the magnetic direction carried the node in towards the surface ; and this may very possibly have been the case in the experiments of Hertz, described above. The breadth of the sheet (in the direction of the magnetic force) was 2 metres, or about the same in effect as a strip 14 cms. broad used with Mr. Trouton's 68 cms. waves. This would give a sensible inward displacement of the node. Reflection of Electric Waves by Mirrors. Refraction by Prisms 541. Experiments were also made by Hertz on the production of plane polarized waves, by means of a linear vibrator consisting of two cylinders placed in line with a spark-gap between their opposed ends. The cylinders were about 12 cms. long and 3 cms. in diameter each, and the ends of the spark-gap were well rounded. The vibrator was placed vertically in the focal line of a parabolic cylindrical reflector made of ordinary sheet zinc nailed on a wooden framework cut into proper parabolic shape. The cylinders were connected with an induction coil ' Electric Waves, p. 16. = Phil. Mag., July, 1881. XI GENERAL ELECTEOMAGNETIC THEORY 419' by insulated wires passing through holes in the zinc behind them. The mirror was about 2 metres in length, and about 70 cms. in depth along the axis of the parabolic figure, as shown in Fig. 147. Ihe exciter thus placed produced waves of electric force, the _ direction of which near the source was parallel to its axis. These were received by the mirror, and reflected into a parallel beam which could be observed by means of a suitable receiver. In most of the ex- periments however the beam was received by a similar reflector facing the former so as to concentrate the radiation on its focal line which in some experiments was parallel in others at right angles to the former. In the focal line of the other mirror was placed a receiver made of two- pieces of thick wire each 50 cms. long, placed in line as shown in Fig. 148, with a gap of about 5 cms. between their ends, and completed by < 70C. ■> Fig. 147. Fig. 148. two thin wires about 12 cms. long led out at right angles to the rods to- the back of the mirror. These were tipped with a knob and point as shown, so as to form an adjustable spark-gap which could be conveniently observed from behind. Polarization of Electromagnetic Beam. Relation of Plane of Polar- ization to Direction of Electric Vibration 542. It was found by this arrangement that electric radiation could be detected at a much greater distance from the source than with the ordinary vibrator and receiver used as described above without reflectors. In these as in all other experiments the knobs of the vibrator have to be repeatedly cleaned, and its spark-gap must be screened from the direct light of the spark in the induction coil. Clearly a parallel beam of plane polarized light was thus obtained, and consisted, as the experiments showed, of electrical vibrations parallel to the vibrator accompanied by magnetic vibrations at right angles to the former and to the direction of propagation. Placing the axial planes of the mirrors in coincidence gave augmentation of the electric effect, crossing the mirrors extinguished the effect at the receiver in the second mirror. E E 2 420 MAGNETISJI AND ELECTRICITY chap. Again, a grating of parallel copper wires placed between the mirrors •entirely stopped the vibration when the wires were at right angles to the vibrator, but allowed it to pass freely when turned through 90° from the former position. Also it was found, in a repetition of these experiments by Prof. FitzGerald and Mr. Trouton,^ that the electromagnetic beam was reflected from a wall about three feet thick when the vibrator was at right angles to the plane of reflection, and not at all at the polarizing angle when the vibrator was in the plane of reflection. This result showed that the electric vibration is at right angles to the plane of polarization. This is a very important result as it settles the question as to the relation of the plane of polarization to the electric vibration. The ■question of the relation of this to the ether vibration is a distinct question to which no answer has yet been given. It may be that there is after all no vibration, in the ordinary sense, of the matter of the ether. 543. Hertz found that such an electromagnetic wave was not only reflected like a light wave,but is also refracted according to the same law of refraction. An immense prism of pitch, having an isosceles triangular section of 120 cms. side and a refracting angle of 30°, was made by melting pitch into a wooden supporting case. The prism was placed with its refracting edge vertical, at a distance of 2'6 metres from the vibrator (also vertical), and the beam was made incident on the face at an angle of 65°. The receiving mirror was estimated as 2'5 metres from the prism on the other side, and showed a radiation beginning, reaching a maximum, and falling ofif to zero, at the respective deviations 11°, 22°, 34°. The experiments were repeated with the focal lines of the mirrors iiorizontal, and practically no difference in the results was observed. The index of refraction for pitch given by the experiments was 1"69, which nearly agrees with the index 1'5 to 1'6 found for pitchy substances by optical experiments. Prof. Oliver Lodge and Dr. Howard have made observations on the concentration of such vibrations by means of lenses.^ Two enormous lenses of hyperbolic cylindrical figure were constructed of mineral pitch, and were placed with their axial planes coincident, and their plane faces, •or bases, turned towards one another. These lenses were so proportioned that the beam produced by a linear exciter in the external focal line of ■one might emerge parallel from the plane face of that lens, and then be •concentrated by the second lens on the corresponding focal line. Transparency of Ordinarily Opaque Substances to Electromagnetic Waves 544. An interesting point noticed in many of these experiments is the perfect transparency to these vibrations of optically opaque substances. A stone wall three feet thick has been found to offer no obstacle to the passage of such waves. In fact in some experiments made by Prof. FitzGerald at Dublin the receiver was placed on a pillar in the garden 1 Naiure, loe. eit., p. 417 above. ^ Phil. Mag., July, 1889. ^i GENERAL ELEOTEOMAGNETIC THEOEY 421 outside while the exciter was in action in the laboratory. This is no doubt a phenomenon of similar character practically to that of the transpar- ency of a thin film of metal to light, and is conditioned by the nature of the material and the relation of the wave-length to the thickness of the stratum. It has been found by ' Maxwell, ii7. and Mag. Vol. II. „ Chap. XIX., that the transparency of thin metallic films is greater than that given by the electromagnetic theory according to the conductivity of the material. [See also Wien, Wied. Ann. 35 (1888).] A form of radiation, which appears to consist of waves of length small in comparison with that of visible light, and to which some kinds of ordinarily opaque matter are quite transparent, has recently been discovered by Rontgen at Wilrzburg, and has attracted much attention. Great additions to our knowledge of electrical radiation have been made during the last few years by an army of investigators in this field in this country and all over the world. Some account of their work will be given in Vol. II. Section II. — Flow of Energy in the Mectromagnetic Field. Motion of Energy across Bounding Surface of a Closed Space. Poynting's. Theorem 545. If we assume, as has been done above, the localisation of the electrokinetic and electric energy in the field, we obtain for the energy per unit volume at any point the value fifP/STr+lcE^jS-jr. We neglect terms which theory shows must exist in the complete expression for the energy, as shown by the phenomena discovered by Hall, and other effects more or less small in amount which, if not yet discovered, must be held for theoretical reasons to have a real existence. Now consider any closed space in the field, and let the total energy within it at any instant be E + T, where E denotes the electric energy and T the electrokinetic energy. Supposing that k and fi do not vary with the time, we have, if dm be an element of volume at which the components of electric intensity are F, Q, B and those of magnetia intensity a, /Q, y, -IT- -slK'^i^^i^''^^. d(E + T)_l^^^^j^d_P^ dQ^^d_R^ dt dtj da .3(^/8 dy\ K-i-"! -'©}*■ ■ • ("> where the integration is taken throughout the closed space considered. But if u, V, vj be the components of the total current, andp, q, r those- of the conduction current (the latter in the general case not merely generating heat in the conductor) we have for those of the displace- ment current d^ 1 /9r °^\ /RON 422 MAGNETISM AND ELECTRICITY chap. with two similar equations. Also we may write the equations of electric intensity (2) above in the form P, Q, R= F + ell - hi, Q' + az - ex, R' + h.c - ay (53) From these we obtain by differentiation da _ d /dll _'^\ 9£' _ 3^' /g^x dt ~ dtKdi/ dzJ dz dij ' ' ' and two similar equations. Substituting from these in (51) and using (53) we obtain on re- .arrangement - \{{cv - bw)x + (aw - cu)y + {bu - av)z}dvs - \{Pp + Qq + Rr)dz:r. The components of electromagnetic force (not those of total mechanical force, which include forces due to electrification), on a part of the medium moving with velocity (tjo, y, i), are X, T, Z=cv — hw, ■aw—cu, hu — av. Hence we get by integration over the closed surface ■of the space and transposition '^^^^ ^^ + {{Xx +Y^j + Zz)dTS + hPp + Qg + Er)d-a = lAiWP - Q'y) + HP'y - ^'«) + '»(<2'« - P'li)}dS (55) ■where I, m, n are the direction cosines of the normal to the surface element dS drawn outwards. If V, m', n' be the direction cosines of a normal to the plane defined by the resultant E' of the component electric intensities P, Q', B', and the resultant magnetic intensity H, and drawn in the direction in which a, right-handed screw would move if the handle were turned round in the plane of E' and H from the direction of H to that of E' (Fig. 149) the element of the surface integral on the right has the value ilE'sm6(ll' + mm'+nn')/4nr, where d is the angle between H and E. The rate of flow of energy per unit of area is therefore represented in magnitude and direction by the vector-product of H and E' (that is the vector HE'sin^ at right angles to the plane of H and E') divided by 47r. The component of flow across unit of area at right angles to the direc- tion (I', m', n') is thus HE'smOiW + mm' +nn')/4f7r. The direction of flow is opposite to that of (l', m',n') as defined above. Thus in Fig. 149 the flow is in the direction xO. GENERAL ELECTROMAGNETIC THEORY 423 H Fig. 149. 546. The terms in equation (55) may be thus interpreted. The first term on the left is the time-rate of increase of the energy within the closed space, the second is the rate at which work is done by electromagnetic forces, and the third com- bines the rates at which energy is dissi- pated and is expended in producing chemical changes, and the equation asserts that the sum of these rates is equal to that of the flow of energy across the bounding surface of the space, as shown hy the expression on the right-hand side. This theorem is due to Professor Poynting (Phil. Trans. R. 8., Part VI., 1885). If energy is conveyed into the system by the action qf impressed forces arising from another system, and producing energy within the space, terms must be added to (55) taking account of the energy so delivered. This point will be fully discussed and illustrated in Vol. II. It is to be observed that the addition of any term (lj> + m')(^ + n-^)dS, of proper dimensions, to the element of the integral would not alter the value of the integral over the closed surface provided (^, ')(^, yjr are func- tions of the co-ordinates fulfilling the condition dcj)jdx+d^jdy + dyfr/dz = 0. It is thus not strictly demonstrated that the flow across an element of the closed surface is that stated above. The results obtained in the following examples (taken from Poynting's paper) agree with known facts, and so far confirm the theory. Examples — 1. Straight Wire with Steady Current. Energy Stream-Lines 547. Take first the case, illustrated by Fig. 150, of a long sti-aight wire of circular section in which a steady current of strength 7 is flowing. The displacement does not vary, and there is therefore no displacement current. The magnetic intensity is tangential to a normal cross- section of the conductor and is of amount 2y/r. The electric intensity is parallel to the conductor and is equal to the current per unit of area of section divided by the conductivity of the conductor, that is to 'yjirr'^K. The rate of flow of energy across unit area is thus by the theorem (27/r x 7/7rA)/47r = 7727rV«, and the direction of flow is by the rule found above, inwards from the surrounding medium to the wire. The rate of flow of energy inward upon unit length of the wire is thus j^/Trr'^K, and across I units of B Fig. 150. 424 MAGNETISM AND ELECTRICITY chap. length is •y-ljirr-K, or rf-R, if R is the resistance of the length I of the wire. This agrees with the result alreadj- several times used above. Thus the conducting wire controls the manner of arrangement of the electric and magnetic equijiotential surfaces in the field. The flow of energy is along the lines of intersection of such surfaces, which may therefore be called energy stream-lines, when the intensities are derived from potentials. In a metallic conductor there is dissipation of energy received from the medium ; and if at any place electrical and magnetic energy are utilised in doing work, this by the theory does not come along the conductor, but from the surrounding medium, along paths perpendicular to the electric and magnetic intensities, the dis- tribution of which is conditioned by the existence of the conductor. 