■<>— — *Mli iiir'irjii*. "^\yt. (Kbrtrirtan" ^rni.». PRACTICAL NOTES ELECTRICAL STUDENTS, VOL. X. A, li. KENKELLY & B. J\ WILKINSON. bought with the income erom" the SAGE ENDOWMENT FUND the gift of Henrg W. Sagis 1891 Ajm^z '^.PhJi^ Cornell university Library QC 533.K36 Practical notes for electrical students, 3 1924 012 333 310 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/cu31924012333310 PRACTICAL NOTES FOR ELECTRICAL STUDENTS. VOLUME I. LAWS, UNITS, AND SIMPLE MEASURING INSTRUMENTS. A. E. KENNELLY H. D. WILKINSON, M.I.E.E. LOXIJOX : 'THE ELIXTRICIAX'- PRIXTIXd AND PUBLISHING COMPAXV, lAMlTEl), 1, SAi.isiirkV Cni'KT, KLiicr Si KKF/r, E.C. NEW YORK : THE \V. J. JOHXSTOX COMPAXV, LIMITED, 167-177, TiMi:s Bitiluing. 1890. [All Eights lUsa-mi] A- 3yo^( CORNELL^ UNIVERSITY LIBRARY \ I'lintfil .ind rnl.lislio.l by ■ THE P.T.KPTI:!ri.VN " I'llIXTfNG AN]i PL'Br,TSI[IN''i TO.. I,tMrTKl\ 1, Siilisliiivy I'oiirt, Flci't Strei't, PREFACE. THIS book is mainly a reprint of a series of articles which appeared in Tlic Electrician under the title of " Letters for Learners and Unprofessional Readers." The series was begun in November, 18S7, by Mr. A. E. Kennelly but subsequently relinquished on his appointment to Mr. Edison's laboratory in America. In the beginning of the following year, at the Editor's request, I undertook the task of carrying them on under the same title, although I found the last two words in the title some^^'hat ambiguous. If a general and popular style of writing was looked for b)y the unprofessional readers, I cannot hope to have pleased them ; but if, while engaged in other professions, they wished to become acquainted with some of the laws underlying this rapidly-growing science, then I may hope that they have found something worthy of their attention and interest. To serve this purpose, I started with the good resolution to exclude all mathematical expressions, and endeavour to put into words anything expressing a relation between quantities. Unfortunately, this good resolution was speedily broken, although not for the want of trying to keep it. The reader may, however, be reassured by the fact that only simple equations are used, and the three simple functions of angles. The only class of men that I could think of in another profession who could be sufficiently interested to follow up such a series were those learned gentlemen who square up some of our little differences for us not very far from Temple Bar. On the other hand, to those who are, in common with myself, students of Electrical Science and Practice, I may hope that, in whatever branch of its application they are engaged, some good ma)' accrue through these pages ; not on account of anything \\ hich might be termed novel, for this will not be found, but on account of the pains I have taken to explain in plain language some of the funda- mental laws and processes, and the more modern terms in use at the present day. The great disparity between the lengths of some of the chapters may strike those who are accustomed to write books, and divide up equally the matter treated upon, as ludicrous. This, however, must stand as it is, the subjects handled having been developed as it was thought necessary at the time of writing. The length of the last chapter somewhat appalled me, but I could not well leave the subject without adding what I considered to be of use and interest under that heading, although at the expense of being somewhat theoretical at times. The experiments mentioned were conducted at the School of Electrical Engineering in Hanover Square, London, and I have to thank Mr. Lant Carpenter, Mr. L. Drugman and the Students there who a.ssisted mc, amongst whom I recollect with great pleasure Mr. Webb-Watts, Mr. Ernest Tidd, Mr. Thomson and Mr. Blakclc}-. It will be observed that the present volume contains several hints of continuation, and, indeed, b)' itself, cannot be called a complete treatise. I hope, therefore, as soon as time and opportunit)' permit, to take the matter up once more, and continue the work under its new title. H. D. Wilkinson. CONTENTS, CHAPTKK I. Introductory 1 Early Ideas. — Electrical Energy. — Electricity Produced by Cliemic^J Energy. — Eleetroiaotive Force. — Uequireuieuts in a Good Cell, — Simple Cell. — Iiifference of Voteiitial. — r)irection of Current. — Chemical Action. — Polarisation. CHAPTEi; II. 1).>,TTBI!1KS 1 laniell Cell. — Local Action. — Porou.^ Partition. — Internal liesistanee. — Types of Daniell. — Gravity Daniell. — Minotto Cell. — Tray Cell.— Leclanche Cell. — Fuller s Cell. — De La Itue'-s Cell. — Standard Cell. CHAPTEi; in. El,l-;CTRCOIOTIVE I\il;CE AND I'nTEN'lTAr Cells in Series. — Leakage. — Potential. — r)isti'iliutiLiu of Potential ii; ; Batterv. — Zero Potential in a P.attei v. — Earth Connection-. CHAPTKl! I\'. Kesistance :i3 Eelative Eesistance of iletale. — Kelati\e Specific I!e.sistance and Conductivity.— Decrea.-e of Conductivity with Impure Metals and Alloy.s. — Use of Copper for Conductors. — Resistance of a Wire at Constant Teuiperatuie Not AB'ected by Strength of Current. — Resistance of a Oiven A\'ire Proportional to its Length. — Resistance of a (iiven Length of Wire Inversely Proportional to its Cross-Section. — Conductivity of a Given Length of Wire Proportional to its AN'eight. — Relation between Length, L'iaineter, and Weight of Telegraph Conductors. — Practical Problem- in Telegraph Conductors. — Multiple and Su)i-ilultiple L'nits of rhe Ohm. — Uesistances in Series. — Resistances in Multiple Arc. X. CONTENTS. CHAPTER V. PAOK- ClTBRENT 33 Effect of "Opening" or "Closing" a Circuit.— Telegraph Signalling on the American Closed Circuit System. — Retardation of Current Slight on Overhead Lines Except at High Signalling Speed. — Retardation Considerable on Submarine and Subterranean Lines. — Velocity of Electricity Variable. — Period of Constant Flow of Current. — Strength of Cuirent Proportional to the Conductivity of the Circuit when the E.M.F. in that Circuit is Constant. — Current Strength Proportional to E.M.F. when Conductivity is Constant. — Practical Unit of Current: The Ampere. — Unit Quantity of Electricity: The Coulomb. — The Milliampcre. — Calculation of vStrength of Current bv Ohm's Law. CHAPTEK VI. Current Ixdicators 42 General Use^. — Direction of Current. — Directicai of I>eilection. — Mounting of Needles in Vertical Detector. —Mounting of Coils. — Direction of Winding of Coils. — Coils Connected "in Series." — Coils Connected " in Multiple Arc.'' — Relation between the Sen- sitiveness of a Detector and its Resistance. — Eiact I'eterminatiort of Best l-iesistance of Coil. — Detectors for Telegrap)h AVork. — Iietectors for Telephone A\'ork. — Indicators for Large Currents. — Effect of Current in Leading Wires. — Indicators with Coils of Different Resistances. — Effect of Loss of Magnetism in Needle of A ertical Indicator. — Effect of Loss of Magnetism in Needle of Horizontal Indicator. — Effect of Friction of Pivot. — Hori- '/outal IndicatcTS. CHAPTEl; VII. SiJirj.E Tests with Indkatuiw 64 Tests for " Continuity." — Detection of ;i Fault in Telegraph Appa- ratus. — Tests for Identity of Wires. — Test for Insulation. — Practical Directions for Taking Insulation Test. — Insulation Resistance per Mile. — Allowance for Leading Wires in Insulation Test.— G. P, 0. Standard Indicator. CONTENTS. xi. "'HAPTER VIII. The . 13 Fig. 2. clear that any metallic impurity in the zinc plate, and at its surface of contact with the solution, tends to set up local action ;. and a plate of zinc containing such impurities con- sumes irregularly in action, and will show signs of pitting after merely remaining plunged in water for some time. Locaf action due to metallic impurity may be got rid of by cleansing the zinc plate and rubbing it well with mercury — amalgamat- ing it, as it is termed. The superficial amalgam or alloy of zinc so formed has practically the same E.M.F. with copper as unprotected zinc, but the impurities are kept covered and separated from the solution by a metallic film of mercury, and no series of metallic contacts in a circuit from which liquids ' are excluded will give rise to an E.M.F. Impurities, however, FOE ELECTRICAL STUDENTS. ,9 that are deposited from without — as, for example, particles of reduced copper from stray copper sulphate in the solution round the zinc plate — are not so readily imbedded in the mer- curic film as to pass out of free contact with the solution, and so local action from this cause takes place in spite of amalga- mation. 13. Porous Partition. — It would, of course, be possible to prevent the copper sulphate solution surrounding the copper plate from straying towards the zinc, by setting a partition or barrier of impervious substance between the plates, thus keep- ing them and their solutions entirely separate ; but if this partition were of non-oonducting material it would effectively prevent the current set up by the cell's E.M.F. from passing between the plates, and if, on the other hand, it were of con- ducting material, such as metal, then hydrogen would be thrown up on its surface, and polarisation would take place, so that the plan of introducing copper sulphate would be rendered useless. The difficulty is met by employing a partition of porous earthenware, a material in itself non-conducting, but containing numerous small channels, which become filled with liquid, so that, while the solutions on each side are kept apart, no great obstacle is interposed to the passage of the current. Cells which thus employ a separate solution for each plate are called double-fluid cells. 14. Internal Resistance. — All such porous partitions do, however, allow the liquid solutions to mingle to a greater or less extent, the process of so doing being termed diffusion. This diflFusion takes place more slowly as the density of the separated solutions becomes more nearly equal, and as the porous barrier becomes thicker ; but by increasing this thick- ness we add to the obstruction which is offered to the current in traversing the partition. This obstruction is termed resistance, and as electricity flows in virtue of E.M.F., so that flow is obstructed and diminished in rate in virtue of resistance. The flow of water presents an analogy, for if a current of water be flowing through a pipe from one tank to another owing to a difference of level at which the water stands in them, the 10 PRACTICAL NOTES rate of flow will depend, assuming this difference of level artificially maintained, upon the dimensions of the connecting pipe. By altering the length and calibre of the pipe, or by altering, as with a tap, the amount of obstruction in it, we can make the current of water as great or as small as we please ; in other words, we can make the quantity of water which passes through the pipe in a given time as great or as small as we desire. Just in the same way, having a certain E.M.F. in a circuit, we can, by increasing or decreasing the resistance in that circuit, make the current generated by the E.M.F. as great or as small as we please ; or, in other words, we can alter at will the quantity of electricity which passes through the circuit in a given time. A good conductor, offering little obstruction to the passage of an electric current, has little resistance ; while a bad conductor is one that has, on the contrary, great resistance. We shall •consider the nature of resistance more fully at a later period, but it will at present suffice to observe that it is measurable in terms of a definite unit, called the ohm, after the German scientist of that name. An ohm is roughly the resistance offered by a column of mercury one metre long and one square millimetre in section, or of 120 yards of ordinary No. 8 over- head iron telegraph wire, or of 200 yards of ordinary cable core, whose conductor weighs 1201b. per nautical mile; in ■other words, each of these three lengths of conductor would offer approximately the unit amount of obstruction to the flow of electricity through them. 15. Types of DanielL— We have seen that the Daniell cell, by the diffusion of its solution, infringes the second condition of para. 5, page 3, by the waste of material which goes on when the cell is not at work, but it has otherwise so many advantages that various forms have been adopted for different purposes and conditions of current supply. 16. Gravity DanieU.— One of the simplest types is the gravity Daniell, shown in Fig. 3. The liquids are kept apart by gravitation only, the solution of copper sulphate being denser than that of zinc sulphate. A flat plate of copper, C, FOR KLECTKICAL STUDENTS. 11 is placed on the floor of the cell as the negative element, and covered with crystals and solution of sulphate of copper up to L, about one-fourth of the cell's height. The zinc solution of acidulated water or sulphate of zinc is then carefully added, and the zinc plate Z supported last of all a little below the surface H. Diffusion of the solutions proceeds rapidly if the cell is out of action, but sustained work tends to keep the diffusion in check owing to the more active consump- tion during current supply. The cell is therefore well suited for yielding continuous currents, and has always been much in favour for telegraphic purposes in the United States, where the circuits are worked on the closed-circuit principle — that is to say, with the current from the battery always flowing to line, except in the spaces or pauses of manipulation separating the signals. ^ m fSffimmfj Fig. 3. Fig. 4. 17. Minotto Cell. — The Minotto cell, shown in Fig. 4, is a derivative of the gravity type, more especially adapted to transport and discontinuous work. The outer cell is com- monly of gutta percha, which is a substance less liable to injury than glass. Above the copper plate and sulphate crystals wet sawdust is packed, and the zinc plate rests above the sawdust with the pressure of its own weight, and with its upper surface about level with the top of the cell. Whereas the gravity cell cannot be moved without disturbing the solutions and assisting their difi"usion, the Minotto sufi'ers little by transport. The sawdust takes the place, in fact, of the porous earthenware partition in the typical Daniell cell. The internal resistance is, however, greater, varying from 10 to 30 ohms per cell, according to the quality and packing of the sawdust. If the 12 PRACTICAL NOTES cell is first set up with sawdust soaked in fresh water, the internal resistance {see para. 14) will be 100 ohms or more, until by working the zinc sulphate and sulphuric acid have time to form and impregnate the sawdust. The cells, if made up in this way, should, therefore, be left on sliort circuit, i.e., with the poles {see para. 8) directly connected for twenty-four hours or so before they are brought into use. If required to be set up for more immediate use, the sawdust may be soaked in acidulated water. They tend to increase in internal resistance after a time, owing to the drying up of the sawdust, and fresh water has to be added when necessary, the right condition of moisture existing when the sawdust is thoroughly saturated, and a layer of liquid just rises round the base of the zinc. Fig. 5. 13. Tray CeE — The Thomson tray cell is another form of Daniell, of the gravity type, adapted for supplying strong and continuous currents. Shallow trays, 22in. square and 2fin. deep, form the outer cell. A square sheet of copper rests on the floor of the tray, and the sulphate of copper in crystal and solution rises above to a height of some 2in. The zinc plate, in the form of a stout grating 16in. square, shown in section at Z in Fig. 5, is supported horizontally on stoneware blocks, B B, at the corners of the tray. Stout parchment paper, p p, covers the lower surface and sides of the zinc grating to asists in separating the solutions, on the principle of a porous diaphragm, and the zinc solu- tion — fresh water, or, preferably, a solution of sulphate of zinc — is poured in above until the upper surface of the grat- ing is covered to the depth of about a quarter of an inch. The resistance of these cells is about one-tenth of an ohm each, or even less when in good working order. They require charg- ing with sulphate of copper at regular intervals, and the FOE ELECTRICAL STUDENTS. 13 density of the zinc sulphate solution should also be kept down to a sp. gravity of 1-12, or about one-quarter saturation, by partly drawing off from time to time, and replacing with fresh water. 19. Leclanche Cell. — A type of cell much in use for discon- tinuous work is the Leclanch6. This is a single-fluid cell in which zinc is the positive element {see para. 8), carbon the negative, and a solution of salammoniac the exciting liquid. The carbon plate is surrounded by crushed carbon in intimate admixture with black oxide of manganese, either by being inserted into a porous jar with those materials, or, in what is Fig; 6. called the agglomerate cell, by having those materials solidly cemented round the carbon plate in a cylindrical form under pressure. The zinc element is generally in the form of a rod, placed in one corner of the cell, which is represented in section at Fig. 6. During the cell's action the salammoniac, which is a compound of ammonia and chlorine, is decomposed, the chlorine attacks the zinc, forming zinc chloride, which, being soluble, disappears into the solution, while the ammonia developing at the surface of the carbon also forms a soluble compound with oxygen, which it seizes from the oxide of manganese. Thus, in action, the zinc is con- 14 PRACTICAL NOTES sumed, its chloride accumulates in solution, the carbon is un- altered, and the oxide of manganese loses some of its oxygen. Practically, however, when the supply of current from the cell exceeds a certain rate, and consequently when the chemical decomposition reaches a certain degree of activity, the above simple set of reactions is not strictly carried out ; secondary and more complex chemical products form, which soon culminate in the development of hydrogen at the surface of the carbon plate, and polarisation {see para. 10) sets in, unless resting time is allowed for the secondary products to dissolve. For this reason the Leclanch6 cell is not suited for strong and continuous current supply, and its E.M.F. {see para. 4), normally about 1-48 volts, soon falls under hard work. For more limited, and especially for inter- mittent current supply, however, the cell has many advan- tages. Its internal resistance is generally between 2 and 5 ohms, and can be made much lower if, with this object, the cell be heated. It requires very little supervision, and when engaged on light work, such as the occasional ringing of bells, will not unusually continue to operate for two years without any attention or recharge. If the zinc rods be of good material, there is also exceedingly little local action. 20. Fuller's.— Fuller's bichromate cell is a double-fluid form of the zinc-carbon combination. The outer cell is a stoneware jar, as shown in section at Fig. 7. In it stands the carbon plate 0, and the porous pot pp, filled with water to the level L, hold- ing the zinc plate Z, which is usually of conical form, and kept permanently amalgamated by dipping into mercury placed for that purpose at the bottom of the porous cell. The solution for the outer jar is of bichromate of potash in water acidulated with one-ninth of its volume of sulphuric acid. During hard work polarisation takes place to some but not as a rule°to any serious extent, and the E.M.F. of the cell is approximately 2 volts, and its internal resistance from about 2 to 5 ohms. 21. De la Rue's.— A very convenient form of cell for testino- and Hght signalling work is De la Rue's single-fluid chloride of silver cell, shown in section at Fig. 8. The outer cell FOE ELECTKICAL STUDENTS. 15 is a glass tube one inch in diameter, with a flat bottom. It contains a solution of 200 grains of salammoniac to the pint of water, and the elements are zinc and silver. The latter is in the form of a wire, w, fused into a cylindrical mass of chloride of silver, which is further protected by a paper tube, pp, to keep it from coming into contact with the zinc rod, z 2. If set up with care, and not subjected to very hard work, the E.M.F. of this cell is remarkably constant at about 1'03 volts, and the average is so well maintained by different cells that in the absence of a standard of E.M.F. more accurate, it may be Fio. 7. generally relied upon for that purpose to a limit of 1 per cent. when comparing measurements made at different places. Its internal resistance in good order is from li to 5 ohms. The dimensions of the plates prevent the use of these cells for hard work, as the zinc would soon be entirely consumed ; but they take up so little space, are so easily insulated, and, once care- fully set up, require so little attention, that they are admirably adapted for testing and signalling, particularly on board ship. As the zinc should never touch the paper tube in the cell, it is a good plan when setting up the latter to first pour in melted paraffin wax to the height (H, Fig. 8) of aboub 16 PRACTICAL N0TE3 balf an inch, and then to lower the elements into their proper opposite positions, so that they remain fixed at their lower ends as the wax solidifies. It is also well, in order to prevent corrosion, to whip with silk the external loop of silver wire •which connects cell with cell. 22. Standard Cell. — A cell which is most useful in testing, but used almost solely for the purpose of supplying a reliable standard of E.M.F., is the Clark standard cell. This is a com- bination of pure zinc and mercury, and its E.M.F. is 1'45 volts at 66° F., diminishing very slightly as the temperature rises. This E.M.F. is very constant, and the amount of its variation between different cells is generally exceedingly small. To maintain its constancy this cell should never be allowed to send a strong current. Its internal resistance is therefore immaterial ; but as it tends ultimately to increase enormously with age through the drying up of the exciting salts, it is necessary to ascertain when employing the cell that its resis- tance is not excessive. FOR ELECTRICAL STUDENTS. 17 CHAPTER III. . . ELECTROMOTIVE FORCE AND POTENTIAL. 23. Cells in Series. — The usual mode of connecting cells together to unite their powers is by joining the positive pole of one cell to the negative of the next all through the series. The E.M.F. and resistance between the terminal poles of the battery so formed will be the sum of the E.M.F. 's and resis- tances of the individual cells. Thus, if the zinc pole of a Fuller cell, whose E.M.F. is 2 volts and resistance is 3 ohms, be connected to the carbon pole of a Leclanch^ whose E.M.F. 13 1"5 volt and resistance is 5 ohms, then between the carbon pole of the Fuller and the zinc pole of the Leclanch6 respec- tively — the positive and negative poles of the two-celled battery so formed — there will be an E.M.F. of 3'5 volts and a resistance of 8 ohms. Similar reasoning applies to any number of cells. So that if we connect up in series, as it is termed, 20 Daniells, each having an E.M.F. of r05 volts and a resistance of 10 ohms, we shall obtain a battery of 21 volts E.M.F. and 200 ohms total internal resistance. 24. Leakage. — It is important to remember that the necessity for the insulation of cells increases with the total E.M.F. accumulated, or, in other words, with the number of cells connected in series. For although a slight film of mois- ture on the outer surface of a glass cell may only allow an inappreciably small current to pass from pole to pole of that cell singly, the loss of current may become serious when that leakage may form the path of escape to a large E.M.F.; in the same way that a small leak at the base of a tank may cause 18 PRACTICAL NOTES only a trifling escape of water when the tank is nearly empty, but might allow of serious loss as the level of water and con- sequently the pressure is raised from within. It should be the object, therefore, with large batteries, especially when they are employed in delicate measurements, to place them on insu- lating supports, and also to prevent leakage from cell to cell by keeping the outer surfaces of the cells clean and dry. A good method is to coat with paraffin wax the outer surfaces of the cells and the trays or brackets on which they stand. 25. Potential — While the poles of any cell are insulated their potentials will be equal and of opposite signs. For example, if an insulated Daniell cell has an E.M.F. of 1-OS volt, the potential of its copper or positive pole will be 0-54 -(-, or positive, and that of its zinc or negative pole 0-54-,, or negative, the difference of potential between them being thus 1-08, equal by definition to the E.M.F. Similarly with an insulated battery of 100 volts E:M.F., the potential of the positive pole will be + 50, that of the negative pole - 50. Consequently, if the battery be homogeneous, the potential at its centre must be zero. This zero potential is really the potential of the earth and of all neighbouring con- ductors in connection with it. So that, like the effect of th& moon's gravitation on the ocean, which produces in the tides a, fluctuation of level above and below the normal level of the sea, the effect of the contact of the different metals of a cell, either singly or as accumulated in a battery, is to produce at the insulated poles potentials equally deviating above and below from the potential of the earth, which, being constant, is reckoned as zero. So that when we speak of potential we tacitly refer to this earth potential zero, but no such allusion is necessarily made when we discuss difference of potential, viz., E.M.F. ; just in the same way, when we speak of the level of a position as being 3,000ft. high, we refer to the mean level of the sea, while difference of elevation is an idea in which the sea level may have no part. 26. The Distribution of Potential in a Battery. — Fig. 9 shows how we may graphically represent the rise of potential in an insulated battery. The thick lines similar FOR ELECTRICAL STXJDENTS. 19 to 23 across the dotted line AB represent the zinc plates, and the thinner lines similar to c c the copper or carbon plates of the battery. Now, holding up the paper so that its plane is vertical, the upright lines similar to A D represent in length and direction the potential in the battery at the points on which they rest. Positive potentials are represented by lines above A B, negative potentials by lines below. Suppose we take a scale of measures such that a difference of potential of one volt shall be represented by one-eighth of an inch. As we may select any horizontal scale we please, it will be con- — +8H -hs — - +* ■■^z H^-'i-'-H-h-F^'^ Fig. 9. venient to make it correspond to the internal resistance of the- battery, and our diagram will then embody both E.M.F. and resistance. Let us take a horizontal scale along the line A B of 8 ohms to the inch. Now let us consider the case of an insulated battery, A B, of nine cells, numbered as shown, the first four being Leclanch6's, each of 1'5 volt E.M.F. and 3 ohms resistance, and the last five Fuller's, each of 2 volts E.M.F. and 2 ohms resistance. Then the total E.M.F. will be 16 volts and resistance 22 ohms, so that the latter will be represented on our scale of 8 ohms to the inch by placing the 2'' poles of the battery — = 2-J inches apart, so that the line 8 c 2 20 PRACTICAL NOTES A B must be 2f " long. Each cell may now be marked off along this line, the distance from one connection to the next being f" for the Leclanoh^s and |" for the Fullers. The plates may then be sketched in their right positions, as shown. The ver- tical scale has now to be considered. We have seen in the preceding paragraph that the points A and B will receive equal and opposite potentials, each equal to half the total E.M.F., ■which in this case is 16 volts ; consequently, B bting the positive pole ■will have a potential of + 8, represented on our scale by a point one inch above A B, ■while the negative pole Earth. ^----'-|--^--f---^----l|-:--"----l|-^-||-"f-"--t|--"-iK^ 'W^ -to -tt-5 -!3 -74 5 - KD Fig. 10. A will have a potential of - 8, indicated by the point D one inch below A. By then carrying horizontal lines from D and II parallel to AB, and altering the potential opposite each junction of plates by the exact amount of E.M.F. that junction generates to scale, we shall produce the zigzag line D E F G H, every pciat of which represents by its distance from A B the potential of the point opposite to it in the battery. 27. Zero Potential in a Battery. — In the particular case of the battery we have been dealing with, we see from the figure that the potential line in the first Fuller cell, No. 5, coincides with the line A B, and since A B is also, FOR ELECTRICAL STUDENTS. 21 as explained in para. 25, the earth potential or zero line, the potential of cell No. 5 must be zero. Consequently, the potential line DEFGH will not be disturbed if this cell be connected with the earth by dipping a wire into it whose other end is deeply buried in the ground, for example ; for by so doing we simply connect two bodies at the same potential, and no current tends to flow under ordinary circumstances between two bodies at the same potential any more than water tends by gravitation to flow between two points at the same level. n. „ E KcurOi, \-----\--^-}-'-\''¥¥\"¥\ +45 +f5 1^ Fio. 11. 28. Earth Connections. — If, however, the earth wire be removed from No. 5, and connected to any other point, the jiosition of the line of potentials will be altered. For example, if the earth wire be connected to B, the positive pole, as in Fig. 10, a current will tend to flow from B to earth, but can- not be maintained because it can find no circuit, the battery being otherwise insulated. The point B and the earth cannot, however, retain difierent potentials after connection, so B's potential falls to zero. But the mere connection of B with earth cannnot alter the E.M.F. of the battery — that is to say, 22 PRACTICAL NOTES cannot alter the difference of potential existing between the poles A and B ; so that as B has lost a potential of 8, A must be augmented by that amount, and will therefore become - 16, as shown, and, filling in the sections of the potential line as before, we find that it has descended bodily from its previous position through the distance H B. Similarly, if B be insu- lated and A instead earthed, as in Fig. 11, the potential of A becomes zero, that of B makes up for the loss by becoming +16, and, all other points retaining their relative potentials to these, the potential line is moved bodily up to the position shown. H . G\ F i: '■'! 1 M Eoj-Ot, +7 ■¥3 +1-£- Fio. 12. Again, Fig. 12 shows the effect of putting to earth a particular point in the battery otherwise insulated. We thus see that in all cases of connecting to earth, the potential of the point earthed is brought to zero, and the potentials of all the remain- ing points in the battery follow from this point by the simple summation of the individual E.M.F.'s as they are met with. FOE ELECTEICAL STUDENTS. 23 CHAPTER IV. RESISTANCE. 29. Relative Resistance of Metals. — We have already seen (see para. 14) that the resistance of any electric path is the obstruction which a current traversing it encounters, and we are now in a position to examine the nature of this obstruction more closely. Of all known substances metals ofTer the lowest resist- ances, and the more common of these may again be compara- tively classed in the following list, which commences with the least resisting and ends with the most resisting metal :— Relative Specific Relative Specific Conductivity. Resistance. 1-000 Silver (hard drawn) 1-000 0-996 Copper „ , 1*005 0-780 Gold „ „ ......... 1-283 0-290 Zinc (pressed) 3-446 0-168 Iron (annealed) 5949 0-131 Nickel „ 7-628 0-124 Tin (pressed) 8-091 0083 Lead „ 12-131 0-046 Antimony „ 21-645 0-016 Mercury (liquid) 62-500 0-0125 Bismuth (pressed) 80-000 30. Relative Specific Resistance and Conductivity. — Since in this list, taken from Clark and Sabine's tables, silver is the metal offering least resistance to a traversing current, it must also be the metal which most facilitates its passage — that is to say, which conducts it best. Silver is therefore said to 24 PEACTICAL NOTES be the metal of greatest condudiviti/. The tabic therefore gives the metals in diminishing order of resistance by ascent, and Id diminishing order of conductivity by descent. The figures in the column on the right hand represent the resistance of each metal relatively to that of silver taken as unity, or, as it is called, the relative specific resistance ; while the figures in th& left hand column give the conductivity of each metal relatively also to that of silver as standard, or, as it is termed, the relative- specific conductivity. Thus iron is shown to have a resistance of nearly six times that of silver, so that if a conductor made of pure hard-drawn silver were of such dimensions as to offer a resistance of one ohm, a conductor of annealed iron of exactly the same dimensions would offer about six ohms resistance. Consequently, since iron resists a current six times as much as silver, the conductivity of silver must be six times that of iron ; therefore, calling the conductivity of silver 1, that of iron must be six times less, or one divided by six — that is, -, or 6 0-167, the more exact number being, by the table, 0'168 = —-——. Similarly, the relative specific resistance of mer- 5*949 cury being 625, its relative specific conductivity is , or 0'016 — that is to say, 16 per cent, of pure silver ; so that in all cases the specific conductivity of any material relatively to a standard material is the quotient found by dividing into unity the specific resistance of the substance relatively to the same material ; and, further, the conductivity of any path is found from the resistance of that path in the same manner and is expressed in terms of a unit, to which the name of rnlio has been given by Sir William Thomson, the term convey- ing the inverse idea of resistance by being an inversion of the word "ohm." Thus, if a certain conductor had a resistance of 0-2, 1, 50, or 7,000 ohms, its conductivity would be , _ 0-2 1, .— , or — --- mhos respectively— that is, 5, 1, 0-02, or 0-000143 mhos. FOR ELECTRICAL STUDENTS. 25 31. Decrease of Conductivity with Impure Metals and Alloys. — The relative specific conductivity of metals is always reduced by impurity, and even a very slight degree of impurity -will suffice to palpably lower the conductivity, or, in other words, add to the conductor's resistance. It is by no means impossible to find a sample of impure commercial copper which will only possess the specific conductivity of zinc, owing to the admixture of foreign and non-conducting material, or of baser and more resisting metal. Also a con- ductor composed of an alloy of different pure metals in known proportions has generally a greater specific resistance than would be furnished by an estimate made on the supposition of its being a bundle of separate and purely metallic conductors- of proportional dimensions. 32. The Use of Copper for Conductors. — The table also shows that pure copper is so close to pure silver in conduc- tivity as to scarcely make it desirable to select the latter and rarer metal for conductors, even were the matter of cost entirely set aside. In telegraphy, therefore, copper is the metal employed as the conductor for submarine and sub- terranean purposes, while both iron and copper are used for overhead wires. 33. Resistance of a Wire at Constant Temperature Not Affected by Strength of Current. — The resistance of a con- ductor of homogeneous material depends only upon its dimen- sions and temperature. So that if the temperature of a fixed conductor be constant, its resistance will be the same for every strength of current. 34. Resistance of a Given Wire Proportional to its Length. — As regards dimensions, the resistance of a conductor of homogeneous material depends upon its cross-section and length. If its cross-section be constant, the resistance will be proportional to the length. Thus, if a mile of a certain wire had a resistance of 10 ohms, then the resistance of 02, 0'4, 7, or 5,000 miles of the same material and cross-section, would be 2, 4, 70, or 50,000 ohms respectively. 26 PKACTICAL NOTES : 35. Kesistance of a Given Length of Wire Inversely Proportional to its Cross Section. — With conductors of the same length and material, the resistance increases as the cross-section is reduced, just as the resistance to the flow of water through a pipe is increased as the pipe is reduced in cross-section. Thus, if two wire?, A and B, are of the same length and material, but B has twice the cross-sectional area of A, then B may be regarded as being made up of two wires very close together, each of A's cross- section, and therefore equal to A in every respect. Obviously the double wire would convey under the same E.M.F. double the current that would traverse the single wire A, and its conductivity being doubled, its resistance would be only half that of A. Similarly, another wire, C, of the same length and material, but of three times A's sectional area, would offer one-third of A's resistance; and in the same way, if C had 0'25, 0'7, 3, or 15 times the cross sectional area of A, it would offer — , — , A, ori-; that is, 4, 1-43, 0-33, or 0061 times 0-25 0-7 3 15 A's resistance respectively. 36. Conductivity of a Given Length of Wire Proportional to its Weight. — Further, since in doubling the sectional area of a given length of homogeneous wire we double the quantity of metal in it, we necessarily double the weight of the wire, so in every case the conductivity of a wire of fixed length and material will be proportional to its weight. Thus, if a wire. A, one mile long, weighing 1001b., offers 10 ohms resistance, then another wire, B, of the same material, one mile long, and weighing 2001b., would have double the cross-sectional area, double the conductivity, and consequently half the resistance of A, namely, 5 ohms ; and in the same way, if B were to weigh 0-51b., 21b., or 5001b., it would offer — , — , or— times : 0-5 2 500 that is, 200, 50, or 0'2 times the resistance of A respectively. 37. Relation Between Length, Diameter, and Weight of Telegraph Conductors. — The conductors employed in telegraphy FOR ELECTRICAL STUDENTS. 27 are wires — that is to say, are cylindrical in form, so that their dimensions are generally — (1) Length (on land in statute miles of 1,760 yards, at sea in nautical miles of 2,029 yards), and (2) Either the Diameter or the Weight per mile. If the diameter is given, the sectional area is found by multiplying the square of the diameter by 0-7854:. For example, a wire O'OSin. in diameter would have a sectional area of 0-03 x 0-03 x 0-7854, or 000071 square inch. The sectional area and weight of a wire are directly connected, for in the case of iron wire the sectional area in square inches multiplied by 17,500 gives approximately the weight in pounds per statute mile. For copper wire the similar multiplier is 20,000, but for cable core, whose conductor is usually a strand of seven copper wires, the approximate weight of conductor per nautical mile is found by multiplying the sectional area found from the diameter as above by 18,100. For example, a copper strand of cable core has a dia- meter of 0-13in. Its area will therefore be approximately 0-13x0-13x0-7854, or 0-01327 square inch, and 001327 multiplied by 18,100 gives 240-2, the approximate weight in pounds of one nautical mile of this strand. 