i = Uq + at
V = Va + \at
d = vj. + \ afl
u) = oi„ + ^ al
Simple Harmonic Motion
F = - 4 ir VjV ■ x I r = -47r%2/.<^
CIRCULAR AND SIMPLE HARMONIC MOTION 69
Problems
1. A stone having a mass of 1 lb. is "whirled, at the end of a string 3 ft.
long, in a circular path at a velocity of 30 ft./sec. What is the tension in
the string in pounds weight?
2. The flywheel of an engine is 5 ft. in diameter and makes 300 revolu-
tions per minute. What is the radial acceleration of those portions of the
wheel which lie farthest from the axis? What radial force in pounds
weight is acting upon each pound mass of the outermost portions of the
wheel ?
3. A mass of 50 g. is whirled at the end of a string 50 cm. long.
Find the tension in the string when the mass makes 2 revolutions per
second.
4. How will the tension in the string of problem 3 change if the velocity
of the body is doubled ? if the length of the string is halved V
5. A certain railway curve has a radius of 400 ft. If trains are to pass
this curve in safety at 25 mi./hr., what should be the angular elevation of
the outer rail ?
6. A body is moving uniformly in a circle of 5 ft. radius. At what
velocity will the radial accelerating force just equal the weight of the body ?
7. A body having a mass of 5 lb. is vibrating in simple harmonic
motion. It makes one vibration per second. What is the force acting upon
it when it is 3 in. from the center of its path ?
8. What must be the length of a simple pendulum in order that it will
make one swing per second at a place where g = 981 cm. /sec. 2?
9. A small sphere of lead is suspended by a thread 25 ft. long. What is
its time of vibration? Assume .
the point B comes to the position B' . Let it be required to
find the work done by the force F. Applying the principle
given above, the distance BB' through which the sled has
moved must be resolved into components parallel and perpen-
72 MECHANICS
dicular to F, namely, BG and GrB' . We have, therefore, work
done by the force F is
W=FBa
It is evident that this gives the value of the work done by the
force F as the sled moves from B to B' from the following con-
sideration : The sled might, for example, be supposed to be
free to move in any direction and to pass from B to B' by first
traveling over the distance BCr and then over the distance
CrB' . As it moves from B to Gr it is moving parallel to the
force F and the force F is doing work. As it moves from G-
to B' it is moving at right angles to the force F. The force F
is therefore not moving in its own direction at all and hence
is doing no work. Calling the angle B'BG, (p, it is evident
^^^^ BG =BB' cos ^
.-. W= F ■ BB' cos^
This expression for the work done may be reached in a dif-
ferent way. Instead of resolving the motion into two compo-
nents, we may resolve the force F into two components, the one
of which is parallel to the motion, and is therefore effective in
doing work as the sled moves the other perpendicular to the
motion, and therefore doing no work. Thus in Figure 40,/ is
that component of F which is parallel to the motion, and
y:
Fig. 40. — Work done = BB' x F cos 0.
which might therefore be called the working component, while
T is the component perpendicular to the motion. We have
therefore for the work done as the sled moves from B to B',
W=fBB'
but evidently
f=F-eos (23)
This relation may be demonstrated as follows. Referring to
Figure 41, let it be assumed that a force F acts upon a body
OB which is pivoted at 0, for ex-
ample, a vi^rench applied to the
nut of a bolt. Call the perpen-
dicular distance of the point
from F, r. Then the torque
action of the force F upon the
body OB is
T=Fr
Let it be assumed that under the
action of the force F (which is
supposed to remain at all times
perpendicular to r) the point B
moves to B' so that the line OB
has turned about through the angle ^
cording to Section 59, is
W=FBB'
BB
Fig. 41.
The Work done in tightening
a Nut.
The work done, ac-
But,
Or,
Therefore,
i.e.
r
BB' = r
Fig. 47. — Balanced Forces on a Block in Uniform Motion on a Table.
librium. If i^and IF are known, the magnitude and direction
of R may be determined by applying the principle of Sec-
WORK AND ENERGY. FRICTION 83
tion 37. See Figure 47, b. The angle between R and Wis called
the angle of friction. It is found that for given conditions, if
Wis increased, # must be increased in the same ratio, in other
words, FxW, or, — = a constant (for a given pair of surfaces,
in given condition as to smoothness, etc.).
The ratio — - for a given pair of surfaces is called the coeffi-
W
cient of friction for those surfaces.
The above relation may be written
F=/j^W (33)
In which the proportionality factor /jl is the coefficient of
friction. Evidently from Figure 47, 5,
The coefficient of friction between two surfaces depends upon
the nature of the materials and upon their roughness or smooth-
ness. It is found to be nearly independent of the area of con-
tact and the velocity with which the one body moves over the
other.
ROLLING FRICTION
75. When one body rolls upon another, the friction is less
than would be the case if sliding took place. The resistance to
rolling motion between two bodies is called
rolling friction.
The friction effects in machinery are
diminished by the use of ball bearings.
Rolling friction is in this manner substi-
tuted for sliding friction. Figure 48 rep-
resents a ball bearing. The shaft A rolls
upon the balls which in turn roll upon the ' ' ~.
bearing B. Thus the sliding friction be-
tween A and B is avoided. In order to secure the best results
in a ball bearing, the balls must be slightly separated. Other-
wise there will be sliding friction between adjacent balls.
84 MECHANICS
Problems
1. How much work in joules is required to lift a mass of 10,000 gr.
(10 kilos) from the floor and place it upon a table 120 cm. high ?
2. A man whose mass is 175 lb. climbs a ladder 20 ft. long. The ladder
is leaning ajainst a wall, its lower end being 10 ft. from the wall. How
much work is done against gra\"ity ?
'^. A man draws a box along a sidewalk for a distance of 100 ft. He
draws it by means of a rope which makes an angle of 30° to the horizontal.
How much work is done in moving the box if the pull in the rope is 25 lb.
weight ?
4. What is the potential energy of a ton (2000 lb.) of water on the
brink of a fall of 150 ft. ?
5. What is the potential energy of a tankful of water 100 ft. high, the
base of the tank being at the surface of the ground, if the capacity of the
tank is 200 tons (approximately 50.000 gal.) ?
6. What is the kinetic energy of a rifle baU having a mass of 0.04 lb.
and a, velocity of 1000 ft./sec. ?
7. What is the kinetic energy of a 40-ton car at a velocity of 60 mi./hr. ?
8. A box of 250 lb. weight lies upon a level sidewalk. The coefficient
of friction between the box and the walk is 0.37. What horizontal force is
requii'ed to move the box ?
9. A man presses an ax against a grindstone with a force of 25 lb.
weight. The radius of the stone is 15 in. The torque required to turn the
grindstone is 12.5 pound-feet. What is the coefficient of friction between
the ax and the stone ?
10. A block having a mass of 1000 gr. slides down an incUued plane.
The height of the plane h is 50 cm. The velocity acquired by the sliding
block is 250 cm. /sec. How much work is done against friction?
11. It requires a force of 100 lb. weight to stretch a certain spiral spring
6 in. What is the potential energy of the spring when stretched in this
manner ?
12. A constant torque of 50 pound-feet is acting upon a rotating body.
If the body makes 2o revolutions, how much work is done by the torque ?
THE SIMPLE MACHINES
CHAPTER VII
DEFINITION OF A MACHINE
76. A machine is a device which facilitates the doing of
work. It is to be regarded as a transmitting device, since a
machine is able to do work upon other bodies only when work
is done upon it. The force which operates a machine is called
the working force; and the force against which the machine
operates, the resisting force.
The simple machines are the lever, the wheel and axle, the
pulley, the inclined plane, the wedge, and the screw.
The mechanical advantage of a machine is the ratio of the
resisting force to the working force.
The efficiency of a machine is the ratio of the work done by the
machine to the work done upon the machine. The efficiency of
a machine is always less than 100 % , since in every machine a
certain amount of work is done against friction.
THE LEVER
77. The lever is a rigid bar, straight or curved, which when
in use rotates about a fixed point called the fulcrum. There
are three classes of levers (see Figure 49) : (a) that in which
the fulcrum is between the working force and the resisting
force; (5) that in which the resisting force is between the
working force and the fulcrum; (e) that in which the work-
ing force is between the fulcrum and the resisting force.
The pump handle is a lever of the (a) class. The oar of a
boat is a lever of the (6) class. A man's forearm is a lever of
the (c) class.
The mechanical advantage of the lever is given by the ratio
85
86
MECHANICS
of the length of the lever arm of .the working force to the length
of the arm of the resistance force. This is evident from the
following : Consider that the lever is without weight and the
torques are in equi-
librium, then (Sec-
B
F
P
(Q.)
(b)
f
^f tion 38),
/■AP = FBP
(See Figure 49.)
F^PA
" f PB
It is evident that
with the lever of
the class (a) the
mechanical advan-
tage may have any
value. In a lever
of the class (J) the mechanical advantage is always greater
than unity. In class (c) it is always less than unity.
\c)
Fig. 49. — The Three Kinds of Lever.
THE WHEEL AND AXLE
78. The mechanical advantage of the wheel and axle is given
by the ratio of the radius of the
wheel to the radius of the axle.
This may be shown as follows : In
Figure 50 let the small circle rep-
resent the axle upon which the
cord c is wound as the wheel is
turned. It is assumed that the
working force is applied at the
circumference of the wheel and it
may be represented by a small
weight /. Assuming the torques
to be in equilibrium, we have
(Section 38),
f.R = F.
Fig. 50. — Wheel and Axle.
THE SIMPLE MACHINES
87
in which R represents the radius of the wheel and r the radius
of the axle. P -n
and
= 1
THE PULLEY
79. The mechanical advantage of the pulley when the
" block," that is, the frame which supports the pulley wheel,
is fixed, is unity. This is evident from the
following considerations : In Figure 51, /rep-
resents the working force applied at one end
of a rope passing over a fixed pulley. The
weight which is being lifted hangs from the
other end of the rope. Let it be assumed that
the torques are balanced (Section 38),
then, f ■ r = F ■ r
or f=F
F
f
A pulley used in this manner is termed a
"fixed pulley."
A pulley when used as
shown in Figure 52 is called a "loose
pulley."
The mechanical advantage of the " loose
pulley " is 2, providing the two ropes which
support the loose pulley are parallel. This
is evident from the following consideration :
If the forces are in equilibrium, then (Sec-
4tion 37),
^ The " block and tackle " is a combination
Fig. 52. — Loose Pulley. j. ^ i , ,
01 fixed and loose pulleys with one con-
tinuous rope running between them. In effect the arrange-
ment of pulleys and ropes in the block and tackle is like that
Fig. 51. — Fixed
Pulley.
^=2
88
MECHANICS
(36)
That is to say, the power developed by a moving torque is
given, by the product of the torque and its angular velocity.
BEAKBS
88. The capacity of an electric motor or a steam engine is
sometimes determined by applying to the pulley or flywheel
what is known as a brake. The power developed by the machine
is used in overcoming the frictional resistance of the brake. If,
therefore, the frictional resistance of the brake is known and
the distance through which such re-
sistance is overcome is determined,
we have at once the data for de-
termining the capacity of the ma-
chine. In Figure 60 is shown a
simple form of brake used for this
purpose. A represents the pulley
or flywheel of the machine. OD is
a strap which is held against the
periphery of the wheel. To the ends
of this strap spring balances are at-
tached. Let F-^ represent the force
indicated by the spring balance on
the right and F the force indicated
by the spring balance on the left.
Let it be assumed that the wheel A is moving in the direction
of the arrow. It will be evident that F is larger than Fy
The difference between them is the frictional resistance be-
tween the surface of the wheel and the strap. It is this fric-
FiG. 60. —The Strap Brake.
98
MECHANICS
tional resistance against which the machine is working.
Therefore, if the wheel A makes n revolutions per second, so
that its angular velocity in radians per second is 2 ttw, the
velocity of a point on its circumference is 2 irrn, where r is the
radius of the wheel. Therefore, the work done per second
(i.e. the power developed) by the machine, against the fric-
tional resistance at the circumference of the wheel, is
P = 2 Trm ( J' - Pj)
In Figure 61 is shown another simple device which is used
for the purpose of determining the capacity of a motor. It is
known as the Prony brake. Let A represent the pulley of the
motor to be tested,
and B and rep-
resent two pieces
of wood which are
clamped on the
pulley A as in-
dicated. If A is
rotating in the
direction of the ar-
row, the frictional
resistance between
A and B and G
This tendency
Q
I
H
B
Fig. 61. —The Prony Brake.
tends to rotate the brake in the same direction,
to rotate, that is, the torque acting upon the brake, is balanced
by means of the spring balance H attached to the end of the
arm B. Let it be assumed that the indication of the spring
balance is F. Then the torque which prevents the rotation of
^^^ T=F-AH
but this torque is equal to the torque action of the pulley on
the brake, since it just balances it. Therefore, assuming that A
makes n revolutions per second, so that its angular velocity in
radians per second is 2 ttw, we have
= 2 TTW X P • AH.
(Equation 36.)
POWER 99
Hence, if the value of F is read from the spring balance, AH
is measured, and n is determined by means of a speed counter,
it is a simple matter to calculate the power developed by the
machine.
Problems
1. What pull must a horse traveling 4 mi. per hour exert upon a
wagon in order to develop one horse power ?
2. A force of 5 x 10' dynes is acting upon a body. If the body moves
40 m. in the direction of the force in 20 sec, what horse power does the
force develop ?
3. What power would be required to raise water to a height of 50 ft.
at the rate of 500 T. per hour ?
4. What horse power would be required to raise a loaded elevator
having an unbalanced weight of 3000 lb. at the rate of 6 ft. per second ?
5. What horse power will a man weighing 165 lb. develop in running
upstairs, if he climbs a vertical distance of 10 ft. in 3 sec ?
6. A strap brake (Figure 60) is applied to the pulley of an electric
motor. The difference in the indications of the spring balances is 5 lb.
weight. The pulley is 12 in. in diameter and makes 2000 R. P. M. What
is the brake horse power ?
7. A Prony brake (Figure 61) is applied to an engine flywheel. The
brake arm ^i^ = 4 ft. and the spring balance reads 40 lb. weight. The
engine makes 300 R. P. M. What horse power does it develop ?
8. An elevator having an unbalanced weight of 2000 lb. starts from
rest and in 3 sec. has acquired a velocity of 9 ft./sec. What is the average
horse power developed during this interval ?
9. What is the work done and the average horse power developed in
the first second in problem 8? in the second? in the third?
10. A shaft making 200 R. P. M. is transmitting 200 horse power. What
is the torque ?
11. A torque of 500 pound-feet has an angular velocity of 10 rad./sec.
What power does it develop ?
ELASTICITY
CHAPTER IX
ELASTIC AND INELASTIC BODIES
89. Experience teaches that solid bodies offer resistance to
a change in form or size. Certain substances when forcibly
distorted exhibit the property of recovery ; that is to say, when
the distorting force is removed, they return more or less com-
pletely to the original form or size which they had before the
distortion took place. Bodies which exhibit this property of
recovery in a large degree are said to be elastic. Those in which
this property is not strongly marked are referred to as inelastic
bodies. Thus elasticity may be thought of as that property of
a body which enables it to recover from distortion.
STRESS AND STRAIN
90. When a force acts upon an elastic body in such manner
as to cause distortion, it is opposed by force actions within the
body. These internal force actions are larger for large distor-
tions than they are for small ones ; and when a distorting force
is applied to a body, the distortion in-creases until the internal
force actions balance the distorting force.
The internal force action per unit of area, across which the
forces are acting, is called the stress. Thus if a column AB,
Figure 62, 5 inches square, supports a weight of 5000 pounds,
the stress is.
^, _ total force _ 5000 /pounds weightN
area 25 V square inches /
_ 200 /' pounds weight N
\ square inch /
For the " total force," we have taken the external force, since,
as stated above, when the various parts of the column are in
100
ELASTICITY
101
equilibrium, the internal force action balances (i.e. is numeri-
cally equal to) the external force.
The distortion of a body, per unit
length, or unit volume, as the case may
be, is called the strain. Thus, if the
column represented in Figure 62 is 20
inches high and is shortened -^^ inch
by the weight supported, the strain is
given by
changfe in lengrth
Strain = ■
total length
■.9^(JI^)= 0.005
20 \inches/
Fig. 62. — Column under
Stress.
THREE KINDS OF STRESS
91. There are three kinds of stress, viz. :
1. Tensile Stress.
2. Hydrostatic Pressure.
3. Shearing Stress.
A body is said to experience a tensile stress when the forces
acting upon the body tend to clwinge its length. The strain
which accompanies this kind of stress is a change in length
and is called a stretch. P'or example, a vertical wire or
string supporting a weight is subjected to a tensile stress.
A body is said to be subjected to a hydrostatic pressure when
the pressure upon it from all sides is the same ; for example, a
small object submerged in a body of water is under hydrostatic
pressure. The strain corresponding to hydrostatic pressure is
a change in volume.
A body is said to be subjected to a shearing stress when the
system of forces acting upon it tends to cause one layer of
particles in the body to slide over an adjacent layer. For
example, in the shearing or punching of plates of metal one
part of the plate, the part sheared off, is made to slide past
another part, that is to say, one layer of particles is made to
slide upon an adjacent layer. The strain which accompanies
a shearing stress is called a shearing strain.
102 MECHANICS
ELASTIC LIMIT
92. If the strain in a body exceeds a certain value, the body
will not recover completely when the distorting force is re-
moved. Such a body is said to be strained beyond its elastic
limit.
hooke's law
93. In 1676 Robert Hooke discovered that, for elastic bodies
under any kind of stress, stress is proportional to strain, that is,
stress , ,
r- = a constant
strain
This is known as Hooke's law.
MODULUS OE ELASTICITY
94. The ratio of tensile stress to tensile strain is called Young's
modulus, the stretch modulus, or the modulus of elasticity.
The ratio of hydrostatic pressure to the corresponding strain
is known as the bulk modulus.
The ratio of shearing stress to shearing strain is called the
coefficient of simple rigidity.
The stretch modulus of a few common materials is given in
the following table :
Dtses pee Cm^, Lb. "Wt. pee Sq. In.
Copper 11 X 10" 16 x 10«
Glass 6 X lO'i 9 x lO^
Wrought Iron 19 x 10" 27 x 10^
Lead 1 x IQii 1.5 x lO^
Steel 23 X 10" 33 x 10^
HOW THE MODULUS IS USED
95. The stretch modulus of any building material is one of
its most important physical properties. Before an engineer
can design a structure, whether it be a bridge, a building, or a
machine, he must have knowledge of the elastic properties of
the material to be used. Having such knowledge, he can
determine how large a rod, beam, or column should be to sus-
tain a given weight or load.
An example will serve to illustrate how the stretch modulus
is used in such calculations. A steel rod in a certain structure
ELASTICITY
103
is 20 feet ( = 240 in.) long. Let it be required to determine
how large this rod should be to support a weight of 5 tons with
an elongation of not more than -^-^ inch. Assume that the
stretch modulus of steel is 33,000,000 pounds weight per square
inch.
Now, _j,^ stress
strain
F
where JE^is the stretch modulus of the steel. But stress =— ,
a
that is, the total force acting upon the rod divided by the
cross-sectional area of the rod. *^~~ -'-—--
the elongation and L the length of the rod, therefore,
e/L
Also strain = ^ in which e is
or.
U =
a = ■
FL
ea
FL
E-e
(37)
Substituting the assumed values for F, L, F, and
10,000 X 240
a =-
33,000,000 X J^
0.727 square inch
FLEXURE
96. When a rod or beam is
bent, the convex side is stretched
and the concave side compressed.
Let AB, Figure 63, represent a
beam resting on supports SS.
Imagine a heavy weight W to
be placed upon the beam at its
center. Under the action of
W the beam will be bent into
the form shown (exaggerated)
in the lower part of the figure.
Since the lower horizontal lay-
FiG. 63. — Beam under Flexure.
104 MECHANICS
ers are lengthened and the upper layers shortened as indicated,
it follows that there is one layer (the neutral layer) the length
of which remains unchanged. The stretch (or compression) in
any layer is proportional to its distance from the neutral layer.
Now since stretch and compression are the distortions which
take place, the deflection depends upon the stretch modulus of the
material of the beam.
Experiment shows that
D
lEBT^
in which I) is the deflection, L the length (^SS^, B the width,
and T the thickness of the beam. IE is tlie stretch modulus of
the material.
IMPACT AND MOMENTUM
97. Closely related to the subject of elasticity is that of
impact, and the effect of impact upon elastic and inelastic
bodies. Consider the case of two bodies coming suddenly into
forcible contact with one another. Each body will receive an
acceleration. The acceleration experienced by either body is
given by Equation (10) in which il!f is the mass of the body, ^is
the force action between the two bodies while in contact, and a
is the acceleration received by the body in question. Evidently,
since action is equal to reaction, therefore we have
MA =
ma
in which M and m are the masses of the two bodies, A and a
are the accelerations received by them during the time of con-
tact. Since acceleration is defined as change in velocity di-
vided by the time in which that change takes place, therefore
we may write ,-rr T^^ ^ ^
t t
or M(V^- V^)=m(y^-v-^^ (38)
The product of the mass of a body and its velocity is known
as tlie momentum of the body. Then M(J\— V^) is the
change in the momentum of the mass M during impact, and
m(y-^ — Wg) is the change in momentum of the mass m. We are
ELASTICITY
105
thus led to the conclusion that in the case of impacting bodies
the change in momentum experienced by each body is the same.
ELASTIC AND INELASTIC IMPACT
98. In elastic impact work is clone in distorting the impact-
ing bodies and energy is stored momentarily as potential
energy. If the bodies are perfectly elastic, they will recover
completely from the distortion and will therefore return all of
the energy which was expended in distorting them. Therefore,
1 MV^^ + } mv^^ = 1- MV^ + \ mv^' (39)
If the impacting bodies are inelastic, the energy expended in
distorting them is not returned, hence the kinetic energy of
the bodies after impact is less than before.