2. Discharge of a Condenser 548. Let a charged condenser have its plates connected by wires to another pair of plates, so that the capacity is increased. The tubes of electric induction formerly existing for the most part in the portion of the medium between the plates of the original condenser, and entirely depending for their arrangement on these plates, move out sideways with their ends on the connecting wires, until the state of strain has been set up between the other pair of plates. While the motion of the tubes is proceeding, magnetic intensity exists in the field, but dies away with the motion of the tubes. If the change proceed slowly, the intensities will be approximately derivable from potentials. There will then be a fall of potential along the wire connecting the insulated plates, and some of the equipotential surfaces must cut the wires, and so there is a flow of energy into the conducting wire, which is dissipated in heat according to the law of Joule. This view of the transference of energy from the medium between the plates of one condenser to that in another, has already been insisted on in Chap. V. above. And the view seems eminently reasonable. It is very difficult to suppose that the energy has been transferred along the conductor to the other condenser, and there inserted between the plates. Step by step with the process of charging the other condenser has gone on the growth of electric strain in the dielectric between its plates, and since the energy is certainly stored up in the dielectric, it is natural to suppose that the strain has been propagated through the medium under the guidance of the conductors. These, as we have seen, localise the electric and magnetic equipotential surfaces, which exist in the case of slow change, and the intersections of which are the stream lines of energy. Let the plates of the condenser be connected by a wire L, M, N, as in Fig. 151. The tubes of electric induction will pass out with their ends on the wire, and will shorten as they advance, being swallowed up at each end in the wire with dissipation of their energy. Finally, XI GENEEAL ELECTKOMAGNETIG THEOEY 425 somewhere about midway between the ends of the conductor the last Iragment of the tube disappears laterally into the conductor. The magnetic tubes of force which encircle the conductor contract down upon it and disappear within it, giving up also their energy as they _ The curves in the figure shown intersecting the conductor are intended to represent the intersection of the equipotential surfaces for the time being with the plane of the diagram. The rate of flow is again along the lines of intersection of the magnetic and electric equipotential surfaces, if these can be said to exist. Strictly speaking. Fig. 151. except in cases of slow discharge, the intensities are not' derivable from- potentials. ' o-Mr-r"";] If the conductor is of such form that the total magnetic induction through it is very great, the electrokinetic energy of the field will become very great, and the flow of energy absorption in the conductor- will be much altered. The tubes of electric induction will then move, with still a certain amount of absorption, though much less, in the conductor until their ends pass one another, when the tube returns- reversed to the dielectric between the plates, and the condenser is charged the opposite way. Then this discharges with reversal as- before, and so the electric oscillation goes on, with radiation and dissi- pation of energy, until all energy has disappeared from the system. [See further on oscillatory discharge and radiation of energy in VoL II. See also J. J. Thomson on "Faraday Tubes of Force," Eecent JResearches in Electricity and Magnetism, Chap. I.] 426 MAGNETISM AND ELECTRICITY chap. 3. Radiation of Energy from a Hertzian Vibrator 549. Close to the vibrator, as we have seen above, the lines of electric intensity alter their arrangement in such a manner that there is flux and reflux of energy across a closed surface surrounding the vibrator. The flow outwards is on the whole greater than the flow inwards, and thus in every complete oscillation a certain balance of energy is radiated. We shall be able to estimate this easily by considering a closed spherical surface, having its centre at the centre of the vibrator, and of radius r containing a very large number of wave-lengths. Taking the expressions given in Art. 620 above for the intensities at a very great distance, and first finding the rate of flow of energy outwards across a zone of breadth rd6 surrounding the axis, the intensities E and H( = a) are tangential to the surface, and at right angles to one another. Thus for the rate of flow across the zone we have $ -?r»-2 EH sin edd = h -r nv^n sin^ (mr - nt) sin^Ode. " k Thus for the total outward flow across the sphere in half a period we obtain r/2 , EH sin Odtde = ^ 5^„ Since we take h in ordinary electrostatic units and the medium is air k is taken as unity. This is the result used in Art. 537 above. Several other examples of the flow »f energy will be found in Prof Poynting's paper loc. cit. See also Absolute Measurements, Vol. II., p. 219. Distribution of Current in Cross Section of Cylindrical Conductor 550. As a flnal example for the present of the results derivable from Maxwell's equations of the electromagnetic field, we give here an investigation of the distribution of the current over the cross section of a long straight cylindrical conductor carrying rapidly alternating currents, and of the resulting resistance and self inductance of the conductor. We follow here Lord Kayleigh's ^ mode of treatment. Let the axis of the conductor be along z, the component w of current parallel to the axis is a function of the time and the distance p of the point from the axis. The components of vector-potential are 1 Phil. Mag. , May, 1886. Another discussion will be found in Absolute Measv/remenls, Vol. II., p. 331, and a third by the Bessel Function Analysis in Gray and Mathews, Bessel i'unclions and their Applications to Physics, p. 157. ^i GENERAL ELECTROMAGNETIC THEORY 427 F = 0, G = 0, and IT, which must be a function of the same variables ■as w. Let ff = S + T + T.j>'- + T^pi + + ^„p2« + (56) in which S, T, T^, . . . . are functions of the time t. Now by the circuital equations (4) 3/8 3a 4:TrW = -^ — - OX 01/ dff „ dll •and therefore _ + _ + 4.^«, = or _+___+ 4.^, = (57) From this and (56) we obtain - TT/tw = ^'i + r-T^'^ + S^rgp* + + H27'„p2«-2 + (58) If « be the conductivity of the material, the component elec- tromotive intensity at every point where the current is w is wJk. Hen6e by (2) K ct dz where .'^ is as before the potential corresponding to that part of the electromotive intensity which does not depend on induction. This by <56) is (59) K dz dt dt ^ dt Equations (58) and (59) give T-^ /3* dS dT "7 - '"'" U ^ ^ ^ dt Putting dSjdt = — d'^jdz, which is here assumed to be a function of the time only, we obtain 428 MAGNETISM AND ELECTRICITY chap. and therefore - ir/xw = ir/iK ^ + ir^^K^ -r-^ p^ + . If 7 be the total current in the conductor f" y = 2:rl Wpdp Jo where a is the radius of the wire. Writing a for itc^k, the conductance of unit length of the wire, we obtain from (60) -/.y = ^ n-O '' dS dy ( d\dT dt'^^irt-'^^rwdt Equation (61) may also be written d\dT ( d\dT Hence elimination of dTldt between the two last equations gives dS ^ dy l'^('^'^d?) ^^dt) XI GENERAL ELECTROMAGNETIC THEORY 429 But since dSjdt = — d^J^jdz, dS/dt is the part of the electromotive intensity at each point which does not depend on the inductive action ■of the current. We have supposed this to be the same at every point, ■and hence, if H be its line integral along a length I of the conductor, H = IdSjdt. The last equation becomes E ^ ^'%-' (64) 551. If the currents be simple harmonic with respect to the time, they are, to a constant factor, represented by the real part of e™* where n = 2-jrjT (where T is now used to denote the period). We have then to replace in (64) djdf by in, and we obtain H ■■ . l^ (ifi.iTn) If X be small '(i(TiJi,n) l+_^2^2,,2 •G' 1 180 /;1*0-2H* - + %'"'''- 18^'"'''' + 8^''''^'''' Thus we obtain H = r(i + 1 fjiHV I iJ.H*ni 180 B* R^ + 13 ixHhi'^ 8640 iJ* ■)} since Ijcr = B, the resistance for steady currents. This equation is of the form wl^ere B' = B[l + E = R'y + inL'y = B'y + L' -j- m 1 ■i.Wn^ 1 li.H'^n'^ ( + ...) 12 B^ 180 B^ 48 BP' "^ 8640 B^ ■)}J (64') (65) (66) (67) 430 MAGNETISM AND ELECTKICITY CHAP. which are the effective resistance and self-inductance of the wire in consequence of the rapid variation of the current. If the frequency of the alternation be very small, the resistance approximates to B, and the self-inductance to 1{A + Jytt), the values for steady currents. This gives the value of A, which depends on the situation of the return current. If the conductor be enclosed in a perfectly conducting co-axial sheath of internal radius b, A = 2log (hja). With increasing frequency the resistance increases without limit, and the inductance diminishes towards the value lA. This result may be obtained from the analytical theorem of Bessel Functions that when x is very great (ji(x) = e^^^/(2y/Trx) so that in this case ■ 1. Taking the depth of the effective surface stratum of a conductor as that surface stratum which would offer the resistance B' to a steady current it has been found from these results that for copper, lead, and iron, its values for different frequencies are as in the following table — Frequency of Alternation. Copper. Lead. Iron (yi = 300.) 80 120 160 200 •719cm. •587 „ •509 „ •455 „ 2 ■49cm. 2-04 „ 1^76 „ 1-58 „ ■0976cm. •0798 „ •0691 „ •0617 „ For further information the reader may consult Absolute Measure- ments, Vol. II. XI GENERAL ELECTROMAGNETIC THEORY 431) Section III. — Moving Electric Charges. Convection Currents 553. It was discovered by Professor H. A. Eowland in 1876 ^ that an electrified ebonite disk turning about its own axis produced a mag- netic field, deflecting for example a needle placed below or above it. We have thus to consider a moving charge of electricity as a current, and to inquire how the idea is to be introduced into our system. Imagine a point charge of amount q moving with speed v va. & straight line in a uniform dielectric. If it were standing still the total electric induction through a circle, the axis of which is the line of motion, and distance of any element of which from the charge is r, would be 2-!rq I ainede = 2-^q{l - cos 6) where d is the angle r makes with the axis. If then the charge be moving with velocity v towards the circle, and the corresponding change of displacement be supposed for the moment to take place instantaneously throughout the field, the value of the electric induction will be altered per unit time by an amount 2Trqsm 0d6/dt, and dOjdt is obviously vsindlr. Therefore the rate of increase of the total electric induction through the circuit is ivqv sinW/r. This then is the total rate of change of the electric induction through the circuit, and is therefore 47r times the measure of the total displacement current through the circuit per unit of time. Hence if H be the magnetic intensity produced at the circle we must have by the first circuital theorem of Art. 506 . ^ sin 26) H . 27rr sin 6 = iirqv .TT sin C7 ,r.n\ lL = qv -^- (69) But this clearly is the magnetic intensity that would be produced,., by an element of a current of length ds, and carrying a current 7, such that ryds = qv. The direction of the magnetic intensity is related to that of the current in the manner already specified several times above [e.g., see Fig. 149, Art. 546). . • 554 The conclusion is suggested therefore that the moving point- charge " should be regarded as a current- element of moment, so to speak, qv, and that wherever there is m'ovmg electrification, there should' 1 Bcr. d. Berl. AJcad. 1876, p. 211. See also Rowland and Hutchinson, Phil. Mag. 27' (1889). 432 MAGNETISM AND ELECTRICITY chap. be taken at each point components of current pu, pv, pw, where p is the volume density of the electrification at the point. Of course in the actual case the changes of displacement in the field are not propagated with infinite speed, and their finite propagation, which we have found above to exist, must be taken account of. This we shall do in the following articles, following a method due to Oliver Heaviside. The equations of propagation as derived from the circuital equations (4) become with the convection currents expressed ,/3P , dQ , dR , \ /dy dB da dy dB da\' H^^*'^''"' ¥ + ^'^'"'' ¥+'^'^n = W-3^' 3^-8^' ^-^j (70) '^"{dt' di' di)~ '\dy~dz' dz ~ dx' dx~dy) The first of these gives 3 /Si? 3<2\ / 3 3 \ ^Tt\d;j-Tz)^'^Adyf"'-dzn = ^''' which. Up be written for djdt, and V^ for Ijkii, becomes by the second (^^_V^). = 4.gp.-|p.). . . . (71) Similar equations can of course be \vritten down for /3, 7. If F, G, H be the components of vector-potential from which a, /3, 7 are derived, we have with two similar equations. These equations are satisfied by putting Hence we obtain the symbolical solution IT'). V -1 (73) with two others for G, H. 555. Now consider the equation A solution is r XI GENERAL ELECTROMAGNETIC THEORY 43S where r is the distance of the poiat considered from the point at which pu is situated at the instant in question. This may be written <^ = - v'Hi^pu) = ^- r Therefore The symbol of summation is used since the whole distribution of moving electricity is concerned in producing F, G, H, and not merely that at the point considered. These equations enable the magnetic- intensity to be found, and hence the whole problem may be regarded as solved. As a particular example, consider the case of a point-charge of amount q moving along the axis of z with velocity w. Then F=Q, G = -11 ^^-/^?-(i-T^)"- 556. It can be verified at once that v2,.n+2 = (n + 2)(« + 3)r", so- that ..11 + 2 y-2(r») = (n + 2) (w + 3) Thus ,1 r _A r' __, 1 r2»-i V-^- = ^, V-'- = 71'----' V 2m_ _ r 2 !' r 4 !' ' r {2n) ! Again, since the origin is at the moving charge, 8 cz Hence the solution becomes (l 1 w2 3V 1 w* SV \ .„ . It is easy to prove that t-^ L = 12 . 