38. Practical Problems in Telegraph Conductors. — The problems which practically arise for solution in the case of telegraphic conductors are generally of the following form : — Having given the metal of which a wire is made, and any two of the three following particulars — dimensions, resist- ance, and conductivity relatively to pure metal, it is required to find the third. Leaving the effect of temperature at present out of consideration — that is to say, supposing that the tem- perature remains constant — we shall find no difficulty in solving any such problem, as the following cases will show : — (a) Having given the dimensions and resistance, to find the conductivity. Example. — A copper strand conductor of cable core 5 knots long and 0-096in. in diameter gives a r;sistance at the 23 TOACTICAL NOTES standard temperature of 48-42 ohms. What is its conductivity compared with pure copper ? The sectional area of the strand is 0-096 x 0-096 x 0-7854, or 0-007238 square inch approximately. Its weight per knot is therefore 0-007238 x 18,100, or 1311b. As may be seen in electrical tables, the resistance of a wire of pure copper one knot (nautical mile) long, and weighing altogether lib., would be 1196-7 ohms at the standard temperature of 75° F. ; con- sequently, the resistance of a wire of pure copper one knob long and weighing 1311b. would be (see para. 36) at the standard 1 -| Qf* .'7 temperature , or 9-135 ohms, and hence its conduc- ioi tivity would be mho — that is, 0-10947 mho. The wire •^ 9135 in question offering 48-42 ohms, is five knots long, so that one knot would offer -, or 9-684 ohms, and the conductivity of one knot would therefore be , or 0-10326 mho. The 9-684 conductivity of pure copper is to the conductivity of this wire, therefore, in the proportion of 0-10947 mho to 010326 mho — that is, as 1 is to 0-9433, or as 100 is to 94-33; so that the wire's conductivity is thus 94|- per cent, that of pure copper. (b) Having given the dimensions and conductivity, to find the resistance. Example. — A copper wire 600 yards long and 25in. in diameter has a conductivity 96 per cent, that of pure copper. What will be its resistance at the standard temperature 1 The sectional area is in this case of a cylindrical wire, not a strand, strictly 0-25x0-25x0-7854, or 0-0490 square inch, so that its weight per statute mile will be 0049 x 20,000, or 9801b. approximately. The resistance of a statute mile of pure copper weighing one pound will be found in electrical tables to be 872 ohms at the standard temperature, which for overhead land lines is generally taken at 60° F. A wire of pure copper weighing 9801b. per mile would therefore have a FOR KLECTRICAL STUDENTS. 29 resistance of ^, or 0-8898 ohm, and its conductivity will thus be ■ , that is, 1-1238 mho; but its conductivity, being only 96 per cent, of pure copper, will be 96 per cent. of 1-1238, or 0-96x1-1238— that is, 1-0789 mho, or L_ 10789' namely, 0-92G8 ohm per mile. But the vrire in question is •only 600 yards long, and will therefore have less than this resistance in the ratio of 600 to 1,760, so that its resistance •11 CUT, 600x0 9268 „ qic v, will finally be , or 0-316 ohm. ■^ 1760 ' (c) Having given the resistance and conductivity, to find the dimensions. Example. — A length of homogeneous cable strand conductor, 0-133in. in diameter, is known to be of 97 per cent, con- ductivity compared with pure copper, and has a resistance at the standard temperature (75° f .) of 37-5 ohms. What is the length of the conductor 1 The area of this wire is approximately 0-133x0-133 X 0-7854, or 0-013893 square inch. Its weight per knot is therefore 0-013893x18,100, or 251-51b. The resistance of a wire of pure copper one knot in length and of this weight 1 1 Qfi "7 ■would be , or 4,759 ohms at the standard tern- 251-5 perature. Its conductivity would therefore be , or 4-759 O-21013 mho. The wire in question, however, having 97 97 per cent, of this conductivity, would have only 0-21013 x — -. IOOj or 020382, and therefore a resistance of - — , or 3 988 20382 ohms, but its total resistance being 37-5 ohms, its length must be greater than one knot in the proportion of 37-5 to 3-988, so that its length will be ^1^, or 9-425 knots. 3*988 30 PRACTICAL NOTES The steps in all these calculations can of course be very much abbreviated when the method is clearly understood. Thus, the example in (J) is capable of being worked out directly by one fraction in the following form : — 872 X 100 X 600 0-25 X 0-25 X 0-7854 X 20000 X 96 X 1760 that is, 0-316. 39. Multiple and Sub-Multiple Units of the Ohm. — Since the resistances which are dealt with in telegraphy may sometimes be exceedingly small (perhaps thatj for instance, of a short length of thick copper wire offering a minute fraction of an ohm), and on the other hand may be very great, (such as that of the insulating gutta-percha covering of a short length of cable core amounting, it may be, to billions of ohms), it is convenient to adopt two secondary units, one a multiple and the other a sub-multiple of the ohm. Thus a million ohms is spoken of as a megohm, and is usually represented for brevity by the last capital letter, omega, of the Greek alphabet (fi)— thus, 227-5Ji means 227,500,000 ohms. The ohm itself is commonly represented by the small letter omega (w). One- millionth of an ohm is called a microhm, and has at present no representative symbol. Thus, 7,560 microhms is the same as 0-007560). 40. Resistances in Series. — Eesistances that are connected in series or simple circuit are always directly additive. This is obvious from the fact that any resistance may be re- garded as an equivalent length of some particular wire, and we have seen that the resistance of a wire is pro- portional to its length. Thus, if a circuit be composed of a battery of 40w total internal resistance, an overhead conducting wire of 50a), and the copper wire wound on the coils of a sounder of 35a), then we may regard that circuit as being equivalent to 135 miles of wire whose resistance is 1 ohm per mile, the resistance of forty such miles being in the battery, fifty in the line, and thirty-five in the receiving sounder. FOR ELECTRICAL STUDENTS. 31 41. Resistances in Multiple Arc. — Eesistances that are not connected in series, but side by side, or as it is ■termed "in mtdtiple arc," cannot of Course be additive, for between the points they connect there is more than one path, and the conductivity is increased by each addi- tional path so introduced. We therefore have in this case the conductivities additive. Suppose ABC and AD C to be two wires connected at A and C, and each offering 5(o, then the conductivities of these wires ere each '^ or 0-2 mho. 5 The double conductor may be regarded as a single conductor of double sectional area, so that we know its conductivity will Fig. 13. be doubled — that is, O'i mlio — and the resistance of the two together, or, as it is termed, their joint resistance, will be —or 2-5w, half that of either branch. So that the joint resistance of two equal wires is half that of one, of three equal wires one-third that of one, of ten one-tenth, and so on. When, however, the wires are not equal, the case, though not so simple, is similar. Thus if in Fig. 13 ABC had lOw, while ADC had 20w, then the conductivity of ABC would be --- or 0-1 mho, that of ADC — or 0-05 mho. 10 20 The joint conductivity is their sum, Q-l -I-0-05, or 0-15 mho — that is, or 6-CC7w, and similarly with any number of ' 0-15 32 PRACTICAL NOTES parallel conductors. For example, in Fig. 14 five resist- ances are shown in multiple arc, of 150a), GOw, 55cd, 2,000u), and 80o) respectively ; their conductivities will therefore be Fig. 14. 1 1 m' 60' 55' 2W ^ '^^°' respectively-that is, 0006G67, O-016667, 018182, 0-000500 and 0012500 mhos. The joint -conductivity will be the sum of all these, namely, 0-054516 mhos, and the joint resistance will be 0054516 , or 18-3430J. FOR ELECTKICAL STUDENTS 33 CHAPTER V. CUERENT. 42. Effect of "Opening" or "Closing" a Circuit. — Having examined the nature of electromotive force as supplied by batteries, and of resistance, we are now in a position to deal with the laws which relate to the current which flows in any given circuit, laws which are among the simplest yet most important considerations of the subject. Reverting to our typical form of simple cell at Fis. 1, page i, we saw that no current flowed in the circuit B E F G H if the wire F G H were severed at any point. When a circuit is interrupted at any point it is said to Be opened or brolcen, and when re-established to be completed or dosed. Every possible circuit, even though it may embrace, as in telegraphy, hun- dreds of miles of conductor, and the earth itself between that conductor's terminals, may be regarded as a simple extension of that wire F G H, and hence a current in a circuit ceases and recommences upon the complete opening and closing of that circuit at any point, even though in the battery itself, as, for example, by withdrawing and replacing a battery plate in its cell. 43. Telegraph Signalling on the American Closed Circuit System. — Fig. 15 represents the simplest form of telegraphic circuit. A, B, and C are supposed to be three stations connected by the overhead conductors A B and B C. At each station there is a key, K, a sounder, S, and a switch, s. At A there is a battery, F, one pole of which goes to the key and the other is 34 TEACTICAL NOTES ^~ o < t — H-1 ^ W'M'W'K 2 FOR ELECTRICAL STUDENTS. 35 connected to the earth, say by a large metal plate buried in the ground. A similar earth connection is made at C. The switches have each two positions : when turned to the right, as at B, they break connection between the key terminals ; and when turned to the left they re-establish it by making con- tact at c. The keys remain up while at rest through the action of a spring. If, now, all the switches are turned to the left or closed, a complete circuit is formed from the battery through s-^ Sj A B, s, Sj B C, Sj S3 and the earth, and a current will therefore flow through all the three sounders. The action of the current on the sounder we shall not at present consider, but it will sufilce to observe that when the current passes through the wire of its coils, those coils attract the soft iron, cross-piece or armature on the beam above, and pull it down against the attraction of the spiral spring p. If, now, B. opens his switch by turning it to the right the circuit is completely interrupted at his key, the current ceases, and the spiral springs reassert their con- trol over the armatures, which rise to their upper limit- ing stops. If, now, B. depresses his key for a single moment and releases it again, he will have closed the circuit during that moment at the key contact, the cur- rent will have passed during that interval, and caused the armatures of all three sounders to be attracted down to their lower stops, only to recoil again when the key Kj rises ; so that whatever contacts B. makes, whether long or short, at his key, will be faithfully reproduced on all three sounders, whose armature levers follow, in fact, all the up and down motions of B's key. When B. has finished sending he closes his switch, and either A. or C. can open their switches and reply to him in the same way. Thus, with one battery, F, any convenient number of stations can communicate with each other along a line of conductor by inserting a key, switch and sounder. This system is called the closed-circuit system, already alluded to in para. 16. It is in almost universal use in the United States, but is not so suitable for the conditions of European telegraphy, and is scarcely employed at all in England, one great disadvantage being the waste of battery power while D 2 36 .. PRACTICAIi NOTES the circuit is silent, as the current is normally always, flowing. 44. Retardation of Current Slight on Overhead Lines ex- cept at High Signalling Speeds. — The instant such a circuit- as that in Fig. 15 is closed a current commences to flow, but it- does not instantaneously arrive at its full ultimate strength, A certain amount of time is required for the current to rise to its constant maximum limit, and this period of time is called the varialle period of the current's flow. The duration of the variable period depends not only upon the nature of the circuit itself, but also upon the nature of the space sur- rounding it ; other things being equal, however, it increases with the length of the conductor. Practically, however, the variable period is in the case of overhead land lines exceedingly fchort. For example, with an ordinary No. 8 overhead wire 100 miles in length, with a battery and earth at one end and earth at the other — that is to say, with no telegraph instru- ments — the current at the end distant from the battery would arrive at 99 per cent, of its ultimate maximum value in 0'0021 second, or about -jj-^th part of a second, so that although the duration of the variable period is increased by the introduction -of receiving apparatus into the circuit, still that period is, with overhead lines, so short that it may be neglected, except on lines worked at a very high signalling speed. 45. Retardation Considerable on Submarine and Subter- ranean Lines. — With submarine and subterranean conductors, however, the conditions are such, for reasons we need not at present enter upon, that the variable period is gene- lally very much greater than that for overhead wires of the same length. The Thomson recorder shows this very clearly, for supposing that the current received on a long cable arrived immediately at its full strength, and that the coil and siphon responded immediately — that is to- say, had no retarding friction or inertia — then the received sigijals would always be as shown in Fig. IG, and the applica- tion of a perriianent current would produce a right-angled out- line like al c; but owing to the gradual rise and fall of the FOR ELECTRICAL STUDENTS. 37 x;urrent during the variable periods preceding and succeeding an application of the battery to the sending end, the slurred signals and curved line, clef (Fig. 16a), are actually received. On a long submarine cable the current which passes from the cable to earth one second after the application of the battery to the distant end may be perhaps only 40 per cent, of the full ultimate current. After three seconds the current will probably be very nearly at its full strength, but many seconds must elapse before the maximum can be said to have been reached. 46. Velocity of Electricity Variable. — It i| clear, there- fore, that it is incorrect to speak of the velocity of electricity. If, for instance, an indicating instrument be «et at the receiving end of a line, and the time which — "^-^y i d Figs. 16 /n> 16a.* -elapses between the application of a battery at the sending end and the indication of this instrument be measured, we can say that this signal required that particular interval of time to be produced over that distance of line under those circumstances ; but by altering the sensitiveness of the instru- ment and the form, dimensions, and position of the line we can make the time required to transmit that signal as great or ■as small as we please ; and, so far as is known, there is no time of transmission over any given distance so short that by ■still further altering the conditions it might not be made yet shorter ; so that, under these circumstances, it is impossible to a.ssign any velocity as inherent to electricity, like that which •we attribute, for example, to light in its passage through space. * The curve cl cf is aircurafcly given on a larger scale at p. 329, Jenkin's •"Electricity and Magnetism," 2nd Edition. 38 PEACTICAL NOTES 47. Period of Constant Flow of Current. — After the variable period has passed, however, the current in a circuit will continue to flow, under the control of much more simple laws, at a constant rate, so long as the conditions of that circuit remain constant. 48. The Strength of Current is Proportional to the Con- ductivity of the Circuit when the E.M.F. in that Circuit is Constant. — This almost follows from our conception of the term conductivity, for if a certain circuit offers a given degree of facility to the passage of a current supplied by a constant E.M.F., then it follows that by doubling that degree of facility we double the current flow, and if we make the conductivity 0-2, 0-9, 15, or 1,000 times what it was at first, we shall with the same E.M.F. obtain 0-2, 0-9, 15, or 1,000 times our first current. 49. Current Strength Proportional to E.M.F. when Con- ductivity Constant. — -When the conductivity of a circuit is constant, then the strength of the current which flows through it is proportional to the E.M.F., just as the flow of water through a pipe is proportional to the difference of level or " head," which drives it. So that by doubling the E.M.F. in a circuit of constant conductivity we double the amount of current flowing through the circuit, and if we make the E.M.F. 0'005, 07, 16, or 1,000 times what it was at first with the same conductivity, we shall get 0-005, 0'7, 16, or 1,000 times the original strength of current. 50. Practical Unit of Current : The Ampere. — The unit strength of current is called the ampere, after the French physicist of that name. It is the current which flows in the circuit of unit conductivity containing the unit E.M.F. — that is to say, one volt in a circuit of one mho generates a current of one ampere. It follows, therefore, from what we have seen in para. 48, that the current in a circuit of one volt and two mhos would be two amperes, with one volt and 50 mhos, 50 amperes ; also by what follows in para. 49 that with two volts and 50 mhos it would be 100 amperes, with 50 volts and 50 mhos 2,500 amperes, and in the same POR ELECTRICAL STUDENTS. 39 way with ^ a volt and J of a mho ^th of an ampere ; so that in all cases the current in a circuit is found in amperes by multiplying the E.M.F. in volts by the conductivity in mhos. This simple and vitally important statement of facts is named, after its discoverer, Ohm's law. Since, however, we have seen in para. 30 that the conductivity of a circuit is the same in mhos as unity divided by the number of ohms in that circuit, we may, to find the current, multiply the E.M.F., not by the number of mhos, but by their equivalent unity divided by the ohms in the cir- cuit's resistance; in other words, divide the E.M.F. by the resistance. Thus we have seen that a circuit containing five volts and of 0"025 mho conductivity would sustain a current of 5 X 0'025, or 0-125 amperes. But the resistance of the circuit would be , that is, 40 ohms, so that we arrive at 0-025 the same result if, instead of multiplying 5 by the conductivity 5 1 0-025, we divide it by the resistance 40, that is — or -, or 0-125. So that, as it is much more usual to speak about a circuit's resistance in ohms than its conductivity in mhos, the common expression of Ohm's law is that the current in a circuit is obtained in amperes by dividing the E.M.F. by the resistance. Thus, in a circuit of 100 volts and 1,000 ohms, the current would be , or 0-1 ampere, and by our previous rule, the conductivity being , that is, O'OOl mho, the current would ^ 1000 be 100 X 0-001, or 0-1 ; of course, precisely the same result. 51. Unit Quantity of Electricity : The Coulomb. — This unit current or ampere is that which transmits the unit quan- tity of electricity in one second. The unit of electric quantity is called the coulomb, and just as the unit flow of water through a pipe might be taken as that which allowed one gallon of water to pass any point in the pipe during one second of time, so the ampere is the strength of current, the rapidity of flow, which allows one coulomb to pass any point in the circuit during one second. So that if a constant current i40 ,■ PRACTICAL NOTES of one ampere has been flowing for 100 seconds in a circuit, then we know that 100 coulombs of electricity have passed any point in the circuit during that time. In the same way a steady current of five amperes flowing for one minute causes a total transmission of 5 x 60, or 300 coulombs, and generally the quantity of electricity in coulombs transmitted by a con- stant current is the product of that current and the number of seconds it has been in flow. 52. The Milliampere. — For convenience in telegraphy a submultiple of the unit, the milliampere, is in very com- mon use ; it is the thousandth part of an ampere. To give an idea of what a milliampere practically is in telegraphy, it may be mentioned that on Morse land line circuits receiving becomes difficult to adjust and sustain when the received working current falls below one milli- ampere, which may be regarded as a minimum working current, while 5 milliamperes — that is, 0005 amperes — is a good average working current. The current which enters the line at the sending end may be greater, and in the case of a long line in wet weather perhaps much greater, than this ; but the difference escapes by leakage at the insulators along the line, just as a quantity of water forced into a long pipe containing many small leaks would be much reduced in amount before its issue at the distant end. 53. Calculation of Strength of Current by Ohm's Law. — The working current in a telegraph circuit whose insulation is good — so that the current escaping by leakage need not be considered — is easily found by Ohm's law. Suppose, for example, that we have an overhead land line 50 miles long, of 13(0 resistance per mile, worked in the manner represented at fig. 17 — that is, on the open circuit principle,where no current passes through the line, except in actual signalling. The stationsA andB are each provided with aMorse key, K,for send- ing the currents, a galvanoscope, G, of SOw resistance, for observing that those currents duly pass, a Morse inkwriter or sounder, M, of 200co resistance, and a battery, E, of 11 volts and 200(1) total internal resistance. The levers of the keys are normally kept back by the spring p on the receiving contacts r. FOR , ELECiraCAL STUDENTS. -.41 Under these circumstances there is a complete circuit from A over the line returning by the earth and both receiving instru- ments, but in this circuit there is no battery. When A. de- presses his key, however, the circuit is completed no longer through the ba^k stop and receiving instrument but through the front stop s and battery E, and a current from the latter therefore flows to line and actuates the galvanoscopes and instrument S. While A's key remains depressed we know from Ohm's law that the strength of current at any point in the circuit — there being assumed no leakage — will be the total B r-r-T-r<. Fia. 17. E.M.F., namely 11, divided by the total resistance, which we can sum thus — A.'s battery B 200a) A.'s galvanoscope G SOco The line A B— Fifty miles at 13a) per mile 650a) B.'s galvanoscope G 30a) B.'s instrument M SOOo) The earth connections and the earth perhaps 1" Total 1,111" The current flowing will therefore be 11 1111' or Y^ith part of one ampere— that is, 0-0091 ampere or 9-1 milliamperes. 42 rilACTICAL NOTES CHAPTEE VI. CUKRENT INDICATORS. 54. General Uses. — The laws of the current, and the relation between E.M.F., resistance, and current having been considered, we shall now examine some of the instruments used to indicate the presence of a current, these being termed detectors or indi- cators, in distinction from galvanometers or current measurers. It will also be shown how detectors ma}'' be calibrated,' so that their indications may be made to show the exact strengths of current passing through them. Detectors are in constant use on land-line telegraph circuits, one being fitted to each Morse instrument, so that when the operator is sending a message he may observe that his current is going out to line all right. And when the distant station calls up, the call is shown on the detector as well as the Morse armature, so that if the latter should fail for any reason there will still be always a visual indication attracting the attention of the operator. Detectors are very useful for tracing a fault on an instrument or line, or for identifying certain wires from a large number. The manner of doing these will be explained as soon as the instrument itself has been considered. 55. Direction of Current. — One of the most useful and common forms is shown in Fig. 18, which will indicate the direction of a current by the needle being deflected one side or the other. It is advisable to ascertain which way the needle moves with a given direction of current through it. FOR ELECTRICAL STUDENTS. 43 as we shall require to know this frequently when testing with the instrument. Suppose that on connecting it to a voltaic cell, as in Fig. 19, the needle moved to the right, then, knowing that the current leaves the carbon plate in the cell and enters the detector by its right-hand terminal, as shown by the arrow, we should know that the needle would always deflect towards the terminal where the current entered, or towards the positive terminal. For if we reversed the wires on the detector, so that the current entered by the left-hand terminal, the needle also would be deflected to the left. Fia. 18. 56. Direction of Deflection. — All instruments do not deflect the same way for the same direction of current through them. The direction in which the needle of an instrument will deflect when a given current is supplied to it depends on three things ; 3. The position of the poles of the needle relatively to the coil when the former is at rest. The direction of winding of the instrument coils. The connecting up of these coils to the instrument terminals. .44 PTLVCTICAL NOTES As an illustration, we might cause the deflection of the above detector in Fig. 19 to bo alvvays towards the negative terminal by altering either of the three conditions mentioned. Cut if we altered the instrument by any two of these conditions, say the position of the needle at rest, and the connecting up of the coils to the terminals, we should not alter the direction of deflection for a given current, since these two changes would neutralise each other. Similarly, if we altered all three con- ditions, we should reverse the deflection. 57. Mounting of Needles in Vertical Detector.— There are two needles to the instiument — one, the smaller, N S, being of hard steel well magnetised and weighted at the end N, this end Fig. 19. being made broader. The use of this is to bring the noodles back to zero after being deflected, or, in other words, to act as a controlling force (viz., that of gravity on the larger end) to the system. Fig. 20 shows the needles and spindle detached from the instrument. The longer needle, ns, which is seen out- side in Fig. 18, has its poles reversed to those of the inner needle K" S. The reason of this is that when a current is passed through the coil a magnetic field of force is produced whose direction of flow is through the interior of the coil and back again round the outside in a complete circuit. Hence the direction of tbe force on a magnetised needle placed inside the coil is opposite to that exerted upon it when placed out- side ; and the needles, which are fixed parallel to each other, must therefore have their similar poles pointing opposite ways, FOR ELECTRICAL STUDENTS. 45 in order that both needles may be deflected in the same direction by magnetic forces in opposite directions. This will be explained more in detail when galvanometers are considered. Both needles should be perfectly parallel, to ensure which each is pierced with a square hole at its centre, which fits over a square key on the brass hubs B B ; and the latter, having a very fine thread cut, permits of tightening up the needles securely in position by a little square-headed nut (shown at A A), which can be screwed on by means of a small pair of pliers. Fio. 20. 58. Mounting of Coils.— Turning now to the interior of the instrument, the two coils are wound on brass bobbins, one of which is shown mounted in position in Fig. 21, the other having been removed to expose the needle, N S, to view. The mounting of the spindle can also be seen, and the freedom of the latter to turn can be adjusted to great delicacy by the screw, T. The interior space of the coils in which the needle moves is very little larger than the needle itself ; which brings as much turning force as possible to bear on the needle when the current is on ; this makes it important to fix the coils 46 PRACTICAL NOTES accurately in position, so that the needle may not touch the sides during any part of its play. The brass bobbins are therefore screwed on to the brass plate P,with little brass pins to act as guides and keep them in position. The screw hole of the removed coil is seen at H and the three guide holes round it. The brass angle pieces B B serve to screw the plate and coils when adjusted inside the wooden box of the instrument. 59. Direction of Winding of Coils.— Eegarding the winding of the coils, it should be remembered that the current must Fig. 21. flow through both coils in the same direction, since the two coils should form virtually one electro-magnet ; in fact, the reason for the electro-magnet being made in two coils is only for facility of construction and for getting at the needle conve- niently. In order to be able to detach each coil separately from the instrument (for convenience of adjustment) the coils are not connected by a wire between them, but the underneath end of each coil is soldered on to the brass bobbin. So that, before starting to wind a coil, the end of the wire is soldered to the brass bobbin, and when the coils are finished and FOR ELECTRICAL STUDENTS. 47 screwed into their places electrical contact is made between the two inside ends of the coils through the brass work. 60. Coils Connected "in Series."— Now, if we intend the coils to be Joined electrically in series, a little thought will show that they should be wound in opposite ways. Referring to Fig. 22, each coil when its winding was commenced had the underneath end soldered at C C^ to the bobbin, the winding on each coil then being in opposite ways. If, now, a current is sent, say from terminal A to B, it wUl flow round the two coils in series, and in the Fig. 22. same direction through each, the points C C^ being connected through the brass work. 61. Coils Connected in "Multiple Arc." — If we required to reduce the resistance of the coils, still utilising both, we should connect them in parallel or multiple arc, their combined resistance being then reduced to one quarter (para. 41). To do this we could not alter the connections of the coils in Fig. 22 without having wires between them to connect them electrically. The inside end of either coil, say that end attached to Cj, would have to be removed and con- nected to the outside end of the other coil, a short wire, w, being used if the inside end exposed was too short (see Fig. 23). 48' PE.iCTICAL NOTES Then the outside end of the first named coil would have a wire connecting it to the brass work at any convenient point, say at C,. Now, passing a current from A to B, it is seen to flow through both coils in " parallel," and in the same direction through each, and meet at the brass work C G^, whence it passes to B. If, when the coils were being wound, they were intended to be used in parallel, we should solder both inner ends to the brass frame, as before ; but the coils would be wound the same way, their outer ends, joined together, forming one terminal, A, and the other terminal, B, being a connection simply to the brass framework {see Fig. 24). It will then be seen that the current traverses both sides simultaneously and in the same direction. 62. Relation between the Sensitiveness of a Detector and its Resistance. — The resistance of the coils is an important point as regards the sensitiveness of the instrument when we come to employ it with batteries of diilerent resistance. For instance, any number of similar voltaic cells connected in series would give the same deflection on the detector as a single one of FOR ELECTRICAL STUDENTS. 49 those cells if their internal resistance was high and the resist- ance of the instrument coils very low. No idea could be formed, therefore, from such a test as to the number of cells connected together. To take an example — a Minotto cell (see para. 17) measures when in actual work about 1 volt E.M.F., and 20a) internal resistance. Now, if wo connect this to a low-resistance detector — say y\ very convenient one for working out problems in magnetic fields. Let a bar magnet (Fig. 69a) be. laid horizontally in th& magnetic meridian, with its north or marked pole pointing northwards, and its south pole due south. The lines of force due to the magnet and the earth are then in the same direc- tion, in a line through the axis of the magnet, and the actual strength of field at any given point p on this line will be the sum of the two above fields at that point. That due to the earth is the horizontal component of the earth's magnetic field, and is denoted by the letter H. Its value is -18 G.-G.-S. units in London — that is, unit pole (para. 108) experiences 'IB dyne force acting upon it horizontally in London. Suppose th& above bar magnet (Fig. 69a) to be 11-06 centimetres long, and each pole to be 82 C.-G.-S. units in strength, the point p rOE ELECTRICAL STUDENTS. 137 being 8-94 centimetres distant from the S pole. "We have, then, at the point p unit magnetic field, for the force on a unit north pole placed at^ would be (para. 108) QO GO (8-94)2 =gQ<3yQes (of attraction) due to pole S, go QO ^°*^ (8-94 + 11-06)2 ^400 ^^""^^ ^°^ repulsion) due to pole N, the net action due to the magnet being, therefore, 80 ~ 15^ "^yoes (of attraction). Fig. 69b. which is equal to •82(li-|) = -82dyno. But the earth's field H is acting on the unit pole at p in the same direction with a force of 'IS dyne; therefore the total action on the unit pole is equal to •82 + -18 = 1 dyne (of attraction). In other words, we have unit magnetic field at p, and a similar reasoning would show that there is precisely the same unit field at p\ a point at the same distance from the N pole. If, now, the bar magnet were turned to the rightabout, but still kept in the meridian (Fig. 69b), the field of the earth would act in opposition to that of the magnet. We should now find that unit magnetic field would exist at a point p, distant from the N pole by 7 '58 centimetres (this unit field being .138 PRACTICAL NOTES opposite in direction to the unit field in the last example). For the force on a unit north pole at^ would be (para. 108) 82 82 (7-58)2 5745 dynes (of repulsion) due to pole N, 82 82 and = dynes (of attraction) due to pole S, (7-58+ 11-06)2 347-45 ^ \ / f , the net action of magnet being, therefore, — — =1-18 dyne (of repulsion). 57-45 347-45 J \ f / Now the earth's field exerts an attractive force on the unit north pole^ equal to '18 dyne. This must, therefore, be sub- tracted from the force due to the magnet to obtain the actual force on the pole at |), which is then 1 -18 - -18 = 1 dyne (of repulsion), that is, we have unit magnetic field at the points p and p'. It will now be understood what is meant by saying that the total intensity of the earth's magnetic field is equal to-47C.-G.-S. units. The total intensity of the earth's field does not come into the calculations with galvanometers, since the needles are not used pivoted in the angle of dip ; all needles moving in a horizontal plane being acted upon or controlled by the earth's horizontal field H. It is evident that, knowing the value of the total force of the earth in the direction of the dipping needle, and the angle of inclination which this force makes with the horizontal, we can resolve it into two compo- nent forces, one acting horizontally, the other vertically Kesolving it, we have the horizontal component = total intensity x cosine of angle of dip. Thus in the year 1879 the total intensity in London was •4736 C.-G-.-S. units, and the angle of dip was 67° 42'. From this we get the horizontal component = -4736 X cosine 67° 42' ..= -4736 X -37945 = -1797 C.-G.-S. units. FOK ELECTRICAL STUDENTS. 139 Similarly resolving for the vertical component we have the vertical component = total intensity x sine of angle of dip. Therefore for the same year the vertical component was = -4736 X sin 67° 42' = -4736 X -9252 = -4381 C.-G.-S. units. The value of the horizontal magnetic field of the earth is on the increase, and is now equal to '18 unit in London, while the total intensity is decreasing gradually in value. 110. The Magnetic Meridian. — At any given place on the earth's surface the magnetic meridian is the vertical plane (through that place) in which the total intensity of the earth's magnetic field is a maximum. Of the two component inten- sities, the horizontal and the vertical, into which, as before shown, the earth's field may be resolved, the former is neces- sarily a maximum in the same vertical plane as the total intensity, viz., in the magnetic meridian; hence the direction of this meridian may be seen by the direction in which a horizontally-pivoted magnetic needle sets itself, while the latter component, viz., the vertical, is the same in intensity in all vertical planes at one place. If, for example, we take a vertically-pivoted magnetic needle, whose horizontal axis of motion is through its centre of gravity, and move it round bodily, so that the vertical plane in which the needle moves is gradually altered until a vertical plane is found, in which the needle comes to rest in an absolutely vertical position, we know that the plane so found is at right angles to that of the magnetic meridian, because it is only in this plane that the horizontal intensity could be nil, and the vertical alone act. Having found this plane in the foregoing manner, we know that the magnetic meridian is in a plane 90 degrees removed from it. This method of determining the plane of the mag- netic meridian is employed at Kew Observatory, in order to place the "dipping needle" correctly in that plane before taking readings of the angle of " dip" or " inclination." Full descriptions of the instruments employed, and the methods of 140 PRACTICAL NOTES measuring the horizontal intensity, and the angles of inclina- tion and declination, with the corrections to be taken into account, will be found in Gordon's " Physical Treatise on Electricity and Magnetism," Vol. I., and in Stewart and Gee's "Elementary Practical Physics," Vol. II. The plane of the magnetic meridian does not always coincide with that of the geographical meridian as determined astronomi- cally. From data accumulated since 1576 it is found that in London the magnetic and geographical meridians coincide about every 320 years. Since their coincidence 231 years ago the magnetic meridian became more and more west of the true meridian, until, after a period of 160 years, a maximum was reached, and it then commenced to return. The angle between the two meridians is called the angle of dedinatwn. The latest determinations of the magnetic elements for the completion of the year 1887 have been kindly supplied to the writer by the Superintendent of the Kew Observatory, and are as follows : Mean Declination, 18° 7' west; Mean Inclination, 67° 37'; Mean Horizontal Intensity, -1810 C.-G.-S. unit. It will also be understood that the intensity of the hori- zontal magnetic field varies according to the latitude ; for instance, on the magnetic equator, or line of no dip, the hori- zontal intensity is equal to the total intensity, and the further north or south we go the weaker does the horizontal intensity become, the difi"erence in its value being noticeable even in Great Britain, where, for instance, its value in the town of Inverness is -15 C.-G.-S. units, while in London it is -18, the difi'erence of latitude being about 6 degrees. 111. Uniform Magnetic Field. — A magnetic field is said to be "uniform" when the force exerted on a magnetic pole of given strength is the same at all points in that field. Throughout the space described by the movement of any given magnetic needle suspended or pivoted horizontally, as in galvanometers and indicators, we may regard the earth's FOE ELECTEICAL STUDENTS. HI horizontal field H as absolutely constant in value ; in other words, such a needle is suspended in a " uniform " field, and the force exerted on its two poles would be the same for all positions it could possibly take up. The number of lines of force per square centimetre would also, by definition, be constant at all points in a uniform field, and therefore the lines of force would necessarily be parallel lines. 112. Comparison of the Intensities of Magnetic Fields.