THE BALLISTIC PENDULUM
99. The ballistic pendulum is a device used for measuring
the velocity of a bullet. The principle upon which its use is
based will be understood from
the following discussion : Let M,
Figure 64, represent a large block
of wood suspended pendulum wise
as indicated. Let it be imagined
that a rifle bullet is fired into this
pendulum. It is required to find
the velocity of the bullet v at the
moment it comes in contact with
the pendulum. Call the mass of
the pendulum M and the mass of
the bullet m. Evidently the pen-
dulum and the bullet swing as
one body after the impact, the
mass being M -\-m. If m is very
small in comparison with M^ it
may be neglected after impacting with M. Therefore from the
principle above enunciated, we have
or MV^ = mv^
m
Fig. 64. — Ballistic Pendulum.
106 MECHANICS
since the pendulum was stationary ( I^ = 0) before the impact
and Vj is negligible in comparison with Dj, that is, the change
in the velocity of the bullet is practically v-^.
Hence v-,= — ■ V„
^ m ^
In the actual use of the apparatus V is determined by observ-
ing the horizontal distance through which the pendulum moves
under the impulse of the bullet. Knowing this distance and
the length of the pendulum, it is an easy matter to calculate
the height through which the pendulum bob is lifted. But the
height through which the pendulum bob is lifted multiplied
by the weight of the pendulum bob is the potential energy of
the pendulum at the extremity of its swing. Under the theory
of the conservation of energy this must be just equal to the
kinetic energy of the pendalum as it started to swing toward
the left. This kinetic energy is equal to
Hence V^ may be calculated.
Problems
1. A rod 1 m. long and 0.2 sq. cm. cross section sustains a weight of
100 Kg. and is stretched so that its length is 100.04 cm. Find the stress,
the strain, and the stretch modulus.
2. A vertical rod of wrought iron 10 ft. in length and 1 in. in diameter
supports a weight of 5 T. What is the increase in length of the rod ?
3. A vertical rod 10 ft. long and 1 in. in diameter is stretched 0.05 in.
by a certain weight. What stretch will be produced in a rod of the same
material 5 ft. long and J in. in diameter by the same weight?
4. A vertical iron wire 3 m. long and 1 mm. in diameter is attached
to a copper wire of the same diameter 5 m. long. A weight of 3 Kg. is
attached to the lower end. What is the elongation of each wire ?
5. The stretch modulus of a certain metal is 2 x lO^^ dynes/cm^. What
force would be required to stretch a rod of this metal 1 sq. cm. in cross sec-
tion until its length is doubled, assuming that the elastic limit is not passed
in the operation?
6. Two vertical wires of steel and copper of the same diameter carry
the same load. What must be their relative lengths in order that their
elongations may be equal?
ELASTICITY 107
7. An elastic ball, M^, moying with a velocity «j, strikes a stationary
elastic ball, M^, of equal mass. What are the velocities of the balls after
impact ?
8. If the mass of M^, problem 7, is twice that of M^, what will be the
velocity of the balls after impact ?
9. A ball of putty having a mass of 700 g. and moving with a velocity
of 10 m./sec. strikes a stationary ball of putty having a mass of 250 g.
What is the velocity of the balls after impact?
10. A rifle bullet is fired into a ballistic pendulum having a mass of
2000 g. The mass of the bullet is 1 g. Under the impulse of the bullet
the pendulum swings so that its center of mass rises 1 cm. Required the
velocity of the bullet.
FLUIDS AT REST
CHAPTER X
THE THREE FORMS OF MATTER
100. We are familiar with matter in three forms; namely,
solids, liquids, and gases. Liquids and gases are commonly
considered together as fluids.
A solid may be defined as a portion of matter which offers re-
sistance to any force action which tends to change either its form
or its size.
A fluid is a portion of matter which offers resistance to a change
of size but none to change of form. A liquid is distinguished
from a gas in that when placed in an open vessel it will present
a free surface, and also by the property of forming itself into
drops.
A gas is distinguished by the property of indefinite extension.
A gaseous body tends to expand until it fills all available space.
Generally speaking, any of the simple substances may exist
in either the solid, liquid, or gaseous state. The most common
example is that of water, which in the form of ice is a solid, in
the form of water is a liquid, and in the form of very hot steam
is a gas. Any other substance, for example, iron, may be made
to pass from one of these states to another; a piece of solid iron,
if placed in a furnace and strongly heated, melts and assumes
the liquid form. If still more strongly heated, it vaporizes and
becomes a gas.
THE GEXEEAL PBOPEKTIES OF THE THREE STATES OF MATTEB
101. As general properties of a solid we may mention density,
elasticity, hardness, ductility. We think of the ultimate
particles (molecules) of a solid as bemg bound together by
some sort of intermolecular force action which causes them to
cohere and resist any force which tends to separate them. They
do not possess any great degree of freedom of motion, that is to
108
FLUIDS AT REST 109
say, we imagine that a molecule in one part of a given solid
continues in that neighborhood. In the solid the molecules are
to be thought of as sliding over one another with great diffi-
culty.
In a liquid, while the ties which bind one molecule to its
neighbors are present as in the solid body, the molecules are to
be thought of as sliding upon one another with great freedom;
and furthermore, it is to be considered possible for a molecule
which is in one given portion of a liquid to wander to an en-
tirely different part of the liquid. In matter in the liquid state,
as in the solid state, the molecules are to be thought of as being
very close together.
In matter which is in the gaseous state the molecules may
be more or less widely separated. The bond of union between
the adjacent molecules is not nearly so strong as in the other
forms of matter. The molecules in the body in this state
move freely about from place to place, the spaces between
molecules being large as compared with the size of the molecules.
DENSITY
102. The density of a body is its mass per unit volume. In
the c. g. s. system the densit}- of a substance is given in grams
per cubic centimeter. In this system the density of distilled
water is for practical purposes unity, since the mass of one
cubic centimeter of water (at 4° C.) is almost exactly one
gram. The density of water in the f. p. s. system is about
62.3 (pounds per cubic foot).
The following table gives the densities of some of the com-
mon substances.
Densities
Aluminum 2.58
Copper . 8.02
Cork . . 0:2i
Glass (Crow n) . 2.6
Ice 0.91
Alcohol 0.789
Glycerine ...... 1.26
Solids
Iron, Wrought . .
7.86
Lead .
. 11.3
Platinum .
21. .5
Silver
10.53
Tin ... .
7.29
Liquids
1 Mercury . . .
. . 13.596
Olive Oil . .
. . 0.91
110
MECHANICS
Gases
At Freezing Temperature and Standard Atmospheric Pressure
Air 0.00129
Carbon Dioside .... 0.00197
Hydrogen 0.000089
Oxygen 0.00143
Fig. 65. — Pressure due to the
Weight of a Liquid.
THE PRESSURE IN A LIQUID DUE TO ITS WEIGHT
103. The freedom of motion possessed by the molecules of a
fluid give rise to certain phenomena which are characteristic of
fluid bodies and which distinguish
them from solids. One of these phe-
nomena is that of distributed pressure
on the walls of a containing vessel.
Consider a cylindrical vessel stand-
ing on end and partly filled with
liquid ; see Figure 65. Let it be re-
quired to find the pressure p on the
bottom of the vessel due to the liquid
contained. If h is the height of the
liquid and r the radius of the base,
evidently the volume of the liquid is
V = TTT^ ■ h
Let the mass per cubic centimeter of the liquid, that is to say,
the density of the liquid, be d. Then the total mass of the
water contained in the vessel is
M=Trr^hd
and the weight of the liquid, that is M- g, is
W = irr^hdg
This weight is supported by the bottom of the vessel. The
force action per unit area of the bottom is found by dividing
the total weight supported by the total area supporting that
weight. This force action per unit area is called the pressure
on the bottom of the vessel. We have, therefore,
weight
area
_ Trr^hdg
.■.p = hdg (40)
FLUIDS AT REST 111
That is to say, the pressure at a point in a liquid due to the
weight of that liquid is proportional to the vertical distance of
the point from the free surface of the liquid, to the density of the
liquid, and to the acceleration of gravity.
One of the important consequences of this law is that the,
pressure is independent of the lateral extent of the body of
liquid. Thus the pressure at the bottom of a well is the same
as the pressure at the bottom of a lake, providing the depth of
water in the lake is the same as that in the well.
THE HYDROSTATIC PARADOX
104. Let AB, Figure 66, represent a wide vessel communi-
cating with a narrow vessel CD by means of the tube BO. A
liquid poured into the vessel AB will rise to the same height
in the two vessels, that is to
say, the narrow column of "
liquid in OD balances the wide
column of liquid in AB. In-
asmuch as these columns of
liquid represent entirely dif-
ferent weights, the condition
of equilibrium seems para-
doxical. However, when we
remember that the pressure at
due to the column DC is
FiQ. 66. — Hydrostatic Paradox.
determined by the height of that column and the pressure at
B due to the column AB is determined by the height of the
column AB, it will be evident that the pressure at C acting
toward the left is equal to the pressure at B acting toward
the right if the heights of the columns AB and OB are the
same.
THE PEESSTJRE PERPENDICULAR TO THE WALLS
105. The pressure on the walls of a vessel due to a contained
liquid at rest is at every point perpendicular to the wall. This
is evident from the following consideration. Consider the
pressure at any point in the bottle represented in Figure 67.
Fig. 67. — Fluid Pressure
is Perpendicular to the
Walls.
MECHANICS
Let it be assumed that the pressure at
a certain point is not perpendicular to
the wall of the vessel. Then this pres-
sure may be resolved into two compo-
nents, one of which is perpendicular and
the other parallel to the wall. That
component which is parallel to the wall
will tend to move those portions of the
liquid whicli lie at this point along the
wall. It is assumed, however, that
the liquid is at rest. Therefore there
can be no component of the pressure
parallel to the wall; that is to sajs the
pressure must act perpendicular thereto.
THE PRESSURE THE SAilE IN ALL DIRECTIONS
106. Consider the vessel represented in P^igure 68. The
points A, B, O, and D are all at the same vertical distance from
the free surface of the liquid. There-
fore, according to Equation (40), the j-
pressure at each of these several I
points is the same. ,
THE PRINCIPLE OP ARCHIMEDES
/^
>_^
Fig. 68. — The Pressure at all
points in a Horizontal Plane
is the same.
107. A body submerged in a fluid
is acted upon at each and every part
of its surface by a pressure the value
of which is given by Equation (40).
It will be evident that for every small
area on the right-hand side of the body
there is a corresponding small area
of the same size on the opposite side of the body upon which the
pressure is the same. In a general way, therefore, it can be seen
that the pressures right and left will neutralize each other. If we
consider, however, points on the upper and lower surfaces of the
submerged body, it will be evident that the pressures acting from
above are less than those acting from below, since the height of
— ^-
— -
^^
—
—
--('
\ )~
J
—
wt
FLUIDS AT REST 113
liquid in the first case is less than that in the second. Therefore
the resultant of all of the force actions of a fluid upon a body-
submerged in it is an upward force action. This upward force
action can be sh^wn to be equal to
the weight of the fluid displaced.
Let A, Figure 69, represent a small B
portion of the liquid which fills
the vessel BO. Assuming that the
liquid is at rest, the forces acting
upon A are in equilibrium. There-
fore, since the weight Woi the por-
tion A is urging it downward, the
resultant of all the pressures upon
., . .. -I i ,, . . Fig. 69. — Archimedes' Friuciple.
the A portion due to the remaining
parts of the liquid must be an upward force action F exactly equal
to the weight of A. Imagine the A portion to be removed and
in its place some other body, for example, a stone having exactly
the same form and size as the A portion of the liquid removed.
Evidently the remaining portions of the liquid will act upon
this stone in exactly the same manner that they acted tipon
the A portion of the liquid which it replaces. The stone will,
therefore, be acted upon by two forces : First, the weight of the
stone acting downward. Second, an upward force action F equal
to the weight of the A portion of the liquid which the stone dis-
places. Thus the stone in this position will apparently weigh
less than it does outside of the liquid by exactly the weight of
the liquid which it displaces. This apparent loss of weight is
found in all submerged bodies. The following is a general
statement of the fact, and is known as the Principle of Archi-
medes. A body submerged in a fluid loses a portion of its weight
equal to the weight of the fluid displaced.
SUBMERGED FLOATING BODIES
108. Let it be imagined that in place of the stone referred to
in the last section the A portion of the liquid is replaced by
a body having not only the same size and shape as A, but having
also the same weight as A. Under these circumstances the body
will float in the liquid BC. The condition, therefore, for the
114
MECHANICS
floating of a submerged body is that it must displace a weight
of fluid equal to its own weight. , If its weight is greater than the
weight of the fluid displaced, it will sink. If its weight is less
than the weight of the fluid displaced, it will rise.
THE CARTESlAlSr DIVER
109. The apparatus shown in Figure 70 is used to illustrate
the eif ect of a change in density upon a submerged floating body.
J. is a small hollow vessel, for example,
a bottle or test tube, with its opening
at the bottom. It is nearly filled with
the liquid in which it is floating, but
contains enough air at the top to give it
an average density equal to that of the
liquid in which it floats. Under these
conditions it will tend neither to rise
Q — nor sink. If the pressure on the liquid
• is increased, the air bubble in the top
of the diver will be compressed, the
average density (and therefore the total weight) of the floating
body will be increased, and the diver will sink. If the pres-
sure is decreased, the diver will rise.
^-.caUJU
1
A
Fig. 70. — Cartesian Diver.
THE BALLOON
110. A floating balloon affords a good example of the appli-
cation of the principle of Archimedes. The combined weight
Fig. 71. — The Balanced Forces acting on a Balloon.
FLUIDS AT REST 115
of the gas bag, gas, car, engine, passengers, etc., tends to drag
the balloon towards the earth. This tendency to fall is opposed
by the buoyant force of the air, which, according to the prin-
ciple above stated, is equal to the weight of the air displaced.
See Figure 71. In other words, in order that the balloon may-
float it must displace its own weight of air. Since the density
of air is small, therefore, the balloon must be large if heavy, in
order that it may displace a sufficient weight of air.
In the dirigible balloon the weight is counterbalanced by the
air displacement as above described, and the balloon is moved
through the air by a propeller driven by an engine.
THE PRINCIPLE OF ARCHIMEDES AS APPLIED TO BODIES
FLOATING ON THE SURFACE OF A LIQUID
HI. A body submerged in a liquid of the same density will
float; in a denser liquid it will rise to the surface and a certain
portion will project. As it begins to project above the surface,
the liquid which it displaces becomes less and less. Evidently
there will come a time when, having projected itself a certain
distance above the surface, the weight of the liquid displaced
will be equal to the weight of the body. The body will,
under these conditions, be in equilibrium under the action of
its own weight and the buoyant force of the liquid.
The statement of Archimedes' principle may be modified to fit
the case of a body floating on the surface of a liquid as follows :
a floating body sinks in a liquid to such depth that the weight of
the liquid displaced is equal to the weight of the floating body.
THE MEASUREMENT OF DENSITY
112. The density of a body as above defined is the mass of
the body per unit volume, that is,
D = f (41)
To determine the density of a body therefore we may find its
mass, and its volume, and the quotient of mass by volume gives
at once the density.
116
MECHANICS
DENSITY OF SOLIDS
(a) In the case of a regular body the volume may be deter-
mined by measuring the linear dimensions of the body. From
these measurements the vol-
ume of the body may be
calculated. The mass of
the body maybe determined
by vi^eighing.
(J) In the case of an ir-
regular solid the volume
may be determined by " dis-
placement." This method
of determining volume is
as follows : A vessel AB,
Figure 72, is filled with
some convenient liquid,
for example, water, up to
the level of the spout as
shown. The irregular body 0, after being weighed to de-
termine its mass, is lowered into the liquid. The volume of
liquid which overflows is evidently equal to the volume of the
body O. This volume is measured by the measuring vessel D.
A
^
.1
r I
= /
^r
Fig. 72.
-Determining tlie Volume of an Ir-
regular Body.
DENSITIES OP LIQUIDS
(. The engineer
would say the vacuum in A is a " vacuum of h
inches," h being the height of the column OD
which measures the difference between the pres-
sure in A and that of the outside air.
The scientific expression for the vacuum is
B— hdg (Compare Equation 46).
p:
pascal's law
Fig. 88.— The
Measure of
a Vacuum.
130. The pressure applied at any point to a liquid in a closed
vessel is transmitted undiminished to every portion of the vessel.
This is known as the Law of Pascal. It has many important
applications, one or two of which will be given.
One of the applications of Pascal's Law is found in the
hydraulic press, a simple form of which is shown in Figure 89.
P G-H represents a large cylinder with a tight-
fitting piston p. IJ is a small cylinder also
/\ provided with a tight-fitting piston K, which
communicates with
the large cylinder
G-H by means of the
pipe JR. The cyl-
inder and pipe are
filled with water. Let
it be assumed that
a force / is applied
to the piston K. This will give rise to a pressure p, equal to
f
-, where a is the area of the piston K. This pressure is trans-
mitted to all parts of the communicating vessels. Therefore,
the total upward force on the piston P would be given by
Fig. 89. — The Hydraulic Press.
134
MECHANICS
F = p-A
in which A is written for the area of the large piston P. That
IS,
or
f=-£a
a
a
(47)
Let it be assumed that the area of piston p is 1000 times the
area of piston K, i.e.
- = 1000
a
.-. ^=1000 x/
That is to say, the application of the force/ at jfiTwill give rise
to a force 1000 times as great, tending to lift the piston P.
The hydraulic press may therefore be used for lifting large
weights, or for any other operation in which large forces are
required. The mechanical advantage (Section 76) is given by
the ratio of the piston areas. Thus in the example given above
the theoretical mechanical advantage is 1000.
In the practical form of the hydraulic press a force pump
similar to that represented in Figure 85 is made use of in place
of the cylinder FF as described above.
THE HYDRAULIC ELEVATOR
= J
H
FiQ. 90. — The Hydraulic Elevator.
131. Evidently, if the cylinder
CrM were sufficiently high the
arrangement shown in Figure 89
might be used as an elevator, say,
for transferring goods from one
floor to another in a warehouse.
This use of the apparatus as de-
scribed is quite common, the cyl-
inder GrH and its piston P being
arranged in some such manner as
that represented in Figure 90. If
the elevator is to be operated from
the ground floor, a deep hole is dug
FLUIDS IN MOTION 135
in the ground to accommodate the long cylinder GrH. The
piston P is a cylinder of metal which is more or less completely
immersed in the water in cylinder G-H, depending upon the
position of the elevator platform. The water for operating the
hydraulic elevator is pumped in through some conveniently lo-
cated pipe J. In some hydraulic elevators the car or cage is
attached to the piston P by means of an inverted set of fixed
and loose pulleys (Figure 53), and the necessity of having a
cylinder in length equal to the total distance through which
the elevator travels is obviated. This arrangement possesses
another advantage in that it is possible to secure a much more
rapid motion of the elevator car than the piston of the press
possesses.
pascal's la"w as applied to gases
132. The law of Pascal applies to gases as well as to liquids,
providing the transmitted pressures are measured after the
compressed gas has come to rest. For example, referring again
to Figure 89, the space GITUhelow the pistons might contain
air instead of a liquid. If, under these circumstances, the piston
IT is forced downward by an external force action, the compres-
sion which the force / brings about just beneath the piston IC
equalizes itself throughout the entire system of communicating
vessels ; and when this pressure has become equalized, the upward
pressure on P is exactly the same as that beneath the piston IC.
HYDRAULIC TRANSMISSION OF POWER
133. By application of the above principle, power may be
transmitted to a distance. Consider a long pipe filled with water
connected at one end to a force pump and at the other to a hy-
draulic engine. When the pump is operated, the engine at the
other end of the pipe will be fed with water under pressure very
much as a steam engine is fed with steam from a boiler. In this
manner power may be transmitted from the pump to the hydraulic
engine.
TRANSMISSION OF POWER BY COMPRESSED AIR
134. The transmission of power by compressed air is ex-
tensively employed in mines and factories. The air compressor
136
MECHANICS
h .
I
I
I
I Z ,
I
A ~^^
hi
maintains the pressure in the mains which are connected to
engines similar to steam engines, which may be located at any
distance from the compressor.
THE SIPHON
135. The siphon is a bent tube which is used for carrying a
liquid from a higher to a lower level over some intervening ob-
stacle. If, for example, it is desired to transfer the liquid in the
vessel A to the vessel B, Figure
91, this may be accomplished by
means of a bent tube O placed
in the position shown in the
figure. Assuming that the tube
C is filled with the liquid, its
action may be understood from
the following discussion: Con-
sider a portion of the liquid at
the highest part O of the tube.
This experiences a pressure urg-
ing it toward the right which is
B equal to the pressure acting on
the surface of the liquid in the
vessel A minus hdg, in which A is
the height of this portion of the
tube above the level of the liquid
in A. This same portion of liquid at C is urged toward the
left by a pressure which is equal to the pressure acting upon the
surface of the liquid in B minus h^^dg in which Aj is the height
of C above the free surface of the liquid in the vessel £. If the
pressure P acting upon the surface of the liquid in A is the
same as that which acts upon the surface of the liquid iri B, we
l^ave Pi = P~ hdg
p^ = P- \dg
where p^ stands for the pressure urging the C portion of the,
liquid to the right and p^ for the pressure urging this same
portion of liquid toward the .left. We have, therefore,
Pi-P-i= Ch - h')'^9
Fig. 91. — The Siphon.
FLUIDS IN MOTION
13'
h
/T^
where p^ — p^ is the excess of pressure acting to move the portion
C of the liquid in the tube toward the right. Thus at all times
there will be an unbalanced
pressure (Aj — }i)dg, urging that
part of the liquid which is at
the top of the tube toward the
right. In other words, so long
as Aj is greater than h, there will
be a flow of liquid from the
vessel A to the vessel B. Evi-
dently, when the liquid comes
to the same level in both vessels
so that Aj is equal to A, the flow
will cease. Furthermore, in the
event of \ becoming less than
A, the liquid will flow from the
vessel B into the vessel A.
The siphon cannot be used in
,, ,. , , . ^ Fig. 92. — The Siphon fails if M3= p.
case the distance h is greater
than the height of the barometric column. In this case the
condition of affairs would be as represented in Figure 92, in
which the liquid is represented as standing at the height of the
barometric column in each of the vertical portions of the siphon,
there being, of course, in the O portion of the tube a vacuum.