32 I2n - 1)2 where is the angle between the direction of motion and the line (of F P 434 MAGNETISM AND ELECTRICITY length r) drawn from the moving charge to the point considered. Sub- stituting from the last equation in the series for IT, we find that 3 w* •} ./^(i -=-:»»'«)■' (76) 557. It will now be convenient to use cylindrical co-ordinates, and to put a; = 0, and y = }i, the distance of the point considered from the axis of 2, along which the charge is moving. The components of magnetic intensity are given by dx ' = and, as in the suppositions just made, x does not enter in ff, and therefore = 0, the magnetic intensity is at right angles to the planes ■of V and h. Thus we have for its value jj. dh qw sin 6 (77) ;sin2^ 558. Fig. 152 shows the direction of the magnetic intensity at P the point considered. The moving point charge is at 0, and has velocity w in the direction of the arrow. It will be observed that the direc- tion of the magnetic intensity is the same as that of the intensity due to a current flowing in the positive direc- tion of the axis of z, and that in both the lines of force are circles round that jr/2 axis. IP Ya Fig. 152. Further, if w be small in comparison with V, the magnetic intensity is pre- cisely that which would be produced by a current element yds = qw situated at the origin and directed along the axis of z. On ' the other hand, if w= V, the magnetic intensity is zero everywhere except in the equatorial plane {d = irj2) through the moving ■charge, where it is infinite. 559. We have now to find the electric intensities. In going back for these to (70) we must modify the equations to suit the cylindrical ^^ GENERAL ELECTROMAGNETIC THEORY 435 «o-ordmates we have chosen. In the first place, since 7 = ^ = 0, we have r- U ; and since v, to are zero at the point considered, we have Since d/dt= —lodjdz, these equations may be written , 3G da ai? a 8a 8a dz' dz h dh The first of these gives at once « = - 7^ a, or. kw gsine " V^ kQ=^'i^- ^^ .... (78) It will be found that the equation for M is satisfied by the value kR-'-^ .^ .... (79) [I - y:,sm^e) 560. The electric intensity is therefore radial. Its intensity is least along the axis, and greatest in the equatorial plane. For very small values of w, however, the field is simply that of a stationary point- charge at 0, multiplied by a correcting factor of value very nearly unity. When however greater and greater values of tu are considered the electric intensity becomes greater and gi-eater in the direction outwards, and smaller and smaller in the direction along the axis. Finally for w= F", the electric intensity is still radial, but is zero everywhere except at points in the equatorial plane, and there it is infinite. When w>- V the solution does not apply, and the case must be dealt with specially. , ' ■ The values of the electric forces calculated above will be found, on integration over a spherical surface with its centre at 0, to satisfy the condition that the integral of electric induction over the surface should be equal to 4<'7rq. 561. We give here two or three applications of the results found above. These applications are also due to Heaviside. The results are here merely stated with an indication of how they may be obtained. The working out in detail is left to the reader. F F 2 436 MAGNETISM AND ELECTRICITY cha1>. If the charge, instead of being a point-charge, is distributed iiniformh' along a line of finite length, Ij'ing along the axis of « and moving in that dii-ection with the speed of light, and the density of the charge be q per unit of length, the direction of a is everj'where in circles round the axis, and the solution takes the form IqV "A (80) while the electric force is radial and given by the equations /.e = y, /iJ-O (81) The field is entirely contained between the two infinite planes at right angles to the axis, and containing the extremities of the line, and at every point of that space is given by these equations. 562. If a perfectly conducting cylinder be placed round the line coaxially the field will be terminated radially by the cylinder, which will have on its inner surface and between the planes just specified a quantity of electricity uniformly distributed and equal and opposite to that on the line. As the line of electrification moves forward the opposite electrification which terminates the displacement tubes on the cylinder will move forward, keeping pace with it. If the tube and wire are parallel, not coaxial, the solution gives phenomena of just the same character. The distribution of forces in the field is in this latter case of course not the same as before. This is the case of a plane wave moving along a wire surrounded by a conducting tube, on the hypothesis that there is no absorption of energy by the conductors and therefore no distortion of the wave. On the hypothesis stated this is the solution of the problem of a rudimentary telegraph circuit. 563. Again let a plane infinite in both directions be charged to uniform density q, and be made to move at right angles to itself with any speed. The solution shows that the magnetic intensity is zero everywhere whatever the speed, provided the plane be infinite both ways. The electric intensity is ^irq, the electrostatic intensity which the plane would produce if there were no motion. The effects of the convection current are in fact balanced by those of the displacement current which exists at the same place as the former and practically neutralises it, so that there is no true current at all. 564. Next let a plane infinite in both directions move in its own plane with steady velocity w. The magnetic force is tirqw, at every point on one side of the plane, and — ^trqw at every point on the other side, in both cases being parallel to the plane and at right angles to the direction of motion. There is here no displacement current, since the displacement at no point changes. Thus the electric field is simply that of the charged plane. ^t GENERAL ELECTROMAGNETIC THEORY 43T All these results can be deduced from the solution given above for a point charge. Integration for a value of w;< V along a line distribution gives the total effect due to the distribution moving either along the line or transversely to it, or any direction compounded of these. Thence by integrating the effects of parallel narrow strips in a plane distribution results for a plane distribution in motion are obtained. The results enumerated above are the particular solutions for the circumstances specified. Further researches on the subject are contained in Heaviside's Collected Papers and his Medromagnetism. See also a paper by C. F. G. Searle, Phil. Trans. R.S. 187, A. (1896), p. 675. The subject will be further dealt with in Vol. II. Radiation in a Mag^netic Field. The Zeemann Effect. Theory 565. It is important to notice that electromagnetic waves are generated by electric charges in periodic motion. Consider, for sim- plicity, a small point-charge, or an ion (an ultimate portion of matter with which is associated an electric charge) moving with definite period in a circular orbit. A periodic change of its electric and magnetic fields is set up which is propagated with speed l/is/k/i. This motion might be resolved into two rectilinear components of which the equations of motion would be X + n'x = 0, if + n^y = . . . . (82) where, if the force towards the centre of the orbit vary as the displace- ment from that point, n is a constant independent of the radius of the orbit. To these might be superadded a vibration in the direction of the axis with equation z + n^z = (83) From the first two components will emanate waves of polarized light, that set up by the a;-vibration being polarized in a plane through the y axis, and that produced by the ^/-vibration polarized in a plane through the x axis. This circular motion may be imagined as that of, say, a pair of negative ions symmetrically placed in the x, y plane with respect to positive charges situated on the axis of z, supposed to be the axis of rotation. If the distance of the revolving ions from the axis be very small in comparison with their distance from the positive charges, and the electric repulsion between them and any other forces can be neglected in comparison with the force towards the axis due to the attraction exerted by the positive charges, the latter force will be of the proper amount to satisfy equations (82). 566. Now let a magnetic field directed parallel to the axis be impressed on the system. There will act, it is easy to see, on each ion 438 MAGNETISM AND ELECTRICITY chap. a force proportional to the current element to which it is equivalent, that is to the charge and to the velocity. These forces will on each moving ion be in the direction at right angles at once to its motion and to the magnetic field. Thus the equations of motion will become X + Ky + c^x = 0, ij - KX + c^y = Q . . . (84) which are precisely analogous to the equations obtained in Art. 255 above for the small motions of the bob of a gyrostatic pendulum. If H be the intensity of the field, m the effective mass, and e the charge of the ion, it is seen at once that « = /ieH/m. The equations of motion found only hold strictly when the velocity of the ions is small compared with 1/s/kfi (see Convection Currents, Vol. II). If we suppose that the motions are harmonic in period ^irjn and suppose the displacement to be proportional in each case to e™' we have the conditions (c^ - n^)x + iKny = 0, (c^ - rfi)y - ixrix = which give at once the relation C2 - «2 ± KW = (85) Thus we obtain two values of n, and there are two modes of vibration, the period of which, if k be small, are given hy n = c ± ^k. The component of vibration in the direction of the magnetic field is not affected by the field, but remains of its original period. 567. The two linear vibrations of period given by c+|/e are equiva- lent to motion in a circle in one direction, the others of period c—^k to motion in a circle in the opposite direction. Thus the radiation consists of a beam made up of two rays, from each of a complex of such rotating molecules, which, if received in a direction at right angles to the magnetic field, will be seen to be plane polarized in planes parallel to the field, and a third ray, the unmodified component of vibration in the direction of the field, which is plane polarized at right angles to the field. The period of this is midway between those of the modified components. Thus, instead of the single line seen in the spectrum when the magnetic field is zero, a triplet of lines is obtained when a powerful field is applied. The polarization is tested in the usual way by means of a Nicol's prism. If the beam is received in the direction of the axis the ray of mean jjeriod is not perceived since the vibration is end on, and, as explained above, there is no transverse component of vibration along the axis. The beam, however, will be analysed into the two circularly polarized rays specified above. The direction of polarization will indicate, as Zeemann has pointed out, whether the motion is that of a positive or negative charge. ^^ GENERAL ELECTROMAGNETIC THEORY 439' Experimental Verification 568. The result thus theoretically obtained was discovered by Dr. Zeemann at Leyden^ in 1897, and was first theoretically explained by Lorentz. To obtain the effect a sodium flame was placed between the poles of a large electromagnet, and the light emitted was examined by means of a diffraction grating of great power. Many observers have since repeated and confirmed the results of Zeemann, and recently Mr. Thomas Preston of Dublin has succeeded in obtaining excellent photographs of doublets and triplets of lines pro- duced by the action of an intense magnetic field.^ In some cases, in the spectrum of iron for example, the line appeared to be quadrupled ; but this result Mr. Preston sees reason to attribute to a reversal of the central line of the triplet. The doubling of the middle line of a triplet has been observed for sodium, magnesium and cadmium by M. Comu,^ but he is very dis- tinctly of opinion that the doubling is real, and not due to reversal. The two middle lines observed were found to be both polarized at right angles to the magnetic field. • The production of a quadruplet of lines is not indicated by the djmamical theory given above. 569. Dr. Larmor * has recently discussed various interesting questions- concerning the Zeemann effect, amoiig them the phenomena to be expected if the moving point-charges are electrons, consisting of ulti- mate indivisible electric charges, without- inertia except that depending on their charges. He has pointed out that the frequency intervals between the double lines and between the outside lines of triplets should be the same for all lines, and the same also for different spectra. This is a result that will no doubt soon be confirmed or disproved by experiment. The order of magnitude of e/m has been determined by Zeemann, and found to be such that m is about 1/1000 of the mass of the molecule of the radiating matter. 1 Phil Mag., March and July, 1897. " Froc. S. S. for Jan. 27, 1898. 3 L'tdairage tkdrique, Jan. 29, 1898. * tliil. Mag., Dec. 1897. CHAPTER XII THE VOLTAIC CELL Volta's Experiments on Contact Electricity 570. We have seen that a current of electricity is generated 'in a circuit by variation of the number of lines of magnetic induction passing through it, whether this variation is produced by the motion of the circuit in a magnetic field or by creating or annulling tubes of magnetic induction. But it was discovered by Volta about the year 1793 that if a chain of different metals is formed the metals are electrified, apparently at least, to different potentials. For example, when zinc was put into ■contact with copper, the zinc apparently became electrified- positively, the copper negatively ; further, he obtained results of experiments showing that the difference of electric potential between the terminal metals of a series was equal to the difference which existed between these metals when put into direct contact, and therefore that when the terminal metals were the same they were at the same potential. The following table (given by Volta) indicates a series of metals arranged in such order that, if each were put into contact with the one next below it in the list, they would be respectively positively and negatively charged. Zinc. Iron. Lead. Copper. Tin. Silver. 