^ The relative intensities of two or more magnetic fields, or the relative strengths of field at two or more different points in a magnetic field which is not uniform, may be determined by the method of vibrations. A small magnetic needle, horizon- tally suspended or pivoted, is placed successively at the points Fig. 70. where the magnetic fields are to be compared, and after having been set oscillating, the number of oscillations made in a given time are counted. The cause of the oscillations is evidently the magnetic force (equal to the product of the strength of one pole of the needle and the strength of the field) acting at both ends of the needle, and tending to bring it to rest in a line parallel to the lines of force of the field. If there were no force there would be no oscillation. We can observe this by annihilating the force acting on the needle. An ordinary compass needle, if turned round out of the magnetic meridian and then allowed to swing, will continue to oscillate about the meridian till it finally comes to rest in it. The force causing these oscillations can be weakened or anni- hilated by destroying either of the two factors composing it. That is, we may do away with the force by destroying either 142 PEACTICAL NOTES the magnetic field or the strength of the needle poles. If the needle ns (Fig. 70) is lying in the magnetic meridian, the earth's field H (shown by the straight arrows) may be neu- tralised by approaching a magnet, N S, which sets up in the neighbourhood lines of force (shown by the curved arrows) opposite in direction to those of the earth. As the magnet is approached, if the needle is turned round its centre and then allowed to oscillate back, it will be found that the oscillations become more and more sluggish, and a point will at length be reached when the needle will not oscillate at all, but remain in any position it is turned to. This principle of weakening the earth's directive force on a needle by means of a magnet placed, as shown, in the neighbourhood is employed sometimes with galvanometers, where it is required to render the instru- ment more sensitive, and increase the amplitude of the deflec- tions with weak currents. Or, again, we may destroy the force by demagnetising the needle — that is, doing away with the other factor on which the force depends. If the needle is demagnetised completely, as may be done by placing it within a helix, or coil of insulated wire, through which an alternating current is passing, or what is the same thing, if, instead of the magnetised needle, we substitute a piece of hard steel unmagnetised, we find that it will remain still in any position in which it may be placed ; in other words, there is no force acting on it, and therefore the needle does not oscillate or take up a definite position. We have a mechanical example of this when a body is suspended vertically at the end of a length of fine thread. The top end of the thread being fastened to some fixed point, if the body is raised a little with the thread taut and then let go we have a pendulum action, and oscillations take place till the pendulum finally comes to rest in the vertical line of the earth's gravita- tion field of force. In a similar manner to the magnetic needle above, we have here a force acting on the body, causing its oscillations, and composed of two factors, viz., the mass of the body and the acceleration of gravity. If we could destroy either factor, oscillation would cease, in consequence of the absence of any FOR ELECTRICAL STUDENTS. 143 force. The mass of the body could only be dispensed with by removing the body altogether from the thread, which would then do away with the force and the resulting oscillations. We could not, however, neutralise the effect of the acceleration of gravity by a counter field of force in the same manner as a magnetic field. Incidentally, here, attention may be drawn to the analogy between a mechanical and a magnetic field of force. Mass in the one is analogous to strength of pole in the other, and acceleration in the one (force divided by mass) is analogous to intensity of field in the other (force divided by pole strength). If there could be such a thing as a separate pole it would move along a line of force in a uniform magnetic field at a constantly increasing velocity or acceleration. We now come to the well-established law of compound pen- dulums, viz., that the value of a force causing such oscillations (when these are of small angle) is proportional to the square of the number of complete oscillations in a given time, or inversely proportional to the square of the time during which a given number of complete oscillations are made. The time of a complete oscillation or "period" is taken from the instant the needle passes a given point to the instant it reaches the same point again, moving in the same direction. Suppose the intensities at two points, A and B, in a magnetic field are to be compared. If at A a magnetic needle makes, say, 60 oscillations per minute, and at B the same needle makes 20 per minute, the fields at those points are respectively in the proportion of 60^ : 20^ — that is, as 9 to 1. 113. Comparison of the Strengths of Poles of Magnets. — By the same method we may compare the strength of pole of two or more magnets. Let a short magnetic needle vibrate freely over a fiat surface till it comes to rest in the magnetic meridian , then draw a straight line on the flat surface in the direction of the needle's length. This line will be the direction of the magnetic meridian. Now turn the needle round its centre away from the meridian, and let it oscillate. Count the number of oscillations, say 4: in 12 seconds — that is, 20 per minute in 144 PEACTICAL NOTES the earbh's field alone. Now, suppose we have two bar magnets, A and B, of equal length, whose pole strengths are to be compared. Place A on the line marked out, with its north pole pointing towards the south pole of the needle ; the field of the magnet is then added to the earth's field H, and the needle oscillates quicker, say 40 per minute. We have here two fields superposed in the same direction in the region of the needle, each field being proportional to the square of the oscillations in a given time. The earth's field is then proportional to 20^, and the two fields together proportional to 40^ ; therefore the field produced by the poles of the bar magnet in the region of the needle is proportional to the difference of these squares = 40^-201 Now remove A altogether, place B in exactly the same position, and count the oscillations, say 30 per minute. In a similar manner the field in the region of the needle due to the poles of B is proportional to 30^ - 20-. Now, referring to Fig. 69a, it has been shown (para. 109) that the intensity of field duo to the poles of a bar magnet at any point 2) on a line passing through the two poles, distant from the nearer pole say r centimetres, and from the further pole say E <;entimetres, is equal to m 'Q^-W)^-^--^- '''''''' where m is the strength of one pole of the magnet. Now, the ■centre of the vibrating needle being at the point p the oscilla- tions take place under the action of the above field. The quantity _ _ — is constant for all bar magnets of equal length placed in the same position each time oscillations are observed, and hence the magnetic fields of the two magnets A and B at the centre of the vibrating needle are proportional to their respective pole strengths, and also to the square of the number of oscilla- tions in a given time. Hence we have pole strength of A _ 40 ^ - 20^ _ 1200 _ 2-4 . pole strength of B 30^ - 20^ "SOO" T ' rOK ELECTRICAL STUDENTS. 145 or, making the calculation by the inverse ratio of the square of the time during which a given number of oscillations are made, we should have — 20 oscillations due to earth in 60 seconds 20 „ „ „ „ and A „ 30 „ 20 „ „ „ „ and B „ 40 „ , field due to A and earth_602_4 _ field due to earth 30"^ ~ 1 ' and therefore field due to A ^^.^^g^ field due to earth Similarly field due to B ^60;_^^5. field due to earth 40^ 4 and, finally, the distance of the magnet poles of A and B from the needle being the same, the fields produced are proportional to their pole strengths. Hence — pole strength of A 3 2"4 , , ^ — ■ ^.. , p = T" = -7- ^^ before, pole strength of B 5 1 4 If the magnets are not of equal length, the distance r between the vibrating needle and the nearer pole is kept the same for each magnet, while the distance E (the magnet's length added to r) is a variable amount for each magnet. The- field for each magnet is its strength of pole multiplied by the- quantity i^-i^, and therefore the pole strength is proportional to the number of oscillations in a given time divided by this quantity. If the magnets whose pole strengths are to be compared are very long the term — - is very small compared to -, and may therefore be neglected, and we return to the simple propor- tion, as at first, that the strength of pole varies as the number of oscillations squared. 146 PRACTICAL NOTES The vibrating needle must be short, to keep it in a uniform portion of the field during its movement. For near to the magnet the field is variable within very limited areas, while at greater distances it is fairly uniform throughout the area moved over by a short needle, and hence it is better to take the oscillations with the vibrating needle at some distance (say not less than 5 centimetres) from the magnet pole. The kind of magnetic needle best suited for oscillation ex- periments is a short thick one — a piece of thick steel knitting needle not more than two centimetres long does very well — which can be magnetised in the ordinary way by stroking it Fig. 71, over the pole of a permanent magnet, or by wrapping a layer of insulated wire round it and passing a current through the same, tapping the needle during the operation. If it is sus- pended, a convenient stirrup for it may be cut from a visiting card in the shape shown in Fig. 71, bending the ends over at the dotted line, then piercing a hole through the four thick- nesses of card and tying to it a fibre of unspun silk. Before the needle is put in the stirrup a somewhat heavier rod of non-magnetic metal, such as brass, should be hung therein and left for some time until all torsion is removed from the fibre. This will be the case when the rod comes to rest and remains so. The fibre should then be turned slightly at its upper point of suspension till the length of the rod lies in the mag- FOE ELECTRICAL STUtiENTS. 147 netic meridian ; the stirrup is then held while the brass rod is removed and the magnetic needle inserted in it with its poles the right way. The needle is then lowered till it is just above a flat horizontal surface, on which a "card is placed with a black line drawn in the direction of the magnetic meridian. It is quite easy then to count oscillations every time the needle passes the black line going in a given direction. For greater delicacy a small circular mirror may be attached by beeswax to the fibre just above the stirrup, and a ray of light projected on to it from a lamp placed about two feet away, the lamp being surrounded by an opaque enclosure in which a narrow vertical sht is cut opposite the flame to emit the beam on to the mirror. The reflected ray thrown on to a white screen then shows the oscillations of the needle, and since these can be seen oy this means when of very small angle, the arrangement is very sen- sitive, the meridian line for counting the oscillations being then marked vertically on the screen. 114. Action of a Magnetic Field on a Magnet. — It has been mentioned before that when a magnetic needle free to turn about its centre is placed in a magnetic field, it immediately sets itself in the direction of the lines of force in that field — that is, the direction of its own internal ' lines coincides with the direction of the lines of the field at the point where it is suspended. It is important to understand fully this action, as it immediately concerns the cause of the movement of galvanometer needles when the field by which they are surrounded is altered by starting a current in the coils. Let us consider a magnetic needle free to move in a hori- zontal plane in the uniform magnetic field of the earth. It will set itself with its marked or north end towards the north pole of the earth, as in Fig. 72, where the straight lines represent the uniform lines of force of the earth's horizontal field H, and the arrow inside the magnet represents the direction of its own internal lines of magnetisation. The force acting at each end of the needle keeping it at rest in this position is equal to the product of H into the strength of the needle pole, m; that is, L 2 148 PRACTICAL NOTES the force at each end of the needle is equal to H m. This force is shown in dotted arrows at each end of the needle, and it Fio. 72. ■will be seen that when the needle is in this position (the meridian) there is no turning force exerted on it. But now. Fig. 73. if we turn the needle through a right angle, as in Fio-. 73^ we shall have to exert some force to keep it there ; in other FOR ELECTRICAL STUDENTS. 149 ■words, the force H wi at each end is now tending to turn it ^ack to the meridian. 115. "Moment" of a Force. — The capability of a force to produce movement, or turning about a centre, is termed the '' moment " of that force. The moment of a force about a point is the product of that force into its perpendicular distance from the point — that is, the distance from the point to where the force is applied must be reckoned along a line perpen- dicular to the direction of the force. To make this clear let us take a rigid bar, A B, capable of turning about its centre, C, iit one end of which a force equal to F lbs. is pulling (Fig. 74) Fig. 74. in a direction makirg an angle of a degrees with the bar. Let the bar be L feet long ; the length A will then be — feet. Draw C D perpendicular to the direction of the force. The " moment" of the force about the point C is then — F X perpendicular C D. Now we can find the length of C D by a simple trigono- metrical ratio. (It may be mentioned here that the only trigonometrical functions of angles made use of in this treatise will be the three most frequently met with — viz., the sine, the cosine, and the tangent. The student should familiarise him- self with these simple functions of angles. A table of natural «ines and tangents will be required for reference. The cosino 150 PEACTICAL NOTKS of an angle may be found from a table of sines by subtracting the angle from 90 and taking the sine of the remaining angle.) CD Now we have the ratio — p^ = sine a ; and therefore AO CD = ACsina= — sin a (feet); whence the moment of the force is equal to F X — sin a (pound-feet). For example, if F = lOlbs. and L = 10ft., and the angle = 30', we have moment ofF=10x — x- = 25 pound-feet (where sin 30°=-\ 2/ It should be observed here that since sin 90° = 1 and F T sin 0° = the moment of F is when the force acts at 2 right angles to the bar, and is nil when acting in the direction of the length of the bar. Hence a given force acts at the greatest advantage as regards turning a lever, or, in other words, its moment is greatest, when it acts at right angles to the direction of length of the lever. When we, for instance, exert force ourselves in order to move a heavy lever, such as the opening or closing of the gates of a river lock, we in- stinctively use our strength to the best advantage by pushing at the extreme end of the lever and at right angles to it. Taking the above figures with the force acting at 90deg., the moment of F becomes — - — = 50 pound-feet. Suppose, now, that an equal force, parallel to the first, acts at the other end, B, of the lever (Fig. 75), so as to turn it round in the same direction about the centre C. Drop the line C D' perpendicular to the direction of the force, FOR ELECTRICAL STUDENTS. 151 CD' L as before. Then — — - = sin a, or C D' = — sin a (angle C B D' OB 2 = a), and therefore the moment of the force F acting at B is FL . — sm a, 2 which is the same as the moment of the force acting at A. Now, as both these forces tend to turn the lever the same wsk^ round we must add their moments to obtain the moment of Fig. 75. the two together, and as their moments are equal, the moment, of the two will be twice that of one — that is, 2 ( sina ) = F L sin a. For example, with equal forces of 101b. each, acting at each, end of the lever at an angle of 30deg., the moment of the two forces combined is 10x10x1 = 50 pound-feet. If the forces were acting at right angles to the lever, as in Fig. 76, the combined moment would be simply .' F L = 10 X 10 = 100 pound-feet. 152 PRACTICAL NOTES 116. "Moment" of a "Couple." — When there are two equal parallel and opposite forces acting on a lever, each force tend- ing to turn the lever in the same direction, we have what is termed in mechanics a " couple," and by what has been said above it will be understood that the " moment " of a " covple " is equal to one of the forces multiplied by the perpendicular dntance between them. The "moment " of the "couple " in Fig. 75 is F X the perpendicular DD' = F x L sin a, as given above. The "moment" of the "couple" in Fig. 76, where the equal Fig. 76. forces act at right angles to the lever, is F L. The unit " moment " in the C.-G.-S. system is the dyne-centimetre — that is, the moment of a force of one dyne acting at a perpendicular distance of one centimetre from the turning centre of a lever would be of unit value. Similarly, half a dyne acting at two centimetres distance, or ten dynes acting at one-tenth of a centimetre distance, would be of unit moment. 117. The "Couple" due to the Earth's Field. — Eeturning now to the magnetic needle (of strength of pole = m) capable of turn- ing about its centre in the uniform magnetic field H, we know that the mechanical force acting at each end of the magnet is equal to H m dynes. These two forces then constitute a " couple," and we can now determine the moment of the couple FOR ELECTRICAL STUDENTS. 153 exerted upon the magnet in any position in which it may be placed in the field. For example, when the magnet is in the meridian (Fig. 72) there is no moment at all, although the forces are acting at each end of the magnet. There is no per- pendicular distance between their directions, since they are acting in the same straight line, like the opposite sides in a " tug of war." Now, when the magnet is at right angles to the lines of force of the field (Fig. 73) the moment of the ■couple is a maximum, and is equal to Urn I dyne-centimetres, where I is the length of the magnet expressed in centimetres. Fig. 77. If, however, the magnet is in some intermediate position (as in Fig. 77) the moment of the couple is H m X the perpendicular B D. Now, the two angles marked a are equal to each other, and indicate the angle between the needle and the meridian, and we have the relation B D = ? sin a ; hence moment of couple = S m Z sin a. Again, if the needle is displaced from the meridian more 154 PRACTICAL NOTES than a right angle (as in Fig. 78), and we still call a the angle between the needle and meridian, we have moment of couple = H m x B D = H m Z sin a, the same as before. Hence in whatever position the magnetic needle is placed, if we call a the angle by which it is displaced from the meri- dian, -we have the moment of the couple tending to restore it to the meridian equal to H m Z sin a. Fig. 78. Example. — A magnetic needle whose strength of pole is 20 C.-G.-S. units and length 6 centimetres is moved 60 degrees away from the meridian ; what is the moment of the couple tending to restore it to the meridian (H = '18 C.-G.-S. units, sine of 60 deg. = -^^ 3 Moment of couple = 'IS x 20 x 6 x V3 = 18-66 dyne-centimetres. What is the moment of the couple when the needle is 90 degrees from the meridian ? Jns. -18 X 20 X 6 = 21-6 dyne-centimetres. FOR ELECTRICAL STUDENTS. 155 What mechanical force must be applied at right angles to this needle and at one end of it to keep it in equilibrium at 60 degrees ? Here the moment of the force F applied at one end is F -, and this must equal the moment of the couple due to the earth, which was found above to equal 1866 dyne-centimetres. Hence Fx- = lS-66, 2 and F = 6-2 2 dynes. If this force was applied by attaching a thread to one end of the needle, passing it over a small frictionless vertical pulley, and hanging a weight on the end of the thread, what weight would this have to be 1 Weight in dynes = mass x acceleration of gravity = mass in grammes x 981 and this must equal the force of 6-22 dynes. Therefore mass required = -— . = •00634 scramme. Now the gramme mass is equal in weight to 15-43 English grains. Hence weight required = -00634 x 15-43 = — grain nearly. 118. The "Moment" of a Magnet. — Looking at the expres- sion, just derived, for the " moment of the couple," acting on a magnet when placed in a magnetic field, it will be noticed that the factors " m " and " I " (strength of pole and length between the poles of magnet) are properties belonging to the magnet itself, and, moreover, the moment of the couple exerted on the magnet depends directly upon the product " m I." For these reasons, the product "ml" is designated the " moment of 156 PRACTICAL NOTES the magnet," and is usually denoted by the letter M. Putting this in the above expression, we have the moment of the couple exerted on a magnet of moment M, when placed in a uniform magnetic field H, is H M sin a, where a is the angle between the magnet and the direction of the lines of force of the field. We are here considering the poles of the magnet as situated at its extreme ends. Although this is not strictly the case, the poles being a little nearer together than the distance between the ends, we may practi- cally regard them as being at the ends, and not introduce any appreciable error. Whatever shape the magnet has, its moment is always the strength of one pole (m) multiplied by the distance between the poles (/), irrespective of what length of metal actually exists between the poles. For example, the moment of a semi-circular magnet is the strength of pole multiplied by the distance between the poles, viz., the diameter of the circle of which the magnet forms part. Or if we sup- pose a circular steel ring with a very short piece cut out, so as to form very nearly a complete circle, such as a piston ring, to be magnetised, the moment is very small, although the ring may be strongly magnetised, because the distance between its poles is very small ; and if the magnetic circle be completed by inserting a piece of soft iron in the gap between the poles, there will be a continuous magnetic circuit through the mass of the metal, and the magnetic moment will be reduced to zero, because there no longer exist any free poles. 119. Two Uniform Magnetic Fields Superposed at Right Angles to Each Other. The Tangent Principle. — It is of great importance in all measurements of magnetic fields and in the measurement of electric currents by means of galvanometers to comprehend the resultant action on a magnetic needle sus- pended in a region where two magnetic fields, whose directions are at right angles to one another, are superposed. The fol- lowing example will be easily understood. Take a bar magnet, FOE ELECTRICAL STUDENTS. 157 N S (Fig. 79), and place it horizontally at right angles to the magnetic meridian. The magnetic field surrounding it is indicated generally by the dotted lines. It will be observed that where these lines cross the lines of force H of the earth's horizontal field we have within a limited space two magnetic fields superposed at right angles. Considering the line through the centre of the magnetj it will be noticed that near the region of the magnet the lines H z:^:^^ h'-.. •-, \ \ \ ; ,-' .■H Fw, 79. of force form small circles, and therefore that portion of a circle which crosses the earth's field H can only be regarded as straight or uniform for a very short length. As we take points further removed from the magnet, however, the circles of force are larger, and that portion of them which cuts H may be taken as straight or uniform for a greater length. At greater distances away from the magnet, however, the intensity of the field becomes lessened. Now at any given point in this magnetic field there is a resultant direction of the Unes of force and a resultant intensity. 158 PRACTICAL NOTES which is, in fact, the resultant of the two forces due to the earth and the magnet acting on unit pole at that point. We shall, however, for the present only consider the resultant intensity and direction of the field at points where the two component fields may be considered at right angles to each other. Let us consider a unit pole placed at the point o (Fig. 80) under the influence of two magnetic fields of H and f units of intensity (para. 109 and 111). Now, these two fields will act on the unit pole with forces equal to H and / dynes Fig. 80. (para. 114) in their respective direction?, and we may find their resultant in direction and magnitude by the parallelogram of forces. Let the length of the lines H and / be proportional to the respective strengths of the two forces. On completing the parallelogram we get the resultant force E by drawing the diagonal, and if we can determine the angle a, we shall ascer- tain the direction of this resultant. From the parallelogram it is seen that tan a = i^, H and, therefore, the strength of the two fields, or the relation FOR ELECTRICAL STUDENTS. 159 between them, being known, we can determine the direction of the resultant field. This is, in fact, the direction of the lines of force due to the two fields H and/, and a freely suspended magnetic needle will always set itself in the direction of the lines of force of a field when the latter is uniform — that is, when the lines of force are parallel in the region of the needle, which is what we are now considering. If the lines of force are in curves, the needle will set itself as a tangent to the curve. Experimentally, therefore, we may observe this resultant direction by setting up a horizontally-suspended magnetic needle, and first allowing it to come to rest under the influence of one of the fields, say H, the earth's horizontal field, and then bringing into action the superposed field / by placing the magnet N S (Fig. 79) at right angles to the meridian line on which the magnet is suspended. The tangent of the angle moved through by the needle is then equal to the ratio of the two fields, i.e., f : H, and if we know the strength of one of those fields, say that of H, we can find that of the other (/) by / = H tan a. This is the tangent principle which comes so frequently in use in the measurements of magnetic fields ; not only those fields due to permanent magnets, but also electro-magnets, solenoids, and galvanometers. We shall, therefore, refer again to this principle, as proved above, when we have these other magnetic fields under consideration. Example. — A very short magnetic needle is suspended by a long torsionless silk fibre, so that it is free to move in a hori- zontal plane. A long light non-magnetic pointer is attached to it, the end of which moves over a scale graduated in degrees, by means of which the angle moved through by the needle may be observed. As soon as the needle has come to rest in the earth's magnetic meridian, a bar magnet is placed at a given distance from the centre of the needle, its position being such that its length is bisected by and is at right angles to the meridian line through the needle. The latter is then observed by the pointer to move through 60 degrees. What 160 PRACTICAL NOTES is the intensity of the field (/) at the centre of the needle due to the bar magnet t Answer. /= H tan 60 = -18 X 1-7321 = -31177 C.-G.-S. units. Similarly, the intensity of the resultant field may be deter- mined from the same parallelogram (Fig. 80), from which it is evident that H _ / resultant field (R) = cos a sm a the intensity of which in C.-G-.-S. units may, therefore, be found when the strength of one of the fields and the direction of the resultant field, or the angle a of deflection, are known. Example. — In the last example what is the intensity of the resultant field — (1) Having given that H = -18 C.-G.-S. units ? XT .-1 Q Answer. Intensity = ■ = — = -36 C.-G.-S. units. ' cos 60 -5 (2) Having given that/= -31177 units 1 Answer. Intensity = -J- = '^^lllL = -36 C.-G.-S units. ' sin 60 -866 It follows, therefore, that the lines of force surrounding a. magnet cannot be symmetrical throughout, since they are really the resultant lines of two fields, that of the magnet and that of the earth. If we only consider the horizontal field of the earth and lay the magnet do'wn horizontally, say at right angles to the meridian, as in Fig. 79, we should not expect to find its magnetic field perfectly symmetrical about its own figure, as drawn in that diagram. When, however, the field is examined by the iron filing method it does appear quite symmetrical, simply because the want of symmetry is only apparent oiitside the area of field made visible by the filings. What the filings reveal to us is really the resultant field, but the earth's field is so weak in comparison to the field in FOR ELECTTKICAL STUDENTS. 161 close proximity to the magnet, that the resultant direction is practically that of the magnet's field. It is interesting, how- ever, to detect by a somewhat more delicate means the state of the field from points near the magnet up to distances so far away that the field is no longer apparent. This may be done very well by observing the angles of deflection of a small compass needle placed in succession at various parts of the field. In Fig. 81 is shown a field so analysed along one direc- tion, viz., the earth's meridian. The small arrows represent the approximate direction of the compass needle when placed at those points, the arrow heads being the needle's north pole, and therefore representing by the direction of the arrows the direction of the lines of force. At a distance from the magnet ■•-.s.' 4444A-WV^^:*'=''-«- H < ! I ■ . Fio 81. equal to about six times its length, the needle took up, as nearly as could be observed, the direction of the meridian, showing that the field of the magnet was at this point prac- tically nil. The dotted curves close to the magnet NS cover about the area of the field usually indicated by iron filings, the field appearing symmetrical within these limits. A line was drawn on a flat surface corresponding with the mag- netic meridian, and divided out into inches. The magnet was then laid down at right angles to the line, being bisected by it. The small compass needle was then placed at each inch mark along the line, and the directions taken up by the needle noted down as nearly as possible as shown in the figure. At the point where the compass needle stood at 45 degrees the two superposed fields were equal, because tan 45° = 1, and there- f fore ^ = 1- 162 PRACTICAL NOTES 120. The Magnetic Field of a Magnet in Terms of its Moment. — Let us suppose a unit north pole at the point n (Fig. 82) on a line bisecting a bar magnet, N S, whose length is I centimetres and strength of each pole m C.-G.-S. units. The unit pole will be acted upon by two forces, one of attraction and one of repulsion, in the direction of a line drawn from S to m and N to n respectively. The magnitude of each of these forces will be — (para. 108), r being the distance between the Fig. 82. unit pole and either pole of the magnet. Now, the resultant of these two forces will be the actual force on the pole, and will be numerically equal to the intensity of field (f) at the point n. On completing the parallelogram, and drawing the resultant/ we find by the simple proposition of similar triangles that I : r: J ■ 1^ whence /._ ml _ FOR ELECTRICAL STUDENTS. 163 and since mlia the moment (M) of the magnet (para. 118), /=^ in C.-G.-S. units. Example. — (1) What is the intensity of the magnetic field of a bar-magnet at a point 20 centimetres distant from either pole, the moment of the magnet being 800 dyne-centimetres ? Answer. — Intensity = — — = J^ C.-G.-S. unit. (2) What is the intensity of the magnetic field due to a bar magnet at a point 30 centimetres away from the centre of the magnet in a direction perpendicular to its length, the strength of each pole of the magnet being 900 C.-G.-S. units and its length 20 centimetres ? Here the direct distance (r) from either pole to the point in the field is not given, and must be calculated from the data given. It will be seen (Fig. 82) that the distance r forms the hypothenuse of a right-angled triangle, of which the other two sides d and ^ Z are known, viz., 30 and 10. The length r is therefore the square root of the sum of the squares of the other sides (Euclid I., 47) — that is, r= 7302-1-102= 71000 = 31-62 centimetres, and the moment of the magnet = ml = 900 x 20 = 18,000 dyne- centimetres. Therefore the intensity of the field at the re- quired point is M^J8000. ^18000^. 539 (3.-G.-S. units. r^ (31-62)3 31620 If the point at which the field is calculated is, at a perpen- dicular distance from the magnet, equal to many times its length, we may take the hypothenuse (r) or pole distance from the point as being equal to the actual perpendicular distance, and obtain a very near approximation. This could not be done in the M 2 164 PRACTICAL NOTES last example without introducing an error of 17 per cent,, the ratio of the perpendicular distance to the length of magnet being only as 3 ; 2. Let us now measure the field at a point situated on the con- tinuation of the axis of the magnet, d centimetres from its centre, r centimetres from the nearer pole, and E centimetres from the further pole, supposing that a unit north pole is placed at n at the point required (Fig. 83). It is evident that the two poles of the magnet act oppositely on the unit pole, the nearer pole (N) having the greater efi'ect, and exerting a force of repulsion equal to -^ dynes on the unit pole. Fio. 83. The other pole (S) acts with a force of attraction equal to — ^ dynes, and the net or resultant force acting at n will be the difierence of these two, viz. : — m _m W W this being also equal numerically to the intensity (/) of field at the point. The direction of the resultant is from the magnet in a line with its axis in the diagram. If the magnet were reversed the direction of the resultant would be reversed, its intensity remaining the same. Now, r = d-~ , 2 and E = cf + -. 2 Putting these values in the numerator of the above, we obtain, after cancelling, r^2mld FOR ELECTRICAL STUDENTS. 165 where mlis the moment (M) of the magnet. Therefore, /=2M ^ . Example. — What is the strength of field at a point on the continuation of the axis of a magnet 30 centimetres (d) from its centre, length of magnet 20 centimetres, and pole strength SOO C.-Gl.-S. units ? Moment of magnet = 900 x 20 = 18,000 units. Distance from nearer pole (r) = 30 - 10 = 20 cms. „ „ further „ (E)= 30+10= 40 „ Hence, Intensity of field = 2 x 18,000 x 30 20^ X 402 36x3 64 = 1-68C.-G.-S. units. Similarly, we may arrive at an approximate value of the field by a little simpler process (when the point is distant from the magnet many times its length) by reducing the formula to the following form (substituting the values of r and R and redu- cing by simple algebra), 2M from which it is evident that when d is great compared to I, the quantity within the bracket approaches unity, and we have . 2M Example. — What is the strength of field at a point situated 2 metres away from the centre of the magnet in the last example ? Here we may adopt the approximation, as the quantity in the bracket is close upon unity, and we have Intensity = 'l2L^^= -0045 C.G.-S. units. 166 PRACTICAL NOTES 121. Polar and Normal Lines of a Magnet. — In a similar manner the magnetic intensity at points in the field lying in other directions may be found by the application of the parallelogram of forces and geometrical principles ; but the two directions in which we have investigated the strength of field are those in which measurements of the magnet and its field are taken, and, therefore, are enough for our purpose. For conciseness we shall hereafter designate these two directions the polar line and the normal line of a magnet. The Fig. 84. polar line will be understood to mean a straight line of indefinite length passing through the two poles of a given magnet, while the normal line will mean a straight line of indefinite length at right angles to the polar line, and which also bisects the distance between the two poles. In Fig. 84 three common forms of magnets are shown, their polar lines being dotted, and normal lines shown plain. 122. The Magnetometer. — We have already seen (para. 119) that when two magnetic fields of intensities / and H are super- FOR ELECTRICAL STUDENTS. 167 posed at right angles to one another at a given point, the ratio ^ IS equal to the tangent of the angle between the direction of the field H and the resultant field — that is, the tangent of the angle moved through by a short magnetic needle from its position of rest in the field H. To measure this ratio, then, we require a suspended magnetic needle moving over a scale of degrees by which its movement may be observed. The conditions which are required in such an instrument are 1. Short, well-magnetised steel needle. 2. Long light pointer or mirror attached to same. 3. Long unspun silk fibre to suspend needle and pointer (the length being required to eliminate error due to stiifness or rigidity of fibre). 4. Glass covering over needle and suspension to avoid dis- turbance by currents of air. 5. Scale divided into natiural tangent divisions, 6. Means of accurately levelling. An instrument for the above purpose is known as a " magnetometer," and the student will find it an excellent introduction to the theory and use of galvanometers to construct a magnetometer of the simple design here de- scribed, and carry out some measurements of magnetic fields with its aid. The following will be found quite simple to make : — A glass funnel, F, of thin glass (Fig. 85) stands inverted on a flat wooden baseboard, B, which should be provided with levelling screws ; or a glass flask, with its lower half neatly cut off, answers the same purpose. Through the cork in the neck at the top is pierced a piece of No. 12 copper wire, bent into a ring at its upper end, so that it may be turned round with the finger and thumb, the lower end being filed flat, and a small hole drilled, to which the silk suspension may be tied (Fig. 86). This piece of wire should fit tight in the cork, so that when it is once adjusted it will not slip. The lower end of the silk is then tied to the stirrup. To make the latter, stifi" note paper does very well, cut to the shape in Fig. 87, the ends being 168 PRACTICAL NOTES folded over as shown to make it stronger, where holes are pierced to which to tie the silk. The three lines across the centre indicate where it should be folded to form a narrow groove at the lower part of the stirrup in which to place the long pointer. This ensures the pointer being in the same line as the needle. The diagram also shows the needle and pointer as mounted in the stirrup. A more durable stirrup may be made in the same way with copper or lead foil. Before mounting the needle and pointer Fig 85. Fig. 86, a brass or copper rod about 6in. long should be inserted in the stirrup, balanced and left for some time till it comes to rest. During this operation the inverted funnel may be lifted off its stand and placed with its edges resting on two supports, so that the hands may conveniently reach underneath to the lower end of the fibre. This process will remove all twist from the fibre. Now the top of the fibre should be turned by turning the ring r until the length of the brass rod lies in the magnetic meridian, ascertained by its direction coincid- ing as nearly as possible with that of a compass needle FOR ELECTRICAL STUDENTS. 169 placed close to the instrument. The brass rod may now be removed by the right hand, while the stirrup is held in the left to prevent it turning, and then the long pointer placed in the lower groove and balanced, and the short magnetic needle placed above it. The magnetic needle should be about a centimetre long, but not more than H centimetre, and of knitting-needle steel. It re- quires a little practice to draw out a good, straight, light pointer. Fig. 87. Some thin glass tubing, about ^^in. bore, should be procured, and a piece held by its ends over a Bunsen burner or spirit lamp flame with the two hands, turning it round to heat it evenly in one place till about three-quarters of an inch of it is perfectly red hot, and evenly hot all round. Then removing it from the flame the two ends are pulled, at first quickly, and then slowly, which draws out a fine capillary tube two or three feet long. After pulling out, the fine tube must be kept stretched tight till cool, for if the pull at the ends is relaxed immediately after 170 PRACTICAL NOTES pulling out, the tube will not cool perfectly straight. The straightest and lightest portion of the drawn tube must now be snipped off to the required length — say, five or six inches — and is then ready for use. A more elaborate instrument fitted as a magnetometer is shown in Fig. 88, in which the suspension can be controlled Fig. 88. with more certainty. The details of the suspension head are represented in Fig. 89, where a tubular brass cap, 0, screw- threaded, is cemented on to the upper end of the glass tube T. Above this is a brass tube screwing on at its lower end to the cap 0, and carrying at its upper end a split ring, R, through which passes the brass rod B supporting the suspension. The latter may therefore be turned round, raised or lowered FOK ELECTKICAL STUDENTS. 