Thus, for mercury, the limiting value of A is 74 centimeters ;
for water about 34 feet.
A
1
I
I
I
(
- h
I
t
i 1 — ni
B
THE PLOW OF LIQUIDS
136. In any case of motion in a liquid, force action must be
present to account for that motion. If the velocity of the liquid
is changing, there is a force acting which is doing work in ac-
celerating the mass of the liquid which is moving. If the
liquid is moving with uniform velocity, there must be present a
force sufficient to overcome the frictional resistance to flow
encountered by the liquid. We may, therefore, conclude that
liquids move only under the action of sufficient force. A more
convenient way of stating the same thing is that a given portion
of liquid will be set in motion when it is acted upon by un-
138
MECHANICS
balanced pressure. Thus, for example, water flows in a pipe
only when there is a difference of pressure between the two
ends of the pipe. If the pressures at the two ends of the pipe
are equal, their tendencies to move water in the pipe neutralize
one another, and the water under the combined influence of the
two pressures remains at rest.
A
A/
EFFLUX. TOREICELLl'S THBOEBM
137. The velocity with which a liquid will escape through
an opening in the side of a vessel when acted upon by the
weight of the liquid alone is given by the following formula :
in which v is the velocity of the escaping liquid, g is the accel-
eration of gravity, and h is the height of the free surface of the
liquid above the opening through which the liquid is escaping.
This is known as the theorem of Tor-
ricelli. It may be demonstrated in
the following manner. Referring to
Figure 93, let AB represent a vessel
filled with a liquid and having a nar-
row opening at B through which the
liquid escapes as indicated. Let us
call the height of the free surface of
the liquid above the orifice h. This
distance is commonly re-
ferred to as the head.
Consider that which takes
place with reference to the
energy of the system when a portion of the liquid escapes
through the orifice B. Let it be assumed that in the interval
considered the level falls from A to Ay It will be evident
that every layer of liquid in the vessel will have fallen through
precisely this distance AAy Thus the body of liquid has lost
an amount of potential energy which is equal to the total weight
of the liquid contained in the vessel, multiplied by the height
AAy But this is equivalent to the potential energy which
would be lost by the laj-er AA^ in falling from the free surface
B
1
Fig. 93. — The Liquid escapes at
B with a Velocity =V2gh.
FLUIDS IN MOTION
139
of the liquid to the orifice B. We have, therefore, for the
potential energy lost by the liquid
Ep = mgh
in which m stands for the mass of the liquid in the layer AAy
On the theory of the conservation of energy, the kinetic energy
of the liquid which escaped in the interval under consideration
must be equal to this loss of potential energy by the layer AA.^,
but since the mass of liquid which has escaped is the same as
that of the layer AA^ we have, for the kinetic energy of the
liquid which escaped, rr _ i a
in which v is the velocity of efflux, that is, the velocity with
which the liquid escapes from the orifice. Equating these two
energy expressions, we have
|2 = rngh
Tgh (48)
\ mv
or
V =■
It will be noted that this expression is the same as that for
the velocity of a body which has fallen freely through the dis-
tance A, under the action of gravity. (Compare Equation 9.)
FRICTION HEAD
138. The loss of effective head due to the friction effect in a
pipe is called the friction head. A simple example will make
the meaning
of this expres-
sion clear. In
Figure 94, A
represents a
tank filled
with water.
Communicat-
ing with this
tank is a
narrow pipe
having three
orifices B^ C,
Fig. 94. — Friction Head.
140 MECHANICS
and D on the upper side as indicated. The water will spout
through the opening 5 to a height which is nearly equal to the
distance of the orifice from the free surface of the liquid. From
the orifice it spouts to a distance considerably less. Thus,
while the orifices B and C are apparently subjected to the same
head of water, the effective head is different. This is explained
by saying that a part of the head (pressure) is used in over-
coming the friction in the pipe between the orifices B and 0.
That this loss of head between the points B and Cis necessarily
present, is evident from the general statements which have been
made above with reference to the flow of liquids. If there
were no difference in pressure between the points B and C there
would be no flow of liquid between these two points. But if a
certain amount of pressure is used up in this way, evidently a
portion only of the original pressure will be available at the
orifice C.
. EFFLUX FEOM AIR-TIGHT SPACES
139. In the discussion given in Section 137, the assumption
was made that pressure on the upper surface of the liquid in
the vessel AB, Figure 93, remains constant and that the liquid
escapes into a region B the pressure of which is the same as
that at A. Evidently the velocity with which the liquid escapes
under these assumptions depends upon the head of the liquid
alone and is independent of the pressure above referred to. If,
however, the pressures at A and B are different, this difference
in pressures must be taken account of in the discussion of the
efflux. It is conceivable that the pressure at B acting from
without might be larger than the pressure acting upon the sur-
face A by just that amount (Jidg'), which is due to the weight
of the liquid. Under these circumstances there would be no
flow of the liquid through the orifice. This condition of affairs
is actually reached in case the efflux takes place from an air-
tight vessel. Consider the flow of liquid from the vessel rep-
resented in Figure 95. The conditions are supposed to be the
same as those represented in Figure 98, except that the vessel
is closed at the top, the closed space above the liquid containing
air at a pressure py Let it be assumed, to begin with, jOj is
FLUIDS IN MOTION
141
Fig. 95. — Intermittent Efflux
from Air-tight Space.
equal to p^, the pressure on the outside at the orifice. Then
the velocity with which the liquid begins to flow through the
orifice is given by Equation (48).
As soon, however, as an appreciable
amount of liquid has passed the orifice,
p^ becomes less than p^, since the air
inclosed must expand to fill that space
which is emptied by the liquid which
flows out of the vessel. Expanding
into the larger volume this air will,
according to Boyle's Law, have a lower
pressure. To determine whether or
not under these circumstances the
liquid will actually flow from the
orifice, we have the following consid-
erations. The pressure acting upon a
given body of the liquid at the orifice,
which tends to move that volume to the right, is jOj + hdg.
The pressure acting upon the same body of liquid which tends
to move it toward the left through the orifice is p2. Evidently,
therefore, when , ,
Pi + ridg=p^
the liquid at the orifice will have no tendency to flow in either
direction. In case , ,
pj^ + hdg>p^
there is an unbalanced pressure urging the liquid at the orifice
toward the right. In case
Pi + Mg < p^
the liquid in the orifice will move toward the left and air will
pass into the vessel from the outside.
Under the conditions assumed, therefore, the liquid will escape
from the vessel at first with a velocity v=^2gh. Afterwards
V will gradually become less, finally reaching the value zero.
It usually happens, however, that the kinetic energy of the
liquid which is flowing towards the orifice carries the efflux
beyond this point, so that
p^ + Mg < p^
142 MECHANICS
Hence, when the stream stops, air will pass into the vessel.
This air will increase p^ so that again
p^ + hdff>p2
and efflux will once more begin, and so on. It is evident that
if it is desired to secure an uninterrupted and steady flow of
the liquid from the orifice, it will be necessary to provide some
means for maintaining p^ constant. This is conveniently done
by making a small hole in the top of the vessel as at A, through
which the air may be allowed to flow in as the liquid escapes
through the orifice.
THE HYDRAULIC RAM
140. According to Newton's second law of motion any body
of mass m having acceleration a is necessarily acted upon by a
force / = ma. If the body is increasing in velocity, the force
producing the acceleration is in the same direction as the veloc-
ity. If the velocity of the body is decreasing, the force which
acts to give the body negative acceleration, that is to say, which
tends to stop the body, is acting in a direction opposite to that
of the velocity of the moving body. This statement will of
course hold for liquids equally as well as for solids. Therefore
a quantity of liquid of mass m moving with a velocity v which
is suddenly brought to rest must be acted upon by a force which
is given by the product of the mass of the liquid stopped and
the acceleration which it experiences. Since action and reac-
tion are equal, we may say that the liquid in stopping exerts
upon the restraining vessel a force action or pressure which is
proportional to the mass of the moving liquid and the accelera-
tion which it has while stopping.
This effect is taken advantage of in the "hydraulic ram,"
the operation of which will be understood by reference to Fig-
ure 96. ^ is a reservoir containing water ; BO is a, pipe of large
dimensions which leads from the reservoir to the point C.
Evidently if the conditions were as represented in the diagram,
the liquid in the reservoir would flow along the pipe JBO, escap-
ing past the valve D. Imagine that D is so adjusted that when
the liquid acquires a sufficient velocity it will carry the valve
along with it, that is to say, it will lift the valve and close the
FLUIDS IN MOTION
143
^^[
opening at B. This, of course, brings the column of liquid BO
suddenly to rest. This will give rise to the " water hammer "
or " water ram " effect referred to above, that is to say, as the
liquid is suddenly checked in its motion it will
exert a large pressure upon the walls of the /'/' — S^T
pipe at DC. Imagine that a vertical tube EF
is attached as indicated. This tube
being enlarged at E is provided with a
valve opening upward. At the moment g
at which the water ram occurs a small
portion of the liquid in BO will be
forced up into EF. This
will take place even though
the head of water in the pipe
EF is greater than that in
the pipe BO. As soon as the
flow of water in the pipe BO
is checked the valve B will
open by its own weight and
the operation will be re-
peated. Each time this op-
eration is repeated a portion
of the liquid which flows
from the reservoir A along the pipe BO is lifted to the reser-
voir Cr. Considered from the standpoint of the theory of the
conservation of energy it will be ixnderstood that the quantity
of water elevated in the pipe EF must be smaller than the
quantity which flows along the pipe BO. If w represents the
weight of water raised through the pipe EF, h represents the
height through which it is raised, and TF"is taken to represent
the weight of water which flows along the pipe BO, and JS the
distance through which it falls, we have, evidently,
wh • t
Spot by Hot Iron. C/, t igure 106, represent a piece of
cloth containing a grease spot 6r. If
a hot iron / is brought near, the surface tension of the grease
will be reduced on the heated side, and the unbalanced surface
tension of the colder parts will draw the grease to the opposite
side of the cloth, whence it may be removed by a blotter B.
Problems
1. How much does the pressure on the inside of a soap bubble 20 cm. in
diameter exceed that on the outside? Assume surface tension is 80 dynes
per centimeter.
2. Two soap bubbles are connected by means of a glass tube. The
diameter of one bubble is o cm., that of the other is 2 cm. AVhich bubble
will increase in size?
3. What is the pressure due to surface tension in a drop of water 2 mm.
in diameter? Assume surface tension of water to be 81 dynes/cm.
4. Th3 surface tension of pure water is 81 dynes/cm. How far will
water rise in a capillary tube ^ mm. in diameter ?
5. What is the surface tension of a liquid which rises 25 cm. in a capil-
lary tube ^-^ mm. in diameter. Density of the liquid = 0.8.
6. If alcohol and water are made to form drops from the end of the same
pipette, it will be found that the water drops are larger than those of alcohol.
Explain.
PART II
HEAT
HEAT
CHAPTER XIII
THE NATURE OF HEAT
151. We have seen that in the operation of any machine or
mechanical device a certain amount of mechanical energy is
always transformed into heat ; that is to say, from the friction
effects, which are unavoidably present in such devices, heat
is developed whenever such devices are operated. We are
tlius led to the conclusion that heat is a form of energy.
It is thought that the molecules of a hot body are in a state
of rapid vibration, and that the hotter the body, the more rapid
is this vibratory motion of its molecular parts. Heat is there-
fore defined as the energy possessed by a body in virtue of the
vibratory motion of its molecular parts.
TEMPEEATTJRB
152. The temperature of a body should be carefully dis-
tinguished from the quantity of heat which it possesses. While
it is true, in many cases, that hot bodies possess relatively large
amounts of heat, it is equally true that a cold body may actually
contain more heat than a hotter body. The general notions
which we have of temperature are derived from our temperature
sense. We determine by feeling a body whether it is hot, warm,
cool, or cold, and it is true that, in a general way, we are enabled
by this means to measure temperature roughly. However, our
temperature sense may very easily lead us into error in esti-
mating the temperature of objects. An experiment which may
be performed for the purpose of illustrating this point is the
following;
Let the right hand be held for a moment in cold water, the
left meanwhile being held in hot water. Then let the hands
159
160 HEAT
be dipped together into a vessel of tepid water. To the hand
which was in the cold water the tepid water will seem hot,
while to the hand which was placed in the hot water, the tepid
water will seem cold. This experiment gives us a hint as to
the real significance of temperature sensation. We experience
the sensation of cold when heat passes from the body to surround-
ing objects. We experience the sensation of heat when heat
passes from surrounding objects to the body. Furthermore, the
degree in which the sensation is felt depends upon the rapidity
with which heat passes to or from the body. The various ob-
jects in a room on a cold morning will appeal to one as if
some were colder than others. If the hand is placed upon a
woolen blanket, and then upon a piece of wood, and then upon
a piece of metal, the metal will seem colder than the wood, and
the wood colder than the woolen blanket, even though the three
objects are at exactly the same temperature. The explanation
is that the heat flows from the hand to the iron and off through
the iron more readily than it does through the wood and more
readily through the wood than it does through the woolen cloth.
Temperature may in a rough way be likened to pressure or
" head " in hydraulics. Heat tends to flow from a region of
high temperature to a region of low temperature, just as water
tends to flow from a region of high pressure to a region of low
pressure. Reasoning from this principle that a flow of heat
can only take place between two bodies having a difference of
temperature, we may say that two bodies have the same tempera-
ture if when they are brought into contact there is no interchange
of heat between them.
THE EFFECTS OF HEAT
153. When heat is imparted to a body, one or more of the
following effects are observed :
1. Rise of temperature.
2. Change in size.
3. Change of state.
4. Chemical change.
5. Electric effect.
THE NATURE OF HEAT 161
The temperature of a body usually rises when heat is im-
parted to it, although this is by no means the invariable rule.
For example, when water is boiling freely, it makes no differ-
ence how much heat is imparted to it or how rapidly, it is not
possible to change the temperature of the boiling water until it
is entirely boiled away. In the same way the temperature of a
mixture of pure ice and water will remain constant, no matter
how rapidly heat may be imparted to the mixture, until all of
the ice is melted. In the first example cited, the heat imparted
is used in evaporating the water or changing it over into the
form of steam. In the second case the heat imparted is used
in melting the ice. It is assumed, in the examples given, that
the pressures acting remain the same throughout.
The general rule as to change in size is that the hotter a
body becomes, the greater is its volume. There are some ex-
ceptions to this rule. For example, water is found to decrease
in size when it is heated from the freezing point to a few
degrees above the freezing point (0° C. to 4° C).
It is a matter of common observation that by imparting heat
to a solid it may be converted into a liquid, as, for example,
in the melting of ice. Also, that by imparting heat to a liquid
it may be converted into a vapor, as for example, in the gene-
ration of steam from water. This is commonly referred to as
a change of state.
Heat facilitates chemical change. One of the best examples
of this effect is in the burning of coal. Before the carbon of
the coal will combine with the oxygen of the air it is necessary
to apply heat. Once the action is started, the heat developed
by this chemical combination is sufficient to maintain the chem-
ical action. Hence chemical change is one of the most im-
portant effects of heat.
It is found that under suitable conditions an electric current
may be caused to flow in a wire by heating it, hence this is in-
cluded among the effects of heat.
THERMOMETERS
154. A thermometer is a device for measuring temperature.
Referring to the last section and noting that the first two
162 HEAT
effects of heat mentioned are a rise in temperature and in-
crease in size, the thought will very naturally occur to one
that it might be possible to measure the rise in temperature of
a body by taking note of its increase in size. The increase in
size of a body would, of course, be an accurate measure of the
rise in temperature provided the one were proportional to
the other. The increase in size of certain bodies is quite
accuratelj'^ proportional to the rise in temperature, such bodies
naturally offer themselves as suitable thermometric substances.
THBBMOMETRIC SUBSTAJSTCBS
155. Solids. In certain devices used for the measurement
of temperature the expansion of a metal with a rise of tempera-
ture is taken advantage of. By knowing the increase in length
of a bar as it is heated, its rise in temperature may be esti-
mated. This form of thermometer is not adapted to ordinary
temperature measurements for the reason that the increase in
length of a bar of metal is very small as compared with its
length, so that very long bars would have to be made use of in
order that the increase in length might be readily observed and
measured.
Liquids. Certain liquids are found to be well adapted to
use as thermometric substances. Alcohol increases in size
quite uniformly with an increase in temperature, so that by fill-
ing a suitably shaped glass vessel, having a large bulb with a
long narrow stem, with alcohol a very convenient thermometer
is secured. Alcohol possesses the advantage of freezing only
at very low temperatures, so that it may be used in cold
climates and in very cold weather.
Mercury is perhaps the best liquid thermometric substance
known. It possesses several distinct advantages as follows :
It expands quite uniformly over a large range of temperatui'e
(- 40° C. to -I- 330° C). It is opaque, so that a fine thread of
mercury in a glass tube is easily \^sible. It does not wet
glass, and hence does not stick to the walls of the containing
vessel. Its expansion for a given rise in temperature is rela-
tively large.
THE NATURE OP HEAT 163
The last topic mentioned in the preceding paragraph is of
some importance, since it will be quite evident that the apparent
expansion of mercury in a glass bottle is in reality the dif-
ference between the absolute expansion of the mercury and
that of the bottle. If, therefore, the bottle and contained
liquid expanded equally with a given rise of temperature, the
apparent expansion would be 0, that is to say, a thermometer
made up in this manner would give the same reading indepen-
dent of the temperature to which it was subjected.
Gases. A gas lends itself very readily for use as a thermo-
metric substance. It is found to expand quite uniformly
through great ranges of temperature. A gas used in this
manner is usually inclosed in a glass or porcelain bulb with a
long narrow stem, the stem being graduated in the customary
way.
THEEMOMETEE SCALES
156. There are two thermometer scales in common use.
That which is most widely used and quite universally adopted
for scientif].c purposes is known as the Centigrade scale. A
thermometer to have such a scale is graduated in the following
manner. The thermometer is placed in melting ice and a
scratch is made upon the stem at the point at which the column
of mercury comes to rest under these conditions. This point
is marked 0° C. The thermometer is next placed in boiling
water, or, better still, in a closed space just above boiling water,
the pressure upon the surface of the boiling water being that
which is known as the standard atmosphere (76 centimeters of
mercury). The point to which the mercury rises under these
circumstances is marked 100° C. The distance between the
".ice point" and the "steam point" determined in this manner
is divided into 100 equal parts. These parts are known as
Centigrade degrees.
Another thermometer scale which is very widely used for
domestic purposes is known as the Fahrenheit scale. To grad-
uate a thermometer according to this scale the ice point is
marked 32° F. and the steam point 212° F. The distance
between the two points in this case is divided into 180 equal
164
HEAT
212 Steam
parts, and these are known as Fahrenheit degrees. In Figure
107 the two scales are compared. To reduce a temperature as
_ p. read upon one of these thermome-
ters to the corresponding reading
upon the other, we have the follow-
ing general considerations. Let it
be assumed that the mercury stands
on the line AB, and that its read-
ing corresponds to c degrees on the
centigrade scale. It is required to
find the corresponding reading on
the Fahrenheit scale. Call this
reading /, then the reading on the
same scale measured from the ice
point will be /— 32. Then since
100 Centigrade degrees =
180 Fahrenheit degrees
c:/- 32 = 100: 180
c _/-32
100
or.
32.. Ice
180
. = !(/- 32) (52)
and
/=-!
c + 32 (53)
Fig. 107. — Centigrade and Fahren-
heit Thermometers.
From this relation a reading on
either scale may be readily con-
verted into the corresponding read-
ing on the other.
SIMPLE AIK THERMOMETERS
157. The air thermometer is used in scientific work requir-
ing temperature measurements of great accuracy or wide range.
When used for such purposes, the air in the bulb is kept at
constant pressure and the changes in volume observed, or at
constant volume and the changes in pressure observed.
In Figure 108 are shown two simple forms of air ther-
mometers. In A the inclosed air is separated from the out-
THE NATURE OP HEAT
165
side air by means of an index of mercury or other liquid. A
motion of the index toward the bulb indicates a fall in temper-
ature of the inclosed air. If the bulb of this instrument is
grasped by the hand, the
index will move away
from the bulb, showing
the expansion of the in-
closed air due to the heat
received from the hand.
-B is a glass bulb at-
tached to a long tube
which dips into a liquid.
The liquid is caused to
stand in the tube at some
convenient height h
above the liquid in the
vessel below. If the bulb
is heated, the pressure of
the inclosed air is in-
creased and the liquid column falls. If the air in the bulb is
chilled, the column rises. Evidently both forms of air ther-
mometer shown in Figure 108 are dependent upon the pressure
of the outside air. They can therefore be used as temperature
indicators only so long as the outside air pressure is constant.
■"?•■
Fig. 108. — Simple Air Thermometers.
LINEAR EXPANSION
158. If a bar of metal is heated, it will increase in length,
the increase in length being proportional to the rise in tempera-
ture and to the original length, that is,
increase in length rx^L^t
= aLff
in which L^ is the original length, t is the rise in temperature,
and a is a constant for that particular substance of which
the bar is made and is known as the "coefficient of linear
expansion."
The coefficient of linear expansion of any substance is the
increase in length for each unit of length for each degree of
166 HEAT
rise in temperature. This definition comes at once from the
above expression if we make L^ and t each equal to unity.
The final length of the bar Lt is therefore given by the fol-
lowing equation :
or A = io(l + «0 (5i)
Zq should be taken as the length of the bar at 0°. The in-
crease in length in a bar as it is heated from t° to t^ would be
io(l + at^)-L^{l+a.e)=aL^(t^- t)
Evidently Equation (54) may be used for the determination
of the coefficient of linear expansion. In this case the initial
and final lengths are accurately measured and the rise in tem-
perature noted by means of suitable thermometers. Then a
is calculated from the equation. In case a is known, Equa-
tion (54) may be used for estimating the length which a bar
would have at a temperature t, its length at 0° C. being known.