571. It is unnecessary to go into details regarding Volta's method of experimenting. The following is a sketch of his procedure. A disk of one metal was soldered on another so as to form a double plate, for •example, a plate of silver was soldered on one of zinc. Holding the double plate by the zinc, he touched the lower plate of his condensing ■electroscope with the silver, and, while this contact continued, touched the upper plate of the electroscope with his fingers. Breaking the second contact first he then removed the double plate, and lifted the «HAP. XII THE VOLTAIC CELL 441 upper plate of the electroscope. The straws diverged, and were found to be negatively charged. Volta then repeated the experiment, holding the double plate by the silver, and bringing the zinc into indirect contact with the lower plate of the electroscope, by pressing between them a piece of cloth or moist paper. It was found that the straws of the electroscope diverged with positive electricity. Interpretation of Volta' s Results 572. Volta's views as to the meaning of his results may be most conveniently expressed, in the language of modern science, by saying that he regarded the differences of potential produced between dissimilar metals in contact as the cause of the flow of electricity in a closed circuit, consisting of a series of metals with a liquid or liquids interposed between its terminals to complete the chain of contacts. On the other hand the current which in such cases flowed was held by Fabroni, and after him by other experimenters, to be due to the chemical action which, it was soon perceived, took place in the circuit, and chiefly manifested itself at the places of contact of the metals with the liquids. Volta's views, on the other hand, were strongly defended by many of the most eminent physicists who followed him, and a modified voltaic theory, which ascribes the production of the current to the differences of potential produced by contact, and accounts for the energy evolved in the circuit by the chemical changes in the circuit, which have now for the greater part been quantitatively studied with more or less accuracy in different cases, is now held by several eminent authorities. With regard however to the actual amounts of these contact differences, and especially as to whether the contacts of metals with one another or of metals with liquids, are more intimately concerned in the phenomena, is still matter of considerable debate. We give here a brief account of some of the chief investigations, and a short statement of the present position of the question.^ Objections to Volta's Method. His further Experiments 573. It was objected by the chemical theorists to Volta's first experi- ment that the effect produced was due to the contact of damp fingers with the metal, and to the second that the moist cloth or paper only was efficacious.' He accordingly repeated the first experiment in the following manner. A large Leyden jar, of which the inner coating was composed of copper and the outer of tin, had its inner coating connected directly to the upper plate of the condensing electroscope, the outer 3 For further information tlie reader should refer to a Report, On the Seat of the Electro- motive Forces in the Voltaic Cell, by Professor Oliver J. Lodge D.Sc, F.R.S., B A. Report 1884 p. 464. This Report has been of much assistance m the preparation ot the present chapter. 442 MAGNETISM AND ELECTRICITY chap. coating was joined to the lower plate through the zinc-silver double plate. The plates of the electroscope being both composed of copper, the upper plate and the interior coating of the Leyden jar had equal and opposite charges, being of course uncharged before being thus- joined up, but were at the same potential, V, say. On the other hand the difference of potential between the lower plate and the exterior coating depended on the contacts of the different conductors in that part of the arrangement. Thus denoting the difference of potential between copper and silver by Cu/Ag, between copper and zinc by Cu/Zn, and between zinc and tin by Zn/Sn, we get for the difference between the lower plate of the condenser and the exterior coating of the jar the value CujAg + AgjZn + ZnjSn = CulSn = V.^ - V^ . . (1) if Vj, Yj denote the potentials of these plates respectively. Hence if Cj be the capacity of the electroscope-condenser and Cj that of the jar, Q the charge of electricity on the upper plate of the condenser, and therefore — Q that of the inner coating of the jar, we get approximately C,{V- ]\) = Q, C,{r,- 7) = Q . . . . (2) These equations give or Q = — %,- t'M/'^" (3) Thus if C.2 is great in comparison with G^, that is if the Leyden jar employed is very large, Q is simply the charge due to the difference of potential Cu/Sn, that is the difference is simply Cu/Sn. On the other hand if the condenser does not exist in any form G^ is zero, and Q is also zero. Thus Volta's second and improved form of the experiment told nothing about Zn/Cu, but indicated a difference between copper and tin in contact. Result for Chain containing Liquid 574. It was found by Volta himself that the law that the difference of potential between the terminals of a chain of metals is the same as it would be if the terminals were in direct contact, does not hold if there are liquids interposed between the metals of the chain. Thus if the chain consist of copper, zinc, water, copper, the two terminal coppers are not at the same potential. This can be proved easilj' enough by experiments with an electroscope. It is only necessary to connect one ^" THE VOLTAIC CELL 443 copper terminal to the lower plate of the condenser, and through It with the gold leaves of the instrument, and the other copper to the upper plate. When the upper plate is raised the gold leaves diverge with positive electricity if the lower plate is connected with the copper which IS m contact with the water, and with negative electricity if the lower plate is connected with the other terminal. A quadrant electrometer may be used instead of an electroscope, and the difference of potential between the terminals directly measured. In the case supposed a difference of potential of about -75 volt is found. According to the ordinary contact theory this is held to be due to the -contact difference of potential between copper and zinc, while the function of the water is taken to be that of simply bringing the copper and zinc plates immersed in it to the same potential. This view is confirmed by the apparent differences of potential found by experiment and quantitatively measured by a great number of experimenters. We give here a short account of some of the methods used in these researches, and shall then consider the interpretation of the results. Experiments of Eohlrausch 575. A number of valuable measurements were made by Kohlrausch,i Avho arranged parallel plates of the metals to be experimented on, so that they formed a condenser. The method consisted in bringing the plates close together, and connecting them for a moment by a wire, then .separating them, and bringing one in contact with the indicator of a Dellman electrometer, the other with the earth. The deflection of the -electrometer was noted. The experiment was repeated with a Daniell's cell interposed between the plates in the connecting wire, then with the cell reversed. The theory of the experiment is as follows. By the first contact the two plates are brought to different potentials, and are correspondingly •charged. They are then separated to a considerable distance, and the diminished capacity of each plate enables its potential to increase so that the Dellman electrometer can show a measurable deflection. When however the Daniell's cell is interposed, the difference of potential set up •by the contact is increased by the difference which exists between the terminals of the cell, and a comparison of the readings enables the former ■difference to be determined in terms of the latter. Thus calling MjM' the difference of potential between the metals when in direct contact, I) that between the terminals of a Daniell's cell when they are formed of the same metal, and putting a, /3, 7 for the three -deflections, we easily find MIM' = ha, MjM' + D = hp, MjAV' - D = ky 1 Fogg. Ann. Vol. 82 (1851). 444 MAGNETISM AND ELECTRICITY CHAIV account being taken of the signs of a, 0, 7. These give M/M' = ^-a D or MjM' D 30 that there is a controlling equation for each measurement. The Daniell's cell afforded a standard of comparison for the experi- ments in different pairs of metals, which could hardly be carried out' always with the same initial distance between the plates. The results obtained indicated the ratios ZnlPt 4-49 ZnlGu 3-99 D 7-48' D ■ 7-07 and therefore ZnjPt 106-4 ZthjCu 100 These it will be seen are considerably lower values than were obtained later for the contact differences of potential of the same metals. Haukel's Experiments 576. The next experiments of note are those of HankeP on the contact differences of potential between metals, and between metals and liquids. The apparatus is shown in Fig. 153. L is the liquid contained in a Fig. 153. funnel connected by a tube with the vessel B, in which dips a strip of a metal M. 4- copper plate G rests a little way above and parallel to the liquid surface, and can be put into contact with ilf by a platinum wire^, and with an electrometer by another platinum wire p'. A third wire p" kept up a permanent connection between M and the earth. The method of experimenting was as follows : — (1) The funnel being full of liquid,|the plates C and Jf were brought into contact by the wire p. The contact was then broken, and the plate G raised so as to come into contact with p'. This gave a deflection a, pro- 1 Fogg. Ann. 115, 126, 131. ^^^ THE VOLTAIC CELL 445, portional to the contact differences in the chain of substances. Thus. K being a constant Cu/Pt + PtlM + MIL = ka or Cu/M + MIL = ka. by Volta's law. (2) The funnel is emptied, and a plate of the metal 31 laid on its^ mouth. This plate is brought into contact with G and with the earth by platinum wires. Then the contact between C and the plate of Jf IS broken, and C lifted and brought into contact with «' as before m ving- a deflection /3. This gives '55 CulM = Ic^. is replaced I ives ZnjM = hy (3) The plate of copper C is replaced by a plate of zinc, and the last experiment repeated. This gives where 7 is the deflection. From these equations eliminating h we obtain MIL ^^SilcujZn. p - y Also they give the relations CulM _ p ZnIM ~ y CulM = —^ CulZn, ZnIM = —5^ CulZn. P - 7 P - y Hankel's results showed very different values for the contact differ- ence between a metal and water according as the test was made immediately upon immersion or some minutes afterwards. Several; metals, such as copper, platinum, silver, gold, iron, and tin, which showed' a positive potential relatively to water, became just as strongly negative after immersion for from 10 to 30 minutes. He also found that the contact differences of metals varied with the state of their surfaces as to polish, and depended also to some extent uponi whether the surfaces had, after having been polished, been exposed; to the air. Experiments confirmatory of Hankel's results were made a little later- by Gerland in 1868 and 1869 {Fogg. Ami. 133, 137). Lord Kelvin's Experiments 577. The next investigation of importance is that of Lord Kelvin,, made about 1862. A charged metal arm was suspended horizontally, by a torsion thread, over the line of separation between two semicircular plates, one of copper, the other of zinc. The arm being positively,. 446 MAGNETISM AND ELECTEICITY CHAP. charged, was deflected towards the copper when the metals were put in contact, showing that the zinc was positively electrified. When, however, a drop of water was made to bridge across the gap between the semicircles they were found to be at the same potential. This seemed to prove conclusively that the water had merely the effect of equalising the potentials of the two metals, and led Lord Kelvin to conclude " that two metals dipped into one electrolj^tic liquid will (when polarization is ■done away with) be at the same potential." By his invention also of the quadrant electrometer, Lord Kelvin put into the hands of experimentalists an instrument immensely superior to any that had ever before been at their disposal, and thus greatly facilitated further investigation. This instrument he has himself applied to the measurement of contact differences, by the method of balancing the contact difference by a known fraction of the electro- motive force of a Daniell's cell, from two points in a resistance connecting the terminals of the cell. Lord Kelvin's Copper and Zinc Electric Machine 578. On the principle of his water-dropping electric machine, Lord Kelvin at this time constructed a machine which acted by aid of the •difference of potential existing between copper and zinc in contact. Fig. 154 shows the arrangement. A copper funnel surrounded by a zinc cylinder contains a quantity of copper filings, which are allowed to trickle out through the mouth, and are received by a copper vessel below. This vessel rapidly acquires a nega- tive charge. Consider a particle of copper j ust leaving the funnel, and therefore breaking away from the mass of filings above it. It is in the middle of a space surrounded by the zinc cylinder, which is positively electrified relative to the copper, and hence, since the particle has the potential of copper, it must be negatively electrified. It falls, carry- ing its negative charge with it, and being received in the interior of the vessel below gives up to the latter all its charge. Thus the negative potential of the receiver con- tinually increases. By connecting the receiver and funnel by a copper vnre, a current of negative electricity may be made to flow round the circuit,. from the receiver to the funnel through the wire, and by convection along the stream of filings. With regard to the source of the energy in this arrangement it is sufficient to notice that the particles fall against electric repulsion. Work is therefore done against electric forces by gravity, and the particles reach the receiver with smaller velocities than they would otherwise have, and the difference of energies is stored up in the electric distri- bution on the receiver. Fig. 154. XII THE VOLTAIC CELL 447 Lord Kelvin's Induction Electric Machine, Founded on Contact Electrical Action On the same principle Lord Kelvin constructed a revolving induction electric machine which is shown in Fig. 155. There are two connected brass inductors T of the shape shown, one of which is lined with one metal, the other with the other metal. A non-conducting wheel with carrier studs which are touched by springs A A' at opposite ends of a diameter is rotated within the inductors, and the springs become oppositely charged. The disk is kept turning and the springs are con- nected to the terminals of a quadrant electrometer which measures the Fig. 155. difference of potential produced. The absolute value of this is obtained by disconnecting the inductors from one another and removing their , linings so as to make them of the same metal, and then connecting them with the terminals of a Daniell's cell and again rotating the carrier wheel, while the difference of potential of the springs is tested by means of the electrometer. Thus the contact difference between zinc and ■copper or between any pair of metals could be evaluated. Of course by keeping the metals in contact and applying the requisite fraction of the electromotive force of a Daniell's cell the method could be made a null one. Experiments of Ayrton and Perry. Clifton's Experiments 579. A very large series of determinations of contact differences of potential was carried out by Ayrton and Perry in 1876.