171 as required, and then clamped fast by the screw on the split ring R. The glass tube at its lower end has also a tubular brass cap cemented to it and screw-threaded, a nut on the underside of the circular glass plate P clamping it to the same. A table about 2ft. long and the same height as the baseboard of the instrument will be required for placing magnets on whose fields are to be measured. Fio. 89. A graduated scale must now be made, by which the move- ment of the pointer, when the needle is deflected, may be ob- served. If this is marked out in degrees, the angular deflection of the needle will be observed, and the corresponding tangent must be ascertained from a table of tangents. It is better, however, to graduate the scale in divisions proportional to natural tangents, the method of doing which will now be given 172 PRACTIUAL NOTES 123. Graduation of Tangent Scale. — It has already been shown (para. 98) that in a figure such as Fig. 90 distances measured from A along the line AB are proportional to Fig. 90. the tangents of the angles which those distances subtend. For example, two equal angles are drawn at C (Fig. 90) ; the tangent of the first angle is proportional to A D and that FiQ. 91. of the two together to A E. It is evident, that when the force by which a magnetic needle is deflected (from a zero position at A 0) is proportional to the tangent of the angle of deflec- tion, it requires more than twice the force to move the FOR ELECTEICAL STUDENTS. 173 needle through twice the angle, for A E is more than double AD. The object, then, of dividing out a scale in tangent divisions is to make the divisions proportional to the force moving the needle. This principle is shown in Fig. 91, where the distance A E is divided into two equal parts at the point F. Now, on drawing F C the scale is divided into divisions pro- portional to the force exerted on the needle ; that is, the force which would move the needle from the direction A C to that Fm. 92. of C E would be just double that required to move it from A C to F, the needle turning, of course, about the centre C. The method is therefore to mark out a number of equal divi- sions on the line A B, starting say from A, and then to draw lines from these divisions to the centre C, dividing the circular scale into divisions proportional to the tangents of the respec- tive angles, and therefore proportional to the forces exerted on the magnetic needle. It only remains to decide as to the size of each division on A B. In the case of tangent galvanometers, 174 PRACTICAL NOTES the tangent scales of which are generally divided out up to 60°, the line CE is drawn, making an angle of 60° with the vertical A C. The horizontal distance A E is then marked out into 100 equal divisions, and the circular scale is graduated by drawing lines from these equal divisions towards C. This process is shown in Fig. 92, where it will be seen that the divisions become very crowded near 100, and it is useless to carry them beyond 60° ; therefore it is usual to divide the scale on the opposite side of the circle in degrees of arc up to 90° on each side, by which means angles of Fig. 93. deflection higher than 60° may be read off on the degree scale, and the tangents found by reference to tables. It is also customary for the simple comparison of currents, where the natural tangents are not required, to read from the tangent scale j but for the measurement of a current, to read from the degree scale, and find the natural tangent of the angle by the tables (see example in para. 100). It will be found, however, very convenient, when the scale of the magnetometer is fairly large, say not under six inches diameter, to select 45° as the starting point (Fig. 93), and to divide into 100 equal parts the distance which this angle cuts out on A B. We have FOR ELECTRICAL STUDENTS. 175 then the advantage that any scale reading divided by 100 — that is, moving the decimal point two places to the left — gives at once the natural tangent of the angle, for the reason that the tangent of 45° is unity. For instance, if we had a deflection of 85 divisions, the natural tangent of the angle correspond- ing to the deflection is -85 and the angle itself need not be known The line A B is marked out on both sides up to about 60° or 70°, the length of each division being the same all along. A straight rule is then placed in line between each division in succession and the centre, and the corresponding scale lines drawn at the edge of the scale circle for each' position. The V m Fib. 94. lines may be drawn from each single division near zero, but as the divisions become closer together they may be drawn from every second or lifth division, the intermediate divisions being judged by the observer when taking readings. The other side of the circle should then be marked out in degrees, up to 90° on each side. 124. Adjustment of Magnetometer. — To complete the magnetometer and set it up for use the wooden baseboard should have two fine lines mm and nn (Fig. 94) cut or marked on it accurately at right angles to each other and passing through the centre ; a circle should also be described 176 PRACTICAL NOTES on the board a trifle larger in diameter than the outside diameter of the scale, to act as a guide in placing the scale symmetrically with the centre. The scale is cut out ring shape and placed in position with its zero marks in line with the line mm on the baseboard. It is better to let the scale lie loose on the board, as once placed there is nothing to move it, and sticking it down causes bulging. Inside the scale there should be a ring of mirror glass as shown in the figure ; this is for the purpose of reading accurately the position of the pointer, the reflection of which is seen in the mirror. At the centre of the baseboard there should be fixed a little brass point standing vertically, only to the height of about ^ inch above the level of the board ; this will be found very con- venient for levelling the instrument, during which operation the eye may be kept on a level with the surface of the board, and the levelling screws turned till the centre of the suspended needle is seen to hang immediately above the centre brass point, viewed from all sides. The previous directions having been carried out for the removing of torsion from the fibre, the baseboard is placed with the line wi to as nearly as possible in the magnetic meridian, and the inverted funnel carrying the suspended needle and pointer is then placed on the board, the rim of the same being placed symmetrically with the scale. The height of the needle can now be adjusted by raising or lowering, not turning, the wire suspension head ; the stirrup should then just clear the brass centre point. It wiU most likely be found that the pointer is very near zero, but not quite. To bring it to zero, the baseboard of the instrument must be turned round, little by little, till the pointer is at zero. This adjustment is somewhat tedious, because the needle must be allowed to come to rest between each adjust- ment, and the slightest movement of the board often sets the needle swinging considerably, and alters the levelling. This inconvenience may be dispensed with in the following way : Mount the ordinary circular baseboard B (Fig. 95) on a second and larger board, B^, countersunk to receive it. Attach the levelling screws to the lower board, and let the glass shade carrying the suspended needle rest on this. FOE ELECTRICAL STUDENTS. 177 Through the centre of the upper baseboard fix a brass spindle, so as to enable the board to be turned by means of the lever handle H. This lever should not be rigidly fixed to the spindle unless made short enough to work inside the levelling screws. It may be made like a spanner and fit on to a nut head at the lower end of the spindle, and can then be removed when once the instrument is adjusted. The scale and mirror are on the upper board, so that these may be turned till the zero point is exactly under the pointer, while the needle itself and the levelling of the instru- ment are not disturbed. The brass spindle might also carry the centre point, as shown. Without this elaboration, how- -ever, once the adjustment is made it will never require repeat- =»// Fig. 95. ing, if the instrument is not moved. No magnets or pieces of iron must be anywhere near the instrument during this adjustment. 125. The Moment of a Magnet Measured by its Field. — The two methods, due to Gauss, by which the moment of a magnet may be determined can now be easily understood. By the first method the magnet is placed so that its normal line (para. 121) is in the same magnetic meridian as the magnetometer needle, the distance between the centres of the needle and magnet being known. In Fig 96 let the circle represent the magneto- meter in plan, properly mounted and adjusted. A. wooden table. A, is placed close to the magnetometer board, its surface being marked out in cross lines indicating distances in centi- metres from the centre of the suspended needle, and its height such that magnets placed on it are in the same horizontal plane 178 PRACTICAL NOTES as the needle. The line drawn longitudinally along the centre* of the table must lie in the magnetic meridian, and be the con- tinuation of the line joining the zero points on the scale (the line m m, Fig. 94) ; the suspended needle will then lie in the same line before any magnet is placed on the table. Now let a magnet, N S (Fig. 96), whose moment is to be de- termined, be placed on the table so that its polar line agrees with one of the cross lines at a certain distance from the needle, and its normal line coincides with the central line along the table. The magnetometer needle is then deflected, and the natural tangent of the angle of deflection is the scale reading divided by 100 (para. 123). Now, in the region of the needle we have two fields (H of the earth and / of the 10 Ift 20 « 30 30 IS 10 1 5 10 1 Fio. 96. magnet) superposed at right angles to each other, and by para. 119 the ratio of / to H is the tangent of the angle a between H and the resultant field. That is, /= H tan a (in C.-G.-S. units), the angle a being that moved through by the magnetometer needle from its position of rest in the field H. Also the inten- sity of field on the normal line of a magnet at a point r centi- metres from either pole has been shown to be the moment o£ the magnet divided by r^ (para. 120) ; that is — /=^ (in C.-G.-S units). From these two values for/ we obtain the relatioa — = H tan a : ^3 whence M = r3H tana. FOR ELECTRICAL STUDENTS. 179 The only observations to be made, then, are the tangent of the angle of deflection and the distance r. Knowing the dis- tance d of the magnet from the needle, we have (para. 120) whence t^ = f ^^y N //// s S WW iUi a Fig. 99. Auechanical energy spent by the experimenter is proportional to the molecular rigidity of the metal. Prof. Hughes has shown that by magnetising a rod in a given direction, and then slowly demagnetising it, and again magnetising it in the opposite direction, that there is no hap- hazard arrangement ; every change takes place with perfect regularity and symmetry. In Fig. 100, at B and C is shown the neutral arrangement of the molecules, and at A a circular neutral arrangement effected by passing a current of electricity longitudinally through the rod. To show the effect of torsion on the rotation of molecules, Prof. Hughes took a strand composed of about ten untempered 188 PEACTICAL NOTES cast-steel drill wires, each 1 millimetre in diameter, and about 1 foot long, of the kind in use by watchmakers, and fastened handles at the ends of the strand, as shown (Fig. 101). Hold- ing one handle fast, he imparted two entire right-handed twists to the strand with the other, and strongly magnetised it on the north pole of a magnet while under this torsion. He then released the two right-handed twists, and gave it two entire twists to the left, and oppositely magnetised it on the soutb pole while under the torsion. On releasing the torsion the strand possessed the remarkable property of a double polarity — that is, either end could be made north at pleasure by simply twisting it one way or the other. In the figure is shown th& Fio. 100. simple bell apparatus constructed by Prof. Hughes to demon- strate the marvellous transformation of one kind of mechanical movement to another through the medium of molecular motion. The magnetised armature n s is held in its position by the direc- tive force of the magnet beneath it, and when the doubly- polarised strand is held near to the armature and twisted con- secutively in alternate directions, the pole of the strand near to the armature changes in name for every alternate twist and pro- duces attraction and repulsion of the armature, which there- fore strikes alternately the rims of the glasses, producing » clear note for every twist given. The notes could be repeated- in this way some six times per second. FOR ELECTRICAL STUDENTS. 189 In the same Paper Prof. Hughes notices a peculiar property of magnetism, viz., that besides the fact that the molecules of iron can be rotated in any direction when mechanical force is applied in the shape of blows, stress, or strain, it is also possible to rotate the molecules through a small arc under the influence of a very weak magnetic field, without the application of any mechanical force. The molecules appear to be surrounded by Fig. 101. an elastic medium which permits of their free rotation through a small distance or arc under very feeble directing fields, and what is more remarkable is that this limiting distance of free motion, while it cannot be augmented, can easily be shifted so that the axes of free motion of the molecules are turned in any direction relatively to the bar. For instance, in many samples of iron this path of free motion may be observed at 190 PRACTICAL NOTES once. The writer has taken a rod of iron 1ft. long and |m. square, and found, when held vertically, strong north polarity at the lower and south at the upper end. On reversing the rod the same polarity has been observed, i.e., north at the lower end. Hence, without the slightest mechanical shaking of the molecules, the latter have rotated under the weak directing field of the earth. If now a few blows be given the rod while held vertically it will be found that the axes of free motion of the molecules have been shifted, for the rod will now exhibit strong polarity one way and neutrality when turned the other way. The limit of free rotation has been found to have a distinct value for each class of iron. The extreme rapidity and sensitiveness of action of the telephone is probably due to the freeness of rotation of the molecules within this- critical limit. Section II. — Electro-Magnetic Fields. 127. Preliminary. — The telegraph systems of Morse and Wheatstone, with their relay and translating apparatus, elec- tric bells and telephone exchange annunciators, systems for the synchronising of clocks, and a large proportion of the indus- trial applications of electricity, depend for their action upon that great discovery made by Arago and Davy nearly seventy years ago, viz., that pieces of iron, steel, and other magnetic metals become magnetised when an electric current is passed through a coil of insulated wire surrounding them. A magnet so formed is termed an electro-magnet. Incidentally here we may remark that metals which are said to be magnetic are not necessarily magnetised, the terms magnetic and non-magnetic indicating simply whether they are or are not capable of being magnetised. The two important principles of the electro- magnet may be thus stated : — 1. Eeversal of the direction of current round the iron causes a reversal of its magnetic polarity. 2. Cessation of the current causes almost complete disap- pearance of magnetic polarity in the case of soft annealed iron. FOR ELECTRICAL STUDENTS. 191 To this might be added the important aid now available for the manufacture of powerful steel magnets magnetised electro- magnetically with the aid of currents obtainable from dynamos and accumulators. Considering the first principle — viz., that of reversal — it is found experimentally that if the current circulates in a clock- wise direction round the iron core, the end of the same nearest to the observer is a south pole, and vice versd. This will be seen to be the case in the two Figs. 102 and 103, although the direction in which the wire is wound on is different in the two cases. The winding of the coil or helix in Fig. 102 is called right- Fio. 102. Fig. 103. handed, the wire being wound on in a clockwise direction ; while that in Fig. 103 is left-handed; and it will be seen that, in order to produce the same magnetic polarity in each, the current must be passed through in opposite directions in each as regards the ends of the coils. The current then circulates round each core the same way, producing the same poles in each. Eeversing the current in either of these reverses the polarity. An iron-filing diagram, taken with the electro-magnet of an ordinary telegraph "sounder," is shown in Fig. 104, a current being passed through the coils. The current was adjusted till the electro-magnet just supported the weight of the arma- 192 PRACTICAL NOTES ture. Here we have precisely the same delineation of field as in the previous cases of permanent magnets. The two upright cores of soft iron are screwed to the horizontal iron keeper or " yoke " (Y). The magnetic circuit is indicated by the dotted lines through the two cores and the yoke in Fig. 105, where it will be seen that this circuit completes itself through the air between the two poles by the curved lines of force, one typical line being shown in the figure. The direction assumed for these lines has already been explained (para. 104), and is marked by arrow heads in the figure. Now, when there is an iron " arma- FiG. 104. •ture " in proximity to the poles of the electro-magnet, most of the lines of force are deflected, or turned from their original direc- tion, and pass through the armature (Fig. 106). Noticing the direction in which these lines pass through the interior of the armature it will be clear (para. 104) that the latter becomes magnetised, its poles, s n, being as shown. Attraction must, therefore follow, and the armature is drawn towards the poles, N S, against the force of the counteracting spring. Every time the electrical circuit is completed by means of a " key " (see Fig. 34) this attraction occurs, and ceases when the cir- FOR ELECTRICAl STTIDENTS. 193 «uit is broken, by means of which, with the Morse code of signals made up of dots and dashes, signals can be sent from one end of a line to the other, and can be read by sound, or toy the ink-writer, as the case may be. --.N'N^V ^.--'i- FiG. 105. ISPRINO !) S ? s 5 1 1 - -- P Fig. 103. If we take an iron-filing diagram of the field when the armature is held away from the poles, or placed far enough away not to yield to the attraction (Fig. 107), the deflection of the lines of force through the armature is made evident ; the position of the armature being seen by the absence of any o 194 PRACTICAL NOTES filings, since the latter only take up the position of the curves^ of force through the air. We have here between the arma- ture and the poles of the electro-magnet what is termed " mag- netic induction" — that is, the magnetism of the armature is " induced " in it by the action of the poles of the electro- magnet ; or we may look at it another way, viz., the armature is placed in a magnetic field, and immediately becomes a magnet itself by "induction" from the field. It will be seen by Figs. 105 and 106 that the current must pass round the two cores of the electro-magnet in the opposite direction to produce opposite poles at the upper end of the; cores. The winding for this is easy to remember by consider- ing the two cores as put end to end where they join the yoke, so as to make one bar ; the winding is then a simple helix wound continuously along the bar, in order to produce opposite poles at the extremities. In the rapid transmission of signals it is very important to overcome, as far as possible, the time taken by the electro- magnet to become magnetised and demagnetised. To over- come this retarding effect of the coils, which is a property depending on their self-induction, the two coils are wound in multiple arc (paras. 41 and 61), as shown in Fig. 108, for the FOK ELECTRICAL STITDENTS. 195 iast-speed Wheatstone receivers. More will be said upon this point when self-induction is considered. If the cores were not connected together by the yoke, the poles of the electro-magnet would be much feebler, because there would then be two air gaps to be bridged over by the lines of force instead of one. Further, if a complete circuit of iron were formed by bridging over both upper and lower pairs of core ends by yokes, there would be many more lines of force created through the interiors of the cores, but there would be practically no external field. An external field is, however, required in the case of a " sounder " to cause attraction of its armature, and therefore the only air gap left is that imme- diately beneath it. The rule is, let there be a magnetic circuit Fig. 108. of iron everywhere except in the particular position where external magnetic action is required. 128. Magnetising Force. — The field of force set up in the interior of a coil of wire through which a current is flow- ing is found both by experiment and theoretical deductions to be proportional to the product of the strength of current and the number of turns of wire. If an iron bar be wrapped round with a given number of turns of wire, and a current of known strength passed through the wire, we have a definite field of force produced by the current in the coil which is capable of " exciting" or magnetising the iron. Hence we speak of the exciting poicer or magnetising force of the current, and in practice express it in ampere-turns. So long as the product of the number of amperes and number of turns is constant for a 2 196 PRACTICAL NOTES given volume of wire, the magnetising force is constant what- ever individual values the current or number of turns may have. This is the same thing as saying that the current must vary inversely as the number of turns for the same magnetising force. For example, if we have one electro-magnet, A, wound with 50 turns of thick wire, and another, B, wound with 1,000 turns of fine wire, the volume or weight of wire being the same in the two coils, the current strengths in the coils must be in the proportion of 1,000 to 50, or as 20; 1 respectively, in order to produce the same magnetising force on the iron in each. It is sometimes more convenient to express the number of turns or length of wire in terms of the resistance. We have already seen (para. 63) that the length of wire wound on a coil of fixed volume or weight varies as the square root of the resistance ; therefore, for a given magnetising force the product of the current and the square root of the resistance of the coil must be constant ; or, in other words, the resistance must be inversely proportional to the square of the current. Hence, in the above example, the resistances of the two coils A and B are respectively as 1 ; (20)^, or as 1 ; 400. Further, since the volume is constant, the cross-sectional area of the wire multiplied by the number of turns is constant, and therefore the sectional area of the wire on each coil is inversely proportional to the number of turns on each, and the dia- meter of the wires are inversely proportional to the square root of the number of turns. Hence, in the above example, the diameters of the wires on coils A and B are in the ratio of ^20 : 1 respectively. This is without taking into account space taken up by thickness of insulation. Similarly, the length of wire on each varies as the number of turns — viz., as 1 ; 20 respectively. We shall consider further on the C.-G.-S. unit of mag- netising force. 129. Variation in Field due to more or less complete Mag- netic Circuit. — Now let us measure with the magnetometer the strength of the external magnetic field when the magnetising force is constant and the amount of iron in the matinetic circuit FOE ELECTRICAL STUDENTS. 197 is varied. For this experiment two bobbins were taken (Fig. 109), each 20 centimetres long, and wound with No. 30 copper wire, to a resistance of 156-5u. This represented, there- fore, about 2,180ft. of wire wound on each (No. 30 measures 13-95ft. to the ohm at 60° F.). Also, the mean diameter of each coil being 4-25 centimetres, the mean length per turn of wire was 4:-257r = 4-25 x 3-14= 13-3 centimetres = the number of turns on each bobbin was 13-3 '30-48 ft. Hence 2180x 30-48 iFs = 5,000 turns approximately. BATTERY FlO. 109. The two bobbins of wire were then connected in series to produce opposite poles at N S when the battery was connected. The cylindrical iron cores were 23'2 centimetres long and 2 '5 centimetres diameter, and the two keepers or yokes, one of which, Y, is shown, were each 11 centimetres long and 2-75 square centimetres in sectional area. The parallel axes of the bobbins were placed 5 -5 centimetres apart, and a poten- tial difference of 42 volts connected to the terminals, main- taining, therefore, a current of 42 156-5x2 •134 ampere through the two coils, and a constant magnetising force each coil of '134 x 5000 = 670 ampere-turns. 198 PRACTICAL NOTES The results obtained from the magnetometer readings are tabulated : — Magnetometer detlection. Magnetic Moment. C.-G.-S. units Strength of Poles. C.-G.-S. units Coils without iron cores . . . Iron cores inserted Yoke added at end A Yoke added at end B Last yoke removed and larger one substituted 2 100 440 180 8,508 38,030 1,547 6,915 The moments were determined in the manner already described (para. 125, 1st method), the perpendicular dis- tance between the needle and the polar line of the electro- magnet being 36 3 centimetres, and the deflection due to the coils alone being subtracted from each reading. The strength of pole was found by dividing the moment by the distance between the pole centres, 5 '5 centimetres. The conclusions to be drawn from the above may be stated thus : — 1. A magnetic field is set up by the current in the coils alone. (The deflection of two divisions at the distance of the coil from the magnetometer corresponds to a pole strength of about 30 C.-G.-S. units.) 2. The field is enormously increased by the insertion of iron cores. Lines of force are evidently created by the presence of the iron in the field, since the magnetising force remains constant. 3. The field is still further increased by bridging across one air space by an iron yoke. Lines of force are again created, both through the interior of the iron and the remaining external air space, by making a more continuous iron circuit. We should, therefore, conclude that, 4. On completely closing the magnetic circuit by adding the second yoke at B, the field in the mass of the iron is stUl further increased, but that this tends to short-circuit the FOR ELECTRICAL STUDENTS. 199 ■external field through the air. The magnetometer, however, liemg still deflected considerably, we should conclude that 5. The mass of iron is not sufficiently large to carry the whole of the field so created, and therefore there is some leakage of lines of force through the air, which aff"ect the mag- netometer. This is borne out by adding a larger mass of iron (15'2x5xl'5 centimetres) in place of the small yoke at B, which effectually short-circuits the field — that is, causes the flow of lines of force to be entirely within the iron mass. 130. Intensity of Magnetisation. — An important branch of research tending to further the improvement of electrical apparatus and machines as well as to offer one of the most interesting fields for experimental investigation is that in which the relation between magnetising force and the result- ing magnetism in different metals is studied. We shall refer later to some of the work which has been accomplished in this direction, but must first concern ourselves with the necessary fundamental bases on which comparisons can be effected in the degree or extent or quantity of magnetism acquired by masses of iron or steel, difiering in dimensions and shape, when sub- jected to the action of magnetic fields of given intensity. The two fundamental bases of comparison may be stated as : 1. The magnetic moment acquired per unit volume of metal, and termed the " Intensity of Magnetisation " (denoted by the letter I). 2. The number of lines of force per square centimetre of sectional area, forced through the mass of the metal, and called the " Magnetic Induction" (denoted by the letter B). These two quantities were written in old English letters |{ and ^ by Prof. Clerk Maxwell, and the student will find them frequently written so in writings on the subject. We shall now consider the first of these means of compa- rison. The conception of the idea of " intensity of magnetisa- tion" is essentially one in which the strength of pole or amount of " free magnetism" is thought of. That is to say, the extent -of the magnetism in the mass is expressed or thought of with 200 PRACTICAL NOTES regard to the force of attraction or repulsion that can be exerted by its poles externally. The "action at a distance," or force exerted by a given pole strength, has been discussed already (para. 108), and it can be easily understood that we may have a number of bars of iron, of very different sizes, all of which exert the same external force, i.e., have equal pole strengths, but in which the magnetism is distributed in very different proportions ; or again, magnetised bars of similar size may have very different pole strengths. To effect comparisons, then, between the densities of magnetism — if one may so use the term — acquired by different bars, the pole strength is re- garded as so much " free magnetism " distributed uniformly over the ends (shaded portions Fig. 110), and a comparison can then be made by finding for each bar what amount of free magnetism, or pole strength, exists at its ends per square centi- Fio. 110. metre of sectional area. This is termed the " intensity of mag- netisation," or sometimes simply " magnetisation, " and is obviously the same as magnetic moment per cubic centimetre. Take a bar 10 centimetres long and 2 square centimetres sec- tional area. If its pole strength is 1,000 C.-G.-S. units, the intensity of magnetisation is found by pole strength 1000 ^„„ r^ r< c i- — : =— ^— = = 500 C.-G.-S. units ; sectiunal area 2 that is, for every square centimetre of end area there are 500 units of pole strength. It is evident that if we multiply both numerator and denominator of the above by the length, we have magnetic moment 10000 ^^„ r^ r^ c^ — 2 . = =500 C.-G.-S. units, volume 20 FOK ELECTRICAL STUDENTS. 201 which is, therefore, the same thing, and has the same value, and gives us the usual definition for the intensity of mag- netisation, viz., the magnetic moment per unit volume (per cubic centimetre on the C.-G.-S. system). Obviously this is the more generalising definition, as it covers irregularity of sectional area. As an example of the comparison of magnets of the same size but of different pole strengths by their intensities of magnetisation we may take the two mag- nets already mentioned (para. 125, examples 3 and 4), which were found to have pole strengths of 554 and 527 C.-G.-S. units respectively. They were flat bars, measuring at the end 2'8 centimetres by 1"3 centimetre, and therefore of 2'8xl-3 = 3'64 square centimetres sectional area, uniform throughout. Their intensities of masrnetisation were therefore 5.54 3 -04 = 152 0. -G.-S. units 527 J-G4 = 145 C. -G.-S. units. and The intensity of magnetisation reached when soft iron is subjected to powerful electro-magnetic fields may be very much hieher than the above, even to values nine or ten times as hich, although the above are average intensities ordinarily met with in new steel permanent magnets. To further illustrate this method of comparison the follow- ing measurements taken with wrought and cast-iron bars are cited. A constant difference of potential of 42 volts was kept at the terminals of the magnetising coil (Fig. Ill), which was one of those previously described (para. 129, Fig. 109), the current, and therefore the magnetising force, being twice as much as that previously given for the two coils together under the same potential difference, viz., 670 X 2 = 1340 ampere-turns, approximately. We shall show in the next section on the magnetic fields of coils without cores, or solenoids, that the magnetising force expressed in C.-G.-S. units (lines of force per square centi- metre) at the central cross section of a long solenoid of length 202 PRACTICAL NOTES I centimetres, and excited by a current C, in C.-G.-S. measure is equal to I ' C.-G.-S. units intensity of iield, the length of the solenoid being great compared to its dia- meter, and forming a long narrow coil. We anticipate, how- ever, for the purpose of showing the relation between ampere- turns and C.-G.-S. units of magnetising force. The C.-G.-S. unit of current is equal to 10 amperes, so that if x ampere-turns are equal to one C.-G.-S. unit we have and therefore ampere-turns n Mt ■ r^ r^ c< — = field in C.-G.-S. uuits, X IQQn 4-Cm X I (C being in C.-G.-S. units on both sides of the equation), , lOZ I whence x = — = , 4- 1-257 and therefore field in C.-G.-S. units = ampore-turns ^ ^,^^^^_ length of coil (centimetres) In the above example, therefore, which is a coil 18-5 centi- metres in length, instead of 1,340 ampere-turns of magnetising force, we may put it 1340x1-2.57 01 n r< a = 91 C.-G.-S. units. 18-."j The magnetising force is usually denoted by the letter H, or in Maxwell's notation by its old English form ^. It will be noticed that the electromagnet is placed, as pre- viously described, for measuring the moment by the first method of Gauss (para. 125), the distance of its centre from that of the magnetometer needle being 50 centimetres. First of all, before putting any core in the coil, its own field with the current on was measured. This deflected the mag- netometer 11 divisions, and therefore in the measurements of FOK ELECTRICAL STUDENTS. 203 -the moments of the cores this deflection was deducted from the readings. Three cylindrical cores similar to C C, Fig. Ill, but of different dimensions, were tried, with the following results : — Cores. Length in cms. ii) Diam. in cms, id) Magneto- meter deflection. Moment in C.-G.-S units. (M) Intensity of Magnetisation. (I) I. "Wroughtlron II. Ditto III. Cast Iron ... 30-7 26-9 30-8 2-3 1-9 2-3 420 280 280 106,300 67,210 69,350 833 881 541 Fio. Ill, Here the dimensions of core differ, and therefore the mag- netometer readings and calculated moments offer no basis of comparison as to the density of free magnetism acquired by each. This is seen notably in the last two cores, where the deflections are equal, and the moments not very widely diffe- rent ; but the intensity of magnetisation, which forms the basis of comparison, shows one little more than half the other. 204 PRACTICAL NOTES It may be of assistance to the student to see one of these examples worked out. Take the first : — tana =4-20- •11 = 4-09 earth's field (H) = -18 ?-3 =(502 + 15-352)'^ and {see para. 125) moment =r^ H tan a= 106,300 U.-G-.-S. units, ■n cP 3-141G X ''•32 sectional area = = = 4-147 square cms. 4 4 ^ volume = area X length = 4-147 x 30-7 = 127-3 cub. cms.,. and intensity of magnetisation j^ moment ^ 106,300 ^333 ^^^ ^^j^^_ volume 127-3 The low magnetisations obtained for the above-mentioned magnetising force are most probably due to the estimation of the number of turns of wire in the bobbin (para. 129) being too high. It was not wound specially for the purpose, and the exact length of wire and number of turns were not known. Further, no mechanical stress or blows were given to the iron while under the magnetising field, which would have increased the magnetisation acquired. When the core is not of uniform sectional area, if its volume cannot readily be computed by direct measurement it may be determined for a large core by immersing it in water, and measuring the volume of water thereby displaced. Or, again, for a small core, such as a piece of iron wire, it may be found by weighing in air and then in water, the ratio of the weight in air to the loss of weight in water being the specific gravity, and, in the C.-G.-S. system the weight of 1 cubic centimetre of water at 4°G. being 1 gramme, we have volume in cubic centimetres = — ^ — \ — ir; in grammes^ specific gravity Substituting for specific gravity, we have volume in cubic centimetres = loss of weight in water (in grammes). FOR ELECTRICAL STUDENTS. 205 The determination of the intensity of magnetisation is not ■confined to methods like the above, which have for their modus operandi the measurement of the external field of force of the magnetised bar, but may be calculated from the value of the magnetic induction B when that has been determined by a -different method (presently to be detailed) and when the mag- netising force can be accurately computed. The importance of the subject of electro-magnetism as regards our knowledge of iron and steel in various stages of magnetisation, may be conceived when it is borne in mind that most of the men whose names are most honoured amongst us, as much in some for their high scientific attain- ments as in others for their practical experience, insight, and intuition, have given to this field of research their careful thought and investigation, and that what we know and make use of to-day is the result of incessant experimental investi- gation extending over, in many cases, years of patient toil. It will be the aim of the writer to point out some of the most salient results arrived at, and to endeavour to render clear to the student the terms used in connection with this subject. 131. Apparatus for Measuring the Relation between Mag- netising Force and Intensity of Magnetisation. — We shall now consider some of the practical details of a simple test to measure the changes in the intensity of magnetisation acquired by an iron or steel rod, when the current in the coil which surrounds the rod is gradually increased. The practical example given below of a test of this kind will serve as a guide, although the effects may be observed equally well with iron wires of much smaller diameter and currents of less strength than those employed below. The apparatus is connected up as in Fig. 112, where the magnetometer M and distance table are shown in plan, the current meter G and adjustable resist- ance E in elevation, and the battery, in this case a secon- dary battery of 21 cells, is shown conventionally. The galvanometer is placed some five or six feet away from the magnetising coil C, and is connected to the rest of the 206 PRACTICAL NOTES circuit by well insulated and twisted leading wires (para. 67), in order that its needle may not be affected by any mag- netic field but that set up by its own coil. A brass tube of about ^ centimetre bore and a little over a foot long was provided with cheeks at the ends, fixed so that the dis- tance between them was exactly 1ft. The tube was then I I I ! 1 1 BATTERY- Fig. 112. wrapped round evenly with No. 16 double cotton-covered copper wire, and the ends firmly secured by twine. This gave a magnetising coil (0 in the figure) 1ft. long and of 135 turns, which was then placed on the table at 20 centimetres distance from the magnetometer needle in the position shown, and held there firmly by a wooden clamp. FOR ELECTRICAL STUDENTS. 