Equation (54) is based upon the assumption that the increase
in length of a bar of metal is strictly proportional to its rise in
temperature. This is not exactly true. The increase in length
is very nearly proportional to the rise in temperature, and there-
fore for most practical purposes the relation may be regarded
as exact. The error resulting from the use of Equation (54) is
negligible if a has been determined from this relation b}' observ-
ing values of ig and L^ with t comparable in value with the
range of temperatures over which the equation is to be used.
Such a determination of a gives, of course, the average value of
the coefficient between 0° and t°.
Table of Coefficients of Linear Expansion
Substance a
Copper 0.0000178
Iron 0.0000116
Glass 0.0000085
Platinum 0.0000085
Lead 0.000028
Tin 0.000022
THE NATURE OP HEAT
167
B
APPLICATIONS
159. The fact that different solids expand at different rates,
that is to say, that different solids have different coefficients of
linear expansion, is made use of in various v^ays. For example,
in the compensated clock pendulum this principle is employed
for maintaining the length of the pendulum constant. We
have seen that the time of vibration of a pendulum varies as
tlie square root of its length (Section 51). Evidently, there-
fore, any change in the length will be accompanied j->
by a change in the period of the pendulum. If
the pendulum is made longer, it will vibrate
more slowly. If it is made shorter, it will
vibrate more rapidly. It will therefore be evi-
dent that a pendulum clock in order to keep
good time would have to be provided with a
pendulum of invariable length. The pendulum
which is not compensated for temperature effects
grows longer in warm weather and shorter in
cold weather. Thus the clock runs too fast in
cold weather and too slow in warm weather. To
obviate this difficulty the " gridiron pendulum "
is sometimes employed. This pendulum is rep-
resented diagrammatically in Figure 109. Let
represent the point from which the pendulum
is suspended, and a a rod of some suitable metal
supporting a crossbar AB. Suspended from this
crossbar are two bars c and d of the same metal
as that used in a. Supported by the bars e and
d are the crossbars Q, D. Standing upon these
crossbars are two bars e, /, of a metal different fig. io9. — The
from that used in a. From the upper end of Gridiron Pen-
these bars hangs the bob as indicated in the dia-
gram. If the lengths of the bars used are properly chosen, it
will be found that the distance OE of the center of mass of
the pendulum from the point of support will remain unchanged
through wide ranges of temperature. Evidently, the lengths
of the rods a, c, d, and g, which may be, for example, of iron,
168
HEAT
must be so related to the lengths of e and /, which may be,
for example, of brass, that the increase in length of a+e +^
is equal to the increase in length of e or/.
Another application of this principle is found in a form of
thermostat which is devised for giving a signal when the tem-
perature of the room in which it is placed rises above a certain
value. In this device two bars of unlike metals are riveted
together and form a single straight bar at ordinary tempera-
tures. When heated the bar becomes curved, since the metal
having the larger coefficient will expand more than the other.
This bending of the bar is made to close an electric circuit and
ring a bell, and in this manner give evidence that a rise in
temperature has taken place.
PRINCIPLE OF THE GRIDIRON PENDULUM
160. The principle of the gridiron pendulum may be demon-
strated by means of the apparatus illustrated in Figure 110. It
consists of two glass tubes Gi and g and a zinc tube Z con-
nected by short pieces of rubber tubing at R and r. Gi and Z
are firmly clamped together at R by means of a wooden
Fig. 110. — Illustrating the Principle ol the Gridiron Pendulum.
block, and Z and g are clamped together in similar manner at
r. Cr is rigidly clamped at (7 to a support attached to the
table. At 5 is a block of wood fastened to g, which rests
upon a small roller, e.g. a knitting needle, lying upon the
table. To the end of the roller is fastened a pointer a, which
serves to indicate any motion of the small roller under the
block B. A small boiler is connected by means of a piece
THE NATURE OF HEAT 169
of rubber tubing to (} at S. When steam is generated in
the boiler and caused to flow through the tubes, the following
results are observed. As the steam flows through G and
causes a rise in its temperature, it expands and moves the
system RB to the right. The pointer moves from a to h.
When the steam passes into Z and causes it to expand, that
portion of the system lying beyond Z, that is, rB, is moved to
the left. The pointer moves from h to some such position as
0. Finally, when the steam enters g, the expansion in this
tube will move the block B once more to the right, and the
pointer will move from o toward a.
Evidently the expansion effects of Gr and g are in the same
direction and opposed to that of Z. It follows, that if the
lengths of Z, G-, and g are properly chosen, the net result will
be zero, that is, the block B will be at the same distance from
when the three tubes are equally heated that it was when
the tubes were cold. In the experiment this will be indicated
by a return of the pointer to the position a when steam is
flowing freely through all of the tubes.
Now the coefficient of linear expansion for glass is 0.0000085
and for zinc 0.000029. Call the length of the zinc tube L.
The increase in length of the zinc tube is therefore
0.000029 X Lt
in which t is the rise in temperature of the zinc tube when
steam flows through it.
The increase in length of the glass tubes is
0.0000085 X L^t
in which ij represents the added lengths of 6r and g, and t
their rise in temperature.
Now if the distance OB is to remain unchanged, the increase
in length of the zinc tube must equal the increase in length of
the glass tubes, that is,
0.000029 X ii= 0.0000085 x L^t
or since the temperature rise is the same for all tubes
290 i = 85 ij
i.e. ij = 3.4 L (approximately)
170 HEAT
That is, the lengths of 6r and ^ together must be 3.4 times
that of Z.
The arrangement of tubes in this apparatus corresponds to
that of the rods in a gridiron pendulum. Q corresponds to the
point of suspension in the pendulum and the block B to the
pendulum bob.
CUBICAL EXPANSION. CHARLES' LAW
161. If a given volume of any substance has its temperature
increased, it is found in general to increase in volume. The
increase in volume is proportional to the original volume and to
the rise in temperature, that is,
increase in volume oc V,^
in which T'g is the original volume (at 0° C), t is the rise in
temperature, and /8 is the coefficient of cubical expansion for
that particular substance.
The coefficient of cubical expansion is the increase in volume
for each unit of volume for each degree of rise in temperature.
This definition comes from the above expression at once if we
make Vq and t each equal to unity.
The final volume Vt is therefore
r, = Fo(l + ^0 (55)
Since the increase of the volume of a body depends upon the
increase of its three linear dimensions, it follows, of course, that
different substances, for example, brass, iron, glass, etc., have
different coefficients of cubical expansion. (See values of a,
Section 158.)
It is found that the coefficient of cubical expansion for all gases
at constant pressure is the same and has a value of
= 0.00367
This fact was first discovered by Charles and is commonly
known as Charles' Law. It is also sometimes known as the
Law of Gay-Lussac, the name of the man who first put the law
to experimental test.
THE NATURE OP HEAT
171
THE GENERAL LAW FOE THE EXPANSION OF A GAS WITH
CHANGE IN PBESSUEE AND TEMPERATURE
162. It will be remembered that in stating Boyle's Law
(Section 122) the assumption was made that the temperature
of the gas remains constant and the statement was made that
Boyle's Law would apply only under
the condition of no change in tempera-
ture. Boyle's Law, therefore, specifies
the manner in which the volume of a
gas changes with pressure when its
temperature remains constant. Charles'
Law, on the other hand, specifies the
manner in which the volume changes
with the temperature so long as the
pressure remains constant. It is con-
venient to have
V
t'
V.
0°
Po
V,
t=
Fig. 111.
- Illustrating the General Law of the Ex-
pansion of Gases.
these two laws com-
bined in one general
statement. This is
readily obtained in
the following man-
ner. In Figure 111 let the rectangle FJ, represent the volume
of a certain mass of gas, the pressure of which is p^ and the
temperature 0° C. Let it be assumed that the gas is heated
until its temperature rises to t° C, the pressure remaining the
same. Accordinsr to Charles' Law the volume will increase
such that Y^ = Fo(l + /30
in which V^ is the new volume. Let it be assumed that the
pressure is now decreased, the temperature being held constant.
Then according to Boyle's Law there will be a still further ex-
pansion of the gas such that the new volume is given by the
following equation : .^ „
V-p= FjPo
In which Via the new volume and p the new pressure. Elimi-
nating V■^ by combining the expression for V^ in terms of Fq,
and that for Fin terms of Fj, we have at once
j^=£flij) (1 + ^t)
P
(56)
172
HEAT
1.00-
f-
0°-
1)
■D
ro
1-
cp
c
(U
(J
-273^
■373
-273++°
- 273°
o
10
<
This equation expresses the volume of the
gas under the new temperature and pressure
in terms of its volume under the old con-
ditions of temperature and pressure. This
is known as the general law of the expan-
sion of a gas.
THE ABSOLUTE ZERO
163. Since /3 for all gases is equal to -^j-^.
Equation (56) may be written in the fol-
lowing form :
:£a
V,
■112,
(273 +
Fig. 112. — Scale of
Absolute Temper-
atures.
The fraction ^^ " is a constant, call it M, and
273
the factor in parentheses represents evidently
a temperature as measured from a point 273°
below zero on the Centigrade scale. Let us
write T=27?j+t
Therefore, we have
pV=RT (57)
This new zero, 273° below 0°C., from which T
is measured, is called the absolute zero. In
Figure 112 the Centigrade and absolute scales
are compared.
On the theory which assumes that in hot
bodies the molecular parts are in a state of
rapid vibratory motion, while in cold bodies
this motion is less marked, it follows that if a
body were cooled down to 273° below zero on
the Centigrade scale this molecular vibratory
motion would cease. That is to say, at this
temperature the body would have no heat.
This result is reached as follows. From the
equation just written it is evident that the
THE NATURE OF HEAT
173
pressure of the gas on the walls of the containing vessel is
proportional to the absolute temperature. That is to say,
pacT
upon the assumption that the volume of the gas is constant.
But the pressure of a gas is supposed to be due to the impact
of its molecular parts as they vibrate to and fro. The pressure
therefore can only be zero when the vibratory motion ceases
entirely. Therefore the temperature 273° below zero on the
Centigrade scale is that temperature at which the molecular parts
of a body are entirely without vibratory motion. It is therefore
the lowest possible temperature.
The absolute zero of temperature has never been reached ex-
perimentally. It is, however, interesting
to note in this connection that a tempera-
ture of — 268.5 on the Centigrade scale
has actually been attained.
THEORY OF THE AIR THER-
MOMETER
r\
(b
164. The essential parts of
a constant volume air thermom-
eter are shown in Figure 113.
The bulb B, which may be made
of glass or porcelain, contains
the thermometric substance, dry
air or hydrogen. This bulb is
connected by a tube of small bore to the
U-tube AQ partially filled with mercury.
The upper end of the tube AQ is, sealed
and contains a vacuum, and the pressure
of the gas in B is therefore measured by
the difference in height h of the mer-
cury columns in A and C. By raising
or lowering the vessel F, the mercury Fig. lis.
in the tube AC may be brought or held
to a marked point in A, and the volume of the gas in the bulb
is in this manner kept constant.
— Air Thermom-
eter.
174 HEAT
Now from the general law of expansion of gases,
pv= BT
it follows that if the volume of any body of gas v is kept con-
stant, its pressure will vary as its absolute temperature, i.e.
pcxiT
so that if the pressure of the gas at an absolute temperature
T is p, its pressure at an absolute temperature T' will be p',
such that rp
p'^Y'
or T' = T-^
P
The air thermometer may be used as follows : First sur-
round the bulb with melting ice, temperature 273° absolute,
and by adjusting the height of V bring the mercury to A and
observe the difference in height h of the mercury columns.
Then place the bulb in the region of the temperature T which
is to be measured, and when the mercury is again brought to
A, observe the difference in height of the mercury columns as
before. Call this difference A' . Then
T'=273.-
h
EXPANSION OF LIQUIDS
165. The determination of the expansion of a liquid by
ordinary methods is complicated by the fact that the contain-
ing vessel expands and contracts with rising and falling tem-
perature, so that accurate knowledge of the change in volume
of the vessel with changing temperature is necessary, in order
that proper allowance for it may be made. In other words,
the observed expansion is an apparent expansion which depends
upon the expansion of the containing vessel as well as upon the
absolute expansion of the contained liquid.
Regnault devised a method for determining the absolute
expansion of a liquid, which is independent of the expansion
of the containing vessel. The apparatus used in this method
consists essentially of two glass tubes, AB, connected as shown
THE NATURE OF HEAT
175
in Figure 114, the connecting tube being of small diameter.
These tubes are filled with the liquid to convenient heights.
Let it be assumed that the tube A is kept at the temperature
of melting ice, and the tube B at 'a temperature t° C. Let h^ be
the height of the column in A and Aj the height of the column
in B. Then
since the pressures right and left balance one
another.
-•. ^0^0= htdt (a)
Now from Charles' Law we have
V,= V,il + ^t}
or since
d
f=f<^-«
flint, \<
d ^"
UlLOi U J.D«
' l + /3<
Combining (a)
and (6), we obtain
^ _ A, - A„
(5)
(58)
D
Fig. 114. — Regnault's
Apparatus for Cubi-
cal Expansion.
from which /3, the coefficient of cubical expansion, may be cal-
culated from observed values of Ag, Aj, and t.
The value of /3 obtained from such calculation is the average
value of the coefficient of cubical expansion throughout the
temperature range 0° to t°.
THE EXPANSION OF WATER
166. Water is remarkable in that it forms an exception to
the general law that liquids expand with a rise of temperature.
If water at the temperature of melting ice is heated, it contracts
as its temperature rises to about 4° C. At this point it reaches
its minimum volume, and therefore its maximum density. If
its temperature is raised above this point, it expands, at first
176
HEAT
slowly and then more rapidly as its temperature rises, until the
boiling temperature is reached.
The curve shown in Figure 115 exhibits the changes in vol-
unie which a given mass of water undergoes when its tempera-
ture rises from 0° C. to
100° C.
THE THERMO COUPLE
167. One of the effects
of heat as given in Sec-
tion 153 is the electric
effect. This effect may
be briefly described as
follows: If two dissimi-
lar metals, for example,
copper and iron, are
joined together in such
manner as to form a
complete metallic cir-
cuit," and the two junc-
tions of such are at
different temperatures,
-20
Fig. 115. — Curve showing Change in Volume of a
Given Mass of Water with Change of Tem-
perature.
it is found that there is present in the circuit, because of this
difference of temperature, an electric current. Such a circuit is
represented in Figure 116. The different parts of the circuit
are indicated in the
diagram. The points
marked " Hot " and
"Cold" are the junc-
tions, that is to say, the
points of contact, be-
tween the dissimilar
metals, (r is a galva-
nometer, that is, an in-
strument for detecting
the presence of the elec-
tric current. In such a Cold not
circuit there is no ten- Fig. lie. — The Thermo Couple.
THE NATURE OF HEAT 177
dency for an electric current to flow so long as the junctions
are at the same temperature. If, however, there is a difference
of temperature between the junctions, a current will flow in
the circuit, and the magnitude of the current is proportional
to the difference of temperature between the junctions. It
will be evident that such a device might be used for measuring
differences of temperature.
Problems
1. Reduce to Fahrenheit readings the following Centigrade temperar
tures: 40°, 21°, -20°.
2. Reduce to Centigrade readings the following Fahrenheit tempera-
tures : 110°, 32°, 0°.
3. At what temperature will Fahrenheit and Centigrade thermometers
give the same reading ?
4. A platinum wire is 5 m. long at 0° C. What is its length at 100° C. ?
5. An iron pipe is 5 m. long at 20° C. What is its length at 0° C. ?
6. The length of a copper wire at 30° C. is 20 m. What is its length at
10° C. ?
7. The area of a sheet of iron is 15 sq. m. at 0° C. What is its area at
40° C. ?
8. A mass of gas at 0° C. occupies 150 cc. What volume would it
occupy at the same pressure if its temperature were increased to 100° C.V
9. A mass of gas at 0° C. and a pressure of 760 mm. of mercury oc-
cupies a volume of 500 cc. rind its volume when the pressure is increased
to 1000 mm. of mercury and the temperature to 40° C.
10. A given mass of gas has a volume of 400 cc. when subjected to a
temperature of 20° C. and a pressure of 100 cm. of mercury. At what tem-
perature will it have a volume of 500 cc. at a pressure of 90 cm. of mercury?
CALORIMETRY
CHAPTER XIV
THE UNIT OF HEAT
168. Although heat is a form of energy and may therefore
be measured in ordinary units of energy, it is convenient for a
good many purposes to employ a unit which is based upon the
effect of heat in raising the temperature of water.
The calorie is the unit of heat in the c. g. s. system and is de-
fined as the quantity of heat required to raise the temperature
of one gram of water from 4° to 5° on the Centigrade scale.
The British Thermal Unit (B. T. U.) is the unit of heat in
the f . p. s. system and is defined as the heat required to raise
the temperature of one pound of water from 60° to 61° on the
Fahrenheit scale.
SPECIFIC HEAT
169. The specific heat of any substance is the quantity of heat
required to raise the temperature of one gram of the substance
one degree.
From the definition for the unit of heat given in the last
section it follows that the specific heat of water at 4° Centi-
grade is unity. The specific heat of water at other tempera-
tures is very nearly unity. In fact, it is so near unity at all
temperatures between the ice point and the steam point that for
most purposes in heat measurement this value may be assumed
to be correct. Strictly speaking, however, the specific lieat of
water is unity only at 4° C.
The specific heat of a substance in general depends upon its
temperature. Therefore, in specifying the specific heat of a
substance, to be rigidly exact we should always give the tem-
perature at which the specific heat is supposed to be measured.
178
CALORIMETRY 179
The specific heats of some of the more common substances
are given in the following tables, in calories per gram per
Centigrade degree.
Specific Heats of Solids
Aluminum 0.212
Brass 0.094
Copper 0.095
Glass 0.195
Ice 0.504
Iron 0.112
Lead 0.031
Specific Heats of Liquids
Alcohol 0.,547 at 0" C.
Ether 0.529 at 0° C.
Mercury 0.033 at 30° C.
Gases have two specific heats, — the specific heat at constant
volume and the specific heat at constant pressure ; that is to say,
the heat required to increase the temperature of one gram of
the gas one degree without changing its volume, and the
amount of heat required to raise the temperature of one gram
of the gas one degree as it expands without change of pressure.
The specific heat at constant pressure is the greater since an
expanding gas does work at the expense of the heat contained
by it.
THERMAL CAPACITY
170. The thermal capacity of a body is the heat required to
raise its temperature one degree. The thermal capacity of a
body is equal to the product of the mass of the body and the
specific heat of the substance. That is,
thermal capacity = M • S
in which ilf is the mass of the body and S the specific heat of
the substance of which the body is composed. This relation
is apparent, since from the above definition the specific heat of
a substance is its thermal capacity per unit mass.
The following experiment is often made to demonstrate the
difference in the specific heats of various metals. A number
180
HEAT
of balls of equal size (equal volume) of different metals are
heated to some convenient temperature and placed side by side
upon a cake of wax, Figure 117. The balls melt their way into
Cu Zn Brass 5n Pb
3 QQ i^
Fig. 117. — Illustrating Difference in Tliermal Capacity per Unit Volume.
the wax ; the depth to which each ball sinks being determined
by the amount of heat it can give up as its temperature falls
to that of melting wax. Therefore, the depth to which a
given ball sinks is a measure of its thermal capacity. Now the
thermal capacity of a body is given by the product of its mass
and the specific heat of the substance of which the body is com-
posed. Equal volumes of two different metals may therefore
differ appreciably in thermal capacity, although having nearly
the same specific heat, unless their densities are also equal.
In the table below are given the specific heat, density, and
thermal capacity per unit volume of a number of metals :
Metal
:^PEr[Fic Heat
Density
Thermal Capacity
PEE Unit Volume
Al
0.21
2.6
0.546
Fe .
0.109
7.6
0.820
Cu .
0.095
8.9
0.801
Zn .
0.093
7.1
0.660
Brass
0.09
8.4
0.756
Sn .
0.056
7.3
0.408
Pb .
0.031
11.3
0.350
An inspection of the table will show that the rank of these
metals in thermal capacity per unit volume is very different
from their rank in specific heat. For example, aluminum, rank-
ing first in specific heat, is fifth in rank in thermal capacity per
unit volume. Brass is fifth in specific heat and third in thermal
capacity per unit volume, while lead, tin, and zinc rank the
CALORIMETRY 181
same in this group of metals wheii viewed from either stand-
point.
THE MEASUREMENT OP HEAT
171. There are various ways in which a given quantity of
heat may be measured. One of the most common methods is
that in which the quantity of heat to be measured is imparted to
a known mass of water and the rise in temperature which takes
place in the water is measured by means of a thermometer.
Knowing the mass of the water and the rise in temperature,
the quantity of heat is calculated in the following manner:
Since it requires 1 calorie to raise the temperature of 1 gram
of water 1 degree, evidently it will require 10 calories to raise
the temperature of 10 grams of water 1 degree, 100 calories to
raise the temperature of 100 grams of water 1 degree, etc.
Or, in general, if the mass of the water is M, then M calories of
heat are required to raise the temperature of the water 1 degree.
If the temperature rise is 2 degrees, 2 M calories will be re-
quired ; if the temperature rises 10 degrees, 10 M calories ;
or, in general, if the temperature of the water rises from ^j to
^2, the total heat required is
H= M(t^ - E is obtained. Evidently, this curve
Temperature
Fig. 130.
VAPORIZATION AND SOLIDIFICATION
199
marks the boundary between the liquid state of the substance
on the right and the solid state on the left. If the substance in
question is of such nature that it contracts upon freezing, evi-
dently the ice line will slope in the opposite direction ; that is,
E will lie farther from the axis of pressure than does the
point D.
In the S9,me way a pressure-temperature curve may be drawn
to represent the boundary condition between the vapor and
solid states. Such a curve is shown in Figure 130 and is
called the frost line. When a substance passes directly from the
solid to the vapor state (sublimation), it crosses this boundary
line. The reverse of this process is familiar to every one in the
formation of frost, from the water vapor of the atmosphere.