^ A diagram of their apparatus is shown in Fig. 156. On a platform A£ were placed in contact the substances to be tested, for example, a metal and s, liquid as shown at P and Z. This platform could be turned round a pivot Jf through 180° on wheels running on a railway R Two well insulated gilded plates 3, 4 were attached to an upper bar, which could be raised or lowered by a parallel ruler arrangement attached to the top of the case. In an experiment these plates were lowered close to the substances to be tested, and were put in contact for a moment by a wire, then raised and connected to a quadrant electrometer. The platform A B was now turned through 180° on its railway, the plates brought down, 1 Proc. B. S. 1878, and PMl. Travs. It. S. 1880. 448 MAGNBTISM AND ELECTRICITY CHAP. connected for a moment, raised and connected to the electrometer once more. The deflections were in opposite directions on the scale, and the double deflection could be taken as proportional to the difference of potential between the plates P, L. To evahiate this difference of potential the plates P, L were replaced by brass plates, which were brought to various differences of potential by appljdng to them certain fractions of the electromotive force of a Daniell's cell produced by placing the terminals so as to include different parts of the total resistance in the circuit. Thus the deflections on the electrometer scale were evaluated, and the contact differences Fig. 156. measured in volts. A table of results abridged from Everett's Units and Physical Constants, for which the table was specially prepared by the experimenters, is given at p. 457, as an appendix. The results of these experiments were not published until 1878, owing to some delay in the communication of the paper to the Royal Societ}', and in the meantime a series of careful experiments had been made by Professor Clifton of Oxford, who measured with great care the contact differences of potential for substances ordinarily used in batteries. Comparison of Besults 580. Both Clifton and Ayrton and Perry found that the electro- motive forces of different cells could be obtained by simply summing all the differences of potential at the surfaces of contact of dissimilar XTT THE VOLTAIC CELL 449 substances in the circuit. Thus the table given below (extracted also n?nL. ""f ^'^ ^'''^'\ ^r^^ *1^^ electromotive forces as calculated and and Grove ™°''^ well-known cells, viz., those of Daniel) DANIELL'S CELL. Difference of buBSTANCEs IN CONTACT. Potential Copper and saturated copper sulphate + "oTO Saturated copper sulphate and saturated zinc sulphate - -095 saturated zinc sulphate and zinc . + -430 Zinc and copper 0'750 Total Observed difEerence . + 1-155 + MO GROVE'S CELL. Substances in Contact. Copper and platinum . Platinum and strong nitric acid Strong nitric acid and very weak sulphuric acid Very weak sulphuric acid and zinc . . . . Zinc and copper . . . . . . Electromotive force observed on open circuit . Difference of Potential (Volts). + 0-238 + 0-672 + 0-078 + 0-241 + 0-750 1-979 1-9 The agreement of the numbers is very satisfactorj^, and other examples of the same kind strengthen the conclusion that the electro- motive force of any cell can be built up in this way. It is to be observed, however, that this gives only the electromotive force with the plates in the condition in which they were when experimented on, and with no current flowing through the battery. When a current is made to flow the electromotive force in many cases, and to a small extent even in the so-called constant cells, falls off in amount owing to the deposition on the plates of the gaseous products of the decomposition of the liquids. Pellat's Experiments 582. In an elaborate set of experiments made by Pellat,i and published in 1881, a plan of compensation, also previously devised by Lord Kelvin and used in some unpublished researches, was employed to convert the condenser method into a null method, requiring therefore only a very sensitive electrometer arranged to show a deflection for the 1 Journ. de Phys. xvi. (1888). G G 450 MAGNETISM AND ELECTRICITY CHAP. smallest possible difference of potential. This plan consisted in applying to the plates an equal and opposite difference of potential to that produced by contact, and so annulling their electrification. This was obtained by using the arrangement shown in Fig. 157. A couple of Daniell's cells B are joined up in series with a rheostat B, and a resistance slide S, S'. The sliding piece, p, is connected with the earth Earth Earth Fig. 157. and with a wire by which contact can be made with one of the plates at M. The same plate (P) is connected with the indicator / of the electrometer, and the extremity S of the slide to the other plate P'. B' is a battery employed to charge the plates of the Bohnenberger electrometer which Pellat used. The method of experimenting consisted in raising the plate P' (the contact between P and M having been previously broken), and observing the deflection produced by the rise of the potential of P. If no deflection took place then P was unelectrified, and the compensation had been complete. The following are a few of Pellat's results — Metal in Contact with Standard Clean, almost unscratched Surface strongly scratched Gold. surface. with emery. Zinc .... . . i '85 1-08 Lead . 1 -70 •77 Tin. . 1 -60 ■73 Nickel . . 1 -38 •45 Bismuth •36 •48 Iron . . . . -29 ■38 Brass . . . 1 -29 ■37 Copper . . -14 ■22 Platinum ... --03 + ■06 Gold . ... --04 + ■07 Silver . -•06 + ■04 ^" THE VOLTAIC CELL 451 Experiments on Metals Immersed in Different Gases 583. Pellat has also endeavoured to measure contact differences of metals immersed m different gases, and so far as his experiments go they corroborate a conclusion previously arrived at by Pfaff that the contact electric difference is practically unaffected by the gas by which the metals are surrounded, so long 'as no visible chemical action, such as tarmshing of the surfaces by oxidation, takes place. Practically the same conclusion has been arrived at by Lord Kelvin and by Von Zahn and others. On the other hand, J. Brown of Belfast has found that copper is positive with respect to iron in sulphuretted hydrogen, while, as stated above, it is negative in air. Later Views on Contact Electricity. Differences of Potential Inferred from Flow of Energy 584. It is impossible here to give a full account of experiments on this subject, and we shall therefore now try to state shortly theoretical views which at the present time appear to find a certain amount of favour. On the experimental results stated above has been built a theory that the seat of the electromotive force of a voltaic cell is at the junctions of the dissimilar substances in contact, and that the differ- ences of potential measured in the manner described are actually contact differences between the metals, which added together, with their proper signs, give the electromotive force of the cell. The electro- motive force being thus accounted for, the energy consumed by the cell is further seen to be furnished by the chemical changes within the cell which accompany the flow of the current, and so the contact and chemical theories, which once were in severe conflict, are in a sense reconciled. On the other hand it is held by several authorities that the contact difference of potential between two metals is only apparent, being the difference between the potential in the air near one metal and that in the air near the other metal, while the potentials of the metals themselves are one and the same. It is clear that all the experiments described above are consistent with this view. The films of air close to the plates adhere to them when they are separated, and the potentials are altered just as if each had itself a real difference of potential and a charge on its surface. Again the view holds for Lord Kelvin's copper and zinc ring- experiment and for a quadrant electrometer, with zinc and copper quadrants. The needle is acted on by the charged air films just as if these were real charges of the plates. According to this view the different effects found by Brown when sulphur became the active substance of the medium are to be regarded as also only apparent, and that the metals themselves are to be taken as really at the same potential. gg2 452 MAGNETISM AND ELECTRICITY chap. Further it can be shown that though the actual values of the individual differences of potential between the metals may still be unknown, yet their sum for a non-metallic chain of substances must be the same as that obtained from measurements of the kind which have been described. 585. The view has been put forward with great skill and force by Dr. Lodge that the difference of potential between two dissimilar substances ought to be measured by the work done there per unit time on unit current flowing from one substance to the other. The total work spent per unit time in causing a current to flow in a circuit is, as we have seen E'y, (where E is the electromotive force and 7 the current) in the case in which the current is produced in moving a conductor across the lines of force of a magnetic field. This is given out again in the circuit either in heat or in some other form of energy. In all cases the rate at which energy is evolved at any part of the circuit, apart from places at which chemical changes take place, is equal to the product of the current into the difference of potential down which the current flows. Energy value of Electromotive Force of a Cell 586. Lord Kelvin pointed out in 1851 that the only source of energy in the circuit is the chemical potential energy used up, and he assumed the time-rate of consumption of this to be equal to Ey, the rate at which work is given out in the circuit. For, as he showed, any voltaic arrangement can be replaced by a magnetic electric generator giving the same electromotive force, and the rate at which energy is given out in the working part of the circuit can be made exactly the same as before. This theory, it is to be observed, is not quite exact, as the value of the electromotive force requires correction for thermo- dynamic reasons. The whole subject will be discussed in Vol. II. ; but it may be stated here, and it is very easy indeed to prove the particular result, that, if E be the electromotive force of a cell, which varies in electromotive force with temperature only, and 'Z{Jde) be the dynamical value of the chemical changes which take place in the cell when a unit of electricity is passed through it [see Art. 589, (7)] E - t~=%{je€), ot where t is the absolute temperature. (See also Art. 601 below.) Hence it is only going a little further to say that wherever the current flows up a difference of potential whether in a voltaic cell or a voltameter at the junction of dissimilar substances, or in a wire moving across lines of force in a magnetic field, there energy is absorbed, and that the energy absorbed per unit current per second measures the difference of potential. It is in fact extending the view already well ^" THE VOLTAIC CELL 453 established for the whole of the chemical action in a cell or voltameter to every element of the difference of potential, and associating with that element, as regards both its amount and its locality, the energy change which takes place in any time. The actual potential difference, at any rate when the cell is generating a current, can be found approximately when the heats of combination are known. [A fuller discussion of the calculation of electromotive forces from such data, with an examination of the limitations and corrections to which the theory is subject, will be found in Vol. II.] Explanation of Volta Effects as Air Metal— Metal-Air Differences of Potential. Thermoelectric Measure of Difference of Potential 587. The differences of potential to which this theory leads are very much smaller in many cases than those obtained from direct observa- tion of voltaic contact effects. The latter are however explicable by regarding the voltaic difference of potential AjB between two metals A and B as really air jA + A/B+B/ air, so that the true A/B may be really very different from this sum, which is the observed or apparent A/B. But it can be shown at once that this in no way interferes with the method of finding the electromotive force of a cell by adding up the con- tact differences in the circuit. Denote the apparent contact-difference between A and B by A'jB', and let there be substances A, B, 0, . . . G- an-anged in order in the circuit. Then we have A'jB' = AirjA + AJ£ + Bj&xr B'jC = air/5 + BjC + C/air F'jG' = &\TJF'+ FjG + e/air G'jA' = air/y + GjA + Ajaiv. Adding up we find A'jB' -t- B'jG' + + F'jG' + G'A' = A/B + BjC + ....