207 The resistance R is made with German silver wire spirals connected in series and fastened to a wooden frame. German silver (an alloy of 2 parts copper, 1 part zinc, and 1 part nickel) is chosen because less length of wire of a given thick- ness is required than if copper or iron were used, owing to the higher specific resistance (para. 29) of this alloy, which is some 12 or 13 times that of copper. Less space is therefore taken up by the coils. The bights between each spiral are soldered to wires running to contact stops on the switch S. When the switch lever rests on the extreme left hand contact the circuit is disconnected, when on the second stop (as in figure) the circuit is closed, and a current will flow from the cells through all the resistance coils, the magnetising coil, and galvanometer. The current may then be increased step by step by moving the lever to the right. It is advisable to make the spirals of very low resistance on the low-resistance side (right side) of the switch, say of a quarter and half ohm, owing to the fact that when the rest of the circuit is of low resistance, very slight changes in the adjustable resistance produce great variations in the current strength. The spirals near this end of the switch should also be of thicker wire, say No. 14 or 16 B.W.G., as they have to carry the heaviest currents, while the rest may be of No. 18, and wound in spirals of 1, 2, and 5 ohms. The best arrangement, where great variation in resistance is required, is to have two separate switches; for example, suppose the total variation in resist- ance required was about 20 ohms. If one switch (A) of seven contacts were connected to six coils, in the manner shown in the figure, three coils being of five ohms each, and three of a quarter ohm each, and another switch (B) of five contacts con- nected to four coils, each of one ohm, the total resistance would be 19f, and it could be reduced by one ohm at a time down to four ohms, and below four by a quarter ohm at a time by working the two switches. Supposing all the resistance in to start, B would be reduced four ohms by one ohm at a time first, and then its switch put back so as to add the four ohms again ; then A would be moved one stop, reducing five ohms, and then again B one ohm 208 TRACTICAL NOTES at a time, and so on. In a test of this kind, where we are measuring the magnetisation of iron by gradual increments of current, care must be taken that if the resistance is reduced too much by mistake in one step, the reading on the magneto- meter must be taken with this resistance in ; it is no use to increase it up to what was intended at first, as the increased magnetisation of the iron due to the larger current is not lost entirely on decreasing the current. The actual resistance in ohms is not required to be known in this test, so that the lower variations in resistance may be obtained very well by two similar metallic plates of, say, a foot square, immersed in slightly acidulated water, whose distance apart can be varied to a maximum distance of about two feet. We must now turn our attention to the galvanometer. At the outset it will be noticed that in a test of this kind we are varying the strength of the current within fairly wide limits. In the test presently to be detailed, taken with the above apparatus the current was a little over half an ampere to start with, and was gradually increased to nearly 40 amperes. Now, with wide variation in current the galvanometer must have different degrees of sensitiveness in order to allow of reading the deflections with the least chance of error. Suppose a large current through the instrument gives a conveniently large deflection, then if the instrument remained in the same state of sensitiveness, as the current was reduced, the deflections would get smaller and smaller till the very smallness of the deflection would prevent accurate readings being taken, the movement of the needle being very slight for given changes in the current either at very low or high deflections from zero. N^ow, if the sensitiveness of the instrument is increased n times the deflections will be n times as large, and therefore the feebler currents can be read with ease and accuracy. The converse holds in the case before us. ^Ye are increasing the current strengths, and therefore must decrease the sensitiveness of the galvanometer as we proceed. We may do this in four ways, viz. : — 1. Shunting the galvanometer at intervals with shunts of gradually decreasing resistance. FOK ELECTRICAL STUDENTS. 209 2. Increasing the magnetic controlling field by bringing the controlling magnet nearer the needle. 3. By using a galvanometer wound with four or five separate «oils, commencing with the fine wire coil of many turns, and changing at intervals on to the thicker wire coils of decreasing number of turns. 4. By gradually moving the needle and scale further from the coil. As we wish to measure the currents accurately, and are working from a half to 40 amperes, the first method would be impracticable. When the current increased towards its larger values the shunts on the galvanometer would require to be so low in resistance that they would be difficult of accurate measurement, to say nothing of providing them of thick wire or stout copper ribbon, so that their resistance should not alter by heat; and to calculate the total current from the deflection of a shunted galvanometer, we must, of course, know accurately the resistance of the shunt (para. 95). As regards the second method of shifting the controlling magnet, the readings might first be taken without the magnet, and when beyond the limit of accurate reading, say 50 degrees, the magnet would be placed at a vertical height over the needle, such that the deflection of 50 was halved (or if a tangent galvanometer, a position found for the magnet, such that the deflection was the angle whose tangent was half the tangent of 50 degrees). The magnet would be kept in this position until 50 degrees deflection was reached again, and then lowered to a position in which the deflection of 50 was halved, and so on. For each re-adjustment, we should know that the same strength of current gave half the de- flection, and therefore we should multiply the deflections (or their tangents) by 2, 4, 8, 16, 32, &c., for each consecu- tive change. The galvanometer would have to be wound, however, with wire thick enough to carry the strongest cur- rent without dangerous heating, and therefore could not hold many turns. 210 PRACTICAL NOTES The third method requires a specially-wound galvanometer. If such an instrument is available it is very convenient to use,, since the constants (para. 100) may all be determined before- hand, once for all. The resistance of the coils is not taken into account, the variation in sensitiveness, as regards current, being effected only by the different number of turns. The- reason why the coils of fewer turns are made of thick wire is that they may carry the large currents sent through them without heating suflBciently to cause injury to the insulation. The last method can best be explained by describing Sir William Thomson's graded galvanometer, the sensitiveness of which can be varied in this way, and, in fact, was the instru- ment used for the test we are now considering. We shall give in our next chapter a detailed description of galvano- meters, and, although it may be considered as deviating widely from usual custom to digress from the main subject in hand in this manner, it is hoped that the reader will not find it altogether without interest to have instruments explained at the time they are used. 132. Sir William Thomson's Graded Galvanometer. — This instrument (represented in elevation at G, Fig. 112, and in plan in Fig. 113) has a wide range of sensitiveness, effected by varying the distance between the coil and needle. The coil, shown at B, is fixed in position and wound with six turns of copper strip 1'2 centimetre wide and 1"5 millimetre thick, each turn being insulated from its neighbour by winding on, side by side with the copper strip, a ribbon of asbestos paper. The two ends of the copper strip are brought out at T, side by side, being kept apart by a wood strip about ^in. thick, gradually tapered down towards the end, to admit of sliding on the connecting clip A, from which two twisted leading wires con- nect the coil to the rest of the circuit. The advantage of this mode of connection is that the instrument may be introduced into or withdrawn from a circuit without causing temporary interruption. The quadrant-shaped box D, with glass cover, contains the pivoted system of needles and pointer, and is fitted with a mirror and tangentially divided scale. The instru- FOR ELECTRICAL STUDENTS. 211 ment being tangential, the deflections on the scale are propor- tional to the currents. There are four little magnets, two of which are seen in the figure ; the other two are beneath these, and the whole are fixed in an aluminum frame which is pro- longed to form a looped pointer. The needle-box may be moved to any position, the V groove cut in the platform guiding it so that the needle is alvrays in a line with the Fig. 113. centre of the coil. The two ends E and F of the controlling magnet can be fitted into the side arms E and F of the needle- box, so producing an additional controlling field on the needle in the direction of the meridian, the value of which and the date when measured are usually painted on the magnet. This value cannot, however, be depended on in practice, unless the p 2 312 PRACTICAL NOTES greatest care is taken to keep the magnet from getting knocked or being placed within range of the influence of foreign mag- netic fields, and it is therefore advisable to measure it before or after use. A pin on the underside of the magnet at E fits into a hole in the arm E, and the point of the magnet at F fits in the grooved wheel carried by the adjustable screw at ihe end of the arm F, by means of which screw the magnet may be accurately adjusted to the plane of the meridian. The arm F also carries a spirit level. Fixed to the outside of the quadrant, opposite the zero point of the scale, is a little prolongation with a horizontal cut in it, the use of which is to enable the observer to bring the needle-box exactly into a given position of sensitiveness by observing that the cut and the given mark on the platform coincide. In the figure this is placed at position 32. At position 1, without the controlling magnet, the needle is at such a distance from the coil that the current producing one division deflection is exactly equal numerically to the intensity of the earth's horizontal field H at the place where the instrument is used, which, in London, is '18 C.-G.-S. units^that is, Current = '18 ampere per division. Hence, for D divisions. Current = D -18 amperes. Now, at positions 2, 4, 8, 16 and 32 the sensitiveness of the instrument is increased twice, four, eight, sixteen and thirty- two times respectively; that is, a current deflecting the needle, say, 10 divisions at position 1 will deflect it 20 divisions at position 2, 40 divisions at position 4, and so on. Or if we put currents through the instrument, to cause equal deflections when the needle-box is at positions 1, 2, and 4, these currents would be in the proportion of 4, 2 and 1 respectively. Hence the currents vary inversely as the position P of the needle-box on the platform for a constant deflection, and we may write — Current = 'IS— amperes. FOR ELECTRICAL STUDENTS. 213 The field F produced at the centre of the needles by the con- trolling magnet, when this is used, must be added to that of the earth H, as the two fields are in the same direction. To find this field it is necessary to pass a current through the instrument, and note first the deflection without the magnet, then notice how many times the defiection is reduced, say n times, by adding the magnet. This is best done by having the needle-box much nearer the coil when the magnet is on ; for example, a constant current is kept on, and at position ^, without the magnet, the needle is deflected, say, 22'5 divisions; the current is then •18 amperes. •5 Now, at position 32, with the magnet on, the needle is de- flected, say, 30 divisions ; the current is then •18 amperes, 32 "^ where n is the number of times the sensitiveness is reduced by adding the magnet. The current being the same, the above two expressions are equal, and 22-.5 2,011 , ,Q = , whence n = 4s. •5 32 The value of the resultant controlling field is then 48 or n times H, and we have seen above that it is also equal to H -[- F, where F is the field due to the magnet ; therefore H-l-F = wH F = H(«-1) = -18 X 47 = 8-46 C-G.-S. unita. It is the value of F thus measured which is usually painted on the magnet, together with the date when measured. Now, it will be seen in the figure that the needle is not within the coil at position 32 ; there is still a position of higher sensitiveness, which does not usually fall in the same geometrical series as the previous grades. Its value in some 214 PRACTICAL NOTES instruments is over 40. If we take it as 45, and suppose the needle-box placed there, without the controlling magnet, we should have — = of an ampere per division. 45 250 ^ ^ And, using the above magnet with the needle-box at position \, we should have (-18 -H 8-46) „, . ,. . . ^- ' = 34-5 amperes per division. Between these two extremes, which include a very wide range, any degree of sensitiveness may be obtained. 133. The Demagnetising Force set up by a Magnetised Rod. — A bar or rod which has been magnetised sets up by the action of its own poles a reverse magnetising force — that is, it tends to demagnetise itself. To study this phenomenon we must consider the quantity of lines of force issuing from a pole of strength to (the action of the complementary pole being con- sidered negligible). At a distance of r centimetres from the given pole in all directions the magnetic field is equal to — C.-C4.-S. units (see paras. 108, 109, and 113) — that is to say, the above number of lines of force pierce every square centimetre of surface of a sphere of radius r centimetres whose centre is the pole m. But on the surface of such a sphere there are i tt r^ square centimetres area, through each of which, as we have seen, pass the above number of lines of force. Therefore, the total number of lines issuing from the pole to is equal to 4 17 r- X — = 4 TT m lines. (The Greek letter tt (j)i) is always used as the ratio of the cir- cumference of a circle to its diameter, which is 3-14159.) FOR ELECTRICAL STUDENTS. 215 Now take an iron bar (Fig. 114) to which is applied a ■certain magnetising force, the diredion of which is indicated by the straight arrows on each side of the bar. Suppose that the iron acquires m C.G.-S. units strength of pole. Lines of force will issue from the N pole and pass in curves through the air to the S pole (para. 104) ; but these are not all the lines issuing from the pole. They issue radially in all directions, and therefore some pass through the metal to the :S pole. It is precisely these lines tcithin the metal which 4nin-Ti \ N S / J n ^ 4Jim '~ Fig. 114. tend to demagnetise the bar. Say there are n demagnetising lines running from N to S through the metal; we shall then have 4:Trm~-a lines issuing from the N pole into the air, Airm being the total number. Both the lines through the air and the demagnetising lines through the metal form closed circuits, and therefore at the S pole the two return circuits of lines are in the same direction and run back through the metal from S to N. These are respectively 4 TT m - n and n. Adding them together, because they are in the same direction, the demagnetising lines are eliminated, And we have 4 jt ?w as the number of lines through the metal 216 FKACTICAL NOTES in the same direction as the magnetising force applied to the bar. This phenomenon is all-important in magnetisation expe- riments. We wrap a bar of iron round with a certain number of turns of insulated wire, pass a known current through the wire, and from these data can calculate exactly what number of magnetising lines of force are produced by the coil, viz., — - — lines per square centimetre (see para. 130). But as soon as the iron becomes magnetised, the action of its own poles causes n deviagnetising lines to pass through the metal and weaken the above magnetising field. Now if the sectional area of the bar is a square centimetres, the demagnetising lines are - lines per square centimetre, a and therefore the resultant, or actual magnetising force H; acting on the bar is - lines per square centimetre. I a Without knowing the value of n, therefore, the magnetising force cannot be exactly estimated. Now, in experiments to determine the magnetic " susceptibility " and " permeability " of specimens of iron, it is essential to know the exact mag- netising force applied, and this is practically accomplished by reducing the demagnetising force to such a small relative value that it can be neglected. Now, the demagnetising force is greater in proportion as the poles are stronger and nearer to each other, and in the case of bars we can only, with strict theoretical exactness, get rid of its effect by using rods of infinite length. Practically, however, by using rods whose length is many times their diameter, the demagnetising effect may be neglected. Prof. J. A. Ewing, of University College, Dundee, who is a high authority on the electro-magnetisation of iion, says in his Paper, entitled "Eesearches in Magnetism" FOll KLECTllIUAL STUDENTS. 217 {Phil. Trans., 1885) : — "My own observations show that it is only when the length of the rod (if of iron) is about 300 or 400 times its diameter that the effect of length becomes insensible." The same authority also states that the demagnetising force due to the ends is unequal along the bar, being much greater at the ends. The demagnetising effect is entirely got rid of if the iron is employed in the form of a ring, since, in this form, there is a complete magnetic circuit, and therefore no poles. The intensity of magnetisation can be found indirectly this way from the value of the magnetic induction B, and the magnetising force H ; but what we are considering now is the direct measurement of the magnetisation by the action of the poles on the magnetometer. We shall select two instances from a number of experiments to illustrate the difference between these conditions. Take, first a short, thick iron rod, in which there is considerable demagnetising effect. AVhen the demagnetising action of the poles is got rid of by employing either very thin iron wires or iron rings, the condition is technically spoken of as that of "endlessness." 134. Electro-Magnetisation of Short Iron Rod. — The neces- sary apparatus having been explained, we shall detail the method of taking observations, to show the gradual magnetisa- tion of a short iron core placed within a coil of wire through which definite currents are passed. Referring to the apparatus, as arranged in Fig. 112, the magnetometer and galvanometer must first be levelled and adjusted in position till their re- spective pointers coincide with the zero marks on the scales. If the controlling magnet is employed with the graded gal- vanometer described in para. 132, it must first be removed several feet away while the instrument is being levelled and adjusted to zero. The latter is effected by turning the instru- ment round until the pointer is at zero, and the levelling is effected by the two screws seen in the figures. The controlling magnet is then placed in position, and if not in the plane of the meridian will deflect the pointer a little away from zero. '218 FK-WriCAL NOTES To rectify this the adjustable screw on the arm F of the needle box is turned until the pointer is restored thereto. By a measurement, taken immediately before the readings, the con- trolling magnet used was found to reduce the sensitiveness twenty-one times (n). Its field at the needle was therefore (para. 132) H (m - 1) = -18 X 20 = 3-6 C.-G.-S. units. The first thing to be done is to put two or three different strengths of current through the coil alone, and read the cor- responding deflections on the two instruments. From these observations, when plotted on squared paper, it can be seen •what must be deducted from the readings taken with coil and oore together to give the true magnetic effect of the core. Having done this, the current is switched off and the core inserted into the coil. The core used in this experiment was a cylindrical rod of unannealed iron, 10 centimetres long and 43 millimetres diameter. Its sectional area was therefore = '7S54 X 'iS'-^^ -1452 square centimetres, and its volume = ■14.52 x 10 = 1-4.52 cubic centimetres. The length of the core being only some 23 times its dia- meter, there was considerable demagnetising effect. The core being shorter than the coil, care was taken by measurement to push it into an exactly central position in the coil. The latter being held down by a clamp this operation could not shift its position. Now it is important to notice at this point whether the magnetometer is deflected ; if it is, the core has some traces of magnetism which should be removed before commencing the test. This may be done by heating it and then allowing it to cool while lying horizontally in an east and west direction ; or a few light taps at the end while held in the same position will generally suffice. The resistance switch is now turned so as to put the greatest amount of resistance in circuit, and the current is then put on. FOR ELECTRICAL STUDENTS. 219 Both magnetometer and galvanometer deflections must now be read as each successive reduction is made in the resistance, and the same jotted down in two vertical columns. Some of the readings are here appended, the galvanometer deflections being divided into columns corresponding to the position of the needle-box on the platform when the respective readings were taken. Magneto- meter Deflections. Galvauometer Deflections. Ampere- 4 8 16 32 Reduced to 32 turns. Coil 17 r 2 1 14 Coil ^l and ■{ ,f,o Core 1^02 126 L131 35i 26i 33 41| 35i 23i 45 5 20 282 5 20 47 90 142 210 264 334 4512 80 320 752 1440 2272 3360 4224 5344 It is convenient afterwards to reduce them all to their value at position 32. At this position, with the controlling magnet in use, the current is equal to (•18 + 3-6) — = amperes, ^ ' 32 8-47 ^ where D is the deflection reduced to position 32, and 3'6 (as given above) is the field of the controlling magnet, which was used throughout. The number of turns on the magnetising coil being 135, we get the number of ampere-turns by multi- plying the current by this number — that is, ampere-turns = x 135 = 16 D. ^ 8-47 The deflections reduced to position 32 and multiplied by 16 give, therefore, the magnetising forces in ampere-turns, these values being worked out in the last column of the above table. The curve in Fig. 115 is plotted on squared paper from the readings. The horizontal ordinate is divided off in divisions, 220 PRACTICAL NOTES each representing 1,000 ampere-turns, and the above value* marked off accordingly. On the vertical ordinate it is more convenient to plot the magnetometer deflections first, which are proportional to the field produced by coil and core together,, as we have to subtract the efi'ect of the coil before we can estimate the magnetisation of the core, and this is most readily done by plotting the deflections. This gives a straight line- -npE -H -.-X K--" CORE (00 +' ^ -■-•'^^ (^ 75 / /' INTENSr -Y OF M, BNETISAl ION CURVt #/ SO V (HIGHZST POINT 1217 C.G.S VA 'ITS) /. *y il 25 ^/ */ ,—- / _CQ}}^— — C^ — — AMPERE TURNS Fio. 115. for the coil alone, and a regular curve (shown dotted) for the variation in external magnetic field of the coil and core together. Taking now a point on the horizontal ordinate for any given number of ampere-turns, the height of the vertical line from this point to the straight line of the coil must be- subtracted from the height of the vertical line from the point to the dotted curve ; this gives one point on a new cui-ve which represents the effect of the core alone. This operation- rOE ELECTRICAL STUDENTS. 221 having been repeated for several points on the base line, we get a corresponding number of points on the new curve, which may then be connected by a line, giving the curve shown. By inspection of this curve we can see what deflection the mag- netometer would have indicated had it been acted upon by the core alone. Calling this angle of deflection a, the tangent of which, as before explained, is equal to the deflection divided by 100, we may find the intensity of magnetisation of the core corresponding to any given point in the curve by , moment r^ H tan a p, ^, „ 1 = — ; = — , ,^^ C.-C4.-S. units, volume 1 -4:52 Now 9-3 H= (20- + 5-p X -18 = 1577, and ^•''"-1086. i-45i; Therefore, I = 1086 tan a. Take the highest point in the curve ; the deflection is 112, and therefore 1 = 1086x1-12 = 1217 C.-G.-S. units. Values of I can be worked out in this way for any other point ©n the curve. It should be noticed, however, that the shape of the magnetisation curve is the same whether we plot strength of pole, moment, intensity of magnetisation, or deflections on magnetometer due to core, since all these quantities are propor- tional to one another. Similarly, the curve will be the same whether the magnetising force is plotted in ampere-turns or O.-G.-S. units. The foregoing example is detailed with the view of furnish- ing the student with material sufficient to guide him in obser- vations and calculations of his own. Experiments made on however small a scale, the results of which are reduced to known units of measurement, are the best means of acquiring a clear conception of terms and their relative values. Near the higher yari of the curve, although the readings were taken 222 PRACTICAL NOTES as rapidly as possible, the current perceptibly warmed the coil and core, and this may have prevented the iron acquiring a higher magnetisation. The fact of coils becoming heated when large currents are used is a bar to obtaining more than a few hundred C.-G.-S. units of magnetising force this way. We have previously shown (para. 130) that the magnetising field in C.-G.-S. units produced in the interior of a long narrow coil is equal to ampere-turns i-of;?. length of coil (centimetres) and therefore, in the case before us, in which the coil is 1ft. (30'48 centimetres) long, the magnetising force per 1,000 ampere-turns is 1257 30-48 = 41-24 C.-G.-S. units; and, therefore, for, say, 6,000 ampere-turns, which is about 200 ampere-turns per centimetre length of coil, and not far from the maximum exciting power possible to use with one layer of wire, we obtain a magnetising force of only 41-24 x 6, or about 250 lines of force per square centimetre in the coil. If the number of layers of wire is increased, the current put through cannot be as strong as through a single layer, since the radiation of heat is slower, and a limit of exciting power is soon reached probably not much above 300 ampere-turns per centimetre length of coil. This prevents the investiga- tion by the magnetising coil method of the direction taken by the magnetisation curve under very high magnetising forces — that is to say, whether it ever reaches a maximum height beyond which no additional force wiU increase it, and if this condition of things (which means absolute saturation of magnetism) does take place, whether the curve continues horizontal or shows an inclination to descend. If these points, which are of the greatest interest, are to be deter- mined, recourse must be had to some other method of mag- netising the iron, by which powerful fields can be brought to bear upon it. FOR ELECTRICAL STUDENTS. 223 Immediately on reaching the highest current considered safe for the insulation, the resistance was increased again, step by step, and readings taken on both instruments as before. This gave the descending curve of magnetisation which fol- lowed pretty closely the shape of the ascending curve, but at some distance higher from it. The residual magnetism in the core was calculated by the above formula from the mag- netometer deflection after the current was switched off. This deflection being 10, the residual magnetisation was I = 1086 X -1 = 108 C.-G.-S. units. This is only 9 per cent, of the total magnetisation acquired, and is due to the demagnetising effect of the core on itself, already spoken of. In order to prove that the magnetometer has worked all right during the set of readings taken, as soon as the residual mag- netism has been measured the core should be withdrawn from the coil and placed at a distance, upon which the needle of the magnometer should return accurately to zero. We shall next consider the magnetisation of a wire of annealed iron, of sufficient length to render the demagnetising effect inap- preciable. 135. Electro-Magnetisation of Annealed Iron Wire. — In order to observe the magnetisation of a piece of iron when there is practically no demagnetising effect from its own poles, it has been stated that a rod or wire must be selected whose length is at least 300 times its diameter. Now, if the magnetic moment is measured by either of the two methods of Gauss already detailed (para. 125), the iron wire tested must be very small in diameter, as it is not convenient to test a very long wire by these methods. And the strength of pole developed in a wire of small diameter is necessarily very small even when the iron has acquired a high magnetisation, in consequence of which the magnetometer can only be deflected to very small angles. With a light mirror attached to the needle, the amplitude of the deflections can be considerably increased by a ray of light reflected from the mirror on to a screen, and i2i PRACTICAL NOTES In this manner very small movements of the needle can be accurately observed. We are here, however, employing a mag- netometer without a mirror attachment, and it is therefore advisable to arrange matters so that the needle is deflected to a considerable angle, say to 50 deg. or 60 deg., when the iron is near its highest magnetisation. The intermediate deflections can then be read with ease and accuracy. This may be efifected conveniently by using a long core and placing it vertically, as shown at C (Fig. 116). As large a diameter as No. 11 B.W.G. Fig, 116. iron wire may be used for a core 3ft. long, and yet be over the necessary ratio of length to diameter. This gives su.fiicient sectional area of iron to develop, at high magnetisations, enough strength of pole to strongly deflect the needle, the top end of the core being within 20 centimetres from the needle. In this diagram, C is the magnetising coil wound round a brass tube, and fixed vertically by passing it through a hole in the table. The iron wire core of the same length as the tube should slip easily into it, and the upper end of the latter be fixed a little above the level FOR ELECTRICAL STUDENTS. 225 of the magnetometer needle, in order that the pole, whicli is formed at a little distance from the end, may act more nearly in a horizontal line with the needle. It has been pointed out by Sir William Thomson (in Papers on Electricity and Magnetism, p. 512) that as the iron becomes more magnetised the poles move up more towards the ends, but are always 6ome slight distance off, in long thin cores, even when most Btrongly magnetised. It is the position of the poles and the variation of field along the length of a magnetised bar that Fig. 117. constitutes what is known as " magnetic distribution." With the magnetising coil vertical the changes in distribution or position of the poles during magnetisation produce minimum effect on the results. The upper pole now acts powerfully on the needle, and with a long core the action of the lower pole is very feeble. If we neglect the action of the lower pole, the intensity of field at the needle, placed r centimetres away from the upper pole, is ^ C.-G.-S. units, 226 PRACTICAL N0TK3 where m is the strength of pole {see paras. 108 and 109). But where accuracy is required the action of the lower pole must be taken into account. Taking the point p (Fig. 117) as the centre of the magnetometer needle and SN as the magnetised core, it will be seen that the lower pole (R centimetres from p) would repel a unit north pole at p with a force of — dynes. R2 -^ But this force is in an oblique direction, and the needle moving in a horizontal plane is only influenced by horizontal forces. It is therefore only the horizontal component of the above force that we have to deal with. Resolving by the parallelogram of forces we have by similar triangles horizontal component : rgj .' : r ; R, ix whence the horizontal force = — - dynes. The resultant force / at the needle is then given by the difference between the forces due tO eaph pole, viz. : — Now, the force in dynes on unit pole at a point is the same thing as the intensity of field at that point, and, therefore, the above value of / represents the strength of field at the needle centre due to both poles of the core. This field also acts on the needle at right angles to the earth's horizontal field H, and therefore (by para. 119) /= H tan a, where a. is the angular deflection of needle. Glancing at the figure it will be noticed that the end of the magnetometer needle is viewed, the earth's meridian being supposed to be at right angles to the plane of the paper, and the magnetising coil C being placed east or west of the needle. FOR ELECTRICAL STUDENTS. 227 Equating the above values of /, and reducing by simple i^gebra, we have H tan o r* m = 1 <0 which gives the strength of each pole in C.-G.-S. units, and from which the magnetisation may be calculated by I = ^ C.-G.-S. units, a where a is the sectional area of the core in square centimetres. The cubed fraction in the denominator is the ratio of the distances of the poles of the core from the needle, and this frac- tion is small when the core is long and placed near to the needle. If the cube of this fraction is very small and is neg- lected it wUl be seen that the formula represents the action of the upper pole alone. The manner of taking the test is precisely the same as that already detailed for the short iron rod. In the figure the adjustable resistance is shown conventionally, its practical form having already been detailed in para. 131 and Fig. 112. The following practical example may serve as a guide to work by : — ^A brass tube of three millimetres bore and 90 centimetres long was first served round with an insulating covering of brown paper and then wrapped round carefidly with No. 18 sUk-covered copper wire. This took 657 turns of wire in all, or 7"3 turns per centimetre. Melted paraffin wax was brushed over the coil, and laid on quite hot so as to allow it time to soak well into the silk before solidifying. The current was obtained from a maximum number of 12 secondary ceUs through two adjustable resistances, one of German silver wire (Fig. 112) and the other of water, the latter being for the smaller alterations in resistance. Measurements of the current strength were taken on a Thomson-graded galvanometer G (Fig. 116), the manner of using which has been described. The controlling magnet was not used in any of the readings, and therefore the current was found by -p. C=*18 —amperes, Q 2 228 PKACTICAL NOTES where D is the deflection and P the position of the needle-box (para. 132). A curve was then plotted on squared paper from the readings in the same manner as described for the last curve, the vertical ordinates being magnetometer deflections and the horizontal ordinates the magnetising forces, calculated in ampere-turns p^r foot. The reason of this is that the last curve (Fig. 115) was plotted from readings taken with a mag- netising coil one foot long, and therefore the magnetising forces in ampere-turns expressed on that curve are actually ampere- turns per foot. Now we wish to compare the rates of magneti- sation of the two cores under test for a given series of magnetis- ing forces, and therefore to be able to compare the two curves together we plot the magnetising forces exerted on the core at present vmder consideration in the same units as those em- ployed on the previous core (para. 134), viz., ampere-turns per foot. We have already noted (para. 130) that the magnetising fore in C.-G.-S. units is equivalent to ampere-turns i-ofc?. length of coil (centimetres) that is, the number of ampere-turns per centimetre multiplied by 1'257, from which it may be noticed that magnetising forces are proportional to the number of current-turns per unit of length. When comparing two or more coils of the same length the strength of field set up by them or their magnetising forces are proportional simply to their respective number of ampere- turns, but when dealing with coils of difierent lengths the simple statement of the number of ampere-turns on each offers no basis of comparison between them ; we must know the number of ampere-turns for a certain definite length of coil in each, (We are referring to cylindrically- wound coils, as generally used for electro-magnets, not ring-shape coils, as in galvanometers.) For example, take two coils, each wound with one layer of wire of similar size and around bobbins of similar diameter ; one bobbin being twice the length of the other. If the same cur- rent is passed through both, the number of ampere-turns in the former is twice that in the latter, and the length of wire wound on is also; twice as much, but the magnetic field developed by FOE ELECTRICAL STUDENTS. 329 each, or their respective magnetising forces, are equal, for we have the same number of ampere-turns per unit of length in each. Again, take two coils, each wound with one layer of wire of similar size and around bobbins of similar length ; one bobbin being twice the diameter of the other. If the same current is passed through both, the number of ampere- turns per unit of length is the same in each, and therefore the magnetic field developed by each is the same, although the length of wire wound on the former is twice that on the latter. And, further, if we take two exactly similar bobbins and wind any size wire on each, the magnetic fields developed are pro- portional to the number of ampere-turns on each ; and if the current is the same in both, the fields are proportional simply to the number of turns or to the length of wire wound on each The remarks in para. 128 allude to this latter and more simple case. We may put these facts very concisely in algebraical form by saying the magnetising force produced by a coil is propor- tional to Current X total wire length . coil diameter x coil length or the same law may be expressed in C.-G.-S. units by deri- vation from that already given in para. 130, viz. : — Magnetising force _ ampere-turns 1.057 in C.-G.-S. units coil length (centimetres) ' where the constant 1 -257 is the ratio — -, as explained in. that paragraph. For, the number of turns in any coil is equal to the total length of wire, divided by the mean length per turn, and the mean length per turn is equal to the mean dia- meter of the coil multiplied by tt (3'1416). Therefore, cancelling out the constant it, we may write the above in the form Magnetising force ^ ^^ total wire length ^ _4 in C.-G.-S. units ~ " coil length x coil diameter 10' all measurements of length being in centimetres. From this 230 PRACTICAL NOTES it is clear that it is only when the denominator of the expres- sion — ^that is, the mean surface of different size cylindrical coils — is the same that we can compare the strengths of field pro- duced by each by the product of the current and the length of wire, in other words, by the number of ampere-feet on each. To return, however, to the magnetisation test. The number of turns per foot was 222'5, and the magnetising force was, therefore, Amperes x 222-5, = -IS — X 222'5 ampere-turns per foot. Reducing all galvanometer deflections (D) to position 32 (P = 32), as in the last example, and multiplying up, we have Ampere-turns per foot= 1:| D ; all that was necessary, therefore, was to add on to each deflec- tion (reduced to 32) one-quarter of its value. These results give the magnetising forces in ampere-turns per foot, due to the current in the coil ; but we have an additional magnetising force constantly acting on the core, which must be taken ac- count of. The core, being vertical, is acted upon by the vertical component of the earth's field. Taking the Kew Observatory data given in para. 110, and calculating out the value of the above, we have Earth's vertical field = H x tangent of angle of dip. The angle of dip or inclination is 67° 37', and its tangent = 2-4282. Therefore Vertical field = -1810 x 2-4282 = -4394 C.-G.-S. units. This must be expressed in ampere-turns per foot, in order to add it to the magnetising forces due to the coil. We can do this from the expression Field in C.-G.-S. units = 1 -257 x ampere-turns per centimetre. Now, 1 foot = 30-48 centimetres, and therefore the number of ampere-turns per centimetre is equal to Ampere-turns per foot "" 30-48 FOB ELECTRICAL STUDENTS. 231 Substituting this value, and dividing out, we have Ampere-turns per foot = field in C.-G.