TRIPLE POINT
188. The three temperature-pressure curves discussed above
may all be placed in the same diagram. When so placed, they
VAPOR
Temperature
Fig. 131. — The Triple Point.
intersect in a common point, which is known as the triple point
for the substance in question. For example. Figure 131 repre-
200
HEAT
sents the triple point and corresponding temperature-pressure
curves for water. The curves here shown are not drawn to
scale and therefore can only be used to show general relations.
The steam line gives the conditions under which the vapor
and the liquid may exist together in equilibrium, the ice line
those under which the liquid and the solid may exist together,
and the frost line those under which the solid and vapor may,
exist simultaneously. It is obvious, therefore, that the triple
point gives the pressure and temperature conditions under
which all three, solid, liquid, and vapor, can exist together in
equilibrium. The triple point for water corresponds to a
pressure of 0.046 centimeter of mercury and a temperature
slightly above 0° C.
PRESSURE-VOLUME CURVES OF A GAS
189. Another diagram much used in connection with dis-
cussions on the behavior of gases and vapors under varying
pressure is the
pressure-volume
diagram. Let A,
Figure 132, repre-
sent the condition
of a given mass of
gas having a vol-
ume V when sub-
jected to pressure
■p. Let it be as-
sumed that the
temperature of
the gas is held
constant at, say,
20° C, and the
pressure is steadily
increased. Then
the point representing the condition of the gas on this diagram
will move along the curve AB. This curve is an equilateral
hyperbola, the equation to which is
pt) = a constant (Boyle's Law)
Volume
Fig. 132. — Isothermals of a Perfect Gas.
VAPORIZATION AND SOLIDIFICATION 201
Since the temperature of the gas remains constant, as the pres-
sure and volume change, as contemplated in this discussion,
the curve AB is sometimes called an isothermal.
If the mass of gas considered in the above discussion is taken
at some other initial temperature and is allowed to expand at
constant temperature, the isothermal corresponding will lie
above or below the curve AB, according to the temperature
chosen. The isothermals shown in Figure 132 are separated
by temperature intervals of 10° C.
ISOTHERMALS OF A VAPOR
190. In the above discussion it is assumed that the substance
under consideration is a perfect gas. If, instead of such a gas,
a vapor, near its saturation temperature, is considered, the iso-
thermal is no longer an equilateral hyperbola, but
assumes a form like that represented by CDEF in
Figure 133. Let C represent the condition of the
vapor at the outset. Let it be assumed that the
temperature is held constant and the pressure is
steadily increased, as before. The points represent-
ing the successive conditions of the vapor as to pres-
sure and volume will lie along the
curve CD. The point I) is as-
sumed to correspond to the satur-
ation pressure of the
C vapor, so that when
- this point is reached,
the vapor will begin to
condense and the pres-
n
in
L.
CL
Volume
Fig. 133. — Isothermal of a Vapor.
sure will remain constant until all of the vapor passes into
the liquid state. This change is represented by the horizontal
line BE. The point E corresponds to the condition in which
all of the vapor is liquefied. An increase of pressure from this
point will be followed by a decrease in volume of the liquid, as
indicated by the line EF. This portion of the curve is very
steep, since the substance is much less compressible in the
liquid state than it is in the vapor state.
202
HEAT
In the above discussion, the relation between pressure and
volume has been considered for one temperature only. Evi-
dently, if the gas had been taken at a lower temperature or at
a higher temperature, a curve similar to ODEF would have
been developed, which would lie either above this curve or
below, according as the initial temperature v^as higher or lower
than that assumed above.
ISOTHERMAL AT THE CRITICAL TEMPERATURE
191. In Figure 134 are shown a series of isothermals for COg
vapor at different temperatures, as determined by actual ex-
periment. It will be observed from this diagram that the hori-
zontal portion, BE, of
the isothermal corre-
sponding to the condi-
tion of saturated vapor
is shorter for the higher
temperatures, and in
the isothermal corre-
sponding to a tempera-
ture of 31.6° C. is en-
tirely wanting. The
meaning of this is that
when COg vapor at a
temperature of 31.6° C.
is subjected to increas-
ing pressure, its volume
decreases, that is to say,
the vapor becomes more
dense, just as when
compressed at lower
temperature, but no
matter how far the
process is carried, the vapor does not pass through the saturated
stage. That is to say, the vapor cannot be converted into a
liquid by the application of pressure at this temperature. This
temperature of 31.6° C. is, therefore, called the critical tem-
FiG. 134.
Volume
-Isothermals of CO2 at Various Tem-
peratures.
VAPORIZATION AND SOLIDIFICATION
203
perature of carbon dioxide. The pressure and volume corre-
sponding to the point H in Figure 134 are called the critical
pressure and critical volume of the given mass of the substance
considered.
If COg vapor is compressed isothermally at temperatures
considerably above 31.6° C, the isothermals obtained are like
those of a perfect gas, since for temperatures far above the
critical temperature the vapor obeys Boyle's Law. If the tem-
perature of the compressed gas is near the critical temperature,
it departs to some extent from the law which is followed by a
perfect gas. This is indicated in the diagram by the form of
the curves. The higher curves in the diagram are like the
isothermals of a perfect gas. For this reason a gas is some-
times distinguished from a vapor by saying that a gas is a
vapor far removed from its critical temperature.
LIQUEFACTION OP GASES
192. It will be evident from the above discussions on critical
temperature that in any attempt to liquefy a gas it is first neces-
sary to secure a lowering of the temperature which will bring
the gas below its critical temperature. When this condition has
been secured, an in-
crease of pressure will
force the substance
through the condition
of saturated vapor into
the liquid state. One
of the earliest experi-
ments in the liquefac-
tion of gases was that
performed by Faraday,
who made use of a
bent glass tube, some-
thing like that repre-
sented in Figure 135. In the long arm of the tube A is placed
a chemical compound, from which the gas to be experimented
upon is driven off by the application of heat. The other end
Fig. 135. —Faraday's Bent Glass Tube Experiment.
204 HEAT
of the bent tube is placed in a freezing mixture which is
capable of lowering the temperature of the gas within the tube
below the critical point. When heat is applied at A and the
gas is driven off, it, of course, spreads to all parts of the
tube, and as more and more gas is evolved, the pressure in-
creases to higher and higher values. The vapor in the B end
of the tube, being held at the temperature of the freezing mix-
ture, will undergo a change in condition, which is very well
represented by one of the lower curves in Figure 134. That is
to say, as the pressure rises, it will eventually reach the con-
dition of saturation and liquid will begin to form in the bottom
of the tube, at B. By means of this simple device, Faraday
succeeded in liquefying chlorine, carbon dioxide, cyanogen, and
ammonia, which, up to that time, had only been known in the
gaseous state.
In liquefying such gases as hydrogen, oxygen, nitrogen, and
air, a more elaborate apparatus is required, since the critical
temperatures of these gases are so low as to be attained only
with the greatest of difficulty. The means resorted to for
reaching the extremely low temperatures required in the lique-
faction of these gases is that of allowing the substance to cool
itself by sudden expansion, after it has been placed under high
pressure and cooled by other means as far as possible.
HYGROMETRY
CHAPTER XVI
HUMIDITY
193. Hygrometry is that branch of physics which deals with
the condition of the atmosphere as regards the water vapor
which it contains. The atmosphere is made up largely of
oxygen and nitrogen in almost constant proportions. In addi-
tion there are other substances present in relatively small
amounts. The most important constituent of the atmosphere
aside from the two first mentioned is water vapor. The
water vapor contained in the atmosphere is altogether variable,
there being at times large quantities of water vapor, at other
times relatively small amounts. The condition of the air as
regards the quantity of water vapor contained is of the greatest
importance in determining the weather or the climate of a
given place. It is therefore necessary in all weather observa-
tions to determine carefully the condition of the atmosphere in
this respect. There are various ways in which the condition
of the atmosphere as regards the quantity of water vapor con-
tained is determined. In one class of determinations the abso-
lute humidity is measured. In the other class the relative
humidity is determined.
The absolute humidity of the atmosphere is defined as the mass
of water vapor contained per unit volume. It is commonly ex-
pressed in grams per cubic meter. The relative humidity of the
atmosphere is defined as the ratio of the quantity of water vapor
present to that which would be necessary to bring about the con-
dition of saturation (Sections 181 and 182).
THE CHEMICAL HYGROMETER
194. A device used for determining the condition of the air
as regards the quantity of water vapor present is called a
hygrometer.
205
206
HEAT
The principle upon which the chemical hygrometer operates
is as follows : A known volume of the air in question is passed
through a series of U-tubes A, B, Q (Figure 136), filled with
— * ^._^^ ^_^_^ calcium chloride
l\\ U\\ (C n or other hygro-
FiG. 136. — The Chemical Hy-
grometer.
scopic material.
The tubes to-
gether with their
contents are
weighed before
and after the ex-
periment. The
difference be-
tween the two
weights is the
amount of water
which has been absorbed by the calcium chloride. This in-
strument measures, therefore, the absolute humidity of the
atmosphere. A convenient means of passing a known volume
of air through the tubes is that indicated in the figure. D is
an air-tight vessel which is filled with water at the beginning
of the experiment. Wlien the stopcock at the bottom is
turned, the water flows out and air Hows in through the con-
nection at the top. The volume of air which passes through
the tubes is therefore equal to the volume of water which flows
out of the vessel D.
THE DEW POINT HYGROMETER
195. The dew point hygrometer depends for its indications
upon the fact that air containing non-saturated water vapor may
be brought to the saturation point by reducing its temperature.
At low temperatures it requires less water vapor to bring about
the condition of saturation than is required under higher tem-
peratures. If, therefore, a given quantity of air with the con-
tained water vapor is sufficiently cooled, this water vapor will
begin to condense. The temperature at which this takes place
is called the dew point. The extent to which the air must be
cooled before the dew point is reached depends upon the rela-
HYGROMETRY 207
tive humidity of the air. Therefore, if the dew point can be
determined, it is possible, by comparing this temperature with
the actual temperature of the air, to estimate the amount of
water vapor present. Tables have been made up from which
the relative humidity may be obtained when the temperature
and dew point of the air are known.
WET- AND DEY-BULB THERMOMETERS
196. Two thermometers which are as nearly as possible iden-
tical in construction, etc., are mounted side by side, the one
being left exposed in the ordinarj^ vvay, the other having
wrapped about its bulb a bit of candle wicking which dips
into a vessel of water below. By capillarity the candle wicking,
and therefore the bulb of the thermometer about which the
candle wicking is wrapped, will be kept wet, and this mois-
ture in evaporating will produce a cooling effect upon the ther-
mometer. This wet-bulb thermometer will therefore give a
lower reading than the dry-bulb thermometer. The difference
in temperatures indicated by the wet- and dry-bulb thermometers
depends upon the relative humidity of the atmosphere in which
they are placed. Evidently if the atmosphere is filled with
saturated water vapor, the moisture on the wet bulb will not
evaporate and there will be no lowering of temperature. Under
these conditions the two thermometers will give the same read-
ing. On the other hand, if the water vapor present in the air
is far from saturation, there will be a rapid evaporation of mois-
ture from the wet-bulb and the difference between the two
thermometer readings will be correspondingly great. By com-
paring the indications of this instrument with the determinations
of the chemical hygrometer tables have been made by means of
which it is possible to interpret the indications of the wet- and
dry-bulb thermometers in terms of relative humidity.
PRECIPITATION
197. When the dew point is reached, the moisture in the
air begins to precipitate in one of several different ways, de-
pending upon the actual temperature of the air and manner in
which it is cooled. If the air is cooled in those layers only
208 HEAT
which come into intimate contact with cold bodies, the result
is the formation of dew or frost according as the temperature
is above or below the freezing point of water. If the chilling
takes place throughout the body of the air itself, precipitation
takes place in the form of fog or cloud, from which rain may be
formed by coalescence of the minute particles of liquid, or in
the form of ice clouds in which the particles are minute crystals
of ice instead of spheres of water. From this cloud, snow may
be formed by the slow growth of ice crystals, or snowHakes.
Hail is formed if rain drops pass through a cold layer of air
and are sufficiently chilled to freeze as they fall.
KINETIC THEORY OP GASES
CHAPTER XVII
THE VIBRATORY MOTION OF GAS ATOMS
198. A body of gas is conceived to consist of a great number
of distinct particles (atoms or molecules), minute in size and
separated by distances which are large in comparison with the
size of one of the particles. It is further assumed that these
particles possess a rapid vibratory motion, rebounding with
undiminished velocity when they strike the walls of the con-
taining vessel. The particles are also assumed to exert no
appreciable attraction for one another and to seldom collide.
This conception of the nature of a gas is almost universally
accepted for the reason that it enables all the principle laws of
gases to be readily explained and understood.
GAS PRESSURE
199. Consider a mass of gas M at a pressure p, volume v,
and absolute temperature T, represented in section by Fig-
ure 137. The molecular kinetic energy
of this body of gas is constant, since the
particles are assumed to rebound from
the walls with undiminished velocity and
to seldom collide. Now the kinetic energy
of a moving particle is J mo)^, m being the
mass of the particle and a its velocity.
Hence, if the molecular kinetic energy of
the body of gas remains the same, it fol-
lows that the average value of 2 (64)
Let the width of the containing vessel, that is, the distance
between the walls AB and BG^ be d. Then the time required
by the average particle to travel from AB to BC or from BO
ri n
to AB is — , hence it will strike — times per second, and the num-
a a
ber of times it will strike either wall will be — . At each im-
pact its velocity changes by the amount 2 a (from + a to —a).
Its change in momentum is therefore 2 am for each impact and
in one second against one wall 2 am x -^—z or -— —
We have seen that the rate of change of momentum of a
body is numerically equal to the force which causes that change.
KINETIC THEORY OF GASES 211
and, since action is equal to reaction, its rate of change of
momentum is equal to its reaction. The total force with which
the gas acts (pushes) on the wall AB is therefore,
F.
d
■p =
S
Bd
sd =
■■ V and a^ —
1
3
0)2
■ P =
1
■3
NmoP'
V
in which N is the total number of particles in the given volume
of gas.
Call the area of the wall S. Then tlie pressure on this wall
is
F Na?m
P =
But
(65)
Equation (65) gives the relation between the pressure and
volume of a gas, the total number of particles, the mass, and
the average square of the velocity of the individual particle.
This relation derived from purely theoretical considerations em-
bodies and explains all of the fundamental laws of gases with
which we have to deal in Physics.
BOYLE'S law and CHARLES' LAW
200. If we assume that the absolute temperature of a gas is
proportional to the average kinetic energy per molecule, in
other words, if we assume that T is proportional to ^ma>^, or
say ^ mafi = K ■ T,\a which jSTis a constant, then Equation (65)
may be written
pv=R ■ T (57 bis)
in which R is written for \ N ■ E. This is the general law of
a gas and includes the laws of Boyle and Charles (Section
162).
AVOGADEO'S PRINCIPLE
201. Avogadro's principle states that under the same condi-
tions as to pressure and temperature all gases have the same
number of molecules per cubic centimeter.
212 HEAT
For W, Equation (65), we may substitute n ■ v, v being the
volume of the gas and n the number of molecules per unit
volume. We have then
p = ^n ■ m ■ co"^
Now consider two gases at the same temperature and pres-
sure. Let Wj be the number of molecules per cubic centimeter
of the first gas, «ij the mass of the individual molecule, and a^
the average square of the velocity of a molecule. For the
other gas, let n^, m^, and (o^^ represent the corresponding quan-
tities. Then since the pressures are equal, we have
But since the temperatures are equal
It therefore follows that
which is the principle of Avogadro.
dalton's law
202. Dalton's Law states that when two gases occupy the
same space, each exerts the same pressure that it would exert
if it occupied the space alone. In other words, the pressure
exerted by each gas is independent of that exerted by the
other, and the total pressure on the walls of the containing
vessel is the sum of the pressures exerted by the individual
gases.
On the assumption that the gas particles are so small that
they do not interfere with one another in their motions, this
condition of the non-interference of one gas with another occu-
pying the same space is exactly that which would be expected
under the kinetic theory.
THE POROUS PLUG EXPERIMENT
203. Joule and Thomson carried out a series of experi-
ments on the expansion of a gas from a region of high
pressure to a region of low pressure through a porous plug.
A general idea of the experiment may be gained from Figure
KINETIC THEORY OP GASES
213
Wi
W2
p,
Pa
A P B
Fig. 138. — The Porous Plug Experi-
ment.
138. A and B are two cylinders connected by a narrow ori-
fice P (a porous plug in Joule and Thomson's experiment),
through which the gas passes from the region of high pressure
p^ to the region of low pres-
sure ^2- "^^^ pistons are as-
sumed to move without friction,
and are weighted with W^ and
TF^, which serve to maintain the
pressure and therefore the density
of the gas in each cylinder con-
stant. It will be understood
that the above is but a schematic
diagram of the apparatus.
If we assume that there is an
attraction between particles,
then as each particle of gas passes through the orifice it is
pulled forward by particles near the orifice in B and held back
by particles near the orifice in A. But the gas in A is more
dense than that in B, therefore there are more particles pull-
ing back than forward on the escaping particle and it will
lose in velocity as it passes the opening. This corresponds to a
fall in temperature.
On the assumption that the particles repel one another, it is
evident a particle would gain in velocity as it passed the opening.
This corresponds to a rise in temperature.
In the experiments of Joule and Thomson, sensitive ther-
mometers were placed on either side of the porous plug. In
experiments upon oxygen, hydrogen, and nitrogen, under ordi-
nary pressures and temperatures only very small effects were
observed. This would indicate that under ordinary conditions
as to pressure and temperature such gases are almost entirely
free from intermolecular force actions.
Oxygen and nitrogen were as a matter of fact slightly cooled
in the experiment, a result which would indicate a slight at-
traction between the molecules of these gases under the given
conditions. On the other hand, hydrogen, was slightly warmed,
a result which would indicate that under the given pressures and
temperatures, hydrogen molecules repel one another slightly.
214
HEAT
If gases having complex molecules are used in the experi-
ment, a more pronounced cooling effect results. Gases at low
temperature and under high pressures (molecules ■ relatively
close together) show a more pronounced cooling under free
expansion. This is exhibited in Linde's liquid air machine.
'^^
B
Fig. 139. — Principle
of Linde's Liquid
Air Machine.
linde's liquid air machine
204. In Linde's method of liquefying air,
dry air at ordinary temperatures and under
a pressure of about 200 atmospheres is led
into a system of tubes, the essential features
of which are shown in Figure 139. The air
at high pressure enters at A and, passing
through the inner tube, expands through the
narrow orifice P into the outer tube, a region
of lower pressure. The air is cooled by the
Joule-Thomson effect, as it passes the orifice
P, and flowing upward about the inner tube
causes a lowering of its temperature. Thus
each succeeding volume of escaping air is
colder than that which preceded it. The es-
caping air becomes colder and colder as the
operation continues, and is eventually lique-
fied. The liquid air accumulates in the
bottom of the outer tube and may be drawn
off at 0.
THE EQUATION OF VAN DER WAALS
205. The equation pv = MT is not rigidly exact. It would
seem that the two assumptions made in the kinetic theory
(1) that the size of the molecules may be neglected, (2) that
the molecules exert no mutual attraction, are not quite justified.
It was suggested by Van der Waals that the actual behavior
of a gas could be more accurately represented by writing the
general law in the following form.
(66)
P + -,)(.^-b} = RT.
KINETIC THEORY OF GASES 215
By writing v — h for v, allowance is made for the effect of the
size of the molecules. If the molecules have appreciable size,
they strike the walls of the containing vessel before their
centers of mass reach the walls and collisions occur more
frequently than they otherwise would. This amounts to a
reduction of the effective volume of the containing vessel.
The quantity 6 is a constant which depends upon the amount
and nature of the gas.
By writing p + -^ for p, allowance is made for the effect of
mutual attraction between the molecules. This attraction
tends to slow down the motion of the particles as they approach
the walls of the containing vessel and, therefore, tends to re-
duce the pressure. This reduction of pressure can be shown
to be inversely proportional to the square of the volume of the
gas. The quantity a depends upon the amount and nature of
the gas.
THE TRANSMISSION OF HEAT
CHAPTER XVIII
CONVECTION
206. There are three distinct ways ia which heat is trans-
mitted from one point to anotlaer, namely, by convection, conduc-
tion, and by radiation. In the first process heat is transferred by
the motion of the heated substance. The motion of the substance
in this process is due to the change in density which takes place in
the heated portions. For example, when a vessel of water is
placed upon the stove, those portions of the liquid in the bottom
of the vessel become heated. Their densities are thereby de-
creased and they tend to rise among the heavier, colder portions,
so that convection currents are set up in the water, the warmer
portions rising and the colder portions falling. If the vessel, in-
stead of being placed upon the stove, is heated by the flame of
a spirit lamp so that the heat is applied to a limited portion of
the bottom of the vessel, these convection currents will be dis-
tinct and easily followed by the eye. The general manner in
which the circulation takes place is indicated in Figure 140 at
A. Evidently the reverse of this effect may be brought about
by chilling that portion of the water which is near the center
of the vessel, for example, by placing a body of ice on the sur-
face of the water as indicated in Figure 140 at B. The con-
vection currents in this case are exactly like those represented
at A, except that they are reversed in direction.
The convection currents in a pond or other exposed body of
water, as the cold weather of winter comes on, are of interest.
Evidently those portions of the liquid which are first cooled
are the surface layers. These portions tend to sink, as in the
experiment illustrated in Figure 140, £, so that the convection
currents will persist so long as the chilled portions at the top
216
THE TRANSMISSION OF HEAT
217
are of greater density than the warm portions below. Now at
a temperature of 4° C. water has its maximum density. If we
imagine, therefore, that the convection currents described above
have persisted until the whole pond is chilled to 4° C, it will
be evident that a further cooling of the surface layers will tend
to arrest this cooling process, that is to say, as the surface
layers become cooled below 4° C. they become lighter and therefore
tend to remain at the surface. The result is that the body of
water in the pond will be cooled by the process described, down to
the temperature of 4° C. Any chilling effect below this will be
confined to the surface layers. It is altogether probable that
this fact makes it possible for certain forms of animal life to
Fig. 140. — Convection Currents.
continue through the winter, which would not be possible if
the chilling action due to the convection currents continued
down to the freezing point, since in this case the pond would
tend to become frozen from bottom to top.