+ F/G + G/A since &\v/A + B/air + air/5 -H . . . . -f- ^/air = 0. Thus the electromotive force has the same value in whatever way it is obtained. It is to be noticed that any one of the electromotive forces of contact between air and the substances in contact may be very great; there is no known method of finding its value. Energy Criterion of Existence of an Electromotive Force 588 It must be admitted that the only electromotive force at the iunction of a pair of dissimilar substances which has been determmed without ambiguity is that measured by the amount of heat absorbed or evolved at the junction when unit current flows across it ; tha.t.is, it 454 MAGNETISM AND ELECTRICITY chap. is equal to the coefficient 11 of the Peltier effect, as explained in Art. 596 below. This was the view held by Clerk Maxwell, and set forth by him in his treatise on Electricity and Magnetism} This determination of the amount and locality of an electromotive force, however, depends on taking absorption or evolution of energy as the test of its existence in the manner just explained, and must stand or fall with the validity of that criterion. But this is the method adopted in all our calculations regarding the supply of energy to an electric system from without, and the evolution of energy in the system itself Thus if we take as the system considered a conductor in which heat is being generated by a current, the rate at which heat is generated by unit current is the measure of the electromotive force which must be impressed on the conductor by the part of the electric system outside the conductor that the current may exist. Again, when a linear con- ductor is moved at right angles to itself and to the lines of induction of a magnetic field with velocity v, the dynamical force which must be applied to each element of it of length ds is precisely B-yt^s, where B is the magnetic induction and 7 the current, and therefore the rate at which work is spent in the element is Syvds. The rate at which work is done upon the conductor is then Svds, which we have seen in Chap X. above is the measure of the electromotive force in the element. The question therefore resolves itself into whether this process can be accurately applied experimentally to each part of the heterogeneous circuit of a voltaic cell. If it can the question would seem to be settled. All that remains is to work the matter fully out by experiment, by making local energy determinations all round a heterogeneous circuit, a work undoubtedly of very great difficulty, though not perhaps in simple cases impossible. As has just been shown, experiments on contact electromotive force cannot be regarded as conclusive, and further investigation, by the method of the absorption and evolution of energy at different parts of the circuit must be awaited, before the conclusions of any theory are held to be definitely proved. According to Poynting's theory energy is thrown out into the medium wherever in the circuit the current flows up a slope of potential under the action of an impressed electromotive force, and flows into the circuit fi-om the medium where the current flows down a slope of potential. Thus, in the original paper in which this theory is set forth, a diagram of the flow of energy is given, according to which the greater part of the flow of energy to the medium takes place at the surface of f ontact of the zinc and acid. Dr. Lodge,^ however, has pointed out that in ordinary arrangements the stream-lines of energy may be crowded together along the zinc, and then pass out into the medium at the copper-zinc junction, as apparently they do on the ordinary voltaic theory in which the electromotive force has its seat at the surface of contact of the two metals. The view that the energy evolved at different parts of the circuit cannot be transmitted along the wire 1 Vol. I. p. 369 (3rd edition). Electrician, April 26, 1879. ^ Phil. Hag., June 1885. ^" THE VOLTAIC CELL 465 without its passage being accompanied by- some physical manifestation of Its presence, but that it travels through the medium guided by the conductor, has been sufficiently emphasised above. Summary of Results of Later Theory 589. The following resume of statements which are consistent with this theory, and of others which are not, is taken from Dr. Lodge's Report. (1) Two metals in contact ordinarily acquire opposite charges ; for instance clean zinc receives a positive charge by contact with copper, of such a magnitude as would be otherwise produced under the same circumstances by an E.M.F. of about -8 volt. (2) This apparent contact E.M.F. or " volta force " is independent of all other metallic contacts wheresoever arranged ; hence the metals can be arranged in a numerical series such that the " contact force " of any two is equal to the difference of the numbers attached to them, whether the contact be direct or through intermediate metals. But whether this series changes when the atmosphere, or medium surrounding the metal, changes is an open question. It certainly changes when the free metallic surfaces are in the slightest degree oxidised or otherwise dirty. And in general this " volta force " is very dependent on all non-metallic contacts. (3) In a closed chain of any substance whatever, the resultant E.M.F. is the algebraic sum of the volta forces measured electrostatically in air for every junction in the chain, neglecting magnetic or impressed E.M.F. (4) The E.M.F. in any closed circuit is equal to the energy conferred on unit electricity as it flows round it. [In the next four statements magnetic and other impressed E.M.F. is to be neglected.] (5) At the junction of two metals any energy conferred on, or with- drawn from, the current must be in the form of heat. At the junction of any substance with an electrolyte, energy may be conveyed to or from the current at the expense of chemical action as well as of heat. (6) In a metallic circuit of uniform temperature the sum of the E M.F.'s is zero by the second law of thermodynamics (see Vol. II.) ; if the circuit is partly electrolytic, the sura of the E.M.F.'s is equal to the sum of the energies of chemical action going on per unit current per (7) In any closed conducting circuit the total intrinsic E.M.F. is equal to the dynamical value of the sum of the chemical actions going on per unit electricity conveyed t(J0e) where e is the quantity of matter affected by chemical change in the passage of unit current for unit time the heat evolved or absorbed in the change, and J the dynami- cal equivalent of a unit of heat), diminished by the energy expended m algebraically generating reversible heat. 456 MAGNETISM AND ELECTEICITV chap, xir (8) The locality of any E.M.F. may be detected, and its amount measured, by observing the reversible heat or other form of energ}- there produced or absorbed per unit current per second. The following statements held to be true b}^ many contact theorists are inconsistent \vith the theory. (1) Two metals in air or water or dilute acid, but not in direct con- tact, are practically at the same potential. (2) Two metals in contact are at seriously different potentials, [i.e. differences of potential greater than such milli-volts as are con- cerned in thermoelectricity]. (3) The contact force between a metal and a dielectric, or between a metal and an electrolyte such as water and dilute acid is small. It is to be remembered that authorities are still divided in opinion on this subject, and that what has been given above is to be regarded only as an attempt to state the opposing views. The discussion will be resumed in Vol. II., where some account will be given of the behaviour of different cells, and of the thermodynamics of the subject. APPENDIX TO CHAPTER XII nnivT^A r.m ^ ^^I'i^ged from Everett's Units and Physical Constants. OUJNTACT DIFFERENCES OF POTENTIAL IN VOLTS (A YRTON AND PERRY j Solids avith Solids in Air. Carbon Copper \ Iron '" Lead Platinum Tin Zinc " Amalgamated Zinc Brass Corper. Iron Lead. Plati- num Tin. Zinc. Amalga- mated Zinc. Brass. ■svo ■486* •868 •lis :795 1^096 1^208* •414* •146 •642 -•288 ■466 ■760 ■894 •087 - -146 •401* - •869 ■813* ■600* ■744* -•064 -■542 - •401* -•771 -■099 •210 •367* -•472 •288 •S69 •771 •ci90 ■981 1^126* •287 -•466 - •SIS* •09fl -•600 ■281 •468 -•872 -•760 -•600* -•210 -■981 -■2S1 •144 -•679 -■894 - •744* - •S67* -1^125* -•463 - -lii -■822* -•087 •064 ■472 -■287 ■872 ■679 •822 n<:t»ri=l.- w„v?^L ■ ^^°''j'"''®.,''*v''^ *"™ °f experimenting -was about 18° C. The numbers without an K,™'fr„^f obtained directly by experiment, those with an asterisk by calculation, using the' well- eiecrromotlv"£" " "" compound circuit ot metals, all at the same temperature, there is no Tlie numbers in a vertical column below the name of a substance are the diflEerences of potential in TOlts between that substance and the substance in the same horizontal row as the number, the two substances bemg m contact. Thus, lead Is positive to copper, the electromotive force of contact being •642 volts. i i ' t, The metals were those of commerce, and therefore only commercially pure. Solids -with Liquid.? in Air. Carbon. Copper Iron. Mercury Distilled water Sea salt solution, sp. j;.) l^lSat 20^6°C )' Sal-ammoniac saturated\ atl6^5°C I Concentrated sulphuric^ acid f 1 distilled water, 5 strong sulphuric acid byweight ■092 '/■Ol to I -1" iW-269,t0| (■•85 86 to| ■56 / -•896 •502 ■148 -■652 Lead. Plati- num. Tin. Zinc. Amalga- mated Zinc. ■166 ■171 (■286 to) I -345 / •177 1 -■106 to + -1X } ■lOO -:267 -■866 -■334 -■565 -■189 ■057 -■364 -■63" /•728toi I 1^252 1 i^etoi^s ■848 -■120 -■260 •281 ■486 Solids -HriTii Liquids and Liquids with Liquids in Aie Amalga- Mer- Dis- Copper sulphate Zinc sulphate Copper. Zinc. mated tiUed solution satu- solution, sp. g. Zinc. water. rated at 16° C. 1^125atl6^9°C. Copper sulphate solu-) tion, sp. g. 1^087 at} ■103 ■090 W6-C. / Copper sulphate satu-l rated at 15° C t ■070 -■048 Zinc sulphate solution,! sp. g. 1^125 at 16^9° C. 1 •238 Zino sulphate solution,! saturated at 16^3° C. .../ -■430 -■284 -■200 -■096 1 distilled water mixed\ with 8 zino sulphate > -■444 saturated ,' 20 distilled water, 1 ■844 strong sulphufio acid 5 distilled water, 1) strong sulphuric acid / -■429 Distilled water with trace) of .sulphuric acid / Mercurous a ulphate paste ■ -■241 •475 The average temperature at the time of experimenting was about 16° C. All the liquids and salts employed were chemically pure ; the solids, however, were only commercially pure. CHAPTER XIII THERMOELECTRICITY Elementary Fhenomena 590. It was discovered by Seebeck in 1822 that if a circuit be formed of two different metals, and the junctions be at different temperatures, a current flows round the circuit. This is illustrated by. heating with a flame the junction of a bar of antimony with one of bismuth, as shown in Fig. 158. To fix the ideas let the circuit be composed of a wire of bismuth and ; a ^vire of antimony. It is found that no current flows if the j unctions of the bismuth and antimony are at the same temperature. If, how- 1 ever, one of the junctions is warmed a current flows round the circuit; and passes across the hot junction from bismuth to antimony. Cooling the same junction below the teriiperature ' of the other also produces a current, but in the opposite direction. The interposition of solder, or even of a chain of one or more different metals in a junction {e.g. a wire, say, of copper joining, as in Fig. 158, the remote extremities of the bars of the two metals) will not affect the electromotive force in the circuit, if the junctions of the chain are all at one temperature. The effect can be much increased by using a chain of two metals arranged alternately, and heating the alternate junctions, as shown in Fig. 159, which is the arrangement of course of the thermopile. It will be seen later that these statements hold only for ordinary temperatures, and are not true in all circumstances. For example, if one junction be below, the other above a certain temperature, dependiiig on the metals employed, no current, or a reverse current, may be produced by making the higher temperature sufficiently high. The intermediate temperature thus referred to will be seen to be such that if the temperature of one junction be lower by a small difference, i and the other higher by the same small difference, no current flows, and is therefore called the neutral temperature for the metals con- cerned. CHAP. XIII THEEMOELECTEICITT 459 591. The elementary phenomena may be conveniently studied by making a circuit of two wires, say of copper and iron, by twisting or soldering the wires together at the junctions, and inserting a galvan- ometer in one of them, say the copper. One junction may be immersed m a beaker of water so as to keep its temperature constant, the other may be gradually heated by a spirit lamp or Bunsen flame (if it is not soldered). As the junction is heated the current indicated by the galvanometer will gradually increase, reach a maximum, diminish to Fig. 158. Fig. 159. zero, and finally be reversed.' The temperature of the hot junction when the current is a maximum is the neutral temperature, that at which the current changes sign is the temperature of inversion for the metals employed which correspond to the temperature of the colder junction. Thermoelectric Inversion 592. The phenomenon of inversion was first observed in 1823 by Cumming, who found that the order of the metals in a thermoelectric .series was not the same at all temperatures. As we have seen in considering voltaic action no current is pro- duced in a purely metallic circuit if the conductors are all at one temperature. To this may be added here the fact that in a homo- geneous circuit or homogeneous part of a circuit no electromotive force is produced by inequalities of temperature provided that in the case of part of a circuit the extremities of the conductor are at the same temperature. Thermoelectric series were formed by Seebeck and others, according to which any substance in