-S. units x 24 j. The vertical force of the earth is, therefore, in this lati- tude, equal to a magnetising force of ■4394 X 24J = 10*66 ampere-turns per foot, *nd is added to the magnetising forces set up by the coil to give the total magnetising force. We say added, because the 5» I 3 Fio. 118. current was purposely passed round the coil in the direction to cause the lower end of the core to become a north pole, and the earth's vertical force has the same effect. Had the current been in the other direction, the earth's force would have been subtracted. The usual method of dealing with this force is to neutralise it by winding another coil outside the first coil. This extra coil is usually of fine wire, and a current is passed through it just sufiicient to set up a field exactly equal and opposite to 232 PKACTICAL NOTES that of the earth, and this current is kept on continuously during the magnetisation test, neutralising the vertical field. The iron core, which was mechanically very soft and care- fully annealed, was 90 centimetres long and 2'6 millimetres diameter. The length was, therefore, 346 times the diametei^ and over Prof. Ewing's minimum limit for negligible demag- netising effect. In consequence of this it will be noticed that the rise in magnetisation is much more rapid at the commence- ment of the curve (Fig. 118) than with the former short iron core. The vertical ordinates of the two curves do not allow of comparison, since the cores are of different dimensions. This comparison must be made by the intensity of magnetisation, which can be calculated by the formula stated above. We must first find the value of m. The distance (r) of the upper pole from the needle was 20 centimetres, and the length (I) of core being known (90 centi- metres), the distance (R) of the lower pole can be found by R being the hypothenuse of a right-angle triangle, of which r and I are the two other sides ("Euclid," I., 47). Working out these values we get for the denominator of the above expres- sion •9898, and for the numerator 72 tan a. We have, there- fore, m.= 72-73 tana. Now the sectional area of core is equal to - Illff! = -7854 X (-26)2 = -053 sq. cms., 4 and therefore the intensity of magnetisation is equal to m 72'73 tan a , „„^ , — = = 1,370 tan a. a -053 We can now find the magnetisation at any given point on the curve. Take the highest point, viz., 97 divisions : — I = 1,370 X -97 = 1,329 C.-G.-S. units. On switching off the current the magnetometer remained steady at 45 divisions ; hence the residual magnetisation was 1,370 X -45 = 616 C.-G.-S. units, FOR ELECTRICAL STUDENTS. 2^3 or 46 per cent, of the maximum magnetisation acquired. When the current is suddenly interrupted in this manner the residual magnetism is not so great as when the current is gradually reduced to zero. In soft annealed iron wires, whose relation between length and diameter was similar to the above, Prof. Ewing found the residual magnetism some 85 per cent, of the maximum ; and as much as 90 and 93 per cent, in other speci- mens of soft iron (" Eesearches in Magnetism," Phil. Trans., 1885). 136. Comparison of Magnetisation Curves. — The two curves obtained have been plotted with magnetometer deflections as ordinates, but they cannot be compared by these deflections, as they were taken by different methods and with cores of diiferent sectional area. We shall now compare them by plotting them with the intensities of magnetisation as ordinates, and at the same time express the magnetising force in C.-G.-S. units instead of ampere-turns per foot. For every 1,000 ampere- turns per foot we have 41'24 C.-G.-S. units (para. 134), and by inspection^ of each curve we find the magnetometer reading (tan a) corresponding to each 1,000 ampere-turns per foot and intermediate! points. The intensity of magnetisation is derived from the magnetometer reading by I = 1086 tan a in the first curve (para. 1 34), and by I = 1370 tan a in the latter. For example, in the latter curve we find by inspection thati for 1,000 ampere-turns per foot (equal to 41 '24 C.-G.-S. units) the deflection is 92, and therefore the magnetisation is 1370 x "92 = 1260 C.-G.-S. units. Similarly, for 400 ampere-turns per foot (equal to 16 '5 C.-G.-S. units) the deflection is 80 and the mag- netisation 1370x-8 = 1096 C.-G.-S. units. Taking several points this way, and plotting them as in Fig. 119, we are able to observe the relative magnetisation of the two cores. The main points of comparison are the relative magnetisations acquired under any given magnetising force and the magnetic " reten- 234 PRACTICAL NOTES tiveness " of each core. It should "be borne in mind that the top curve is that taken with a core whose dimensions satisfy the condition of "endlessness," and is a soft annealed wire, while the lower curve is that from a short iron rod of soft iron, but unannealed. The marked difference in the rise of the two curves is partly due to the slightly different nature of the iron, but chiefly due to the dimensions of the cores. This is very clearly shown by Prof. Ewing (to whose results the above are similar) in his experiments. He cut from the same piece of iron wire (of 1'58 millimetre diameter) different lengths, whose ratios of length to diameter varied from 50 to 300. The demagnetising effect and slow magnetisation were, therefore, proved to be due to the ends in short lengths. < 40O In the above there was practically no demagnetising effect set up by the poles of the long wire core, and hence the fuU magnetising force of the coil could take effect upon it. With the short rod core, however, the poles caused a considerable reverse field of force, through the metal (para. 133) neutralis- ing part of the field of the magnetising coil, and preventing the iron from becoming magnetised. This accounts for the very little residual magnetism left, being only about 9 per cent, of the maximum, for, when the current in the coil was reduced to zero the poles of the core exerted their reverse field, tending to neutralise the magnetism acquired. On the other hand, with FOR ELECTEICAL STUDENTS. 235 the long wire core the poles could exert no demagnetising effect, and hence the iron, though softer, retained a much greater per- centage of magnetism. These facts indicate clearly that mag- netisation and residual magnetism depend as much on the shape as on the kind of iron. While it is always best to use :8oft iron to obtain the highest magnetisation for a given mag- netising force, it is advisable in the construction of electro- magnets where the residual magnetism is to be minimised to make the core short and thick rather than long and thin. The cores of electro-magnets of fast-speed telegraph instruments have of late years been constructed more of the " squat " shape, this having been found to allow of more rapid signalling than the original tall, thin cores. 137. Magnetic Susceptibility. — As Science moves onward she ■draws in her train an ever-increasing number of newly-coined or adopted words and names, to each of which she assigns a definite role — that of representing with rigid exactness some new idea or phenomenon, or perhaps of being the more concise represents^ tive of some already known but hitherto undefined property of matter. Perhaps in no paths of discovery has she so far per- petuated in this manner the names of those who have worked laboriously in her cause as in electrical and allied researches, in which such famous names as Volta, Ampfere, Ohm, Coulomb, Faraday, Joule, and Watt are constantly in use by those who are engaged in developing the many practical appli- cations turning on their discoveries. It sometimes happens that Science presses into her service words which have been more or less in the familiar use of common parlance. When, however, such words are " told off " to serve her, she requires that, whatever variety of garb they may choose to wear in their capacity as civilians, one distinctive uniform only must be "worn when they are enlisted in her corps and on duty in her •service. In other words, Science requires that definite ideas -shall be represented by special terms set apart exclusively fot their use. As instances, we might cite such words as force, strength, mass, intensity, power, work, energy, capacity, &c., which all have considerable flexibility of meaning in ordinary 236 PRACTICAL NOTES use, but only one distinctive meaning in their scientific application. The word " susceptibility," which we are about to consider, has been told oflf to express a state or condition of iron in which it is more or less " susceptible " of becoming magnetised under the action of electro-magnetic force. In the familiar use of the word we speak of " susceptibility to pain," " suscept- ible of impressions," " susceptible to charms," &c. ; and, turning to see what Johnson says, we find he quotes a sentence from " Wotton " to illustrate its meaning as follows :— " He moulded him platonically to his own idea, delighting first in the choice of materials, because he found him susceptible of good form." A noted judge, prominent in electrical litigation, is reported to have said on one occasion, " No doubt the words used are susceptible of two constructions." Put into the simplest language, we may say that great or little susceptibility conveys the idea of the ease or difficulty with which an acting cause produces a corresponding effect. When the subject operated upon possesses great susceptibility a given acting cause can produce a great effect, but when it has Uttle susceptibOity the same acting cause can produce but little effect. The subject operated upon in our present case is iron, the acting cause magnetising force, and the effect the mag- netisation produced. Iron is said to have a greater maximum magnetic susce2}tibiliti/ tha,n steel, because under a given magnetis- ing force it acquires a higher intensity of magnetisation. Also, the magnetic susceptibility of iron or steel varies according to the state of magnetisation. On applying a gradually-increasing magnetising force to iron its susceptibility is at first small, then increases very rapidly to a maximum, and then falls again almost to zero. This may be noticed in the curves already worked out. We may now define magnetic susceptibility (usually denoted by the Greek letter k) as the ratio between I, the inten- sity of magnetisation acquired, and H, the acting magnetising force, and write it I FOR ELECTRICAL STUDENTS. 237 For example, taking the higher curve in the last figure it will be noticed that for a magnetising force of 10 C.-G.-S. units the magnetisation acquired is 800, and therefore the susceptibility is 80. This is at the steepest part of the curve, and is there- fore the maximum susceptibility in this specimen. As the iron becomes practically "saturated" at the sharp bend of the curve the susceptibility is reduced to = 40, and at the highest magnetising force employed (170 C.-G.-S. units) it becomes very small, viz., = 8 nearly. Prof. Ewing found the susceptibility of a specimen of very soft iron to reach up to 280, and when mechanical vibrations were applied to the same specimen while under the magnetising force its susceptibility increased to as high a value as 1,600. One important point must be noted. Any piece of iron, the susceptibility of which is to be determined, must comply in dimensions with the condition of " endlessness " spoken of in para. 133, otherwise the exact magnetising force brought to bear upon it cannot be computed. For instance, from the lower curve in the last figure the susceptibility of that specimen of iron cannot be derived, because the dimensions of the core are such that a demagnetising force of unknown value is acting on it in addition to the force due to the coil. 138. Further Developments of the Subject. — Before leaving this subject we shall indicate in what direction the student may experiment with advantage. Besides the acquaintance he will gain with the subject by trying cores of various specimens of iron and steel of difierent dimensions, he should carry the mag- netisation of some specimens through a complete phase or cycle ; that is, starting by increasing the current from zero, the ordi- nary curve of magnetisation rises from (Fig. 120). After carrying this on until the core is fairly saturated, the current is gradually reduced to zero. The curve, as we have seen, will 'not return to the point of origin by reason of the residual magnetism A. The current in the magnetising coil is then reversed, and precisely the same operation repeated — viz,. 238 PRACTICAL NOTES gradual increase of current to fair saturation, and then decrease to zero. This brings us to the point B, the residual magnetism being B. Now the current is reversed again, and gradually increased until the highest point in the curve is again reached. The core has then been led through a complete cycle of magneti- sation. The valuable suggestion made by Mr. Sidney Evershed {The Electrician, Nov. 9, 1888) for eliminating the effect of the magnetising coil on the magnetometer, simultaneously with the test, by winding a few turns of the said coil backwards round the magnetometer needle at such a distance as to just neutralise its own effect, should also be taken advantage of. This would allow of seeing at once by the magnetometer deflections when the core approached saturation, and also save the time of plotting two curves. The core having been led through a cycle of magnetisation in the manner indicated, we have a completely closed curve, the area enclosed by which is proportional to the amount of work spent per unit volume of iron in conducting it through the cycle. FOR ELECTRIOAL STUDENTS. 23^ It is evident that the cause of the cyclic curve being of this- general shape and enclosing an area is due to the fact that the descending curve does not coincide with the ascending curve^ and this latter fact is due to residual magnetism of the iron. In other words, the magnetisation of the iron follows at a dis- tance or "lags behind" the magnetising force acting on it. Prof. Ewing, in a set of exhaustive experiments on this pro- perty of iron, carried out some five or six years ago at the Physical Laboratory of the University of Tokio, Japan, calls- the phenomenon "Hysteresis," from va-repito, to lag behind, and defines it as " the lagging behind of changes in mag- netic intensity to changes in magnetising force " (" Eesearches in Magnetism," Fhil. Trans., 1885). In the same Paper we find the pregnant remark, " The existence of residual magnetism,, when the field is reduced to zero, is, in fact, only one case of an action which occurs whenever the field is varied in any way, viz., a tendency to persistence of pi-evious magnetic state." In all electro-magnetic apparatus where the magnetising force is alternating in direction, or where an iron mass moves periodically through alternate fields, the iron undergoes this- cyclic change, and we have the phenomenon of hysteresis. In dynamo and motor armatures, and in the cores of " transformers " or " converters " used in distributing electrical energy by the alternate current system, the loss of energy by hysteresis is an important point affecting more or less their working efiicienoy. The dissipated energy due to this cause takes the form of heat, resulting in a rise of temperature in the iron. In the earlier forms of " transformers " the iron cores became perfectly hot from this cause, showing the necessity for working with a weaker field through the iron. Mr. Gisbert Kapp, in his Paper on "Alternate-Current Transformers," before the Society of Telegraph-Engineers and Electricians (The Electrician, February 10th, 1888), in describing the experiments conducted by himself and Mr. W. H. Snell in the design of their well-known transformer, observes that, owing to the rise of temperature of the iron due to hysteresis, the " induction " through the iron had to be reduced to somewhat less than 10,000 C.-G.-S. units {see paras. 240 PRACTICAL NOTBS 139 and 140), which may be considered as about one-third the maximum induction at which it is practically possible to work dynamo armatures constructed of the softest wrought iron. It is not within the scope of the present work to treat of the above- mentioned machines ; the student who is pursuing the subject should, however, consult the works already referred to, in addi- tion to which he will find the subject very clearly treated in a " Note on Magnetic Hysteresis," by Dr. J. A. Fleming {The Electrician, September 14, 1888). In furtherance of the sub- ject of magnetic susceptibility, a Paper by Mr. R. Shida (Proc. Koyal Soc, 1883), and one by Mr. A. Tanakadati on the " Mean Intensity of Magnetism of Soft Iron Bars of Various Lengths in a Uniform Magnetic Field," read before Section A of the Bath meeting of the British Association (The Electrician, November 2, 1888), will be found valuable sources of infor- mation. 139. Electro-Magnetic Induction. — We shall conclude this section with a consideration of electro-magnetic induction and magnetic permeability, both of which are closely allied to the foregoing subject of magnetisation, and in such constant use in descriptions of, and discussions on, electro-magnetic appliances as to make it important for the student to have a clear concep- tion of their meanings. At the same time, consistently with making the subject intelligible, we shall dwell on it as briefly as possible. At the outset, the relation which " magnetic induction " bears to " intensity of magnetisation " should be clearly under- stood. We may state this concisely by saying that intensity of magnetisation is proportional simply to the number of lines of force per unit area passing through the iron due to its own acquired magnetism, while magnetic induction is proportional to the number of lines per unit area passing through the iron, due to its own acquired magnetism, plus those dice to the magnetising field. The latter is the more practically useful means of measurement, since, in practice, we necessarily concern ourselves with the combined magnetic effect of the magnetised iron and the electro-magnetic field causing its magnetisation. Recalling FOE ELECTRICAL STUDENTS. 241 ■what was said in para. 133, it will be remembered that through the mass of any piece of iron which has been magnetised to a istrength of pole equal to m C.-G.-S. units, and whose dimensions satisfy the condition of " endlessness," there are 4 tt m lines ■running from pole to pole (south to north). The condition of "endlessness" ensures the absence of any demagnetising lines .(running from north to south, or in a contrary direction through the metal) which would make the actual number of lines through such a piece of magnetised iron less than 4 tt m. Such being the case, we have in a piece of iron of a square centimetres (uniform) cross-section lines per square centimetre passing through its mass and due to its own ■ acquired magnetism. Now, the intensity of magnetisation (I) is found by I _™ a putting which value in the above we find that 4 TT I = lines per sq. cm. due to acquired magnetism; and, therefore, I = lines per sq. cm. due to acquired magnetism x 4 IT Tliis proves the statement made above, that the inten- sity of magnetisation is proportional to the lines of force per square centimetre through the iron, due alone to the magnetism acquired. The same fact will have been noticed in the practical examples already detailed, in which the mag- netising force of the coil was subtracted from the combined magnetic effect of the coil and core together before calculating the intensity of magnetisation. Similarly, we have seen that the strength of a permanent magnet may be expressed in units of intensity of magnetisation, since the term only involves the lines of force due to its own magnetism, or tlie strength of pole of the magnet itself ; but the same cannot be expressed in ■units of magnetic induction, since this term implies the E 242 PRACTICAL NOTES presence of a magnetising force, and must include the value of the same. Now, to understand the precise meaning of magnetic induc- tion, let us take a simple cylindrical coil of wire as at H (Fig. 121). On passing a known current through it, a magnetic field of a definite intensity is set up in its interior. We have- Fig. 121. then a magnetising coil — that is, one capable of magnetising' any magnetic metal placed within it. We have shown that the field intensity in the coil, or, what is the same thing, the lines per square centimetre, can be easily calculated. Suppose there are H lines per square centimetre through the coil. Now, on putting an iron core inside, as at B, we find that the intensity of field becomes very much stronger than before ; in FOB ELECTBICAIj STUDENTS. 243 fact, the presence of iron in the coil has enormously in- creased the number of lines. Now, what is termed " magnetic induction," or sometimes simply " induction," and denoted by the letter B, is the strength of field which exists when the iron is put in — that is, the number of lines per square centimetre. It must be remembered, however, that the part of this field B which we called H existed hefore the iron was put in, and there- fore B - H lines have been created by adding the iron. We say created because they did not exist in the iron before it was put in the coil, neither did they exist in the coil space before. The lines B — H are therefore those due alone to the magnetism acquired by the iron, and these we have found above to be equal to 4 t I. We therefore arrive at an equation expressing the fundamental relation between " magnetisation " and " induc- tion," viz., B-H = 47r|, from which, if we know the magnetising force H and the mag- netisation I, we can find the induction by B = 47r|-t-H. For example, by inspection of the upper curve in Fig. 11& we find that the magnetisation was 1,280 units for a magnetis- ing force of 54 units. Hence the magnetic induction for this force was 4 IT 1,280 -1- 54 = 16,144 lines per square centimetre, of which 54 are due to the magnetising coil and 16,090 to the magnetism acquired by the iron. Working out a few values in the above manner we obtain data to plot a fresh curve (Fig. 122) for the same core, the vertical ordinates being now in C.-G.-S. units of magnetic induction or lines of force per square centi- metre through the iron. This shows the connection between induction and magnetisation, and how one may be deduced from the other. It is possible, however, that the student may find some difiiculty in conceiving the physical meaning of the formula as applied to the reverse operation, viz., that of deduc- R 2 2U PRACTICAL NOTES ing the magnetisation from the magnetic induction. From what we have shown above, the formula is evidently : — 1 = B-H 47r Now, suppose the value of B in a certain ring of iron under a given magnetising force H has been ascertained. It is simple enough to put these values into the above formula and so cal- culate I ; but the physical meaning is not so simple to follow, for by I is meant the pole strength per unit area of metal, or ' x ISOOO f CURVE or MAGh ETIC IHC UCTION 17.000 o / HIGHEST FOR sort POINT lei IRON 190 LINES. o MOO. ■Si 1 1 y 4.000 1 / w GMETISIHa FORCE W C.G.S. i NITS. o JO eo so 130 ISO J60 FiQ. 122. moment per unit volume, while the piece of iron under test is ring-shaped, and, therefore, has no poles and no magnetic moment. The explanation is that for a given pole strength per unit area, the mass of the iron is in a certain definite condition magnetically, Avhich can be represented as equivalent to a definite field or number of lines per square centimetre, running- through its substance. Now, it is clear that this identical magnetic condition of the iron, as regards the number of lines through it, can exist equally well when the iron forms a closed FOR ELECTRICAL STUDENTS. 245 circuit, and exhibits no polarity at all ; indeed, we have already shown that the internal field equivalent to a magnetisation I is represented by 4 tt 1 lines, and therefore the formula is quite intelligible. With a ring-shaped iron core the magnetisation cannot be determined by a magnetometer, as there are no poles to act on the needle, but it can be calculated in the above way from a direct experimental determination of the magnetic induction. 140. Practical and Absolute Magnetic Saturation. — It will be noticed in the above curve that magnetic induction in the iron, up to 12,000 lines, takes place very rapidly under the action of 10 or 12 C.-G.-S. units of magnetising force, and that as the force is increased beyond this point induction occurs more slowly. For instance, double the force (24 units) only produces 14,700 lines, or an increase of a little over one- fifth the field. Again, three times the force (36 units) only produces 15,700 lines, or an increase of about one-third the field. And it will be seen by the curve that any further increase in the force produces but slight increase in the induction ; in fact, for practical purposes, it would be very uneconomical to expend 180 units of magnetising force to produce 16,900 lines, when 16,000 could be produced with less than one-quarter the force. It is for tliis reason that the top bend of the curve just before it becomes almost hori- zontal is taken as the practical limit of saturation, and in soft iron may be taken as about 16,000 lines. In the best quality of iron now obtainable, such as Swedish charcoal iron or well-annealed wrought iron, the induction curve rises quicker, and reaches considerably higher values. For dynamo armatures built with iron of this quality a fair average saturation met with in good machines ranges from about 18,000 to 20,000 C.-G.-S. lines, or sometimes higher. The point of " absolute saturation " is of purely theoretical value, and, of course, means the point at which the curve ceases altogether to rise. It would appear that there could be no absolute saturation point for magnetic induction, because the magne- tising force is one of its constituents, and, tlierefore, it would 246 PRACTICAL NOTES seem probable that induction would always increase mth an increase in magnetising force. To determine tliis question evidently requires the employment of exceedingly high mag- netising forces, to obtain which the magnetising coil, as applied directly to the specimen of iron under test, is out of the question. The researches of Prof. J. A. Ewing and Mr. William Low in this direction furnish by far the most advanced experimental data on the subject, inasmuch as these able observers have for some time brought to bear upon the specimens of iron under test far higher mag- netising forces than have been hitherto employed. Their method consists in placing the piece of iron between the poles of a powerful electro-magnet, so as to form a com- ]jlete magnetic circuit, and thus force a large number of lines through it. To increase the density of lines through the test piece it is thinned at the centre by turning down to a very small diameter, while its two sides extend outwards, cone-like, to fit the pole-pieces. This gives the test piece the shape of a reel or bobbin, of which the thin central portion is called the neck. By this means, as commu- nicated to the Royal Society in their recent Paper (Novem- ber 22, 1888), the authors have forced the magnetic induction in wrought iron to the enormous value of 45,350 C.-G.-S. units, and in cast iron to 31,760, and find that there is no appearance of a limit to the extent to which magnetic induction may be raised. To show what an advance this is, a remark of Drs. .J. and E. Hopkinson in tlieir able Paper on Dynamo-Electric Machines in 1886 (Phil Trans.), may be quoted, "No pub- lished experiments exist giving the magnetising force required to produce the induction here observed in the armature core amounting to a maximum of 20,000 per square centimetre." The direct measurement of the induction was made in the usual way ; the central neck, where the induction is greatest, was wound with a coil of wire connected to a ballistic galvano- meter, and the " swing " on the latter observed when the iron bobbin was turned round through half a revolution, so as to face the opposite poles of the electro-magnet. As this com- pletely reverses the direction of magnetisation, the deflection FOR ELECTRICAL STUDENTS. 247 is proportional to twice the induction in the bobbin. We shall refer again to the ballistic galvanometer and the method of -calibrating it in absolute units by the use of earth-coils. The measurement of the exact value of the magnetising force ■ (necessary in order to calculate the intensity of magnetisation) was attended with great difficulty, and it was not until the Paper referred to above that the authors were able to speak with confidence of its determination. The method was to wind a second coil at a certain measured distance concentricallj' round the first, and to subtract the deflection obtained with the first coil from that obtained with the second, the difference being proportional to the field within the air space between the two coils. To investigate how nearly this field approximated to the magnetising force acting on the metal, was a determination of great difiioulty, but the authors say in the above Paper that by modifying the form of the cone-shaped sides of the iron bobbin a uniform field was obtained in the central neck, and the magnetising force within the neck was sensibly the same as that in the air immediately around it. Having established the correctness of the determinations of the magnetising forces employed it was found that for wroiiglit iron the intensity of magnetisation remained stationary at 1,700 C.-G.-S. units, while the magnetising force was varied ibetween the enormous values of 2,000 to 20,000 C.-G.-S. units, showing that apparently a point of " absolute " saturation as regards magnetisation may be reached. This saturation point ior cast iron is found by the authors to be 1,240 units. The point of " practical " saturation for soft iron will be seen by the curve obtained above experimentally (Fig. 119) to be ;about 1,300 C.-G.-S. units. 141. Kapp Lines. — At this stage it may be well to allude to •the English unit of magnetic induction introduced by Mr. Gis- bert Kapp, and now largely in use by manufacturers of elec- trical machinery. A new unit line of force, equal to 6,000 •C.-G.-S. lines, is adopted, and the sectional area of iron is taken in square inches instead of square centimetres. The new unit .of induction is therefore one of these assumed lines per square 248 PEACTICAL NOTES inch section, and is generally known as one Kapp line per square inch, or one English unit of induction. The number 6,000 is composed of two factors, viz., 60, which enables the formula for the E.jNI.F. produced in a rotating, armature to be expressed in revolutions per minute instead of per second ; and the factor 100, which brings the measurement of induction down to figures in the tens instead of thousands. Now, if we divide 6,000 by 6-4514 (the number of square centi- metres in one square inch), we have the number of C.-G.-S, lines per square centimetre equivalent to one Kapp line per square inch = 930-03, or say 930; therefore, 1 Kapp line per sq. in. = 930 C.-G.-S. lines per sq. cm., or 1 English unit = 930 C.-G.-S. units of magnetic induction. For the purpose of comparison the vertical ordinates of the last curve have been reduced to Kapp units by dividing the values of B by 930 (Fig. 123). The highest induction arrived at in this specimen of soft iron is seen to be 18 lines, and the practical saturation 17-2 lines (16,000 C.-G.-S. lines). In the previous paragraph we noted that there was no absolute limit. FOR ELECTRICAL STUDENTS. 249' to induction, and that 20,000 C.-G.-S. units ( = 21-5 Kapp units) might be considered a fair average induction in machines whose armatures were of well-annealed wrought iron. The extremely high value reached by Prof. Ewing of 45,350 C.-G.-S. lines with wrought iron is equivalent to 48'7 Kapp lines. One or two examples quoted from descriptions of dynamos in which the induction is expressed in these units may render their prac- tical application intelhgible. For instance, in Tlie Electrician, Vol. XX., page 35, we find : — " The cross-sectional area of the magnet bars is 30 square inches, and the total cross-sectional area of actual iron in the armature is 20 square inches. The total excitation of field- magnets at full load is 12,130 ampere-turns, and the total strength of useful field is 378 lines (English measure). The density in the magnet cores is therefore 12 '6 lines per square inch, and in the armature core 18'9 lines per square inch." Again, on page 162, in the description of another machine : " The field strength in the core, according to the figures sup- plied us, works out at a very high figure, viz., to 23'6 lines per square inch." These calculations are on the basis of Mr. Kapp's formula given in his Paper on " The Predetermina- tion of the Characteristics of Dynamos " {Journal Soc. Tel. Eng., X"o. 64), and take into account magnetic resistance and leakage. In that Paper Jlr. Kapp gives the average practical saturation values of induction derived from numerous data and experiments on machines, pointing out that the figures given are by no means the highest obtainable with exceptionably good iron. These are as follows ; — Armatures built of well-annealed charcoal iron wire 25 lines per square inch, armatures built of discs of the same iron 22 lines, field- magnets of hammered scrap iron 18 lines. Multiplying by 930 we find these are equivalent to 23,250, 20,460, and 16,740' C.-G.-S. lines of induction respectively. Considerable exception to these English units was taken by scientists at the time of their introduction two years ago, chiefly on the ground of the necessary translation of coefficients used with formulte based on the inch-minute system into their value on the centimetre- 250 PRACTICAL NOTES second system. Mr. Kapp's reply to this and other objections (Society of Telegraph-Engineers' meeting, Deo. 2nd, 1886) was as follows : — " I use minutes because everybody is accustomed to counting revolutions per minute, and anybody quite unacquainted with French measure can work my formulte, and would at the ■same time see what he is doing. The figures are of reasonable magnitude, and present to those who use them a definite mean- ing. They know what is meant by 17 lines to the square inch, but if we talk of 15,800 to the square centimetre a greater mental efibrt is required to gi'asp the meaning. . . . The translation from it toC.-G.-S. units is really not such a diiBcult matter as we are told, and with a little training one could get to manage either system equally well ; but we have to talk to our workmen, and then we could not use grammes, centimetres, and seconds. AVe must give them figures in the usual English measure, and therefore it saves labour if we make the calcula- tions in a system where the figures are of reasonable mag- nitude and directly applicable to the various purposes of the worksliop." 142. Magnetic Permeability — To denote the degree to which iron and other magnetic metals permit lines of magnetic induc- tion to ■permeate through their substance, under the action of a given magnetising force, the term " permeahility " has been applied. The permeability of a specimen of iron is said to be high if it will permit the flow of a large magnetic induction through its mass for the e.xpendituio of a small uuignctisiug force, and vice versd. In other words, magnetic permeabilitj' is the ratio between magnetic induction B and magnetising force H, and is iisually denoted by the Greek letter /x. By inspection of the curves it will be seen that this propert}^, like that of susceptibilitj', is not a constant quantity for any given kind of iron, but depends on its magnetic condition. For instance, in the curve given (Fig. 122) when the induction is between the limits of 4,000 and 12,000 C.-G.-S. lines, the permea- bilitj' is fairly constant at about 800, but gradually falls to 300 at 16,000 lines, and idtimately to 100. In softer specimens of iron the permeability may reach 3,000 or 4,000. FOR ELECTRICAL STUDENTS. 251 The meaning of these figures may be better appreciated by ■considering them as indicating the permeabihty of iron with reference to that of air. In a magnetising coil, with no iron present, the induction produced in the air space in the interior is identically the same as the magnetising field, and, therefore, the ratio of the two, i.e., the magnetic permeability of air, is equal to unity. The number indicating the permeability of iron is therefore the number of times its permeability exceeds that of air. When iron is acted upon by an alternating or increasing and diminishing magnetising force, its permeability, when not near saturation, is constant. The parallelism of the ascending and descending lines in the cyclic curve of magnetisation (Fig. 120) shows the susceptibility to be constant for low magnetising forces, and it follows that the permeability is constant. The exact relation between permeability and susceptibility may be found by taking the fundamental formula given in para. 139 — B = 47r| +H, and dividing each side by H ; we then have yu. = 4 ?r K + 1, ■or Permeability = 1 + (Susceptibility x 1 2'57). We may regard the term permeability as meaning the con- ductivity of iron to lines of force, in the same way as we speak of the conductivity of iron to the passage of electric currents ; with this important difference, however, that while the electrical . conductivity of iron at a given temperature remains constant for all intensities of curi'ent, its magnetic permeability decreases to a very marked extent as the intensity of magnetic-field or number of lines through it is increased. If we pass an electric current through a piece of iron wire, and gradually increase the intensity of current until the wire is red-hot, the conductivity of the metal will undergo a considerable change. As soon as the intensity of current is sufficient to cause the slightest rise in temperature of the iron, vthe conductivity of the latter commences to diminish, and 252 PRACTICAL NOTES beyond this point any increase in the flow of current causes a rapid diminution of the conductivity. This decrease of con- ductivity in an electrical circuit, on increasing the intensity of current through the conductor, is somewhat analogous to the decrease of permeability in a magnetic circuit, on increasing the intensity of magnetic flow, or induction through the iron. Section III. — Magnetic Fields of Coils and Solenoids. 143. Preliminary. — The phenomenon which may be con- sidered the most important and fundamental to be appreciated- BATTERY J Fig. 124. in dealing with this subject is that discovered by Oersted 70' years ago, viz., the magnetic character of and effects produced by currents of electricity. A simple experiment is sufficient to exhibit the magnetic field set up in the neighbourhood of a con- ductor carrying an electric current. A piece of stout copper wire, say No. 14, is pierced through the centre of a sheet of stiff cardboard (Fig. 124), and carried vertically for two- FOR ELECTRICAL STUDENTS. L'jo ■ or three feet before it is bent round to the terminals leading to a battery or other source of current. The card should be supported uniformly underneath, so as to present a flat, rigid surface, and the battery sliould consist of a few cells ■ of large plate area, such as secondary batteries, in order to furnish a strong current. If the latter are used there should be an adjustable resistance between the battery and the experimental wire, which can be gradually reduced until the current is of sufficient strength to exhibit the effects, otherwise the experimental wire, being of neligible resistance, short-circuits the cells. Iron filings sprinlvled over the card while the current is passing arrange themselves in circles round the wire. This exhibits the form of the magnetic field sur- rounding a straight conductor conveying a current, and demon- strates the magnetic character and properties of the current. What most concerns us in the use of this principle is the measurement of the field ; in other words, we must be able to calculate the intensity and direction of the field at any point in the vicinity of the conductor. The direction of flow of the circular lines of force according to the usual assump- tion (para. 104) may be readily observed by exploring the same with a small magnetic compass-needle as shown in the figL^.e. Suppose a line drawn across the card in the direc- tion of the magnetic meridian, and two compass-needles placed on the line, one on each side of the conductor. Before the current is switched on to the wii-e the two needles will point along the meridian line, both, of course, with their north poles pointing towards the north. Now, if we connect the battery to the terminals, so as to send a current upwards through the wire, we shall observe on switching on the current that the needles move sharply into positions nearly at right angles to the meri- dian line, each pointing the opposite way. The north pole of the one on the right-hand side of tlie wire will point aieay from the observer, and the same pole of the other needle will point toivards the observer, as shown. And if we move one compass needle round the wire, starting from one point and coming back to the same again, we shall observe that the needle turns on its pivot through a complete revolution — in other 254 PRACTICAL NOTES woixls, it shows the direction of flow of the cui-ves of force to be continuous. According to the assumption that the north pole of a needle points to the direction of flow of the lines (para. 104), we find that, in this case, the flow is round the wire in the direction in which one would unscrew a screw (a. Fig. 125). To remember the direction, perhaps the screw is the best analogj-, for in unscrewing a screw we are moving it bodily upwards, i.e., in the same direction as the current flows in the wire. The converse case (b, Fig. 125) is with a descending current in the wire, the direction of flow of lines being reversed, aud coinciding with the direction of turning a screw when screwing it downwards. Again, suppose we are using a spanner to screw ' ' 1 ^ ~^ /' -^ ;. r Fig. 125. a nut off or on a bolt, and we take the bolt to represent the- conductor, and the direction in which the nut moves bodily along the bolt to represent the direction of the current in the conductor, then whichever way the bolt is placed, or in what- ever direction the nut is moved, the direction of the circular lines of force round the conductor will always be the same as that in which the spanner is turned. The student will find it quite easy to recollect the direction of field for a given current by these practical analogies. In moving the needle round the wire it does not quite set itself as a tangent to the circle of force, because it is also affected by the earth's horizontal field,. FOE ELECTRICAL STUDENTS. 