The trade winds, which are winds experienced in regions a
few degrees north and south of the equator, and which persist
for long periods of time almost without change in direction,
are convection currents on a large scale. They are due to the
heating of the earth's atmosphere in the equatorial regions,
which causes those portions of the atmosphere to rise and air
to flow in from north and south to take the place of the ascend-
ing volumes. These currents coming from north and south
218
HEAT
t:
o
-Cold
Fig. 140 a. — Hotwater Heating
System.
constitute the trade winds. If the earth were stationary and
perfectly smooth on its .surface, these trade winds would come
in from the north and the south. Owing to the influence of
the earth's rotation, however,
c- these incoming currents are de-
flected toward the west, so that
the trade winds south of the
equator come from the southeast,
and those north of the equator
come from the northeast.
A hot-water heating system,
like that employed in dwellings,
affords an example of the appli-
cation of convection currents.
Such a system is represented in
Figure 140 a. F is the boiler
in which the water is heated, R
a radiator, and E the "expan-
sion tank." The expansion tank
allows for the increase in volume of the water when it is
heated. The heated water rises through the pipe connected to
the top of the boiler, passes through the radiators, where it
gives up a portion of its heat, and then returns through the
pipe connected to the bottom of
the boiler to be again heated. The
convection current in the appara-
tus is indicated by the arrows.
Another application of the con-
vection principle is found in the
" water-cooled " gasoline engine.
Unless some means is taken to
prevent it, the cylinder of a gas-
oline engine when in operation
becomes excessively heated. In
the water-cooled type the cylin-
der is cooled by means of a
"water jacket." This consists of a hollow chamber filled with
water which surrounds the cylinder. Figure 140 5 illustrates
Cold
Fig. 140 6. — "Thermo-siphon" for
cooling Cylinder of Gasoline En-
gine.
THE TRANSMISSION OP HEAT 219
the arrangement in its simplest form. AB is the water jacket.
The water, heated by contact with the hot cylinder, rises through
a pipe at the top of the cylinder, and is replaced by cold water
which flows in through a pipe at the bottom. In the automobile
the hot water is made to pass through a radiator R, at the front
of the machine, where it is cooled by the air which flows among
the thin-walled pipes which makes up this part of the apparatus.
After being cooled in this manner it returns to the water jacket
and again passes around the circuit. In some machines the
convection current alone is depended upon to maintain the cir-
culation, in others the water is circulated by means of a pump
placed in the circulating system and operated by the engine.
CONDUCTION
207. It will be evident from the very nature of the process
that the transfer of" heat by convection is possible only in
liquids and in gases. The process of heat transference known
as conduction takes place most readily in solids, although it is
possible in liquids and in gases. In this process heat is handed
on from particle to particle in the substance heated in the fol-
lowing manner : The first layer of the substance which is in
contact with the source of heat becomes heated, that is, accord-
ing to the kinetic theory its molecular parts are thrown into a
state of rapid vibratory motion. Since, however, this layer is
bound by the forces of cohesion to the adjacent layer it will be
impossible for its parts to vibrate to any extent without drag-
ging the adjacent layer also into motion; and this layer, be-
cause of the bonds which bind it to the next layer, will impart
some of its motion to that layer ; and so on, until every layer
of the body participates in the motion. It is evident that in
this process as well as the convection process the entire medium
through which the heat has been transferred becomes heated.
Some substances possess the property of transmitting heat in
this manner much more readily than others. These substances
are spoken of as good conductors of heat. Those which do not
transmit heat by this process so readily are spoken of as bad
conductors.
220
HEAT
THERMAL CONDUCTIVITY
208. Experiment shows that the quantity of heat conducted
through a layer of any substance in a given time is proportional
(a) to the area across which the
heat flows, (&) to the difference in
temperature between the two sur-
faces, (c) to the time, and (c?) va-
ries inversely as the thickness of
the layer. Consider the case rep-
resented in Figure 141. AB is a
vessel of water standing on a stove
OD. The water is assumed to be
boiling and remains therefore at
constant temperature, call it t^.
Let the temperature of the stove
be ij. Call the area of the bottom
of the vessel A, and its thickness d. Let Q represent the quan-
tity of heat conducted from the stove to the water through
the bottom of the vessel. We have, therefore, from the above
general statement, —
Q oc ^^^1 ~ ''2^ X time
a
t/
A
B
U ta
c
t, t.
Fig. 141. — Flow o( Heat from a
Stove to a Vessel of Water.
or.
K-A(t,~t,:)
d
If it is desired to find the rate at which heat is conducted from
the stove to the water, we have, —
Q^KA.^i^^
T ^f
(67)
The quantity j5"is called the thermal conductivity of the sub-
stance. It is evidently equal to the heat transferred in one
second through a layer of the substance one centimeter thick,
the area of each face being one square centimeter, the difference
of temperature between these faces being one degree on the
Centigrade scale.
THE TRANSMISSION OF HEAT
Thermal Conductivities
221
SlTBSTASCE
TR.III'EHATUKE
(Centigkadk)
TlIERMAI-
CnMlL'CTIMTIES
Silver
Copper
Aluminiim
Till
0°
15°
0° to 100°
0° to 100°
15°
0° to 100°
1.3°
18°
7.56
7.56
1.096
0.713
0.343
1.52
Iron .
Lead . . .
Alcohol
Water
Hydrogen
Air . . .
0.149
0.083
0.00054
0.00124
0.00033
0.000051
THE MEASUEEMENT OF COStDUCTIVITY
209. The thermal conductivity of a given substance is usu-
ally determined by some form of experiment involving the use
of the relation given in Equation (67). The thermal conduc-
tivities of metals may be roughly
compared by the following ex-
periment : a, b, e, and d. Figure
142, represent wires of copper,
zinc, iron, and lead. If the junc-
tion point A of these several
wires is heated by means of a
Bunsen burner, heat will be con-
ducted along each of the wires.
It will be conducted quite readily
along the copper wire, less readily
along the zinc, and so on, so that
after the lapse of a given time
the extremities of the wires a, b, c, and d are heated to an extent
which depends upon the thermal conductivities of the metals
of which they are composed. If an unlighted match is applied
to the end of each wire and moved slowly along toward the
source, as soon as a point is reached for which the temperature
is equal to that of ignition, the match will be lighted. If this
Fig. 142. — Conductivity Experiment.
222
HEAT
Fig. 143.-
-Illustrating the Low Conductivity
of Water.
point is found to be far from A, the conclusion is that the sub-
stance of the wire in question is a good conductor, that is, its
thermal conductivity is high.
If the point of ignition is
found to be near the source
A, the thermal conductivity
is low.
The determination of the
conductivity of liquids and
gases is a matter of extreme
difficulty for the reason
that it is almost impossible
to arrange an experiment in
such manner as to eliminate
the efPects of convection and
radiation. In fact, in the
most satisfactory determi-
nations which have been
made, these effects have not
been eliminated, but have been corrected for. Thus the state-
ment might be made that it is not possible to measure the con-
ductivity of liquids and gases directly. That liquids, for
example, have low thermal conductivities in general may be
demonstrated by experiments like the following.
If a test tube is partly filled with ice, the ice being held in
the lower part of the test tube, and the test tube is then filled
with water, it may be placed in the flame of a Bunsen burner
or alcohol lamp in the manner indicated in Figure 143, and the
water will boil in the upper part of the test tube without
melting the ice below. This, of course, indicates that but
small amounts of heat are transmitted by conduction through
the water.
In Figure 144 is shown another simple experiment for demon-
strating the low value of the thermal conductivity of liquids,
for example, water. A is an air thermometer, the bulb of
which is placed within a suitable quantity of water contained
in the vessel B as indicated. Upon the surface of the water in
£ a small metallic vessel C is caused to float. This vessel con-
THE TRANSMISSION OF HEAT
223
tains a small quantity of alcohol. If the
alcohol is ignited it will burn at high tem-
perature and develop large quantities of
heat. The upper layers of the water in B
will thus be subjected to a high tempera-
ture. If any appreciable amount of heat
were conducted by the upper layers of the
liquid in B, evidently the effect would be
apparent by the falling of the column in
the air thermometer. It will be found,
however, even after the lapse of consider-
able time, that the column in the air ther-
mometer remains immovable, thus indicating
that practically no heat has been received
by the bulb of the air thermometer, al-
though it is located but a short distance
from the burnincr alcohol.
Fig. 144. — Another ex-
periment showing that
AVater is a Poor Con-
ductor of Heat.
THE TEMPERATUKE GEADIENT
210. Consider a wall one side of which is at a temperature
ITj and the other at a temperature T^. If T^ > T ^ there will be
a flow of heat from the side having a temperature T-^ to the
other. It is interesting to inquire how
the temperature varies from point to
point within the wall. In the case repre-
sented in Figure 145, the point a has a
temperature Z^ and /a temperature I'j-
The intermediate points h, e, d, e are at
different temperatures between the limits
T^ and T^^. It must be evident that the
temperature of I is lower than that of a,
but higher than that of e. The tempera-
ture of c is lower than that of h and
higher than that of d, etc. In other
words, there is a fall of temperature from
point to point in the direction of the flow of heat. That such
is the case follows at once from the principle that heat can
flow only from a higher to a lower temperature. The ratio of
Hot
T2
f
Cold
Fig. 145. — Temperature
Gradient.
224 HEAT
the difference in temperature between two points to the distance
between them is called the temperature gradient.
RADIATION
211. If a thermometer is held a short distance below a hot
body, it will receive heat from the hot body. The heat received
in this manner is evidently not to be accounted for by con-
vection, since convection currents would tend to convey the
heat in an upward direction from the hot body, as we have seen.
Furthermore, air being a poor conductor, the amount of heat
received by the thermometer in such an experiment cannot be
accounted for by conduction. It is still more evident that the
heat transferred from the hot body to the thermometer has
taken place bj' neither of these processes when we determine by
further experiment that this heat transfer takes place even
though the hot body and thermometer are placed in a vacuum.
The heat received by the thermometer in this experiment is said
to be radiated from the hot body, and the process of heat trans-
fer involved is known as radiation.
Attention has been called to the fact that in both the con-
vection and conduction processes the medium of transfer is a
material substance. Radiation is distinguished from conduction
and convection by the fact that it may take place in a vacuum.
In radiation the medium of transfer is the ether, a medium
which is supposed to extend throughout all space and to fill
those portions of space which are occupied by ordinary matter
as well as those which are vacuous. Under the study of light
and electricitj' reference will be made to this medium, which is
supposed to transmit light and electric disturbances as well as
heat, and a more complete discussion of its properties will be
given when those subjects are taken up. It is sufficient for our
present purpose to refer to the existence of this medium and to
show that upon the assumption of the existence of such a
medium the phenomena of radiation are easily explained.
Another fact which distinguishes radiation from the other
processes of heat transfer is that the medium through which
radiation takes place is not heated in the process. The earth
receives vast quantities of heat from the sun, although the
THE TRANSMISSION OF HEAT
225
space which separates the earth from the sun remains very
cold.
The transfer of heat by this process is supposed to take place
by means of a wave motion in the ether. A hot body is capable
of setting up a wave motion in the ether which is in contact
with it, its ability to do this depending, of course, upon the
vibratory motion of its molecular parts. This wave motion
spreads through the ether, and when it falls upon a material
substance is able to impart vibratory motion to the molecular
parts of that substance, that is to say, it is able to raise the
temperature of that substance. Thus the transfer of heat from
one body to another by the process of radiation is to be thought
of as a double process: first, the conversion of the heat energy
of the hot body into wave motion of the ether; second, a recon-
version of the ether wave motion into heat in the body warmed.
prevost's theory of exchanges
212. The various phenomena of rise and fall of temperature
in bodies due to this process of radiation are best explained by
Prevost's theory of exchanges. Briefly stated, this theory is
as follows : That all bodies, cold or
hot, radiate heat. Other things being
equal, hot bodies radiate heat more
rapidly than cold ones; but whatever
the temperature of the body and what-
ever its surroundings, it is at all times
radiating heat. All bodies are also to
be thought of as receiving heat or ab-
sorbing heat which has been radiated
from surrounding bodies. Therefore,
the condition of a body as to tempera-
ture is determined by the ratio of the
heat which that body radiates to the
heat which it absorbs. If the heat
radiated by a body is just equal to that
absorbed, its temperature will remain constant. If it radiates
more heat than it absorbs, its temperature will fall. If it ab-
sorbs more than it radiates, its temperature will rise.
Q
Fig. 146. — Illustrating Pre-
vost's Theory of Exchanges.
226 HEAT
This theory affords an explanation for what appears to be the
radiation of cold. For example, if an air thermometer is placed
near a cold body, as indicated in Figure 146, the thermometer
will immediately indicate a fall of temperature, as if " cold "
had been radiated from the cold body to the bulb of the ther-
mometer. Under Prevost's theory the explanation of the fall
in temperature is as follows: The air thermometer is both radi-
ating and absorbing heat. Since it is warmer than the cold
body near it, it is radiating heat more rapidly than the cold
body. Therefore the quantity of heat radiated by the ther-
mometer to the cold body is greater than that radiated to the
thermometer from the cold body. This results in a net loss of
heat in the air thermometer and therefore a fall in temperature.
DEPENDENCE OP RADIATION T7PON THE CHARACTER OF THE
SURFACE OF THE RADIATING BODY
213. It is found by experiment that the amount of heat
radiated by a body depends first of all, as indicated above, upon
its temperature ; second, upon the character of its surface.
Certain surfaces seem to facilitate the process of radiation,
while others are not so well adapted to the process. Generally
speaking, a rough black surface radiates well. Lampblack,
for example, is an excellent radiating surface as compared with
other substances. In fact, in comparisons of this kind it is
customary to consider lampblack as a perfect radiator, and the
amount of heat radiated from a lampblack surface, other things
being equal, is taken as 100 per cent. It should not be thought
in this connection that the color black is especially significant,
for it can readily be shown by experiment that certain white
substances are almost as good radiators as lampblack, for ex-
ample, ordinary white unglazed paper radiates almost as well
as lampblack.
Polished surfaces in general are poor radiators.
RADIATION AND ABSORPTION
214. The facility with which a given surface absorbs heat
is found to be in every case proportional to the facility with
which it radiates. That is to say, a good radiator of heat is a
THE TRANSMISSION OF HEAT
227
good absorber, and a surface which radiates heat slowly will
absorb it slowly, other things being equal. The equality of
radiation and absorption as de-
pending upon the character of
the surface is shown by the ex-
periment illustrated in Figure
147. ABQD represents a hot
body. The surface AB is coated
with lampblack. Opposite this
face is placed an air thermometer
E having, for convenience, a flat
bulb. The side of the thermom-
eter which is turned toward the
hot body is of polished metal.
The side DC ot the hot body is of
polished metal like that used in
the thermometer E. Opposite
this face stands an air thermometer F similar to I] except that
the face which this thermometer presents to the hot body is
coated with lampblack. If the distances which separate the
thermometers E and F from the hot body are the same, the rise
in temperature indicated by the two thermometers will be the
same. The lampblack surface AB radiates more heat than the
polished metal surface DO. On the other hand, the polished
metal surface of the thermometer E is capable of taking up but
a small fraction of the large quantity of heat radiated to it from
the lampblack surface, while the lampblack surface of the ther-
mometer F absorbs most of the heat which falls upon it from
the polished metal surface DC. Evidently if the rise in tem-
perature indicated by the two thermometers is the same, this
experiment affords a proof that the radiating and absorbing
properties of a given surface are equal.
Fig. 147. — Apparatus for demon-
strating Equality of Kadiation and
Absorption.
THE TRANSMISSION OF RADIATED HEAT BY MATERIAL
SUBSTANCES
215. The wave motion in the ether which constitutes radia-
tion is found by experiment to be able to pass through certain
material substances with more or less facility, and to be quite
228 HEAT
completely intercepted by thin layers of other substances. We
have, therefore, to distinguish between those bodies which allow
this wave motion to pass through them and which are therefore
" transparent to radiated heat," and those which are in this
sense opaque.
The readiness with which this wave motion passes through
a given substance depends upon the wave length of the dis-
turbance. We have already referred to this radiation as being
made up of waves. It should also be borne in mind that the
waves given out by a radiating body are not all of the same
wave length ; in fact, a radiating body is to be thought of as
giving off short waves and long waves and intermediate waves
of various lengths. If the heat waves given off in this manner
are of a certain wave length (about the 50,000th part of an
inch), they affect the retina of the eye and are called light
waves. If they are too long to affect the optic nerve, they are
called dark heat waves. A given material like glass is found
to transmit radiation in the form of short waves (light waves)
and at the same time to be quite opaque to the long, dark heat
waves. This explains how a " hotbed " which is covered with
glass is warmed in the early spring. The energy which is
passed into the hotbed in the form of light is absorbed in part
by the surface of the soil upon which it falls. The surface of
the soil gradually becomes heated, and would tend to cool by
radiation, but the heat waves given off by the soil under these
circumstances are long waves, that is, dark heat waves. These
are intercepted by the glass, and radiation of this kind of heat
from the soil is prevented. Thus the heat is, as it were, en-
trapped and the result is a rise in temperature of the soil.
REFLECTION OP RADIATED HEAT
216. The wave motion of the ether which constitutes radia-
tion is reflected more or less completely by certain surfaces in
exactly the same manner that light is reflected. The law of
reflection, namelj', that the angle of reflection is equal to the
angle of incidence, which applies in the case of the reflection
of light, applies to the reflection of heat. In general, the
amount of this wave motion which is reflected depends upon
THE TRANSMISSION OF HEAT 229
the angle of incidence and upon the character of the surface.
Polished surfaces of course reflect better than rougher surfaces.
In general, the radiated heat which falls upon the surface of
a body is divided into three parts :
(a) That part which is absorbed and tends to raise the tem-
perature of the body upon the surface of which it falls.
(b) That part which is transmitted, that is to say, which
passes through the body.
(c) That part which is reflected.
Evidently (a) + (6) + (c) must be equal to the total heat
which falls upon the surface of the body in question. In cer-
tain cases the part (a) will be large, and (6) and (c) relatively
small. In other cases (J) may be large, or (6) and (c) rela-
tively large and (a) small, and so on.
Problems
1. A sheet of copper has an area of 100 sq. era. aad a thickness of
6 mm. The temperature of one side is 100° C, that of the other, 0° C.
How much heat is conducted through the plate per second ? Thermal con-
ductivity of copper = 0.713 o. g. s. unit.
2. Water is hoiled at atmospheric pressure in an iron vessel 6 mm.
thick. The heating area of the vessel is 1 sq. m. If the surface exposed
to the fire is kept at 2.^0° C, how much water will be evaporated per hour ?
Thermal conductivity of iron = 0.149 c. g. s. unit.
3. A roan is clothed in a fabric 3 mm. thick, the thermal conductivity
of which is 0.000122. If the temperature of his body is 30° C. and that of
the air is 0° C, how much heat does he lose from 100 sq. cm. of the sur-
face of his body per hour ?
4. A circular tank of water 2 m. in diameter is covered with ice 4 cm.
thick. The thermal conductivity of ice is 0.0023 c. g. s. unit. If the air is
at a temperature of — 20° C, how much heat is transmitted through the ice
per hour?
5. A wall is built of a material having a thermal conductivity of 0.0072
c. g. s. unit. If 360 calories are conducted through the wall per square
meter per second, what is the temperature gradient in the wall?
6. When the temperature gradient in a metal is 20 degrees/cm. it con-
ducts 840 calories per square centimeter per minute. What is the thermal
conductivity of the metal?
THERMODYNAMICS
CHAPTER XIX
CONVERSION OF WORK INTO HEAT
|J
B
217. Thermodynamics is that branch of physics which treats
of the transformation of mechanical energy into heat and the
transformation of heat into mechanical energy.
That mechanical energy may be transformed
into heat is demonstrated by the familiar phe-
nomena of friction. When a metal button is
rubbed on cloth or wood, it becomes heated.
The journals or bearings of a car are some-
times strongly heated by friction, causing a " hot
box." Certain tribes of savages start iires by
rubbing sticks of wood together. A simple ex-
periment for illustrating the transformation of
mechanical energy into heat is the following : In
Figure 148, B represents a hollow cylinder hav-
ing a tight-fitting piston A. If the piston is
forced into the cylinder, the air contained in the
cylinder will be compressed and heated, the work
done in moving the piston being transformed into
heat. If the cylinder is filled with air at ordinary room tem-
perature and pressure and the piston is very quickly forced into
the cylinder, the temperature attained by the compressed air
may be high enough to ignite a bit of inflammable material
(tinder) attached to the piston, which will continue to burn
after the piston is withdrawn.
An interesting illustration of the transformation of heat into
mechanical energy is afforded by the simple form of Hero's
steam engine, illustrated in Figure 149.
5 is a small boiler suspended from a suitable support by means
of two cords, as shown. This boiler is provided with two small
230
Fig. liS.— The
Fire Syringe.
THERMODYNAMICS
231
B
tubes extending radially from opposite sides, their outer ends
being bent horizontally at right angles and in opposite directions.
A small amount of water
having been placed in the
boiler, steam is generated by
placing a Bunsen burner or
alcohol lamp beneath it. The
reaction of the steam escaping
from the side pipes gives the
well-known Hero engine ef-
fect and the boiler is set into
rapid rotation about a verti-
cal axis. This rotation twists
the suspending cords, and
the boiler is steadily lifted
from the flame. When it has ^
reached a certain elevation,
steam vi'ill no longer be gen-
erated and the motion will
cease. The boiler will now
descend under the action of
gravity and the cords in untwisting will cause the boiler to
revolve in the reverse sense. Descending in this manner
toward the flame, the boiler will eventually reach a position
in which steam will again be generated. The escaping steam
by its reaction will stop- the backward rotation of the boiler,
reverse its motion, and cause it to revolve as in the first in-
stance. Once more it will " climb " out of reach of the flame,
and the action will be repeated.