the series if made into a circuit with any metal following it in the series, gives a current from the former to the 4G0 MAGXETISil AND ELECTKICITV cHaP. latter across the warmer junction. The following table gives a few of the substances in one of these scries : — Bismuth Copper Siher Nickel Mercury Zinc Cobalt Lead Iron Palladium Gold Antimony. Thermoelectric Power 593. It was found experimentalh' by E. Becquerel that the total electromotive force for any two temperatures of the junctions is the sum of the electromotive forces of the coujile for any differences of temperature making up (not b)^ mere addition, but bj^ actual position in the temperature scale) the range of temperature between the junctions. Let the temperature of one junction be t — ^dt, that of the other t+^dt, where dt is a small difference of temperature, and let the electromotive force of the couple for these two temperatures be measured. The ratio of this electromotive force to the difference of temperatures dt is called the thermoelectric power of the couple at temperature t. We shall here denote it by F. By Becquerel's experimental result the electromotive force U, when the temperature of one junction is t^ and that of the other i, is given t E = [pdt (1) where P is the thermoelectric power at the intermediate temperature ;it which any elementary difference of temperature dt is taken. Thus we see that -f « 594. The thermoelectric power of a pair of metals can easily be found by making up a circuit of two wires of the metals in series with a high resistance galvanometer, and observing the current when one junction is kept at a constant low temperature, and the temperature of the other is varied by small steps until any required range has been covered. Any required temperature of the junctions can be produced by immersing them in baths of water or oil, of which the temperatures can be observed by means of thermometers. The current through the galvanometer obtained in absolute units and multiplied by the resist- jmce in circuit gives the electromotive force for each observation. The temperatures of the hot junction are now laid off as abscissjE, and the corresponding electromotive forces as ordinates of a curve. If ^in THERMOELECTRICITY 46 tangents then are drawn to the curves at different points, the inclina- tions of these tangents to the axis of abscissae will measure the thermo- electric powers of the couple at the temperatures corresponding to the ordinate s drawn to the same points. If now a curve be plotted with thermoelectric powers of the couple as ordinates, and temperatures as abscissse, the area of this curve taken between the axis of abscissae and auy pair of ordinates will measure the electromotive force of the couple when the junctions are at the tempera- tures corresponding to these ordinates. If one of these ordinates be at the point corresponding to the fixed temperature of the cold junction in the experiments referred to above, the areas of the curve of thermo- electric powers will for different positions of the other ordinate, of ■course be the ordinates of the curve of total electromotive force from which the second curve was derived. The curve of thermoelectric Fig, 160. powers, drawn as described, is generally found to be a straight line, as represented in Fig. 160. Here OA denotes t^, the lower of the two ■extreme temperatures, 0£ the. higher iS, while Oilf represents the neutral temperature. The areas AMF and £MQ are to be taken with opposite signs, and when they are equal the electromotive force in the current is zero. If OB = t^, the temperature of inversion, and the curve of thermo- ■electric power be either as here represented a straight line, or be skew- symmetrical with respect to M, it is clear that T — t^ = t^- T, or T = '-J-^« (3) In such cases the neutral temperature T is the mean of t^ and t^. A very important experimental result with respect to thermoelectric power is the following: If at any temperature P^n be the thermoelectric wwer of a couple composed of two metals A and B, and Fbc be tue thermoelectric power of B and another C, then at the same tem- P^'^^*"^' P.c=P.s^Psc. . ^ . ■ . ■ W In reckoning the sum here it is to be observed that the sign of the thermoelectric power is to be observed, and taken account of Gare is 462 MAGNETISM AND ELECTRICITY CHAP. to be taken to place the metals in the proper order, so that with respect to the hot and cold junctions the positions of AG, AB, BG may be the same in the verifying experiments. Hence if the curve of thermoelectriq powers of the two metals AB be represented by the line QB, Fig. 161, and that of ^C by SR, the thermoelectric powers of BG will be repre- sented by the difference of the ordinates of these two curves, and their neutral temperature will be that corresponding to R, namely the temperature represented by OM. A table of thermoelectric powers of different substances with respect to lead is given at the end of this chapter. Total Electromotive Force 595. It follows from Fig. 161 that the curve drawn having tempera- tures as abscissae and total electromotive forces as ordinates is a Fig. ]61. parabola, having its maximum ordinate at M and cutting the axes at A and B. For the area between PQ (Fig. 160) the axis, and the ordi- nates corresponding to temperatures t^ and t, is C (IP where Pq, P are the thermoelectric powers at t^ and t respectively. But, by Fig. 160, P=Po{T-t)j(T~to) and therefore H ^"{'- y- t (i q} (5) which is the equation of a parabola of which t is an abscissa and B the corresponding ordinate. The observation that the curve of total electro- motive force is generally a parabola was first made by Gaugain. The curve of total electromotive force between any other initial temperature t'^ and the corresponding temperature of inversion t\ is ^'"i THERMOELECTRICITY 46» obtained from the general curve by drawing a dotted line parallel to the axis of abscissae through the top' of the ordinate for the temperature «'o- The part of the curve lying above the line so drawn is the curve of total electromotive force for the range of temperatures stated. Peltier Effect 596. In 1834 Peltier discovered that if a current of electricity was sent from a battery through a circuit of two metals initially at the same (ordinary) temperature one junction is cooled and the other heated. The difference of temperature thus set up would by itself send a current in the opposite direction to that producing it. This heating and cooling effect is known as the Peltier thermal effect. It is reversible, inasmuch as it depends on the direction of the current whether the effect is a heating or a cooling. Since one junction is heated and the other cooled, heat disappears at one junction and is evolved at the other, and as a result an electro- motive force (called the Peltier electromotive force) opposing the current is developed. These facts may be experimentally verified by joining up a battery with a galvanometer G^ and wires of iron and copper as shown in the diagram Fig. 162. By means of a rocker the battery can be at any time cut out of the circuit and the galva- nometer G^ inserted, by withdra\ving the contact piece ab, and inserting cd. It will then be found that a current through the galvanometer G^ opposed to the steady current which flowed before will now be set up, and will gradually die away as Fio. 162. the difference of temperature dis- appears. It is found by experiment that these heating and cooling effects are at any one temperature directly proportional to the current. Thus if a wire of resistance i2 contain a junction the rate at which heat is deve- loped in it by a current -y is B^f + U^f where 11 is a quantity which may be positive or negative according to the direction of the current and the nature of the metals in contact at the junction. Hence if between the extremities of the portion of the circuit considered there be produced by a battery or otherwise an applied difference of potential V, the electrical activity in this part of the circuit is F7 and we have Vy = Ry'^ + ny or the electromotive force actually available for working through the resistance i? is V - n = Ry (6) 464 MAGNETISM AND ELECTRICITY chap. n thus assists or opposes the applied electromotive force according as it is negative or positive, that is, according as heat is absorbed or given •out at the junction. Thus in a circuit of two metals there are in general two Peltier <'lectromotive forces acting at the junctions. In an ordinary circuit {generally throughout nearly at one temperature) in which a current is kept flowing by a battery these do not cause any perceptible alteration of the current as they are equal and opposite, unless the current is so \ery great as to cause serious heating or cooling at the junctions. The Peltier effect may obviously be evaluated by careful calori- metrical experiments on the heat evolved in a conductor containing a junction. The same current being sent in opposite directions through the conductor will give opposite thermal effects at the junction, and the difference between the heats generated per unit of time will be twice the value of Uy. Some values of 11 are given in a table at the end of the present chapter. Source of Energy in Thermoelectric Circuit. Peltier Effect is Zero at Neutral Temperature 597. We are led by the Peltier phenomenon at once to a partial answer to the question, What is the source of the energy expended in the circuit of a thermoelectric couple, when there is no battery or magnetic generator in the circuit ? W^e have seen from Peltier's dis- covery that heat is taken in at the hot junction and given out at the other, when the current is produced by purely thermal action. The source of the work done in the circuit is thus, in part at least, the excess of the heat absorbed at the hot junction over that given out at the cold. We say in part, for, as we shall presently see, another thermal effect remains to be considered. Thus if IIj be the Peltier electromotive force at the hot junction .and IIj that at the other we have U.y - U.y = Ry^' + A (7) where R is the total resistance of the circuit, and A is the activity (ither than Joulean generation of heat which goes on in the circuit. 598. It is found, as has been stated, that the thermoelectric power of two metals is zero at the neutral temperature, that is if one junction be slightly above that temperature and the other just as much below it, there is no electromotive force. Hence, according to the view of the matter just taken, there ought in this case to be neither abso]!ption nor evolution of heat, and this is found to be the case as nearly as it can be tested by experiment. That there is no Peltier thermal effect at the neutral temperature has not, however, been quite conclusively settled by direct observation, as all experiments on this subject are greatly ^'" THERMOELECTBICITy 465 complicated by effects of thermal conduction. A theory, however, of the thermoelectric circuit which assumes the vanishing of the Peltier effect at a junction at the neutral temperature will now be developed, and will be found to agree well with experimental results. Thomson Effect— Electric Convection of Heat 599. Suppose one junction to be at temperature T (the neutral temperature) and the other at a lower temperature, there is neither absorption nor evolution of heat at the higher temperature, and there is evolution at the lower. Further there is Joulean evolution of heat through- out the circuit. Hence it is clear that energy must be obtained elsewhere than at the junctions, and it was proved by Lord Kelvin ^ that there is absorption or evolution of heat when a current flows in a conductor along which there is a gradient of temperature. For example, when a current flows down a gradient of temperature in a copper wire, it evolves heat, and absorbs heat when it flows up the gradient. The reverse is the case in an iron wire. Thus the source of energy in the case supposed is clear. The heat absorbed by the current in flowing round the circuit, along the unequally heated conductors, is greater than that evolved in the same process by an amount which is the equivalent of the energy electrically expended in the circuit in the same time. Now consider a copper and iron circuit with junctions at tempera- tures to and t, and. suppose the current to flow as it does at ordinary temperatures when produced by pure thermal action, from iron to copper at the cold junction, and therefore from copper to iron at the hot. Let (T^-^dLt denote the heat absorbed by the current in the copper in ascending through the difference of temperature dt, at temperature t, and (Ti'^dt that absorbed in the iron in ascending through the same difference of temperature. Thus, since the current descends the gradient in iron, the total heat absorbed in the circuit per unit of time is t Dy - Hoy + y (o-c - K E ^\" 1 b B ^ H^ A D ^^M\^ ~^^ C y^ ^^ F ^ ,'CL i t, L Fig. 163. Further, the theory given has also shown that the difference (7 — a of the specific heats in the metals is given by (15) d - i-r dt where t and 11 denote any values of the temperature and the Peltier electromotive force intermediate between the values at the junctions. If GH be drawn in the diagram parallel to BG, CrH=Tllt. Thus it is clear from the diagram that l((r — rr'^dt = area j8.4a6 + area I) Cod (16) and this is the whole heat absorbed per unit of time by unit of current in consequence of the Thomson effect. The whole heat absorbed is thus represented in the diagram by the area BADGcb, and the heat given out by BGcb. The excess of the former above the latter, namely, the area of the quadrilateral ADGB, is the amount utilised in electrical work in the circuit, and measures also the total electromotive force in the circuit. The diagram also shows that if lines parallel to OX be drawn through E and F to meet GH produced both ways in K and L, KG 470 MAGNETISM AND ELECTRICITY represents by its length the value of the specific heat of electricity in one metal, and LH that in the other. For clearly we may take d(n.lt)JcU as the algebraic sum of the inclinations of the straight lines HM, FM, to the axis of X, and the specific heat of electricity in either 1<'IG. i64. metal is the rate of variation with temperature of the thermoelectric power of that metal with reference to any other whatever, multiplied by the absolute temperature. Thus in the one case we get for the product KG, and in the other HL. XIJI THERMOELECTRICITY 47,1 one metafn.?./?w'' fl'^' l^^ ■^^'' ^^^^^'^ ^^ thermoelectric power of ?f th^tnct on, I .1*^' "*^"'" ^" *^"° °^ ™°^^ P«i^t« ^^- ^' &°-> then, points^savr A?',? *^' temperatures corresponding to tw^) of these heS and nn^.H- ^^T 7''^ ''" "° ^^^^^^^ absorption or evolution of Sace 7n onnf.f electromotive force ; and the whole action will take absorbed bv3''''' °1 *^' ^^"^"^^^^ ^ff^°*- ^hus the whole heat rnThtclc'u'.t"Sirr:ire^^^^ f' *^-iftr°*^^^ '°^-^^ between ^I/and iV". "'P'^'^'^^*^^ ^^ the loop formed by the two curves F,V^lir*^ r''°'§-' °f *^^'> ^°™'"^ ^y*h« diagram of nickel (see l^ig. 164). According to Professor Tait, the specific heat of electricity in nickel changes sign about 200° C, and again near 320° C. Anothex IS furnished by the diagram of iron and one of Professor Tait's alloys of platinum and n-idium. The curves of thermoelectric powers of iron and the alloy intersect no less than three times at higher temperatures, showmg therefore three neutral points. Hence, if the functions of a couple formed by iron and the alloy be at any of these two points, the Fig. 165. current must be almost entirely supported by the Thomson effect in iron, since there is little or no effect in the alloj'. In such a' case as this, in which the Thomson effect is zero in one of the metals, there is only absorption of heat in the other metal, and no electromotive force can have its seat in that in which there is zero thermal effect. 