2o-5' but with the strong current through the wire necessary to exhibit the circles of iron fiHngs the earth's field has, in com- parison to the circular field, very slight influence on the needle. It is this principle which enables us to find the direc- tion of flow of a cuiTent in a conductor by the direction in which a compass needle, placed near it, sets itself (para. 66). When a force acts along the circumference of a circle the direction of the force at any given point in the circumference is a tangent to the circle at that point, and this tangent is a line at right angles to the radius of the circle drawn from the centre to the point. If, therefore, we consider any one of the circles of force surrounding a conductor conveying a current, it is clear that, whatever point we choose in that circle, the magnetic force is acting there at right angles to the radius of the circle (drawn from the conductor to the point) and at a tangent to the curve. Further, if we con- sider two straight conductors side by side (Fig. 125), in which equal currents are flowing in opposite directions,, it is evident that those portions of the circles of force which are between the conductors coincide in direction. The mag- netic field at a point equidistant from both conductors, and in the same plane, is therefore twice the intensity of that which would exist if one of the conductors were removed — in other words, there would be twice the force acting on a magnetic needle placed at the point. But a still greater intensitj' of field may be produced at the point by bending the conductor into the form of a circle (Fig. 127), of which the given point is the centre. Here all the circles of force act in the most direct and efficient way, so as to add up their efTects in one direction. This direction, by what has been said, will be understood to be entering the ring from the observer, the current being in the direction indicated by the arrows in the figure. The relative strengths of field prodviced (1) by a ring whose centre is the point p, and (2) by two very long straight conductors carrying the same strength of current, each of which is separated from the point by a distance equal to the radius of the ring (the conductors and the point being in the same plane), is in the proportion of 2-jr to 4, or as 6'28 : 4. -256 PRACTICAL NOTES The same phj-sioal effect of multiplying the magnetic action ,by carrying the conductor completely round the point where BATTERY Fig. 126. the field is required, is obtained in the same way, whether the conductor is in a circular form or otherwise. We shall. Fig. 127. however, confine ourselves to the circular form of conductor or coil, this being the general form for standard galvano- FOB ELECTRICAL STUDENTS. 257 meters, and that which is the best for quantitively determining the magnetic field due to a current. The appearance of the lines of force, as shown by iron filings when the conductor is bent into a semi-circle, is shewn in Fig. 126, where the direction of the current and lines of force are indicated. The next step will be to point out the law by which the intensity of field in C.-G.-S. units due to a current may be determined, and to instance its application to solenoids and galvanometer ■coils. 144. Direction of Magnetic Field in the Begion of a Cir- cular Cnrrent. — Only a few months back the attention of the scientific world was directed once more to the life and labours of that great French mathematician and physicist, Andr6 Marie Ampfere, the occasion being the unveiling of a statue in his memory erected by his fellow citizens in the city of Lyons. Immediately upon the news reaching Paris of Oersted's dis- covery of the action of a current of electricity upon a magnetic needle towards the close of 1820, Ampfere keenly followed up the subject, and very soon after gave to the world his electro- dynamic theory, embracing the laws governing the mutual action between conductors carrying currents of electricity, based upon which we now have some remarkable current and electricity meters. Among these may be cited Siemens' electro-dynamometer, in which a current circulating in two coils, one fixed and the other movable, is measured by the amount of mechanical force required to counteract -their mutual attraction. Again, the Ferranti meter, in which the current to be measured passes radially through a mercury bath, and then through a fixed coil outside it, causing rotation of the mercury, and setting in motion a train of clockwork indicating the quantity of electricity passed through the meter in a given time ; and more recently the beautiful ampere balances of Sir William Thomson, in which the movable coils are at each end of a balance arm, and are attracted, one up and the other down, by a pair of fixed rings, embracing each movable coil. In this instrument a sliding weight is moved along the bar until a position is found, such 258 PRACTICAL NOTES that the balance arm is restored to equilibrium, and this position, indicates on a scale the strength of current passing through the coils. Measuring instruments such as these, designed on the principle of the attraction between fixed and movable portions of an electric circuit, have the invaluable property of being, capable of measuring the strength of alternating as weU as direct currents. But, further, Ampere gave a precise explanation and theory of the magnetic character of electrical circuits ; in fact, he showed that when a conductor carrying an electric current is bent into the form of a single ring or a series of rings side by side, such as a spirally-wound coil, that the behaviour of such is identical with the behaviour of a magnet. The spirally- wound coil, in which a current is circulating, acts, as regards^ attraction, repulsion, and external magnetic eifects, exactly like a cylindrical bar magnet, presenting, as it does, at its extremities north and south magnetic poles. Similarly, a ring- shaped coil, throxigh which a current circulates, presents oppo- site magnetic poles at its two faces, being, in fact, identical in magnetic effect to a thin slice off a cylindrical bar magnet. Such a thin slice or disc of magnetised iron is called a magnetic shell, and the ring-shaped coil carrying a current which is equal in magnetic action to such a shell is termed its equivalent mag- netic shell. Further, every plane closed circuit of whatever external shape can be shown to have its equivalent magnetic shell. It is with this part of the subject that we have now more particularly to deal, and to make the subject as easily under- stood as possible we shall first consider only the direction of the magnetic field in the neighbourhood of a circular current, and then show how its exact value at different points may be deter- mined by Ampere's laws. Suppose, first, a portion of a con- ductor carrying a current, bent into a ring as in Fig. 128, the arrows representing the direction of the current. What is most useful to know with reference to the action of galvano- meter coils is the direction and magnitude of the magnetic field along what is called the axis of the coil. The axis of a coil is an imaginary line in the same position with regard to a coil FOR ELECTRICAL STUDENTS. 259 as a shaft is in with regard to a pulley keyed to it — that is, it is a line (such as the line A B in the figure) whose direction is at right angles to the plane of the coil, and which passes through its centre. Lines of force will enclose the conductor Fia. 128. in circles at every point throughout its length. Now let us consider separately for a moment those circles of force which lie in the sanie vertical sectional plane of the ring, one at the Fig. 129. top and the other at the bottom, and let us examine the mag- netic field they create along the axis A B. Having found this we can add up the fields set up by every sectional plane of the ring, for every one of these planes acts alike. The direction s2 260 PRACTICAL NOTES of the circles of force will be understood to be as indicated in the figure, because the current is flowing towards us in the top of the ring, and away from ua in the lower portion. Take, first, the action of the upper portion of the ring and let us con- sider it separately, the section of the wire being represented at W (Fig. 129). On the axis A B let us mark out a series of points (1 at the centre and 2, 3, 4 on each side of it) at which to determine the field set up. Circles of force emanating from the conductor W as centre will pass through all these points. Now we have already stated that when a force acts along a circle, its direction at any given point is at a tangent to the circle. Fio. 130. We have familiar examples of this tangential action in centrifugal separators. For example, if the dotted circle in Fig. 130 repre- sents a body revolving in the direction shown, the forces acting at any moment when it passes the points A B, CD, are tan- gential to the circle, or at right angles to the radii at those points, and this is the case for any point we may take on the circle. Applying this principle to the circles of force round the conductor, we find the direction of the magnetic forces by drawing lines at right angles to the radii of the circles passing through the various points, and assigning to them a direction agreeing with the direction of the circles of force. It will be FOR ELECTRICAL STUDENTS. 261 noticed, then, at the point 1, which is at the centre of the ring, the direction of force is along the axis of the ring, but as we get further away from the plane of the ring the forces become more and more nearly at right angles to the axis. Now, FiQ. 131. consider the action of the circles of force emanating from the lower portion of the ring. Fig. 131 represents this action, where 1, 2, 3, 4 are the same points as chosen above, and the circles of force are in the opposite direction. Proceeding in Fig. 132. the same way, we find that at the point 1 (the centre of the ring) the force acts along the axis of the coil in the same direc- tion as the force due to the upper portion of the ring, and the forces at the other points act at the same angles relatively to the axis as those above. Now combine the two sets of forces by 262 PRACTICAL NOTES setting them out together along the axis (Fig. 132), the upper and lower sections of the conductor composing the ring being indi- cated at W W. Here we find that although the forces diverge more and more from the axis as we proceed further away from the plane of the coil, yet each pair of forces has its resultant along the axis, and all resultants are in the same direction as the two forces at the centre due to both portions of the conductor. Further, the resultants of these pairs of forces become less and less as we take them further from the plane of the coil, because the forces themselves act more and more in a direction opposing each other, and therefore tend to reduce the intensity of the horizontal resultant. By reversing the current in the coil the whole of these forces would also be reversed in direction. 145. Intensity of Field in the Region of a Circular Cur- rent. The C.-G.-S. Unit of Current. — In the preceding para- graph we investigated the direction of field along the axis of a ring conductor carrying a current, and that direction we con- sidered in one plane only of the ring. What is true for one plane is true for all sectional planes of the ring, and a little consideration will show that the entire magnetic field (or force on unit pole) set up at a point on the axis, that is, the field due to the current in the ring conductor, as a whole, will be acting at the point round a cone-like surface. This effect of the entire field at the point may be understood by the perspective view of the ring and the forces set up by it at the point p in Fig. 133. Here A B is the axis of the ring as before, and if we imagine a unit north pole at the point p, it would be urged outwards from the ring along the axis towards B. In this direc- tion there must evidently be one resultant force equal to the combined forces acting round the cone at the point, and it is this resultant that we have to find. Now, any point on the axis A B is equidistant from every portion of the ring conductor ; and when this is the case the total intensity of field at the point is directly proportional, by Ampfere's law, to the length of conductor multiplied by the strength of current, and inversely proportional to the square of the distance between FOR ELECTRICAL STUDENTS. 263 ■the point and conductor. The distance between any point on the axis and the conductor is simply the common radius to all ^those circles of force which pass through the point. If we take K as this radius, C as the current, and L as the length of con- ductor composing the ring, the total intensity of field at the point is proportional to Now, to find the resultant of this field along the axis, let us consider one plane through the cone, as before. Here "we have two diametrically opposite forces, each, say, equal to F (Fig. 134), acting at the point p, as tangents to the circles Fig. 133. of force through the point, or in other words, at right angles to the common radius R, of the circles of force through p. Completing the parallelogram of these forces, and calling / the resultant of the pair, we find by similar triangles that /:F::2r-:E, where r is the radius of the ring conductor whose opposite sectional portions are at W W, that is, /= -V- Now it is evident that the resultant of the cone Of forces bears the same relation to the combined intensity of those 264 PRACTICAL NOTES forces as the resultant (/) of the pair (2 F) bears to that pair^ Therefore we have the relation, Resultant field _ / Total field 2F' Substituting the value of /as obtained above, and cancelling,, we have Resultant = total field x —, WW Fio. 134. Now, the length of conductor (L) in a ring of radius r is- equal to 2 tt r, and therefore we have, Resultant along axis = ■ x — , R^ R _27rCrg R3 ■ This is a very important expression, as it is proportional to- the field at any point along the axis of a coil. We have not FOR ELECTRICAL STUDENTS. 265 yet taken the quantities in any fixed system of units, so tliat the expression as it stands must be regarded as simply propor- tional to the field. Now in this expression we notice that the numerator is con- stant for a given current flowing in a ring of given size, and the field at any point along the axis of the ring is inversely proportional to the cube of the radius of the circles of force through that point — ^that is, the cube of the distance of the point from the conductor. It is interesting to represent graphically the rate at which this field decreases as we take points further and further from 1000 Fig. 135. the coil. All that is necessary is to take measured distances along the axis A B of the coil (Fig. 135), starting from at the centre of the coil, and for each of these distances calculate the radius E, then cube it and take its reciprocal, the resulting number being proportional to the strength of field at the points taken. For instance, taking the radius of the ring as unity, and the strength of field at its centre as 1,000, we find that at the distances one, two, and three the field is propor- tional to 350, 89, and 32 respectively. A few more points nearer still to the coil should be worked out, as the curve slopes more gradually at very short distances from the coil. '266 PRACTICAL NOTES For example, at distances ^, ^, and J from the coil the field is proportional to 977, 913, and 715 respectively. These six points on the axis of the coil are sufficient to show the general rate of decrease of field along the axis. It will be noticed in Fig. 133 that R forms the hypothenuse of a right-angled triangle, of which one side is the radius (r) of the ring, and the other is the distance (d) of the point on the axis from the centre of the ring. We have, therefore, the relation (Euc. I., 47) W = r^ + d^ whence n^ = (r'^ + d'')i, from which the cube of the radius R may be found for any dis- tance from the centre of the ring. Raising vertical ordinates proportional in height to the field at the several points chosen, and connecting them, we obtain the curve in Fig. 135, show- ing the rate of decrease of field on each side of the ring along its axis. It will be noticed in working out the above values that the field at every point we take depends on two things, viz., the radius of the coil itself, and the distance away from the coil along the axis, and it will be found that whatever size coil we consider (i.e., of whatever radius), and whatever current flows through it, the field will always decrease gradually (as in the above curve) at short distances from the coil on either side, then suddenly the fall will be rapid, and then gradual again. And the fall in strength of field is always most rapid at a distance away from the coil equal to half its radius. We might, there- fore, call this the critical point on the axis, because a little further away than this point the field is considerably feebler, and a little nearer to the coil than this point the field is con- siderably stronger. Returning to the first part of this para- graph, it will be seen that by Amp&re's law the field at the centre of a coil is proportional to CL because the radius (R) of the curve of force through the centre is now equal to the radius (r) of the ring. And of these three FOB BLBCTEICAL STDDEXTS. 267 •quantities, viz., field, current (C), and length (L and r^), which are related to each other in the above proportion, there are two, ■viz., field and length, of which we have already defined units ; it only remains to define unit current. Now, if we take a coil of unit radius (r), and consider only the magnetic field at the centre due to unit length (L) of the wire (i.e., any part of the coil equal in length to the radius), and, further, suppose that a certain current flows in the coil which causes the field at the centre (due to unit length of wire) to be equal to unity, then the only value we can give the current on the basis of an absolute system of units is to take that current as the unit current. Supplying, then, the units on the C.-G.-S. system, we can at once define the absolute unit of current as that current which, flowing in a ring conductor of 1 centimetre radius, produces at the centre of the ring a magnetic field equal in intensity to 1 C.-G.-S. unit /or every centimetre length of -wire in the ring. Further, since the ring must be 2ir centimetres in circumference, the total intensity of field at the centre is equal to 2jr C.-G.-S. units. The practical unit of current, to which is assigned the name of Ampere, is chosen as equal in value to one-tenth of the absolute unit, and therefore one ampere flowing in a ring conductor of one centimetre radius produces a field of —, or - C.-G.-S. units, 10 5 If the ring or coil consists of more turns than one, say n turns, the field is increased n times, and the complete expres- si<)n for the field at any point on the axis becomes C.-G.-S. units, W which at the centre of the coil (where r = B) becomes l^H^ C.-G.-S. units, r where r is then the mean radius of the coil in centimetres. Example. — The coil in Sir Wm. Thomson's graded current galvanometer (described para, 132) consists of six complete Aums of stout copper strip, the inside diameter being 6 centir 268 PRACTICAL NOTES metres and the outside 10 centimetres. What is the intensity of field produced by a current of 30 amperes flowing through the coil, at a distance of 10 centimetres from the centre of the- coil along its axis, and what is the field produced at the centre of the coil } Mean radius of coil (r) = 4 Current in C.-G.-S. units (C) = 3 Number of turns (») =6 2,rC»ir2 =1809-5 log of 1809-5 =3-25755 Il3 = (r2 + c^2)l =(16 + 100)1 log of 116 =2-06445 logofR3 ^2;06445x3 =3.99667 Subtraction gives log of answer — •1608& Therefore, Intensity of field = 1-448 C.-G.-S. units. Field at coil centre = — — - C.-G.-S. units, r = 11^ = 28-25 C.-G.-S. units. 146. Magnetic Fields due to a Coil and Magnet Super- posed. — We have already discussed the case of two uniform magnetic fields superposed at right angles to each other (para. 119). Referring again to the figure (Fig. 80), it was shown that the ratio between the two superposed fields / and H was the tangent of the angle which H, the total controlling field, makes with the resultant. Let us now take the- practical case which occurs in every tangent galvanometer, viz., the superposition at right angles to each other of the controlling field due to the earth or a permanent magnet, and the field due to the current in the coil. A little experiment carried out with a Thomson potential galvanometer coil and a battery of secondary cells may serve to make this action clear. This instrument is of the same external appearance as the current galvanometer described in para. 132, but instead of at POR ELECTRICAL STUDENTS. 269 few turns of stout copper strip the coil is wound with some 8,000 turns of German silver wire, amounting to a resistance of 9,830 ohms. The needle-box, movable along the wooden platform, bemg removed, the magnetic fields produced in the region of the coil were examined by iron filings sprinkled on a sheet of stiff card placed on the platform. First, without any current through the coil, the controlling magnet N S was placed over it in the usual position as used with the instrument (Fig. 136). The iron filings then revealed the Fig. 136. kind of field produced in the region of the coil. The lines of force here pass straight from one pole of the magnet to the other, producing at the centre of the coil a field parallel to its plane. When in use the coil and magnet are put in the plane of the magnetic meridian so that the earth's horizontal field is added to that of the magnet. The earth's field alone is far too weak to be exhibited by iron filings. The central line in the figure represents the axis of the coil, and is the line along which readings are taken with the instrument. Along this axis we may consider the controlling field produced by the magnet and 270 PRACTICAL NOTES the «arth to be everywhere in the direction of the arrow, viz.^ at right angles to the axis. For even at positions along the^^ axis remote from the coil where the lines are in curves, the direction of each line of force as it crosses the axis is a tangent to the curve, and, therefore, at right angles to the axis. Secondly, the controlling magnet was removed, and the ends of the coil connected by leading wires to the terminals of 21 cells, giving a diiference of potential of 42 volts at the terminals of the coil. A vigorous effect was then produced on the filings, which arranged themselves into the form in Fig. 137. It will be noticed that as regards the axis of the- Fig. 137. coil, this field is at right angles to that set up by the control- ling magnet. The current was sent round the coU in such a direction as to cause the direction of the lines to be towards the coil, as shown by the arrow. Lastly, the controlling mag- net was placed over the coil, as at first, the current being kept on the while, resulting in the change in direction of the filings represented in Fig. 138. Here we have an actual representation of the resultant field, that is, the character of the field resulting from the combined effect of two distinct fields at right angles to one another. The arrow in this figure indicates the direction of the lines of the- FOB ELECTEICAL STUDENTS. 271 resultant field at the centre of the coil. As we take points- on the axis further away from the coil the effect of the latter becomes rapidly decreased, and the controlling field predominates, so that the lines become nearer and nearer at right angles to the axis. The direction of this resul- tant field is precisely that taken up by a magnetic needle free to move when placed at a given position on the axis of the coil. If the needle is at the centre of the coil we have an ordinary tangent galvanometer, and if the box containing the needle can be shifted along the axis of thor Fig. 138. coil, as in Sir William Thomson's graded galvanometers^ we have instruments which still follow the tangent law, a.nd which have, in addition, the advantage of great variation in sensibility, as before explained. Let us form a practical idea of the strengths of the two fields at the centre of the coil, their resultant, and its direction in the above case. The outside dia- meter of the coil was 14 centimetres, and the inside 6 centi- metres; the mean radius was therefore 5 centimetres. The- current was equal to 12?^ = -iL = -00427 amperes. ohms 9830 272 PRACTICAL NOTES Dividing this by 10 we have the current in C.-G.-S. units = -000427. The number of turns of wire being 8,000, the field at the centre of the coil due to the current was : — 2:rx •000427x8000 ,^., ^.j. .g ^^^^ 5 Now, the controlling field at right angles to the above con- sisted of the field of the magnet, which was 10-56 C.-G.-S. units, plus the earth's horizontal field -18 unit, total 10-74 units. The ratio of the coil field to the controlling field is the tangent of the angle which the resultant makes with the controlling field, that is ^'l = -4003 = tangent of 22° nearly. 10-74 A small magnetic needle freely suspended at the centre of the coil would therefore turn through this angle with the above potential difference applied at the terminals of the coil. 147. General Character of Field Due to a Solenoid. — In con- sidering the way to determine the strength of magnetic field at any point on the axis of a ring conductor carrying a current, we gave ourselves a more extended and more general view of the subject than when determining the field at the centre of the coil only. In other words, the latter case is involved in the former, and the student will find it best to master the more general case, because he will then be able to apply it to any special case required. The determination of the field at any point along the axis of a solenoid is the most general case of all, and it will therefore be the last case we shall examine in this section. At the same time, this most general case is the most difficult one to be followed and understood, and the writer hopes, at any rate, to lead up to the final result by a series of easy steps. It will be well here, as previously done, to get a general idea of the character of field exhibited by a solenoid; next, to consider the direction of the field, and finally its intensity in C.-G.-S. measure at any point along the axis. This being done, the FOE ELECTRICAL STUDENTS. 273 fitudent will find that he is in a position to intelligently under- stand the various forms in which galvanometers are constructed. A single low-resistance cell connected to the terminals of a solenoid wound with a few layers of No. 18 or No. 16 double cotton-covered wire will be sufficient to exhibit the general con- tour of the magnetic field set up in and around the coil. It is better to mount the solenoid on a bit of wood, as in Fig. 139, and carry the two ends of the coil through the wood and along two slots underneath leading to their terminals. Done in this way you can have two sensible tenninals to connect your wires to, instead of the flimsy little terminals sometimes screwed on to Fig. 139. the cheek of the bobbin, which frequently cause splitting of the same, or, at all events, make bad contact with the coil. After winding, the bobbin may be fixed rigidly to the wooden base by countersinking a hole in the latter deep enough to receive the lower cheek of the bobbin flush with the level of the base. A couple of semi-circular pieces of brass plate covering the cheek and the contiguous part of the base may then be screwed down to the latter, holding the coil firmly in position. Or a piece of brass sheeting may be cut so as to fit over the top cheek and come down in a half-inch strip each side, and be firmly screwed on to the base below. In any case, T 274 PEACTICAL NOTES screws should not be fastened into the cheeks themselves. The solenoid can be fixed horizontally in much the same way on to a base board, but its position will depend on what kind of experiments are to be made with it. When finished, it can be used for a variety of experiments, such as the mechanical pull it exerts upon an iron core to draw it into the coil. This can be varied by trying different kinds of core, such as, for instance, a core built of a bundle of small iron TO BAireni Fio. 140. wires and cores of diflferent shapes with pointed and blunt ends, &c., measuring the pull on a spring balance, when, say, a given core has different portions of its length introduced into the coil. The outside layer of wire on the coil should be well brushed over with thick shellac varnish (shellac dissolved in alcohol). This soaks in and hardens the cotton covering, protecting it from abrasion in handling. Curves made from the plotting of the observations above indicated are very useful in showing the action of cores working in solenoids in arc lamps, FOB BLBCTBICAL STUDENTS. 275 especially the difference between the pull on a core which moves with the carbon rod, as, for instance, in the Pilsen lamp, and that on a core which generally maintains a certain position relative to the coil, as in the Brush or Siemens. The foregoing remarks are merely to indicate what useful experiments and tests may be carried out with a solenoid ; to go into detail in this direction would more befit a separate treatise on electric lighting, and is outside our present limits. Taking, then, such TO BKTTER1 Fig. 141. a solenoid and laying it down horizontaUy, we can examine the character of field by iron filings sprinkled on a card supported close to it, as shown in Fig. 140. When the current is on and the filings are sieved dovm gently they will be found to arrange themselves as indicated in the figure, and it is easy to see that the solenoid exhibits the same kind of field as a permanent magnet whose length is in the same direction as the length of the solenoid. Further, we can better under- stand what goes on in the interior of a bar magnet by noticing t2 276 PRACTICAL NOTES the kind of field that we get inside the solenoid. This is best examined with a small compass needle, which we shall do shortly. Taking an iron core, fitting into the solenoid, and of about the same length, it is interesting to observe that when the core is very nearly out of the coil, as in Fig. 141, the shape of the field is considerably altered. The reason that no iron filings adhere to the card just over the iron core is because the core itself absorbs those lines of force which otherwise would TO DYNAMO Fio. 142. pass through the air and attract the filings. So easy a path does the iron core offer in comparison to the air that the lines pass straight from the interior of the coil into the iron core for some distance before passing outward in the usual curves to complete the magnetic circuit, and this causes the alteration in the shape of the field. The projecting end of the core is strongly magnetised, and its magnetism will be reversed if the direction of current in the coil is reversed. Also, if the core is right inside the solenoid, we have virtually an electro-magnet, FOR ELECTRICAL STUDENTS. 277 and with a direct current we get the same field as we have before shown for electro-magnets. With an alternating current through the coil, however, such a vibration is set up that imme- diately the iilings touch the card the greater part of them dart at once to the core. This efifect is shown in Fig. 142. 148. Direction of Field Due to a Solenoid. — ^A solenoid, mounted as in Fig. 143, is very suitable for observing the direc- tion of field set up. One layer of stout wire, say. No. 12, ia first wound round a cylindrical surface such as a battery jar, to Fig. 143. give it the proper shape which it will be found soft enough to retain itself after slipping off. It is better to leave a little clear space between each turn, so that afterwards when experiment- ing you can look down into the middle of the coU. If this is done there is no need to have the wire insulated. Let the two ends be passed through the wooden base block B carrying the two terminals, and solder the wires underneath this block to the extremities of the terminals. Clamp the coil down by a flat piece of wood screwed down to the base block, as shown. This keeps the solenoid firmly in position, and affords a little table whereon to put your compass needle when examining the 278 PRACTICAL NOTES field in the interior. A convenient size of solenoid is about from four to six inches in diameter, and, say, one foot long. Before passing the current through let the solenoid be placed so that the direction of its length is at right angles to the magnetic meridian, and then put a compass needle inside, or, better still, place the compass inside first and then turn the whole apparatus bodily round till the needle points at right angles to the direction of length of the solenoid. If another compass needle be placed outside the solenoid in the position shown in the figure, both needles will point the same way, north and south, before the current is passed into the coil ; but immediately the current is switched on the needles will turn A Fk. IDA. sharply in opposite directions, and when at rest will lie very nearly in the direction of length of the solenoid. Notice should be taken of the direction of the current and the direction of movement of the needles. In the figure the needles are shown as they are deflected when the current passes through the solenoid in the direction of the arrows. We now see the direc- tion of the field by the direction towards which the north poles of the needles point, and we find by exploring all round the solenoid that the lines seem to flow along the interior space of the coil and then to separate into two streams passing hackioards along the exterior sides of the coil, and then turning in again to the centre. This can be FOR BLECTRIOAL STUDENTS. 279 best observed by cutting out a piece of card as in Fig. 144, bending it upwards at the dotted line, and slipping it over the coil in the manner shown in Fig. 145, the centre portion. A, being then bent downwards again so as to lie along the interior of the coil. By passing the current through the coil again, and letting fine iron filings fall on the card, the character of the entire field is delineated. Supposing the current in the coil to be in the same direction as in Fig. 143, we shall find that wherever we place small compass needles the direction in which their north poles turn is along the arrows marked on TO BKTTERY Fig 145. •the card. Now the card only exhibits the field in one of the sectional planes of the coil, but it is obvious that the field is the same for all sectional planes ; and, hence, when we think of the field of a coil or solenoid we must carry in our minds the conception that the field entirely envelopes the coil. In other words, the lines of force fully occupy the interior space of the coil, and after passing through this space divide evenly and flow hadaoards round the outside ; and with this conception it will be easily understood that if we attach two magnetic needles rigidly together, as in Fig. 146, 280 PRACTICAL NOTES with their north poles pointing opposite ways, and freely suspend the two by a silk thread, so that one is inside the coil and the other outside, as shown, both needles will turn in the same direction when a current is sent through the coil. The two needles together, or, as the combination is technically called, the " system of needles," is hardly at all affected by the earth's horizontal force, because what force it exerts on one needle is counteracted very nearly by the force on the other ; in fact, if the two needles were of exactly the same magnetic moment, the " system," when allowed to swing freely in the earth's field, would not come to rest in any definite position^ © Fig. 146. and hence the " system " is called an " astatic " system. It will be seen that as the earth's field has very little controlling effect on it (only just enough to keep it at zero) the astatic system is very sensitive. It is usual to have two separate coils, one above the other, and each containing and acting upon one of the needles, but when the idea was first developed one coil only was used. Faraday mentions in his renowned "Experi- mental Researches " the use he made of an astatic system with one coil (Fig. 146) in carrying out some induction experiments. And the writer has introduced it here believing it to be of interest in connection with the direction of field in and around one coil alone ; and, indeed, in an earlier part of this series. FOR ELECTRICAL STUDENTS. 281 (para. 57) reference was made to the subject, the explanation of which now will make the case given there better understood. Lastly, if a compass needle placed inside the coil be moved outwards along the axis either way, it will be found that its deflection keeps pretty nearly in line with the axis until it emerges from the coil, when the deflection rapidly falls off; and ultimately, by removing the needle further and further away, we find the field of the coil exerts practically no more force on it, and it rests in the meridian once more. It is precisely this variation of field along the axis that we wish to determine for any kind of solenoid ; and further than this, to determine in C.-G.-S. units the exact intensity of field at any point. 