The energy transformations are as follows : The chemical po-
tential energy of the gas or alcohol vapor in the presence of the
oxygen of the air is, by combustion, transformed into heat. This
heat is transformed into the potential energy of the hot steam, the
energy of the steam being again transformed into the kinetic en-
ergy Qf the rotating boiler. Finally this kinetic energy is trans-
formed into the gravitational potential energy of the lifted
boiler.
As the boiler descends, its potential energy is partly trans-
IlJJ
Fig. 149. — Hero's Engine.
232 HEAT
formed into kinetic energy of rotation and a part is used in
doing work against the resisting forces caused by the reaction
of the escaping jets of steam as the boiler slows down to its
position of momentary rest just above the flame.
Ordinarily, the oscillations of the boiler as it rises and falls
will become less and less until it comes to rest at such elevation
above the flame that the reaction of the escaping steam will just
balance the torque action of the twisted string.
THE FIRST LAW OF THERMODYNAMICS
218. Mechanical energy may be transformed into heat and
heat may be transformed into mechanical energy, and in every
case of a transformation of this character the ratio of the quantity
of heat to the quantity of mechanical energy involved is constant.
That is,
W=JB: (68)
This is known as the first law of thermodynamics.
It has been found by experiment that whenever mechanical
energy is wholly converted into heat, for every 4.187 joules of
mechanical energy that disappears one calorie of heat is devel-
oped ; that is, . . • . 1 07 • i
1 calorie = 4.187 joules
i.e. J" =4.187
This number 4.187 (the factor J, Equation 68) is called the
mechanical equivalent of heat.
In the f. p. s. system of units the mechanical equivalent of
heat is 772, which means that, —
1 B. T. U. = 772 foot-pounds.
The method used in determining the mechanical equivalent
of heat was to churn a given quantity of water by means of
paddle wheel rotating in a suitable vessel. The amount of
work done in turning the paddle wheel was measured, and the
rise in temperature of the water and containing vessel, which
served in this experiment as the calorimeter, was noted. The
amount of heat developed was therefore known; and by com-
paring this quantity of heat with the work expended in turning
THERMODYNAMICS
233
the paddle wheel, proper allowance being made for radiation
and other soui'ces of error, the above ratio was determined.
THE SECOND LAW OP THERMODYNAMICS
219. It is impossible for heat of itself to pass from a cold to a
hot body. This is known as the second law of thermodynamics.
Having in mind the analogy which was employed at the
beginning of this subject, namely, that heat flows from regions
of high temperature to regions of low temperature in much the
same manner that water flows from a high level to a low one,
the significance of the second law becomes at once apparent.
It is conceivable, of course, that heat may be made to pass from
a cold body to a hot one, just as we may pump water from a
lower to a higher level. Thus, in the operation of the ammonia
refrigerating machine, heat is continually being abstracted from
the cold brine in the brine tank and transferred to the relatively
hot water which fills the cooling tank, so that the apparatus
constitutes a heat pump which transfers heat from the cold
brine to the warmer cooling tank.
It should be carefully noted, however, that this transfer of
heat from the cold brine tank to the warm cooling tank goes
forward only so long as mechanical energy is supplied from
some outside source to operate the pump. As soon as the pump
stops, as soon as the supply of mechanical energy from the out-
side is cut off, heat will begin to pass in the opposite direction
and by conduction, convection, and radiation will pass from the
warmer to the cooler parts of the apparatus.
THE STEAM ENGIBTB
220. The simplest form of
steam engine cylinder is that
shown in Figure 150. A and B
are two pipes connected to the
cylinder 0, each of which serves
alternately as inlet and outlet
for the steam. When the piston
P is moving in the direction of
H
m
B
Fig. ISO.
-Simple Steam Engine
Cylinder.
234
HEAT
the arrow, the steam is entering at A and the "cold steam,"
which on the preceding stroke pushed the piston to the upper
end of the cylinder, is "exhausting" (flowing out) at B.
When the piston has reached the bottom of the cjlinder, steam
is admitted at B, and A is connected to the exhaust. Thus the
hot steam from the " boiler "' is admitted alternately at A and
B and pushes the piston to and fro in the cylinder.
The steam may be allowed to flow throughout the entire
stroke, but in that case the exhaust steam is nearly as hot as
the steam in the boiler. The engine working in this way is
inefficient because a great deal of heat energy is carried away
in the exhaust steam and is lost. To increase the efficiency of
the engine, steam is admitted during a part of the stroke only,
the stroke being completed by the expansion of the steam within
the cylinder. During this expansion the steam cools, that is,
continues to give up heat energy. A larger part of the heat
energy of the steam is
thus made available and
the efficiency of the
engine increased. It is
advantageous also to
close the exhaust ahead
of the piston before it
quite reaches the end
of its stroke. By this
means the pressure and
temperature in the cyl-
inder is raised before
steam is admitted to
nearly that of the boiler.
Intake
Compression
Explosion
Exhaust
Fig. l.Tl. — Diagram showing the Successive Oper-
ations in a Gasoline Engine Cylinder.
THE GASOLIXE EXGIXE
221. The source of
energ}' in tlie gaso-
in the
line engine is the heat
evolved in the combustion of a mixture of gasoline vapor and
air. During the combustion, which is made to take place
within the cylinder of the engine, a gas at high pressure is
THERMODYNAMICS 235
evolved. This gas in expanding does work upon the piston
very much as the steaca does in a steam engine. lu the single
acting engine there is one explosion for two revolutions of .the
engine or four strokes of the piston. During the first stroke
the explosive mixture is dravfn into the cylinder. In the sec-
ond stroke the mixture is compressed. The explosion occurs
on the third stroke, and during the fourth stroke the products
of combustion are forced out of the cylinder. These succes-
sive operations are represented in order in Figure 151.
MECHANICAL EEFKIGERATION
222. The process of mechanical refrigeration is essentially
the reverse of that employed in the steam or gasoline engine.
In the steam engine the steam passes from the boiler at high
temperature to the cylinder, there giving up a portion of its
heat energy as it does work on the piston. In the refrigerating
apparatus the working substance is drawn into the cylinder of
a compressor at low temperature and is heated as work is done
upon it by the piston. After giving up its excess of heat
it is allowed to expand, and by the cooling effects of expan-
sion and vaporization it reaches a refrigerating tempera-
ture. One of the common forms of refrigeration apparatus
is that in which ammonia is used, the cooling effect being
secured by the vaporization of the liquid ammonia. A simple
device of this character is represented diagrammatically in
Figure 152. ABO is a force pump or compressor. The pipe
connections at A and B are provided with valves, that at A
opening into the cylinder, that at B opening out from the
cylinder so that on the upstroke of the plunger the pump acts
as an air pump, A being the intake. On the downstroke it
acts as a force pump, B being the outlet. In the operation of
the pump ammonia gas is drawn in at A, compressed in the
cylinder, and forced under relatively high pressure (about 10
atmospheres) into the coil of pipe represented at I. This com-
pression results in a strong heating of the compressed ammonia
which is partly in the liquid state and partly in the form of
vapor as it enters the coil I. The coil /is surrounded by cold
water, the tank UF being supplied by a constant stream of
236
HEAT
water. This cools the ammonia in the coil I to, let us say, a
few degrees below ordinary room temperature. When cool the
ammonia, being still under high pressure, is allowed to escape
through a regulating valve D into the coil J. This coil J is
continually being exhausted by the pump ABC so that within
this coil there is low pressure. The liquid therefore, as it
passes the valve D in the liquid state from the region of high
pressure to the region of low pressure vaporizes in much the
same way that the COg does in the experiment described in
Section 185. The result is that the ammonia vapor in the coil
Fig. 152. — Kefrigeratiug Machine.
J together with the coil and its surroundings are lowered in
temperature. This cooling effect is sufficient under the cir-
cumstances described to reduce the temperature of the coil J
and its surroundings considerably below the freezing point of
water. The vapor which is thus formed in the coil J is again
taken up by the pump and is made to pass once more through
the cycle as described, and so on. For the convenient utiliza-
tion of this low temperature the coil J" is usually immersed in a
tank of brine which, being in contact with the coil J", is cooled
down to a temperature below the freezing point of water.
This cold brine is then pumped into coils arranged much, the
same as radiators are arranged in a steam heating plant. In
THERMODYNAMICS 237
this manner the rooms in which the cooling coils are placed are
cooled.
Ammonia lends itself with advantage to the refrigeration
process described above because of the fact that it can be con-
verted from the vapor state to the liquid state at ordinary
temperatures by the application of pressure alone. If an at-
tempt were made to use a gas like air, oxygen, or hydrogen in
place of the ammonia, it would not be found possible to secure
the same result; since, no matter how much pressure is applied
to oxygen or hydrogen at ordinary temperatures, it is impossible
to change them over into the liquid form (Section 186).
However, a similar effect in smaller degree may be secured by
using an ordinary gas in the refrigerating apparatus described.
In this case the lowering of the temperature of the tank Jis due
to the cooling effect of expansion in a gas. One of the disad-
vantages of the ammonia process is, that ammonia gas is
dangerous to life in case it escapes from the apparatus.
THE PR0DtTCTI02Sr OF " AKTIFICIAL ICE."
223. In the production of artificial ice, brine from the brine
tank (t/, Figure 152) of a refrigerating plant is caused to circu-
late about pans filled with the water to be frozen. The water
gives up heat to the cold brine with which the pans are in
contact, and thereby becomes lowered in temperature until
it freezes.
watt's diagram
224. It has already been pointed out that the physical state
of a gas is completely represented by a point in the pressure-
volume diagram. In the same way the successive states through
which the gas passes, because of changes in its pressure, or its
volume, or both, are represented by the successive points on a
curve. The volumes represented in such a diagram may be
either total volumes of a given mass of gas or volumes per unit
mass. When such a diagram is drawn to represent the rela-
tions between the pressure and total volume of the gas, it is
called Watt's diagram.
238
HEAT
V,
J
AREA IN "watt's DIAGEAM REPRESENTS WORK
225. In Watt's diagram the area under a process curve, that
is, the area bounded by the curve, its end ordinates, and the
axis, represents the work done on the gas, or by the gas, during
the change of its physical state
represented by that curve. This
will be readily understood from
the following illustrations : In Fig-
ure 153 let CO represent a cyl-
inder fitted with a frictionless
piston and containing a given mass
\j of gas at pressure p and
volume Vy The state of
the gas may be repre-
sented by the point A in
Watt's diagram in the
upper part of the figure.
Let it be imagined that
the gas in the cylinder
expands without change
of pressure until its volume is increased to v^. Its condition
will now be represented by the point B.
The work done by the gas during its expansion is deter-
mined as follows : Call the area of the piston s. The total
force with which the gas pushes upon the piston is then ps.
This force moves the piston through the distance d as shown
in the figure. The work done is therefore
W=Fd
= psd
But sd is the change (increase) in volume of the gas. It
follows therefore that the work done by the expanding gas is
numerically equal to the product of the pressure and change in
volume. That is w=pCv^-v^) (69)
Now, — Oh represents v^ to scale and Oa represents v^. Thus
Also aA represents to scale the value
. WxaA X ab.
C i. d
FiQ. 153. — Watt's Diagram, Constant Pressure
ah represents (('2 — Vj^').
of the pressure.
THERMODYNAMICS
239
But aA X ah is the area of the rectangle aABb. Therefore,
the work done by the expanding gas during the process
described is represented by (is proportional to) the area under
the corresponding curve AB.
If the process is one in which both pressure and volume
change, the same relation holds, since evidently in this case the
work done is equal to the
product of the change in
volume and the average
pressure during the vol-
ume change, while the
area under the curve is
given by the product of
ah and the average ordi-
nate of the curve AB,
Figure 154.
In a process like that
represented by the curve
AB, Figure 153 or Fig-
ure 154, the volume of
the gas increases and the
gas does work. If the
process is reversed, work
must be done on the gas. Thus the area aABb, Figure 153,
represents the work which would have to be done upon the gas
to reduce its volume from v^ to v-^ without change of pressure.
In Figure 154 aABb represents the work which would have
to be done upon v^ cubic centimeters of gas at a pressure ^2
dynes per square centimeter to reduce its volume to v^ cubic
centimeters, its pressure rising during the change of condition
to jOj dynes per square centimeter.
p
J"
1
1
A
1
1
p.
1
1
^
B
1
1
1
1
1
.-V,-- .
■^/////
^
Pa
V
^
a-Va-
13
(.— -d-
Fig. 154. — Watt's Diagram, Pressure Variable.
ISOTHERMAL AND ADIABATIC PKOCESSES
226. There are two distinct processes by which a gas may
expand to an increased volume and diminished pressure, (1) by
isothermal expansion, (2) by adiabatic expansion.
An isothermal process is one in which the temperature of
gas remains constant. In order that a given body of gas may
240 HEAT
undergo such a process, heat must be imparted to it (if the gas
expands) or abstracted from it (if the gas is compressed).
Consider the gas inclosed in the cylinder of any form of heat
engine. As the gas expands, it pushes the piston back and does
work. The source of this work is the heat energy of the gas.
But if the gas gives up some of its heat energy, it cools. If,
therefore, the gas is to expand without cooling, it must be supplied
with heat during the process of expansion. Similarl}-, if the gas
in such a cylinder is compressed, its temperature will rise, that
is, its heat energy will increase. The source of this increase of
heat energy is the work done in compressing the gas. If, there-
fore, the gas is to be compressed without rise of temperature, heat
must be abstracted from it during the process of compression.
An adiabatic process is one in which there is no interchange of
heat between the gas and its surroundings. Such a process is
always accompanied by a change in the temperature of the gas.
Consider the gas inclosed in the cylinder of any form of heat
engine. As the gas expands, it pushes the piston back and
does work. The source of this work is the heat energy of the
gas. Hence, if the gas receives no heat from its surroundings
during the process, its temperature will fall as it gives up the
heat energy which is transformed into work. Similarly, the
work done in compressing the gas in such a cylinder is trans-
formed into heat, and if the gas loses no heat to its surroundings
during the process, its temperature will rise.
These processes may be approximately realized in a cylinder
filled with compressed gas. (1) Imagine the gas to expand
very slowly. This will be an isothermal process, since by heat
conduction from the walls of the cylinder the temperature of
the gas will be kept constant. (2) Imagine the gas to expand
very quickly. This will be an adiabatic process, since no
appreciable amount of heat can flow from the cylinder walls to
the gas during the expansion.
CAKyOT'S CYCLE
227. In order that work may be obtained by repeated expan-
sion and compression of a given bodj' of gas, it will be evident
that the expansion and compression processes must be different.
THERMODYNAMICS 241
Consider the process represented by the curve AB, Figure
154. When the gas expands from the volume f j to the volume
v„, an amount of work, represented by the area aABb, is done
by the gas. If now the process is reversed and the gas is com-
pressed from the volume v^ to the volume v^, an equal amount
of work, represented again by the area aABb, is done on the gas.
Evidently an engine working in this way could do no external
work, since all of the work done by the gas during the expan-
sion stroke would be required to compress the gas during the
compression stroke. It follows, therefore, that the expansion
and compression processes, through which the gas in the cylinder
of a heat engine is carried, must be different if the engine is to
be capable of doing external useful work.
When the gas in a heat engine is carried through a number
of processes and returned to its initial condition, it is said to
pass through a cycle of operations.
An ideal cycle for the heat engine was suggested by Carnot.
Carnot's cycle consists of four processes as follows :
(1) Isothermal expansion (temp. T^y
(2) Adiabatic expansion (from temp. T^ to temp. T^')
(3) Isothermal compression (temp. T2)
(4) Adiabatic compression (from temp. T^ to temp. 2\)
This cycle is represented in Figure 155. AB represents
(1) isothermal expansion at the temperature Ty BO repre-
sents (2) adiabatic expansion, during which the temperature
of the gas falls from T^ to Tg- ^^ represents (3) isothermal
compression at the temperature T^. DA represents (4) adia-
batic compression, during which the temperature of the gas
rises from T^ to 2\. The work done by the gas in process (1)
is represented by the area aABb, and in process (2) by bBOc.
The total work done by the gas is therefore represented by
aABQc, that is, the area under the line ABO. The work done
on the gas in process (3) is represented by the area cODd and
in process (4) by dBAa. The total work done on the gas is
therefore represented by cODAa, that is, the area under the
line QBA. The work done by the gas exceeds the work done
on the gas by an amount represented by the area ABOB.
Call this work W.
242
HEAT
During the process (1), a certain amount of heat must be
supplied to the gas (Section 226). Call this heat R^. During
process (3) a certain aniount of heat is rejected by
. the gas. Call this heat H^. Then the 4ifference
between the heat received and the heat rejected by
the gas during the cycle is H^ — H^. The im-
B portant results of the cycle are,
therefore, as follows :
H-^^ heat units are taken up by the
gas at the temperature T^
M^ heat units are rejected by the
gas at the temperature T^.
W units of work have been
done by the gas.
In other words, -9j — ff^
units of heat have disap-
peared in the operation, and
W units of mechanical energy have made their appearance. It
follows, therefore, from the first law of thermodynamics, that.
Fig. 155,
d b i
— Carnot's Cycle.
EFFICIENCY OF AN IDEAL HEAT ENGINE
228. Carnot imagined an engine in which this theoretical
cycle might be realized. This he called the ideal heat engine.
The study of Carnot's theoretical cycle and ideal engine leads
to a number of important principles of thermodynamics.
The efficiency of a heat engine is defined as the ratio of the
heat transformed by the engine into work to the total heat
received by the engine. For Carnot's ideal engine, we have,
therefore,
R
^1
(70)
in which R is the efficiency.
The most important property of Carnot's cycle is that it may
be reversed. That is, the ideal engine may take a quantity
of heat R^ from a source at a temperature T^ and deliver a
quantity of heat Rj^ at a higher temperature T^, providing an
amount of work represented by the area ABCD, Figure 155, is
THERMODYNAMICS
243
done on the gas during the cycle. Garnet's cycle is therefore
called a reversible cycle.
The more important principles deduced from a study of
Carnot's cycle are as follows :
An engine having a reversible cycle has the greatest possible
efficiency ;
All engines having reversible cycles, whatever the nature of
the gas or working substance, have the same eflaciency ; and
The efficiency of a reversible engine depends only upon the
temperatures Tj and T^^ between which the engine works. As a
matter of fact it may be shown that the expression for the
efficiency of a reversible engine given above is equivalent to
(71)
This relation leads to the conception of a new scale of tem-
peratures, depending only upon Carnot's cycle and independent
of the nature or properties of any particular kind of matter.
Lord Kelvin devised such a scale, called the thermodynamic
scale, and found that it did not differ materially from that of
the hydrogen thermometer.
THE INDICATOR CARD
229. The indicator card is a Watt diagram extensively em-
ployed by engineers for determining the conditions under
which a steam en-
gine is operating.
A device is at-
tached to tlie en-
gine cylinder
whereby the dia-
gram is automati-
cally drawn by the
moving piston and
the varying pres-
sure of the steam.
In Figure 156 a diagram of this kind is shown. The ordinate
of the point A represents the pressure at which steam is admitted
A
B
^
^
^Y///
y?7^
E
L)
Fio. 156. — The Indicator Card.
244 HEAT
to the cylinder. During a portion of the stroke corresponding
to AB steam flows into the cylinder at boiler pressure. At £
communication with the boiler is cut off. During the rest of
the stroke the steam expands (adiabatically, nearly). At the
end of the stroke a valve (exhaust port) is opened and the pres-
sure falls to that of the outside air (or condenser). During
the return stroke the pressure remains constant from D to U.
At U the exhaust port is closed and the steam remaining in the
cylinder is compressed. At F the valve admitting steam from
the boiler is opened and the pressure at once rises to that of
the boiler.
The area of the diagram ABCDEF is proportional to the heat
energy transformed into work during one stroke of the engine.
If, therefore, the length of stroke of the engine and the boiler
pressure are known, the work done per stroke may be calcu-
lated from the measured area of "the card."
Problems
1. How much heat can be developed by a weight of 1 kg. in falling
5 m.? Assume the transformation to be complete.
2. If 5000 ft. -lb. of work are expended in stirring a half pound of water,
what will be the rise in temperature of the water? Assume no heat is lost
in the operation.
3. What is the theoretical efficiency of a steam engine taking steam at a
temperature of 160° C. and exhausting into a condenser at 40° C?
4. The temperature in the cylinder of a gasoline engine at the moment
of explosion is 1800° C. and at the moment of exhaust is 800^ C. What is
the theoretical efficiency of the engine ?
5. Assuming a refrigerating machine to be a perfect engine, how much
work is required to take 1000 calories of heat from a room at — 10° C. and
deliver it to the cooling pipes at 60° C. ?
PART III
ELECTRICITY AND MAGNETISM
ELECTRICITY AND MAGNETISM
CHAPTER XX
ELECTROSTATICS
230. It was discovered about 2500 years ago that a piece of
amber rubbed with silk acquires the property of attracting to
itself small, light bodies, for example, bits of paper, chaff, etc.
This condition of the amber, after being excited by frictional
contact with the silk, is known as electrification. The amber
while in this condition was said to be electrified. Electrifica-
tion was for centuries considered to be peculiar to amber. It
was only about 300 years ago that the discovery was made
that other bodies may be electrified. It is now known that any
substance may be electrified by frictional contact with a dis-
similar substance.
That branch of physics which deals with electrified bodies
and the force actions between them is called electrostatics.
POSITIVE AKD NEGATIVE ELECTRICITY
231. An electrified body is said to possess a charge of electric-
ity. Experiment shows that there are two kinds of electricity,
which are distinguished as positive and negative.
If a dry rubber rod is stroked with cat's fur, it becomes
strongly electrified. A dry glass rod rubbed with silk also
acquires this property in a marked degree. The rubber rod
and the glass rod under these circumstances both behave like
amber in the experiment referred to, in that they exhibit
marked attraction for small, light bodies. Examination will
show, however, that the electric charge possessed by the glass
rod is in some important respects different from that possessed
by the rubber rod. For example, if the rubber rod, after being
electrified, is hung in a stirrup which is suspended by a thread,
247
248
ELECTRICITY AND MAGNETISM
as shown in Figure 157, so as to be free to turn, and the glass
rod is brought near as shown in the figure, the charge on the
glass rod exhibits strong attraction for
the charge on the rubber rod. If now, in
place of the glass rod, a second rubber rod
be used, the suspended rubber rod, instead
of being attracted, will be repelled, thus
showing that there is a difference in the
nature of the electric charges on the glass
and rubber rods.