604. Fig. 165 illustrates a case of the flow of energy, described by Poynting (loc. cit., Art. 546). A circuit is formed of two metals of the lead type, joined by a metal of the iron type, in -which a current flowing from hot to cold absorbs heat. ■ i? is a hot junction, G a cold, each sup- posed at the neutral temperature for the pair of metals there in contact. Thus if a current flow from a battery in the circuit, there is no con- vergence of equipotential surfaces upon B or C, and no thermal effects take place there. If the resistance of BC be sufficiently small, there will be a gradual rise of potential in the metal from B to C, and the gradient of electromotive force being opposed to the current, heat will "P'^ition. PL. I.- Earth's Magnetism as shown by the distribution of lines of equal total force in absolute measure (British units, foot-grain-second-system) ; the position of the magnetic Doles and Equator. (Approximately for 1875.) " ^ S "fe. -H t> ^ ^ ^ "& w 'rt p:i g g ^ a 53> h-l 'b s •w '^ ai a o fcq ws la s * 0° LONDON W 22 20 18 IS 14 12 10 lit 90 \s» zt US o^ J 1 \ \ ^ L \ \ jea 9 \ >^ V s. Vf Pl. V. Sectlae Curves foe Dipebkent Latitudes (Bauee.) From Nature, December 23, 1897. En I !=■ -S n 00 o . _»< m H IN o ,0 *^ « N W ' 41 (0 ^ y 3_^ 40° N. n jnn^ \ \ \ \ \ ./■3 20 I 1 ^ inn -f H .'\3 I U 40 s^- o\ s. X;^ ±-■^6 60 \ \.. I -1^ 3\ Ve 80 * — \i- Pl. VII. Veotob Diagrams foii Dipfekent Latitudes. INDEX Activity in network of wires, 174 ^olotropic medium, 113 Alternating current, 365 Ampere, equivalence of current and magnetic sliell, 278, 291 electromagnetic theor3', 284 experiments, 317 Ampere, defined, 343 Attraction, electric, 101 Attraction and repulsion, magnetid, 8 Ayeton, 447 Barlow's wheel, used as dynamo, 331 Battery, arrangement of, 167 Bayly, H., 368 BiOT and Savabt, theorem of, 285 Blue sky, electromagnetic tlieory, 402 boltzmann, 348 Bryan, G. H., 211 Cable signalling with inductance, 369 Capacity, electrostatic, 120 unit of, 344 Cavities, magnetic induction and inten- sity in, 43 Characteristic equation, 131 for surface, 132 for magnetic potential, 46 function, 183 Charges, electric, subdivision of, 108 Circuit heterogeneous, 163 Circuits, reducible and irreducible, 223 Circuital equations, 388 Circulation, 221, 225 Clerk Maxwell, 180 dynamical model of current system, ■ 344 dynamical analogies, 371 Clifton, 448 Coercive force, 8 Collet, Captain, 100 Compass, mariner's, 2 heeling error of, 97 Compass, compensation of, 98 semicircular and quadrantal' ^rrors. Condenser, splierical, 114 Condenser, oscillatory discharge of 3r,'( Condensers or leydens, 120 Conductance, 169 Conductors and insulators, 105 Conductors, system of, energy of, 121, 127 reciprocal relation, 123, 125 capacities, 125 Conjugate conductors, 172 Conjugate functions, 230 Connectivity, 246 Constraint, forces of, 199 work done by, 201 Contact electricity, 440 Continuity, principle of, 168 equation of, 215, 231 Controllable and uncontrollable co-or- dinates, 205 Convection currents, 431 of heat, electric, 465, 467 CoENiT, 439 Coulomb, magnetic experiments of, 8 torsion balance of, 9 Govlomh, defined, 343 Curl, 259 of magnetic force round current, 281 287, 289 Current, electric, newer theory of , 102, 158 analogue of, 160 distribution of, in cross-section of conductor, 426 steady, 160 direction of, 161 magnetic potential of, 286 most general specification of, 296 and magnetic distribution, energv of, 297 ^^ ' current elements, 299, 309, 315, straight, embedded in conducting medium, 299 radial, magnetic action of, 301 476 INDEX Current, specification of, 388 unit of, 343 Currents, magnetic fields of, 276 equations of, 288, 340 energy of, 293 Current-induction, 324 dynamical theory of, 337, 381 Cj'clic constants, 248 Cyclosis, 249 Damping of vibrations, 417 Daniell's cell, 162 E.M.F. of, 449 De Magnete, 4 Demagnetizing forces, 54 Dielectrics, 104 Dip, magnetic, 2 Dissipation function, 205 Displacement, electric, 112 currents, 310 Divergence, 216 Doublet, vibrating electric, 395 verification of theory, 409 Dynamical theory of current-induction, 337, 381 Efficiency of battery, 167 Electric attraction, phenomena of, 101 field, 109 energy of, 112 exploration of, 128 Electrified bodies, forces on, 106 Electromagnetic action, experimental illustrations of, 305 force, 295 Electrostatics, general problem of, 119 Ellipsoid, uniformly magnetized, 52, 54 Energy, electric, 112, 129 in seolotropic medium, 113 of system of conduction, 121 electrokinetic, 294, 339 flow of, 387 location and transfer of, 102 radiation of, 426 stream-lines of, 423 Equipotential curves, 19, 118 EULER, 214 Faraday, ice-pail experiment, 107 experiments on current induction, 324 theory of current induction, 327 experiments on unipolar induction, 329 disk dynamo, 331 experiments on self-induction, 333 Fblici, experiments on current-induction, 325 Field, electric, 109 energy of, 112 exploration of, 128 Field, magnetic, 4 intensity of, 1 2 FitzGerald, 417, 420 Flinders, Captain, 100 Flow of energy, 421 Fluid, hypothesis of incompressible. 111 nature of 312 acceleration of, 214 equations of motions of, 226 kinetic energy of, 228 stream-lines of, 228, 232 motion of, in two dimensions, 229 motion of minimum energy, 239 Force, electromotive, 162 in conductor, 163 theorem of in circuit of network, 168 unit of, 343 inductive, 331 energy value of, 452 criterion of existence of, 453 magnetic, lines of, 4 equations and graphical descrip- tion of, 13, 14 for small magnets, 15 Freedom, degrees of, 181 Gadss, theory of terrestrial magnetism, 56 Gaussin, 100 Generalized co-ordinates, 180 velocities, 180 momenta, 187 forces, 199 Generators, electric, arrangement of, 166 Gerland, 445 Gilbert, Dr., 4 Greeu, his problem, 139 his function, 141 for spherical conductor, 142 his theorem, 235 deductions from, 241 in multiply-connected space, 249 Grove's cell, E.M.F. of, 449 Gyrostatic terms in Lagrange's equations, 190 pendulum, 193 Hamilton, Sir W. R., 180 characteristic function, 183 equations of motion, 203 Hahkbl, 444 Heat, production of, by currents, 164 minimum dissipation of energy in, Heaviside, 304, 305, 432, 435 Heeling error of compass, 97 Hbljiholtz, von, cyclic systems, 209 energy theory of current-induction, 336 Henry, J., experiments on current-in- duction, 334 INDEX 477 Henry, defined, 344 Hertz, solution of Maxwell's equations, electric doublet, 395 field of, 398, 405 receiver, 410 vibrator, 409 refraction of waves, 418 Howard, 420 Ice-pail experiment, 107 Ignoration of co-ordinates, 188, 251 Images, electric, 142 Impedance, 366 Induced distribution due to point charge, derivation of from equilibrium dis- tribution, 148 Inductance, 309 in network of conductors, 355 unit of, 344 and capacity, 366 in parallel conductors, 373 Induction, electric, 103 tubes of, 103, 115 specification of, 112 surface integral of, 112 tension along tubes of, 116 coefficients of, 125 magnetic, 35 solenoid al condition, 37 in cavities, 43 surface conditions for, 43 Induction, experiments ' on current, 324 coil, action of, 376 condenser in, 379 mechanical illustration of, 380 Inductivity, magnetic, 12, 41, 45 Infinite plane, induced distribution on, 144 Insulators and conductors, 105 Intensity, magnetic, 35 surface conditions for, 46 electric, 103 specification of, 112 magnetic, for currents, 304 motional electric, 385 impressed electric, 387 Inversion, geometrical, 147 electrical, 148 Irrotational motion of fluid, 215, 218 kinetic energy of, 245 Jacobi, C. G. J., 184 Joule, law of, 164 Kelvin, Lord, magnetic deflector, 100 electrical invfersion, 147, 148 ignoration of co-ordinates, 191 energy theory of current-induction, 336 copper and zinc electric machine, 446 theorem of fluid motion, 225 induction electric machine, 447 KiRCHHoiT, principles of current flow, lOo theory of rectilineal vortices, 275 KOHLRAUSCH, 443 Lagrange, dynamical method of, 180, 186 theory of fluid motion, 214 Lagrangian function, modified, 189 Lamellar distribution of magnetism, 32 solenoidal distribution of magnetism. 34 * Laplace, equation of, 46 Larmor, 192, 439 Least action, 180 theorems of, 183 Lenz, his law, 332 Liquid, cyclic motion of, 193 kinetic energy of, 249 Lodge, 0: J., 363, 420, 441, 452, 455 Lodestone, 1 LORENTZ, 439 Magnet, axis of, 22 couple on in magnetic field, 23 potential energy of, 24 Magnetic field of moving point-charge, 434 radiation in, 437 Magnetic matter, 10 potential, 29 shell, 31 Magnetism, permanent, 1 unit of, 11 elementary theory of, 29 surface distribution of, 21 lamellar distribution of, 32 terrestrial, 58 Magnetization by induction, 3 processes of, 7 particular cases of, 39 Magnets, action between, 27 Mariner's compass, 2 Maupertuis principle of least action, 180 Medium, action of, 106 Model, dynamical, 344, 347 Moment, magnetic, 22 Momentum, electrokinetic, 339 Motional forces, 205 Kaleidoscope, electric, 156 Kelvin, Lord, mariner's compass, theory of magnetism, 10, 46 Network — see Wires of conductors, 355 Norman, Robert, 2 INDEX Oersted, his expeiiinent, 276 Ohm, his law, 161 Ohm, defined, 343 Orbii i-irtufif, -t Parallel conductors, resistance and in- ductance of, 169 planes, induced charge on, 1.50 inversion of, 154 Pellat, 449 Peltier, 453, 463 effect, 463 at neutral temperature, 464 Perforated solids, motion of, in liquid, 251 Periphraotic spaces, 233 Permeability, magnetic, 12 Perry, 368, 447 POINCAR]^, 417 Point charges, mutual energy of ; ap- parent repulsion between, 40 Polarization of electromagnetic beam, 419 plane of, 419 Poles of magnet, 3 magnetic, 26 terrestrial magnetic, 59 Potential, electric, 117 coefficients of, 125, 135 propagation of, 403 magnetic, 17, 29 characteristic equation for, 46 of current, 286 PoYNTixc, 421, 423, 471 Preston, Th., 439 Primary and secondary circuits, 348 alternating currents in, 372 Quadrantal error of compass, correction of, 99 Radiation, electromagnetic, 395, 415 Rayleigh, Lord, dissipation function, 205 dynamical model of current system, 347 distribution of currents, 426 Reciprocal action between two currents, 341, 348 Rectilineal vortices, 269 Reflection of electromagnetic waves, 416, 418 Refraction of electromagnetic waves, 418 Resistance, electric, 162, 164 Resistance, unit of, 343 Resonance, 370 multiple, 417 RoniH, 192 Rowland, H. A,, 431 Sarasix and De la Rive, 417 Searle, 437 Screen, conducting, 109 Sebbeck, 456 Self-inductance, 353 Self-induction, Faraday's experiments on, 333 his theory of, 334 Semicircular error of compass, correction of, 99 Shell, magnetic, 31 Small magnet, lines of force and equi- potential surfaces for, 15 Sphere, uniformly magnetized, 55 field of, 56 Spherical conductor, energy of charged, 114 induced distribution on, 142 electric image in, 142 Spherical conductors, force between two, 146 distribution on two in contact, 154 inversion of uniform, 150 Stationary action, 183 SuMPNEB, W. E., 363, 369 Superposition, principle of, 41 Susceptibility, magnetic, 45 Suspension for magnet, 3 System of bodies, dynamical theory of, 181 Tait, 466, 467, 469 Tangent galvanometer, 291 Terrella, 4 Terrestrial magnetism, 58 magnetic poles, 59 Thermodynamic relations, 207 Thermoelectric inversion, 457 power. 457 E.M.F., 459 circuit, energy in, 462 diagrams, 466 Thermoelectricity, 456 thermodynamic, theory of, 464 Thermokinetic principle, 207 Thomson, J. J., 211 Thomson and Tait, 21 1 Torsion balance, 9 Trouton, 417, 418, 420 Uniformly magnetized magnet, 11 Unipolar induction, 329 Vector potential, 48, 259 specification of, 50 Velocity potential, 215 permanence of, 219 of system of vortices, 263 Vibrator, 409 period of, 414 Volta, 440 experiments of, 441 INDEX 479 Voh, defined, 343 Voltaic cell, 440 Vortex motion, 215, 218, 255 lines and tubes, 255 surface, 256 determination of velocities, 257 electromagnetic analogues, 265 vortex sheet, electromagnetic ana- logue of, 265 kinetic energy, 267 rectilineal vortices, 269 filament, 260 velocity due to, 262 Wave, electromagnetic, 390 in Eeolotropio dielectric, 391 velocity of propagation of, 393 reiiection of, 416, 418 Wave, standing, 416 determination of length of, 417 Weber, W., electrodynamic experi- ments, 321 experimental verification of Ampere's theory of electromagnetism, 322 Wires, network of, 168 bridge arrangement of, 170 analytical treatment of, 171 cycle method for, 174 activity in, 174 reduction to bridge arrangement, 176 effect of wire joining two points, 178 Zeemans, 437, 439 effect, 437 END OF "\0L. I. Hli:HABD ' .„„ S.3N-S, LIMrrEU, l,ONDON AND EUNOAV