149. Intensity of Field at the Centre of a Solenoid. — In the treatment of this subject there is involved the use of a branch of mathematics not frequently understood either by those who are commencing their electrical studies or even by old and tried practitioners. But while it is very serviceable to be able to sum up a series of extremely small quantities by the usual mathematical process, it is very much more important to have a common-sense view of what has to be done to attain the result aimed at. It is a cause of disappointment and dismay to many students who have been carefully following up a subject to find themselves, after much labour spent in trying to understand it, handicapped by a mathematical process which is said to attain the desired result, but of which they do not comprehend the meaning. The said mathematical treat- ment would be quite out of place in these Letters, and if the result were put down accordingly without any explanation of the process by which it is arrived at, the reader would be no better off than before. Hence, it seems desirable to steer a middle course between these two extremes. But the task so proposed has in it somewhat of the difficulty experienced by Ulysses on his voyage from Troy, when, to save his men from the grasp of Scylla, he ran the risk of getting wrecked in the whirlpools of Charybdis ; and it is possible that in trying to avoid one alternative we may in some way run foul of the other. The student will find, however, that if he gets a common- 282 PEACTICAL NOTES sense view even of the manner of arriving at the result, or clears up any previously foggy ideas on the subject, he will certainly have scored another point to the good. To fix our ideas, let us suppose a solenoid six centimetres long, wound with five layers of wire, 40 turns to the layer, and therefore making in all 200 complete turns. Let the depth of the winding — that is, the thickness of the five layers — be half a centimetre, and the reel on which the coil is wound 3'5 centi- metres diameter ; this makes the mean radius (r) of the coil two centimetres. Suppose that a current of three amperes is passed through the wire, it is evident that it circulates 200 times round the bobbin, and the total current passing round the bobbin is therefore 3 x 200 = 600 amperes. Although each turn is really in series with the rest, yet if we look at a section lengthwise through the coil the ends of the wires appear like so many separate conductors, each in "parallel" or "multiple arc " with the rest, and the total current is therefore the sum of the currents carried by each conductor, viz., 3 x 200; or we may regard the winding as equal to 200 x 3 = 600 ampere- turns, which may be replaced, without in any way altering the field, by a single turn, occupying the same volume as the 200 turns, and carrying 600 amperes. This is the most con- venient, and, at the same time, the most accurate way of looking at the action of a solenoid. We shall therefore think only of the total current circulating uniformly once roimd the bobbin and occupying the same space as the winding. Or, putting the same thing in a general form, to which any figures may be applied, let n be the number of turns of wire on the solenoid, and C the current in C.-G.-S. units passed through the wire. Then we can regard this as a current of C n C.-G.-S. units making one turn round the bobbin. Now it is convenient to consider this single turn or belt of current as divided up into a large number of thin rings of current, because then we can calculate the field produced at the required point by each narrow ring. In short, the expression which was derived in para. 145, viz. : — 2 TT C -, E,3 FOR ELECTRICAL STUDENTS. 283 -for finding the field in C.-G.-S. units produced at any point on the axis of a ring of mean radius r centimetres, assumes that the current C is in the exact centre of the wire form- ing the ring. To make tliis clear, suppose we take a ring of loire and a wide flat rihhon of copper, which, when bent round, makes a single turn of cylindrical shape, of the same diameter as the ring. Now, although the same current be passed through the single turn of wire and the single cylindrical turn, the field produced by each at their respec- tive centres is not the same ; for in the wire the current is concentrated into a narrow space immediately surrounding the point where it is required to determine the field, whereas in the wide cylindrical coil the current is spread over the whole cylinder, and it is therefore only a small portion of the current which flows immediately over the centre point. Further, it is obvious that the remaining portions of the current which are flowing round parts of the cylinder further and further away from the centre must have less effect in producing field there -than would be the case if the whole current was massed together right over the central point, as in a ring of wire. And therefore we can only apply the above expression to rings of wire or coils of several turns whose breadth is very small compared to their diameter. But it follows that the field produced by the entire cylindrical coil can be esti- m.ited by considering its current as divided up into a large number of very narrow, thin rings, the field due to each of which at the required point can be calculated by the above expression, and the several results for each ring so obtained finally added together. This gives the field produced at any required point on the axis due to the current in every part of such a cylinder, and is, in fact, the way in which the total field due to a solenoid is calculated. For the whole coil of wire of a solenoid is cylindrical in shape, and we have seen that it may be considered as one turn occupying the same volume, which is therefore also cylindrical. The question naturally occurring next is, how many rings are we to consider the cylindrical turn or belt of current -divided into ? Well, first let us divide it into rings of one 284 PRACTICAL NOTES centimetre width. Now if the solenoid is I centimetres Icng^ there will be I such rings, and the current (C,) carried by each will be the total current in the cylindrical belt divided by the length I ; that is, ^ = C, in C.-G.-S. units. This is usually spoken of as the current per unit length of solenoid. We must carry this subdivision of the rings to a still I; ■* / c * s e u Fig. 148. greater extent ; but this will be best understood by plotting a curve over the solenoid (Fig. 148) showing the relation between strength of field at the point produced by different current rings, and the distance of these rings from the point. The axis of the solenoid is along the line A B, and it is at the point on this axis that we will first set ourselves to determine the strength of field due to the entire solenoid. We might at once take this point at the middle of the coil ; but it is better, as a FOR ELBCTRICAL STUDENTS. 285 more preliminary step, to consider the field at one end of the «oil first. Any point on this axis between the beginning and end of the solenoid will, of course, be the centre of a narrow ring carrying a certain fraction of the current per unit length. It is not necessary to consider now what this fraction may be, for the curve will be of the same shape whatever we take it. In fact, as this fraction of the current would be the same for each narrow ring we take, it may be regarded as constant, and the curve plotted by simply working out the reciprocals of R^ for a few points on the axis. Eaising perpendiculars proportional to the figures so obtained, and joining together the summits of these perpendiculars, we get the outline of the curve as shown. The few points on the axis selected, with corresponding numbers proportional to the field they produce at 0, are given below. These few points are quite sufficient to show the general shape of the curve. T, -, • , ■ Values of — propor- Position of nng D,') ^ "^ on aids. tional to field. -125 \ -12 1 -1125 i -09 2 -0425 4 -Oil 6 -004 Now, by inspection of the curve it is clear that in any cur- rent ring of one centimetre width, that part of it nearer to the point produces a much stronger field at than its more remote portion. For instance, in the belt or ring of current between the points 1 and 2 the perpendicular line propor- tional to the field is much higher at 1 than 2, and if we imagine this ring subdivided into a very large number of small rings, it is easy to see that the field of each (at the point 0) gradually becomes more intense as they are further from 2 and nearer to 1, the increase being in exact propor- tion to the rise of the curve. In fact, every perpendicular line drawn from the axis between 1 and 2 to meet the curve is 286 PRACTICAL NOTES proportional to the field produced at by a narrow ring of current in the plane of that line ; and in order to estimate the total field produced by the belt of current between 1 and 2, it is clear that we must add together the values of as many perpendicular lines as can be crowded into that space. But this is the same thing as finding the area of the strip included between the two vertical lines at 1 and 2, the curve, and the axis. Now the area of this strip we may consider as made up of two parts — a rectangle and a space like a triangle (shown separated by a dotted line). It would be very easy to find the area of the rectangle, but not so easy to find the area of the triangular space, because one side is curved. But by dividing • the space between 1 and 2 into a series of smaller strips the curved side of the triangle in each might be considered straight and the areas found by inspec- tion, adding them together afterwards to get the area of the whole strip. Fortunately we are not obliged to do anything 80 laborious as that, for by considering any very narrow strip, such as that drawn on the off-side of the figure 2, it ■will be seen that the little triangle becomes very diminu- tive, and finally disappears altogether if we take the strip smaller and smaller xoithout limit. Just in the same way as the field produced by the belt of current between 1 and 2 is found by measuring the area of the strip contained between these limits, so the total field at the point due to the whole solenoid is found by calculating the area of the whole curve between the limits and I. By taking such extremely small strips, then, we may regard them as rectangles of minute breadth, and adding them aU together from to Z gives the total area of the curve. It is not necessary to understand the exact process by which this is done so long as the student understands what is done and why it is done. It wOi be understood that in adding any series of numbers the process of addition is much simplified if we first separate any factors which are constant from those which vary. For instance, sup- pose it is required to sum up all multiples of some number which we ■wUl call a, between the limits of 5a and 9a, it is easier to perform the summation by adding the factors which FOR ELECTRICAL STUDENTS. 287 vary first; thus :— 5 + 6 + 7 + 8 + 9 = 35, and then taking the product 35a, than to add together 5a + 6a + 7a + 8a + 9a. The difference in the labour involved in doing it these two ways is at once apparent by taking a as some fractional number, say \\. In other words, a=\\ is constant throughout, and it is only necessary to sum the factors which vary as above^ and then multiply their sum by the constant, giving the final summation of the series, viz., 14-x35 = 40. Similarly, in the summation of the little rectangles representing the fields pro- duced by each narrow ring of current there are two factors which go to make up the value of each rectangle, one being constant and the other varying with the position of the current ring. The constant factor in this series is the product 2 tt Ci r^, and the factor which varies comprises the reciprocal of E^ (which Fig. 149. varies with the position of each ring), and a fractional quantity which diminishes the width of the current rings (and therefore of the rectangles) down to an infinitesimally small amount. This latter fraction would be a constant quantity were it not for the fact that it is expressed as a fraction of the distance of each ring from the point where the field is to be determined, and this distance of course is different for every ring. We have now only to take the factors which vary and sum up all their possible values between the limits and I (the length of the solenoid), the result of this summation being equal to I where R has its greatest value, viz., the distance from the point to the mean depth of the furthest ring (see Fig. 149). 288 PRACTICAL NOTES The depth of the winding is represented in the figure by the space (partially shaded) between the two bounding lines of the coil, and the line R is from to a point at the side of the coil central to the winding — in fact, to the point which limits the mean radius r. Multiplying the constant quantity by the sum of the factors which vary, we have the final summation, giving the total field at due to the entire solenoid, viz. : — 2 TT d r2 X -i- C.-G.-S. units, 7-2 R which, after cancelling out r^, becomes 2 7rCixi C.-G.-S. units. R Putting for Cj its equivalent (C)i divided by I), and cancel- ling out I, we have for the field at either end of the solenoid ^-^' C.-G.-S. units. R Let us now apply this to the numerical example given above. The length of R will be seen to be the square root of the sum of the squares of the mean radius and length of the solenoid, which are two and six centimetres respectively. Hence R= ^4-f36 = 6'32 centimetres. The current is 3 amperes (or '3 C.-G.-S. units), aud the number of turns of wire 200 ; therefore the field at either end of this solenoid is equal to 2 X 3-1416 X -3x200 377 6-32 6-32 = 59-6 C.-G.-S. units. Now suppose we take a solenoid (Fig 150) exactly double the length of the above, but in all other respects the same, that is having the same depth of winding and the same current. Evidently the field at the point 0, taken now at the centre, will be twice that at the end of the former solenoid, and therefore we can write it down at once as 4 TT Cj X -^- C.-G.-S. units. FOE ELECTRICAL STUDENTS. 289 But I is now only half the length of this solenoid, and there- fore it is advisable to multiply both I and K by 2 (which does not alter their ratio). Now the value 2 R is equal to the line connecting opposite corners of the solenoid (Fig. 150), which we will distinguish by calling it the diagonal (D). And instead of writing 2 I for the length of the present solenoid we shall write it I, keeping this letter to represent the length of solenoid whatever it may be. Therefore the field at the centre becomes I 4 IT Cj X ^ C.-G.-S. units. Now let Cj (the current per unit length) be expressed in terms of C (the current passing through the wire). It will be understood that the current per unit length of coil is the same Fig. 150. value as before, and is expressed by the total current flowing round the bobbin (C n) divided by I, n being the number of turns and I the length of the present solenoid. CanceUing out I the field at becomes 47rCw C.-G.-S. units. This is applicable to any dimensions of solenoid, and gives the field produced at the central point on the axis where it is most frequently required to be known. We have now the means of determining the strength of field at either end of a solenoid and at its centre on the axis. It wiU be understood that, taking any size solenoid whatever, and D 290 PRACTICAL NOTES denoting by the letters R and D the two lines drawn and so marked in the figure (Fig. 151), that the field at either end on the axis is equal to 2wCn , E and that at the centre is 4 JT Cra C.-G.-S. units. D C.-G.-S. units. It is worthy of note that when the solenoid is very long E be- comes practically equal to D, while if the solenoid is very short, more like a galvanometer coil, E is practically equal to half of D. Applying tlie latter expression to find the field at the centre of FiQ. 151. the solenoid whose dimensions are given above (Fig. 148), we find that the numerator is twice the previously calculated amount, and is therefore equal to 754. The length of D is the square root of the sum of the squares of the mean diameter and length of solenoid, which are 4 and 6 centimetres respec- tively. Hence D= 716 -f 36 = 7-21 centimetres, and the field at the centre is therefore equal to t^ = 104'5 C.-G.-S. units. 7-21 being somewhat less than double that at each end. TOR ELECTRICAL STUDENTS. 291 It will easily be seen that the longer the solenoid the nearer does D become equal to I, and that for a long narrow solenoid we may consider it the same as I without any appre- ciable error. The field at the centre of such a solenoid is then — C.-G.-S. units, to which expression reference has already been made (paras. 130 and 133). Again, the shorter the length of the solenoid the more it approaches the shape of a current ring (Fig. 152), and we should therefore expect the field at the centre to become very nearly equal to that due to a thin ring of current. This is so, for the diagonal line is very nearly the same as the mean diameter of the coil, and does become exactly equal to it when the coil is made shorter and shorter till it forms a narrow ring. When such is the case, instead of the diagonal D in the above expression we can put the mean diameter of the coil or twice the mean radius (2 r). Cancel- ling out the 2, the field at the centre becomes 27rCM C.-G.-S. units. which is the same as previously derived for a Ll narrow ring-shaped coil of n turns (para. 145). Fig. 152. The next point to consider is the strength of field at any point on the axis either inside or outside the sole- noid, and this case will necessarily include those already taken. The expressions, however, which have been derived above for finding the field at the centre and at either end of a solenoid will be found useful not only as steps leading up to the most general case, but as easy expressions to deal with, showing also at a glance the relation between the field at these points and the diameter, length, and depth of winding of a solenoid. 150. Intensity of Field at any Point on the Axis of a Sole- noid. — The results already derived will afford sufiicient ground to work upon in order to show how the field at any point on u2 292 PRACTICAL NOTES the axis of a solenoid is determined. In the two expressions for the field at the centre and at either end, if we consider only those factors which alter in value according to the position of the point on the axis, we shall be able to trace the general law for any point. On examining these expressions we find that the factor 2 tt Cj occurs in both, and therefore the remaining factors depending on the position of the point considered are -- at the centre, D and R at either end. Now the ratio of the length of the solenoid (I) to the diagonal Fig. 153. (D) is the cosine of the angle (0) between them (Fig. 153), and it is evident that the angle is equivalent to the angle a. Further, if a line is drawn from the centre to the other end of the solenoid, as shown, it is evident that the included angle fi is symmetrical with the angle a and equal to it. We have, therefore, 2 I — =2 cos a = COS a + cos (i. This is for the field at the centre of the coil ; and now if we consider the field at one end we find in a similar manner that the ratio of ? to R (Fig. 149) is the cosine of the angle between these lines. Generalising these results, it is not difficult FOE ELECTRICAL STUDENTS. 293 to see that the field along the axis within the coil depends upon the sum of the cosines of the angles formed on each side of the point wherever that is situated on the axis. For instance, as the field is considered at points further away from the centre one angle becomes less and less while the other becomes greater ; and finally, when the point is taken at the end of the coil, as in Fig. 149, the angle (i has increased to 90deg. where its cosine is zero, while a has become equal to the angle enclosed between the lines I and R, as we have seen above. Replacing the constant factor 2 r Cj, and putting for Cj its equivalent (C n divided by I), we have the result that the r ^ .__ u '~ — - , Fig. 15a. intensity of field at any point on the axis vnthin the coil is equal to r (cos a+ COS /6) C.-G.-S. units, where at the end of the coil the cosine of /3 is zero, and beyond this, viz., outside the coil, its cosine changes in sign. The field at any point outside the coil on its axis (such as at^. Fig. 154) is, therefore, proportional to the difference of the cosines of the two angles instead of their sum as above. (The angle a in this figure is shown on the other side of the axis for convenience. It is evident that it is identically the same as when indicated above the axis, in the same way, at the point p.) This may now bo applied to some practical example. It is best to take some definite 294 PRACTICAL NOTfiS size of solenoid and calculate the intensity of field at various points along the axis, and from these results plot a curve showing the way in which the field varies. By this means an insight is obtained into the relative strength of field at various points on the axis of such a coil. Take a solenoid eight centimetres long and wound with 100 turns of wire, the mean radius of the coil so wound being one centimetre (Fig. 155). Say a current of 12-7 xr"^ "n 1 / .-■ ■^ K \ \ \ Mji / /r \ 1 '• ^\ i I ! 1 ; ^ e 5 4 3 2 1 1 2 3 4 I . Fig. 155. amperes is passed through this winding, then the constant quantity 2 TT C» _ 3x 3-1416 X 1-27x100 ~l 8 100, the current being expressed in C.-G.-S. units. The factors which vary according to the position of the points on the axis have now to be determined, and as the distribution of field is symmetrical on each side of the centre it is only necessary to make the calculations for one side. The angles themselves need not be determined, but simply their cosines, and this only involves in each case finding the value of the hypothenuse of a FOR BLBCTRICAL STUDENTS. 295 right angle triangle of which the other two sides are known. We have at the centre two equal angles, the cosine of each of which is J^^ = -97, the denominator being the hypothenuse, and found by ex- tracting the root of the sum of the squares of the other two sides. At the centre of the coil the field is, therefore, 100 X 2 X -97 = 194 C.-G.-S. units. Now calculate the field at points on the axis one centimetre apart. At one centimetre distance from the centre the two angles are different, their cosines being found in the same way as the above, and added together, viz. : — _L + -i^ = l-93. The field at this point is, therefore, 100 X 1-93 = 193 C.-G.-S. units. Similarly, at two centimetres from the centre the field is 100 C ^__ + -^ \ = 188 C.-G.-S. units. At three centimetres the field, similarly calculated, is equal to 169 units. Now at four centimetres from the centre we arrive at the end of the coil where one angle becomes 90 deg. and its cosine nil, the cosine of the remaining angle being - = •99. n/65 The field at the end of the coil is, therefore, 100 X -99 = 99 C.-G.-S units, or a little more than half that at the centre. It will be observed that the field diminishes very rapidly at this part of 296 PRACTICAL NOTES the axis. At five centimetres from the centre the point is out- side the coil, and the difference of the cosines is therefore taken. The field is therefore 100 ^-L: _ -!- ^ = 28 C.-G.-S. units. V ^/82 n/2/ Similarly, at six centimetres from the centre the field is equal to 10 _ 2 7l01 J5> 100 ( ^" - _L '\ = 10 C.-G.-S. units. V 7l01 J5/ These values plotted give the curve shown in the figure, which therefore graphically represents the variation in inten- sity of field along the axis. It will also be noticed by the fore- going example that the variation of field along the axis depends only upon the geometrical dimensions of the coil ; that is, on the relation between the mean diameter and length of the solenoid, and not on its actual size as a whole. For example, the curve above plotted (Fig. 155) is for a solenoid whose length is four times its mean diameter, and there would be precisely the same distribution of field and consequently the same shape of curve for solenoids of 2, 3, 10, or any number of times the acttial size of the above, provided they have the same relation between length and diameter ; and this is irrespective of the strength of current, number of turns, and size of wire composing the coil. The above does not profess to be a proof of the law stated, but the endeavour has been to trace up in a rational manner the derivation of the formula, so that it may be used with an intelligent appreciation of its meaning. TABLE OF NATURAL TANGENTS. The tangent of 45° being exactly equal to unity, the tangents of angles under 45 are fractional quantities, while the tangents of angles above 45 are greater than unity, but not in exact ■whole numbers. The fractional quantities are given in the vertical columns, each such fraction corresponding to the degree marked in the same horizontal line in the first column and the number of " minutes " marked at the head of the column in which the fractional quantity is placed. Instead of 60 vertical columns, viz., one for each minute, it is only for every sixth minute that the fraction is given in full. The last two columns supply figures for the intermediate five minutes by which the fractional nunibers must be increased. If the angle is simply in degrees its complete tangent is at once found in the second column. In all other columns the whole numbers are omitted, but are understood to be the same as those marked in the second column in the same horizontal line, except when the first figure of the fraction has a mark over it, when the next higher whole number is to be taken. With the aid of a few examples no difficulty will be experi- enced in using the table. Required the tangent of 36° 45' : — By the table— tangent of 36° 42' = -7454 Difference in Minutes = 3 Number under 3 = 14 Add •7468 = required tan. Required the tangent of 71° 50' : — By the table— tangent of 71° 48' =3-0415 Difference in Minutes = 2 Number under 2 58 Add 3-0473 = required tan. "298 PRACTICAL XOTES Required the tangent of 83° 54' : — By the table— tangent of 83° 54' =9-3572 Example of the reverse operation : Required the angle whose tangent is 2'3090 : — Nearest lower fractional number opposite whole number 2 = -2998 = tan 66° 30' DifTerence between this and 3090 - 92 In difference column 92 corresponds to 5' Add 66° 35' which is the required angle. ViiR ELECTRICAL STUDKXTS. 299 LO I-H rH 12 15 12 15 12 15 12 15 12 15 12 15 12 15 12 15 12 15 LO eg rH ■-'J LO LO eg CM eg 12 16 13 16 13 16 rH 10 CM 05 C35 O!) sO rohOK) CD CJ) C35 \0 sO ■••O ro ro fO CD ex Ol -O vO to K) K) 05 Oi o^ CT- -O ^0 po hO ^-: O) oj cr. ^O ^ '^ rH ro rH St LO I-H CM l> 03 ro CO rO LO vo 000 LO fO •— ' OD eg i-H rH CO LO CO sO ^ ro LOO oo LO CD !>. rH eg ^ eg eg cd rH OICO vD -^ K) s CO rH ■^ O) <3- I-H CO Ki tg- vjD 000 03 •^ rH CD CO rH rH rH oco [> -^ eg rO LO I-H O) eg ^ CO LO eg ^ eg eg eg egoc35 •d- ro rH CO eg CO re 1-^ PO CO 03 CM ca I— 1 03 CX3 LO OJ CD CO CD rH t-H cg a> 10 roo rOLOO CD CO i-H eg ^ eg eg eg ro rH eg rH ^o CO eg eg ro ^ LO I-H C3J "^ a> LO OJ §S8 LO rHO CO lO CO CD rH 00 rH -^ eg i-H rO rH CD rO LO ^0 I— « rH rH rH KJ LO CD LO fO rH eg <^ eg eg eg LO eg — 1 CD CO \o i:^ CT) CM eg eg eg [> rH ro O to CO C3 CM [>CM vD rO .-H oq ^ 000 ro c^ CO vD rO [> CD rH 00 1-H t>LOK) rH CJ) |> hO ^ vJD LO CO LO l> rH rO rH eg ^ eg eg eg vjd K3 eg CO i> LO CD eg eg eg ro LO ro ro S s ^ CD =:t ■O- i-H C7) Oa C3- 10 000 CD lO eg vo ^ cq 1> 05 T-H 00 i-H O) [> LO CD l> LO eg ^ vD ^-i T-l l-\ LO K) CO rH CD eg rH CD CO I-H rO eg oq eg CO <::t ro LO ^ LO CD eg eg eg ro CO i-H 03 LO OCl -^ lO 00 OCl CO -^ sis -H CDCO CO LO rO eg ^ sD rH cc rH CO OJ eg eg eg LO fo CO 63 I-H CO ro 0^ '^ g:> c:j CO LO oa K) LO 000 •^ ^ hO rH CO l> CD rH K3 rH =:f eg eg ^ y~i r-^ f-i cn CD i> Cv] LO CO .0 t^ Oi —t Ki 1-. egeg or-LO KD rH LO t:^ CD eg eg eg CDi ro eg i) I— ( CM l> C CO I-H ro eg ^ eg eg t^d- ^o rH I-H I-H CO [> rH CM "^ -c- eg CD I-H ro rH eg eg eg CO rH CD CO 10 'JD CO eg eg oc

LO \0 CO Cp rH COLO ^ eg CO 03 -^ LQ r-H rH tH 3 ^ CD =3- eg 03 rH K) tH CM C>q rH eg ro O) -o ^ ^D CO eg CM eg g 0. r^ CM hO =d- LOO CO rH t:j- lO vO t> 300 PRACTICAL XOTES LC or- t> O O O CO CO CO o O O I-H ^ oo LC i-( .— 1 i-t T-H i-H r-l I-H I-H --H I-H I-H i-H CM Oq (M03 csioq ^ to rO l-H i-H lO rO ^ "^iij- 1— t ,-H i-H ^ ^lO lO lO lQ I-H ooo I-H I-H I-H O CO tH 1-H ■^ to o o o oo o O ^ r-i >— 1 I— 1 I-H .-H I-H CM I-H rH I-H O] cs] oa ro to to ro CQ O vO o r^ [>o ooo l> 00 CO CO CO CO CO 05 (j:> oo 1— 1 rO rO to ro roto ■d- ^ 'd- =^ ^ -^ ■d- ts- d- ^ •^ ^ *"• =3- O CD o ^ ^ OJ O ro lC O O LO ■d- en o O O) CS3 CM T— 1 CS) O] lO cj- LO r- CX> CQ UJ CO s ■=3- O CO o oa =3- O CO o 03 10 to to to -^ ^ ^ O 'd- T-H -H e3- I— ( oq CO l> I-H O LC ^ O 03 C3 O oa to uo [> CDC^I vO O ^ C33 LO I-H CO ooa ^d- O CO o oq ^ o CD rO rO to ==f "d- -a- ■d- =3- uo LO lO LC ^ \D ■■£) ■■T) o o uC —1 <-7^ ai ro o cr. to ex O LO -d- 00 oo 05 CNl r^ oa JO LO i> LC CO o CD — « to to ro to ^ -^ ^ ^ LO lO LO lO O vO O O CO o CO I-H CX) ro ro O ■^ tj- K) ro ^ -^ uO O CO O to LO CD 03 O ^ o to rO to to rO ^ ^ ■cr <:::^ cT LO LO LO LO ooo oo C-OJ m O) Cv] l> l> S OdCsJ 1— ( -H 03 C LO [>-O0 I-H <^ 03 =d- O CS] C3i Cv] O CO o o CD CX) o o .-H CM ^ nO 00 r-l l> CO CO CD o oq ro O CO oa CT) LO CM OCl =^ •JD CO o oq d- 1> CD I-H ro LC 00 lO ro to to ^ o o l> C<1 ro O) I-H CO o o LTD uO O ^O [> CO Cn m ■■O CO O CNl ^ ^ CO •o ro ro =^ 'd- ^ ^ ^ LC LC LC LC sO o o o o O) ro o O) O 1X3 OJ ro l> LC l> ro o o CO a: O w CQ ro ^ u:) o C-COOi o 03 03 Cq ro ro ro ro ro ^0 FOB ELECTRICAL STUDENT,^. 301 ^ ?S '^ S S "-^ '-' oa K3 3 cMoaoa o^oac^ oaoa5j S S?oho "^ed-Ln Lo -oor- t>coco oo--* •-H i-i ,-1 i-H ,-t ^ ^ ,_) ^ ^ ^ 03 OJ 0)00 O OrHi-H ^oqw r-OhOia- lQlOlO lo lClCO vOvOO vCt>I> lO O) f-H CO o ^o L-^ O ro [> CO CO 00 ro ro O O) (M en 00 ^o CO O) O) LO CTi o vO ^ cc o:) ro o CD O CD c^ ro LO o o i> o <^ CO r-o eg ro LO Oa T-H C-CM O Oa K) hO o ocv: LO "Tl- ro [> O ro t> CO OD .-HOO ■cf O 0:1 C7) CM LO CO CD CTj O K> OJ r-D CO ca- C^ OJ o C7i O O CO rO ro ^- 00 oa ro o o -^ cc oa r-t I-i .-t CNl CO LO to o c:^ o O I-i sO 00 K) ro CD O] CO eg -H o C- O rO C- CO CO O CO ^0 t-i CO LO CD CO LO CO 0:1 CD ^D [> C<] CD ^ .— I OD 0:1 O (J) O O o ho oa oi CO aj CD h'D O O .-H n-l 1-H hO rO •:3 CO o t> o: O] O l> CO CO LO rO C~ CTj O-l CO —- LO CO CTj CJj I-H eg LO ^JD 1-1 t>- co eg uo O) O O I-H rO CD LO -^ LO CD rO O O I-H r-H O C3i 1^ r-l [> ^O O O LO oa ro ro hO ^ to O LO ^ o Oj oq [> l> CO C^ rO O «d- -JD CD CO ■-( I> CO o I-I l> .-( to lO CO I-H ■:? CO CD CD rO I-H r-t Oj -^ o O I-H LO CJi o o LO 10 l> CO eg \D O I-H I-H r- LO LO OJ CO o LO CD CT eg eg 10 CO CO LO I-H CD CO O CC I-H l> t- CO LO CD ^ CO CD eg CO Oi CD O; LO <3- LO o o O I-H «:?- CD O O ro eg eg CO eg o eg CO o CO ro I-H eg eg ro O CD O C7) O LO LO CO I-H [> O CO ■^ i> I-H LO o cr> l> CJ (^-3 CO CD O- LO O CO eg i> eg L- O tC- CD O O CD ^10 Ol CC CO t^ I-H LO O r^ r-« o eg o 10 01 o ^ CO to eg eg to ro I-H O ^o o CO -^ vO CO eg t-o Lf J I> O ro CO CD CD I-H 10 eg O: ro CT3 sO O ro Oj CD Cj O rH 1— I I-H rO \C CD CD "^ 1-H fo CO 10 eg ^Nj to 'D 10 CO fO .-H OJ LO CO o t> L-- CO ro =3- LO CD o eg MD O ro CO CO Oj O O LO LO O LO vJD O to a> o o ■::+ ^ ^ eg o o O I— I LO O rH r-i CD CD O td" CD O ro i> eg eg eg fO 302 PHAOTIOAIj Nlll'EH LO to LO CO o ro o o LO CO .-1 LO CO r^r) CO ^ C7. l>LOt =3" lO ^ ro ro ro «^ -^ ^ -^ LO LO LO O 03 [> ococr. o rH f-O ^r O oi ocg ^ vO c^ t-t 5d- l> rH LO O LO nH 03 o to oj cj oa rO r-j hj l-O rc ^ r- cc CO CD O — ' ro -^ vo r- CD --1 "^ I> t^\r~ CO CO oz oa CO Od 03 oj oj oa hO ro ro -T '^ <:d- LO LO ,_, C7> CD O O 1-1 .-H CO K5 i^ lO O t> CO O CO -cr ■o CD ,_t CO OCl CO CO CO d o o .— 1 C- I—I o cc 03 ro CO LO lO c- o o cc LO c ro en lo CO O LO -a- to i~o ^ sD Ot) ro CG LO ^ t^ uo LO ^ l> L^ CO CD lO w CSl rO =T LO l> cc ;o O LO OJ O CO CO ro O CO hO >-^ i-iCSi ^ CQ Oi o LO § l> .-1 CO CO rH CO Oi to u J CO LO LO hO O COC- CO .-) O 03 CO lO — < CO ■jD 1^ rO CO c-n ro LO I> f— 1 l> ■^ ^ ^ la uj >ri o o CO CD ,0 i-l O: K) ^ lO O CO ;o C3- Oi t3- CO oro o Ca LO K) lOCOO ro o: ■:d- o o =3- C<] LO CO .-H -^ 1-1 CO C-O K) lO^ CM CO ^ TO lo (O CO rH O CO CO ^ !-• CO LOrO CJ r-t r-l OJ K) O O LO l> CO Oj ,0 ,-H03 ro ^ LO O LO "^ ^ LO O [> KMJD E> LO CO LO O LO CD CO o cr. o l_l r- o o LO CO ^ ■^ O^ CD =d- o ^ o vO .-1 CC (T. o to O M3 rH O rO O [> ■^ oo >-< O CJ t-H CO lO CO H-) c^ ■a- CO Oi lO .-H CO rO ^ LO vJD CO o OiO CO l>Oi|> LO COO t^ LO r-0 CO oa o o c: l> O t— 1 u-D <_j cn I— 1 o o vL' "^ CJ> ■^ cc ■^ re cc o ro O LO i-H -OrOOi o 5 c^ c5 CD Oi CD .-H ror- CO CO to 53- coo: o O rHCv: ■a- LO cc o. CO \ID rH i>iocn ro .-H CO o o: oa CD hO E> LO K) o d; uo re LO O o ^ ca 1> C^ ^ CO CvJ LC O CO 2 ^ CM CT) ^ O ■O CO Q'i o rO ■-< Oi CO CO CO C CO ^ o !>■ ro ^ lo LO %0 O !> cog; Oi CD CO ^ =3- Lc >x; O CD t^ CD CD •d- vo o;3 ^ l> i-H C^ oq iOI> K) CO 1-t cd- CJl i> o: ^ K3 O =^ CO t- c- CD ,-H '^ en LO ro ^ '^ LOO o [> CO CD CD O T-l 03 O CO fO rO LO O CD CO to r^ o CD LC LC CC 03 l>CO LO T-H O [> [> C vO o:) O CD CO -^ c^ CO O OJ c^ I> CO CO ro CD lO .-H C^ «T .-1 CD O ^O O O L-- O K) 1^- rO ro ^ ^o- LQ vO O [> CO CO CD O 7-. D3 fO LO O 1> CD' •d" LC CO CO O CD T-H lt r~- i-H ^ ro O K5 O C^ LO CO rH ro CO lO o o CD t-i CO i-i CD ^d- sO I> l>CC CO o O CO K3 CO -a- O [> •o --< CO o UO LO LO ooo .-< o CO o rO ^ «3 LO O O r- CO CO CD o ^co rO =3- O o O) td" 1-1 o CD ro ro ^ O 1> o K> LO O a. .-1 1— lC CO O CO 03 CD O ^ 03 ■^ o o: O tiT o LO LC LC [> ^ o r- CO CO rO O ^D ro O CO o LO cd" "^ LO 1> O =d- C3 b hO -vt -^ ^ V "P l> CO CO CD CJ r-i 03 hO CD I— 1 T— 1 t-t iH tH T^ '"' rH^ ^ CO eg CO CO O3C0 CO CO ■^ LO O l> CO a:. o r-4 0a K> trf LO mT: !> CO C^ c 1— ' 1 LO LO LO LO LO LO o ^D O O o ^o o o o o o o FC0 to rH LO oo c^jcaoa ooo CM <::^tr)c^] olco c^co 0<^ O^O ^t^=1- rO vDCOI> CMCfi^ CMoO LOO OCOCD OCnii-H ^ CMCXJLO _,^^, rC^A-,,^, CM ^ !> CTj C<] O.OO KM lO lO |hO j^I ^ ^ c3 S C^ LO CO lOC> OOlO Cvl^O (M LOCOCO ODOJCn rori- o CM o (M Kj looi> ^ 03 ir> LO CD o CO o r- lo CM ^ O CJ) OO O O lO IrH C3i ICJ) iCM ^ I5 W ^^OT CO ~^ OCn LOKTjO O-^CS] sO "d-01>- l>rOK3 CDOO 6a ?§ Ot> u:it>0 rO>^vO to OvOCQ COOrO OCM i-*l> 0 COOiCO CMCDLO LOOCO COCDCM C3iCJi rOC:>Cjl ctlO<^ ^ ODCDrH Af^>^ i^Ai^-i i-H ro CO —1 lo 03^ c- a> 2 ^ ^2 c3 § ^ g g en >— 1 LO C35 vocn cnr-K:) f>cv!LO co cmcocd ro[>KJ ct),-io i-HtO OCO.-I LOCDrO CJ) OloD- '--l.-li-H CS]fO^ g OlO ^ 0300 OrOOl 030CJ) CM^ ^CnrO hOi-ihO CO OQ-^O^ CN]^CO OC0<:t SfS §SC2 5BS S S??S ooau, o.^g s CO ■=d-CM OCOOO -^CO-^ (M OCMO CMOO OC7)iO roro i>rHCM OC003 Loooa o.-i'^ cooco :2R sss s§s § sss oc.^ ^H,g CO i-H 03 I— 1 vOCQ O) CO ro i0 1>09 3 vO 95 to S^j-, inoara 3S tSSp: g^?? § S8S SS§ 5SS F-ihC Lor^o ■d-[>c "^rofo cdi-hlo 0.-11-1 O rH c^ CO LOCM CD LO ro S^^KT S S ^S 003 looo roi>i-i [> rocgoj cn^c^- a>Of^ ^1-1 1-H fO vO o r-oi «:fTHCO lOvDO to co0 t^CMO .-H^'^t rH rOLO^ ^^^ ^_u^ C-C- COhO-^ fO ^ I> -1 .-i ^ ^t^Q SS'^S^ ocsi «a-[>o rpc-rH ^^T* ^'?"'P pocM K)to rbto-i- «^ -^lO lb CO °^ ;:; ^ 2 S 5 C3 c^D^ ^D^r^ D-^^r- CO CO CO CO co co od co co co INDEX Aerial Lines, 71 Ampere, The, 38, 267 Ampere-feet, 230 Ampere-turns and Field, 202 „ on Solenoid, 282 Area of Curve, 286 Astatic Instrument, 94 „ Needles, 86 „ System, 280 Batteries (see Cells). Calibration by Comparison, 117 „ Constant, of Tangent Galva- nometer, 118 „ of Indicators, 87 ,, Simultaneous, 105, 106 Cell, Chemical Action of, 5 „ Daniell, 7 „ De la Rue's, 14 „ Elements and Poles of, 5 „ Fuller's, 14 „ Gravity, 10 „ Internal Resistance of, 9 „ Leclanch^, 13 ,, Local Action in, 7 „ Minotto, 11 ,, Polarisation in, 6 „ Porous Partition for, 9 „ Standard, 16 „ Tray, 12, 88 Cells, Diffusion in, 9. „ in Multiple Arc, 96 „ in Series, 17 „ to Insulate, 18 Circuit, Closed and Open, 35 „ Short, 12, 62 Coils, External shape of, 85 „ for Resistances, 99 „ in Multiple, 47 ,, in Series, 47 „ Mounting, 45 „ Winding, 46 85 Commutator, 58 Conductivity and Weight, 26 „ Current and E.M.F., 38 Unit of, 24, 81 Conductors, Copper, 25 , , Length, Diameter, and Weight, 26 Telegraphic, 27 Continuity Tests, 64 Controlling Field, Measurement of, 213 Magnet, 128, 209 Cosine and Sine, 149 Coulomb, The, 39 " Couple," 162 Current, 1, 9, 10 Absolute and Practical Units, 2|57 Belt, 282 Direction of, 4, 42 Retardation of, 36 Unit of, 38 Curves of Force, 126 Daniell Cell, 7 Gravity, 10 Declination, Angle of, 140 Demagnetising, 142, 218 Effect, 234 „ Lines, 215 Detectors {see Indicators). Difi'erence of Potential, 4 Diffusion in Cells, 9 Dip, 133 Dyne, 134 Earth Plate, 71 Earth's Field, " Couple " due to, 152 „ Horizontal Intensity, 136, 138 ,, Magnetic Field, 133 ,, Magnetising Force, to Neutralise, 231 Total Intensity, 134, 138 „ Vertical Intensity, 139 306 INDEX. Electro-MagnetSj 190 „ Ampere-turns, 195 „ Poles of, 191 Retarding E£fect, 194 Electromotive Force, 2 Element, Positive and Negative, 5 Ewing, Prof. J. A. : Experiments on Mag- netic Induction, 246 Fibre, Suspension, 146 Field at Centre of Coil, 267 „ at Centre of Solenoid, 289 „ at End of Solenoid, 288 „ Controlling, Measurement of, 213 „ Intensity of, in Circular Conductor, 262 ,, of CoU and Magnet Superposed, 268 „ of Force, Mechaaioal and Magnetic, 143 „ on Coil Axis, Critical Point, 266 „ on Axis of Solenoid, 293 ,, Resultant, 157 Fields, Superposed, 144, 156, 268 Strength of, 145 Figure of Merit, 103 Force, Magnetic, 137, 138, 141 „ Moment of, 149 „ on Needle, 147 „ Unit of, 134 Galvanometer, Sir AVm. Thomson's Graded, 210,269 Galvanometers, Decrease of Sensitiveness of, 208 ,, Increase of Sensitiveness of, 142, 212 Tangent, 113, 118 „ with Several Coils, 58,209, 210 German Silver for Resistances, 98 Horizontal Component, 136 Hysteresis, 239 Inclination, Angle of, 133 Indicator, Astatic, 94 „ Horizontal, 61 Needle of, 44, 59, 60 Standard G.P.O., 82, 101 Indicators, Best Resistance of, 48, 50 „ Current, 42 „ for Large Currents, 55, 119 ,, for Telegraph Work, 53 Indicators for Telephone Work, 54 ,, Sensitiveness of, 48, 63, 91 with Different Coils, 57 Induction, Electro-Magnetic, 240 Insulation of Overhead Mains, 75 „ Resistance per Mile, 77 Test, 70, 76 Insulators, Porcelain, 71 Shackle, 73 Intensity of Magnetisation, 199, 241 Inverse Squares, Law of, 135 Johnson and Phillips' Insulator, 72 Kapp Lines, 247 Latimer Clark's Insulator, 71 Leading Wires, Correcfion for, 80 „ ,, Effect of Current in, 56, 206 Lines of Force, 126, 136, 141 „ „ Demagnetising, 215 Direction of, 127, 131 „ ,, Round Circular Conductor, 257 „ „ Round Straight Conductor, 254 Local Action in Cell, 7 Magnet, Controlling, 128, 209, 211, 213 Electro, 190 ,, Polar and Normal Lines of, 166 Magnetic Circuit, 196 ,, Distribution, 225 „ Field Round Circular Conductor 257 „ „ Neutralisation of, 142 „ of Earth, 133, 138, 139 „ of Magnet, 157, 161, 162 „ „ Round Straight Conductor 253 „ Uniform, 140 „ Unit, 135 Fields, 123 „ „ Comparison of, 141 „ „ Superposed, 144, 156, 268 „ Hysteresis, 239 „ Induction, 199, 240 ,, Leakage, 199 ,, Meridian, 139 ,, Moment, Measurement of, 177 ,, Permeability, 250 Pole, Unit, 135 INDEX. 307 ilagaetic Poles, Comparison of, 143 „ Saturation, 245 Shell, 258 ,, Susceptibility, 236 Magnetisation, Apparatus to Determine, 206 „ Curve, 220 Intensity of, 199, 241 Magnetise, To, 13?, 146 Magnetising Coil, 2J6, 224, 227 „ Force, 195, 222, 228 „ ,, and Ampere-turns, 202 Magnetism, Free, 200 „ Hughes' Theory of, 185 Nature of, 182 Residual, 252, 234, 239 Magnetometer, 166 ,, Adjustment, 176 Mass and Weight, 135, 155 Megohm, 30 Microhm, 30 Milliampere, 40 Mho, 24, 81 Moment of a Force, 149 Couple, 152 „ Magnet, 155 Morse, Simple Circuit, 66 Mounting of Coils, 45 Needles, 44 Multiple Arc, 31 Multiplying Power of Shunt, 107, 109, 110 Needle, for Magnetometer, 169 ,, for Tangent Galvanometer, 114 ,, Vibrating, 146 Needles, Astatic, 86, 94 „ Checking OscUlations of, 101 „ Loss of Magnetism in, 59, 60 „ of Detector, 44 „ Oscillations, and Strength of Field, 141 ,„ Reversal of, 59 „ Size, in Comparison to Coil, 114, 121 Ohm, 10 •Ohm's Law, 40, 89, 103, 110 Ohms, Multiples of, 30 Oil Insulator, 72 Oscillations, Law of, 143 ■Overhead Mains, 73 „ „ Insulation of, 75 Parallax, Error of, 85 Pendulums, Force Actuating, 142 „ Law of Compound, 143 Period, 143 ,, Variable, 36 Permeability, Magnetic, 250 Pivot, Friction of, 60 Platinoid, 98 Platinum-Silver, 98 Pointer, Glass, 169 „ Vulcanite, 84 Polarisation, 6, 101 Pole, Unit Magnetic, 135 Poles, Consequent, 131 ,, Force Between Two, 135 „ Free, 127, 143, 156 „ North and South, 125 ,, Number of Lines from, 214 ,, (of Cell), Positive and Negative, 5 „ Position of, 156, 225 Porous Partition, 9 Potential, 18 „ Difference of, 4 ,, Distribution of, in Battery, 18 Quantity of Electricity, 39 Resistance, 9. Adjustable, 207 ,, and Cross Section, 26 ., and Length, 25 „ Box, 97 ,, by Reproduced Deflection, 82 ,, Insulation, 75, 77 „ Joint, 31, 111 of Cell, 9 ,, Relative, of Metals, 23 Resistances in Multiple, 31, 111 ,, in Series, 30 „ of German Silver, 98 of Platinoid, 98 of Platinum Silver, 98 Resultant Field, 157, 270 „ Direction of Field, 158 Sensitiveness, Increase of, 142, 212 ,, of Indicator, 91 Series, Cells in, 17 Shackle Insulator, 73 Shackling Telegraph Lines, 74 Short Circuit, 12, 62 Shunt, 106 Multiplying Power of, 107, 109 110 ,, Resistance of, 111 x2 308 INDEX. Sine and Cosine, 149 Soft Iron, Magnetic Saturation of, 245 Solenoid, ]Jirection of Field due to, 277 Field along Axis of, 293 „ at Centre of, 289 „ at End of, 288 Mounting of, 273 Variation of Field along Axis, 284, 294 Specific Conductivity, 23 „ Resistance, 23 Steel Main Suspender, 74 Stirrup for Needle, 146 Straight Line Law, 92, 102 Submarine Cables, Retardation on, 36 Susceptibility, Magnetic, 236 Tangent Curve, 115 ,, Galvanometer, 113 ,, Principle, 156 „ Scale, Graduation of, 172 Temperature and Resistance, 98 Terminal Volts, 89 Tests for Continuity, 64 „ Faults in Apparatus, 65 „ Identity of Wires, 68 Tests, Insulation, 70 ,, with Indicators, 64 Volt, 3, 7 Volume of Gore, 204 Weight and Mass, 135, 15&