Fig. 157. — Illustrating the Attraction of Unlike Charges.
It should be
carefully noted
that the force
actions referred
to are between
the electric
charges possessed by the glass and rubber rods and not between
the rods themselves.
The charge on the glass rod is called a positive ( + ) charge,
and a body is said to be charged positively when it has a charge
like that which appears on a glass rod when it is rubbed with
silk.
The rubber rod in the above experiment is said to possess a
negative ( — ) charge and a body is said to be charged negatively
when it possesses a charge like that which appears upon a rubber
rod when it is rubbed with cat's fur.
The above experiment may be repeated, placing the charged
glass rod in the stirrup. Under these circumstances the sus-
pended rod is attracted by the charged rubber rod, but is re-
pelled by a similarly charged glass rod. As a result of these
experiments we are led to the conclusion that like charges repel,
while unlike charges attract.
THE SINGLE FLUID THEOEY
232. In the discussion above given no reference has been
made to the nature of electricity, and the experiments referred
to, together with all of those with which we have to deal in
the present discussion, may be made without reference to the
ELECTROSTATICS 249
real nature of that which we call electricity. Nevertheless
various attempts have been made to explain its nature and to
formulate a theory which will account for the various phe-
nomena of electrostatics. One of the theories advanced is the
so-called single fluid theory. This theory assumes that elec-
tricity is a fluid, that all substances have a certain affinity for
this fluid, and when normal in this respect, possess a certain
amount of electricity which renders them neutral as to electric
force actions on other bodies in similar condition. A negatively
charged body, under this theorj', is one possessing less than the
normal amount of the electric fluid. A positively charged
body is one which possesses an excess of the fluid. This theory
was advocated by Benjamin Franklin.
THE TWO FLUID THEORY
233. The two fluid theory assumes that there are two kinds
of electric fluid, the positive and the negative. A positively
charged body is one which possesses an excess of the positive
fluid. A negatively charged body is one which possesses an
excess of the negative fluid. An uncharged (iieutral) body is
one which possesses equal amounts of the positive and nega-
tive fluids. This is the theory which is commonly adopted in
explaining the various phenomena of electrostatics. The two
fluid theory affords the simplest explanation of these phenom-
ena, and providing it is borne in mind that we make use of it
simply as a means of facilitating discussions of this character,
it may be used without hesitation.
THE DIELECTRIC THEORY
234. Another theory of electrostatics is known as the dielec-
tric theory and has been championed by such noted physicists
as Faraday and Maxwell. This theory assumes that electric
charges are simply manifestations of a certain kind of strain in
the ether (Section 211). Under this theory to charge a body
is to strain the ether near the body and to discharge a body is
to relieve existing ether strain in its neighborhood.
250 ELECTRICITY AND MAGNETISM
THE ELECTRON THEORY
235. The most modern theory of electrification is the
electron theory. This is really a fluid theory and is somewhat
analogous to the single fluid theory of Franklin (Section 232).
It differs from the old single fluid theory in that it assumes
that the fluid is negative. It maintains that electricity has
atomic structure, and that small particles called electrons are
associated with the atoms of matter. These electrons may,
under certain conditions, be separated from the atoms with
which they are normally associated. When a number of elec-
trons have been removed from a body in normal (neutral) con-
dition, the body is left "positively charged." When a body
possesses more than its normal amount of electrons, it is nega-
tively charged.
THE CLASSIFICATIOX OF BODIES WITH RESPECT TO THE
CHARGES WHICH APPEAR UPON THEM
236. It is found that certain substances acquire positive
charges under almost all circumstances of frictional contact
with other bodies. Certain other bodies appear to take on a
negative charge under the same circumstances. There are
again other bodies which acquire sometimes positive and some-
times negative charges, depending' upon the nature of the body
with which they are brought into contact. Generally .speak-
ing, it is possible to tabulate the various substances in such
manner that if a substance at the top of the list is brought
into frictional contact with one lower in the table, the upper
one acquires a positive charge and the lower one a negative
charge. Such a table is given below :
Cat's fur
Polished glass
Woolen stuffs
Feathers
Wood
Paper
Silk
Thus it is possible by stroking feathers with cat's fur to give
them a negative charge or by stroking them with silk to give
ELECTROSTATICS 251
them a positive charge. It will be evident, therefore, that the
charge acquired by a body when brought into frictional contact
with a second body depends, not only upon the nature of the
body itself, but also upon the nature of the body with which it
comes into contact.
CONDUCTORS
237. If an electric charge is imparted to one end of a long
wire, a portion of this charge immediately spreads to the more
remote extremity of the wire. The wire, under these circum-
stances, is said to conduct the electricity from the nearer to the
farther end. A substance which is capable of doing this is
called a conductor of electricity. It is found that certain sub-
stances conduct electricity with great readiness, others less
readily, and certain substances with the greatest difficulty.
Hence the various substances are divided in a general way
into two classes : good conductors or simply conductors, and
very poor conductors or insulators. Below are given tables
of the more common conductors and insulators :
CONDUOTORS
Insulators
All metals
Shellac
Charcoal
Amber
Plumbago
Resins
Concentrated acids
Glass
Metallic ores
Mica
Water
Ebonite
Moist earth
Silk
Dry paper
Porcelain
The electron theory explains conduction by assuming that in
conductors the electrons have considerable freedom of motion,
while in insulators they have little or none at all.
ELECTROSCOPES
238. An electroscope is a device for detecting the presence
of an electric charge. There are several kinds of electroscopes,
of which the following are the most convenient for such
studies as are undertaken in this course.
252
ELECTRICITY AND MAGNETISM
J
'II
_Llll
Fig. loS. — Pith Ball Electroscope.
(a) The pith ball electroscope consists of a very light ball,
conveniently of pith, suspended by a silk thread, as shown in
Figure 158. If this pith ball
is given a positive charge, it
will be attracted by a nega-
tively charged body or repelled
by a positively charged body.
It can therefore be used, not
only to detect the presence of
charged bodies, but will also
distinguish a positive from a
negative charge.
(5) The stirrup and charged
rod. The arrangement repre-
sented in Figure 157 consti-
tutes an electroscope by means
of which the presence of a
charge upon any body may be readily detected and identified,
(c) The gold leaf electroscope consists of two slender strips
of gold foil suspended from a metal
rod which terminates at the top in
a knob, as represented in Figure 159.
For convenience the instrument is
mounted in a glass vessel, as shown
in the figure. When so mounted, it
is protected from outside disturbances
such as air currents. Care should be
taken to insulate the rod where it
passes through the stopper of the
glass vessel. This is conveniently
done by surrounding the rod at this
point by shellac or amber or some simi-
lar insulating material. The indica-
tions of this instrument depend upon
the fact that like charges repel, so that
two bodies which carry like charges
tend to separate. Suppose, for example, the knob in the
electroscope represented in Figure 159 is stroked with cat's fur.
Fig. 159. — Gold Leaf Electro-
scope.
ELECTROSTATICS 253
The rod together with the leaves will acquire a negative
charge. Those portions of the charge which reside on the
leaves repel one another with the result that the slender gold
leaves separate by a certain amount. This separation of the
leaves is an indication of a charged condition in the knob and
the attached gold leaves.
THE EQUALITY OF THE POSITIVE AND NEGATIVE CHARGES
DEVELOPED BY FEICTIONAL CONTACT
239. Experiment shows that in any case of the development
of electricity by the frictional contact of two bodies equal amounts
of positive and negative electricity are developed. Thus when
a rubber rod is stroked with cat's fur a certain amount of nega-
tive electricity appears upon the rubber rod. An examination
of the cat's fur will reveal the fact that an equal amount of
positive electricity has been
developed upon it. In the
case of the glass rod rubbed
with silk, the silk acquires
an amount of negative elec-
tricity equal to the positive
which appears upon the
glass rod. A simple experi-
ment for demonstrating this Fig. 160.
fact is the following : A,
Figure 160, represents a rod of sealing wax. Over the upper
end B is fitted a cap of flannel to the top of wliich is tied a
silk thread C. If now the rod A is rotated in the cap B, both
the rod and the cap will become charged, the rod with negative
and the cap with positive electricity. So long as the cap re-
mains on the rod, however, these charges will be unable to
manifest their presence upon outside bodies, since the force
action due to the charge on B is neutralized by the force action
due to the charge on A^ the one being positive and the other
negative. If for example, the rod A with the cap B is pre-
sented to the suspended rubber rod represented in Figure 157,
no force action will be apparent. As soon as the cap is re-
moved, which is conveniently done by means of the silk thread
254 ELECTRICITY AND MAGNETISM
O, it is found that both the rod A and the cap B are in condi-
tion to influence the electroscope, thus showing that they are
both charged. They influence the electroscope oppositely, thus
showing that their charges are unlike. This and similar ex-
periments lead to the conclusion that in every case of fric-
tional contact equal amounts of positive and negative electricity
are developed.
Under the electron theory this result follows as a matter of
course, all electrons removed from the cap are added to the rod.
INDUCTION
240. Since the electrons in a conductor have a certain free-
dom of motion, it follows that a neutral conductor when brought
into the presence of a charged body will show charge, since the
electrons, responding to the influence of the charged body, will
be attracted or repelled according to the nature of the charge,
and the neutral condition of the conductor will be disturbed.
For example, in Figure 161, let B represent an uncharged con-
ductor. Let the body A, charged positively, be brought into
the presence of B. Then the two electricities which, before A
was brought up, neutralized each other at all points on B, will
now be separated
through the influ-
ence of the charge
on A in the manner
indicated in the fig-
ure. The electrons
in B will be drawn in large measure to the nearer end of B,
thus giving that end a negative charge, and the farther end,
because of the deficit of electrons, will exhibit positive charge.
Evidently this condition of charge on .B is a temporary one ;
and if the body A is removed, the electrons on B will distribute
themselves over the entire body, thus reducing it to the neutral
condition in which it was assumed to be at the beginning of the
experiment.
The body B may be given a permanent charge by induction
in the following manner : While B is in the presence of A and
the charges upon it are separated as indicated in Figure 161, let
Fig. 161. — Charging by Induction.
ELECTROSTATICS 255
B be placed in communication with the earth by means of a
wire, or by touching it with the finger. Under these circum-
stances a number of electrons will flow from the earth to the body
B in response to the attraction of the positive charge on A.
Evidently the group of electrons on B near A will have no
tendency to flow to ground, since it is held or " bound " by the
attractive influence of A. Since there is now an excess of
electrons on B, it is evident that when the connection between
B and the earth is broken, B will have a permanent negative
charge. This process is known as charging by induction. It
will be observed that the charge which the body B acquires in
this process is opposite in sign to that of the inducing charge
upon A. Had A possessed a negative charge, then upon con-
necting B to the earth a number of electrons would have been
repelled, by the charge on A, to the ground, and B would have
acquired a permanent positive charge (deficit of electrons).
An instructive method of showing that the charges upon B
are separated as indicated in Figure 161 is that illustrated in
Figure 162. B and C are two conductors which are placed in
contact with one another as shown. In this position they con-
stitute a single
conductor. The
inducing charge
upon A is now
brought up as
in the former Fig. 162. — Method ol securing Permanent Charges by
Induction.
experiment, and
the separation of electricities upon the body CB takes place as
indicated. If now C is separated from B and then the body A
with the inducing charge is removed, C will have a permanent
positive charge and B a permanent negative charge. The
charges upon Cand 5 may be identified in the usual way by
causing them to approach the charged electroscope.
THE ICE PAIL EXPERIMENT
241. A very instructive experiment in induction is the fol-
lowing : Consider a hollow conductor A, Figure 163, to which
is attached an electroscope B. Let a charged body C be slowly
niubiiBi as Huuwu. xii i/iiiH pusitiuu uiey uou-
256
ELECTRICITY AND MAGNETISM
lowered into the hollow conductor. Evidently the conductor
AB^ consisting of the hollow vessel with attached electroscope,
will become charged by induction upon the approach of the
body Q. The negative
electricity will be attracted
to the inside of the hollow
vessel A, and the positive
repelled to the outside of
A and to the electroscope.
If the body C is removed
without coming in contact
with the body A, the tem-
porary charged condition
of A and B will disappear
exactly as in the former
experiment in induction.
If, however, before being
removed, the body C is
brought in contact with the
body A on the inside and
then removed, the system
AB will retain a permanent charge. In carrying out this ex-
periment the following important observations are made :
(1) That once the body Q is well within the body A it may
be moved about from point to point without altering the charge
on B.
(2) It may be brought in contact with the inner wall of A
without affecting the charge on B.
(3) When the body G is taken away, it is found to be entirely
discharged.
(4) If the uncharged body C be again lowered into the ves-
sel A and brought in contact with it, it will come away a second
time without charge ; whereas if it is brought in contact with
the outside of the vessel A, it will take away a portion of the
charge upon A.
These facts tend to show, first, that the induced charges upon
the system AB are exactly equal in amount to the inducing
charge upon C. If this were not so, then upon bringing the
Fig. 163. -
+ +
- Ice Pail Experiment.
ELECTROSTATICS 257
body Q in contact with A there would be a little more positive
charge on G than is necessary to neutralize the negative charge
on the inside of J., and this surplus would either come away
with the body C, which experiment shows is not the case, or it
would flow to the outside of A, thus increasing the positive
charge upon the system AB. This would be evidenced by
wider separation of the electroscope leaves. No such increase
in the charge on B is shown ; therefore the negative charge in-
duced on the body A must be exactly equal to the inducing
charge upon the body Q. Second, a free charge on any body
is confined to the surface of that body. This is evident from
the fact that in the operation referred to as No. 4 above, when
the uncharged body (7 is lowered the second time into the hol-
low vessel and brought in contact with its inner surface, it
comes away without charge. If there were any charge within
the hollow conductor, it would be shared by the body C.
The location of an electric charge upon the surface of a con-
ductor follows from the fact that the different parts of a charge
tend to separate from one another as widely as possible, and this
separation is affected in the largest degree, of course, by the
spreading of the charge over the surface of the body.
This experiment on induction may be repeated, making us6
of several hollow conductors, one within the other, each care-
fully insulated from the others. Under these circumstances,
when the body Q carrying the inducing charge is lowered into
the inside vessel, all of the hollow vessels become charged by
induction, — negative charge appearing upon the inside of each,
positive charge appearing upon the outside of each and on the
electroscope attached to the outside vessel. It is found that
once the inducing charge G is well within the inner vessel, it
may be moved about from point to point or brought in contact
with the inner wall of this vessel without altering the charges
upon the other vessels. Furthermore, the vessels may be
brought in contact with one another without altering the
charge upon the outside of the outside vessel. These facts
tend to show that the induced and the inducing charge in the
process of charging by induction are equal.
It is here assumed that the influence of the inducing charge
258
ELECTBICITT AXD IMAGXETISM
C does not extend beyond the vessel into which it is lowered.
In other words, its entire influence is confined to the vessel A.
This is true only when the vessel A quite completely surrounds
the body C. Thus in the process of charging by induction re-
presented in Figure 161, the charges induced upon S are al-
ways less in amount than that upon the body A, since a part of
the influence of A extends right and left to other bodies, and
its influence is not limited to the body 5 as is true in the ice
pail experiment in which it is assumed that the body A quite
completely surrounds the body C.
THE ELECTROSTATIC FIELD
242. The electrostatic field in the neighborhood of a charged
body is that region of space into which the influence of the charge
extends. We have seen that when one charged body is brought
into the presence of a second charged body, there is a force
action between them. This force action, whether it be of
attraction or repulsion, becomes greater the nearer the two
bodies are brought together, and becomes smaller as the bodies
are more widely separated. While this force action between
the two bodies falls off rapidly as the two bodies are carried
farther and farther apart, it becomes zero theoretically only
when the bodies are separated by an infinite distance. Theo-
retically, therefore, the
electrostatic field which
surrounds a charged body
extends to infinity, assum-
ing that there is but the
one charged body. Prac-
tically, however, the field
about a charged body is
quite limited in extent.
ELECTKOSTATIC LI^'ES OF
FORCE
Fig. 164. — Field surrounding an Isolated
+ Charge.
243. It is convenient for
many purposes to represent
the electrostatic field about
ELECTROSTATICS
259
a charged body by a series of lines. These lines are so drawn
that they represent at each point the direction of the force action
which a small charged body would experience if placed at that
point, or the direction in which a small charge would tend to
move if placed at the point in question. It is customary to place
arrowheads upon these
lines pointing in the
direction in which a
small positive charge
would tend to move.
The radial lines drawn
in Ij'igure 164 afford
a picture of the elec-
trostatic field which
surrounds the isolated
positive charge on the
sphere A. The lines
drawn between the bodies A and B in Figure 165 represent the
electrostatic field surrounding the two charged bodies A and B.
These lines, which we may imagine to extend from any
charge, are called electrostatic lines of force.
Fig. 165. — Field surroundiag Two Unlike Charges.
THE FOKCE ACTION BETWEEN TWO CHARGES
244. It may be shown very readily by experiment that the
force action between two electrostatic charges depends upon
the magnitude of the charges, and upon the distance which
separates them. It is easily shown that when a rubber rod, for
example, is strongly charged, i.e. when it carries a large charge,
it will exert a larger force action upon a second charge than
when it carries but a small charge ; and in the same way the
force action may be shown to depend upon the magnitude of
the second charge. Experiment also shows that the force varies
inversely as the square of the distance between the charges.
This dependence of the force action between two charges upon
the magnitudes of the charges themselves and upon the distance
which separates them, as determined by experiment, may be
expressed algebraically as follows :
(22
(72)
260
ELECTRICITY AND MAGNETISM
in which Q and q represent the magnitudes of the two charges,
and d the distance which separates them.
The force action between two charges depends upon the
medium which fills the space between the charges. Equation
(72) gives the force when the charges are in a vacuum. The
force action between two charges in air is practically the same
as in a vacuum.
THE ELECTROSTATIC UNIT OF CHARGE
245. The electrostatic unit of charge is defined from Equa-
tion (72) as follows : Let it be assumed that two equal charges
are chosen of such magnitude that when they are placed one
centimeter apart in a vacuum, the force action between them
is one dyne. Then these charges are said to be unit charges.
In other words, the c. g. s. electrostatic unit of charge is that
charge which placed at a distance of one centimeter from a simi-
lar charge will experience a force action of one dyne.
ELECTROSTATIC FIELD INTENSITY
246. The electrostatic field intensity at any point is the force
action per unit charge placed at that point. This may be stated
in another way. Re-
ferring to Figure
166, and having in
mind the point p at
) a distance d from
the charged body^i,
it is found that if we
bring to the pointy
different charges one
after another, the
forces which these charges experience while at the point p are
proportional to the magnitudes of the charges, that is to say,
Fxq
in which F is the force experienced by the charge q when
placed at the point p. This may be written as follows :
F=f.q (73)
Fig. 166.
ELECTROSTATICS
261
in which /, the proportionality factor, is the field intensity at
XT
the point p. From Equation (73), / = — , i.e. the field in-
tensity equals the force action per unit charge.
Comparing Equations (72) and (73), and remembering that
Equation (72) is the general expression for the force action
between any two charged bodies and that it will therefore be
applicable to the case under discussion, evidently.
(74)
That is, the field intensity in the neighborhood of a charge Q
and at a distance d therefrom is equal to the magnitude of the
charge divided by the square of the distance of the point in
question from that charge.
In the above discussions on the force actions between charged
bodies it has been assumed that the bodies upon which the
charges are supposed to rest are very small as compared with
the distance d involved in the expressions.
THE SCREENING EFFECT OF A HOLLOW CONDUCTOR
247. It will be evident from the foregoing discussions that
lines of force terminate upon
charges. A line of force may
be thought of as beginning
upon the surface of a posi-
tively charged body and ex-
tending to the surface of a
negatively charged body.
For example, the lines of
force involved in the ice pail
experiment described in Sec-
tion 241, before the body
is brought in contact with
the inner wall of the vessel,
would be something like
those represented in Figure
167. As the body (7 is caused Fig. i67.
262
ELECTRICITY AND MAGNETISM
to approach the inner wall of the vessel more closely, the con-
ditions would be more like that represented in Figure 168, in
which the lines of force extend-
ing between A and Q are now
grouped together in the narrow
space which separates these
bodies while the lines of force
without the vessel A remain
unchanged. Finally, when the
body G comes into contact with
the body A, the lines of force
within A disappear entirely,
the lines of force on the out-
side of A still remaining un-
disturbed ; that is, there is no
electrostatic field within A after
G is brought in contact with
its inner wall, Figure 169.
This effect does not depend upon the magnitude of the charge
upon the outside of A. Therefore the conditions repre-
sented will be true for any
value of charge upon the
outer walls of the vessel A.
This discussion shows that
the effect of the charge upon
the surface of a hollow con-
ductor is limited to those
regions which lie without the
conductor and does not extend
to the space inclosed by it.
In other words, it is possible
to screen a given region
from electrostatic effects by
surrounding it with a metal-
lic conductor.
Fig. 168.
Fig. 169.
ELECTROSTATICS
263
LINES OF FORCE AKE PERPENDICULAR TO THE SURFACE OF
A CHARGED CONDUCTOR
248. It will be noticed that in all of the electrostatic fields
which have been represented in the preceding figures, the lines
of force drawn in each case leave the conductor at right angles
to its surface. That is, the direction of the electrostatic field
in the neighborhood of a charged body is represented as being
normal to its surface. That this is true follows at once from
the nature of the conductor. Let it be assumed that the field
near the surface of a charged conductor is not perpendicular to
the surface. There will then be a component of this field
parallel to the surface. Under the influence of this component
of the field the charge on the surface of the conductor will tend
to move along the surface. This motion of a charge on the
surface of the conductor will continue until the electrostatic
field is everywhere at right angles to the surface of the charged
body, that is, until the field component parallel to the surface
disappears.
THE DISTRIBUTION OF CHARGE ON A CONDUCTOR
249. Except in one or two special cases the electrostatic
charge upon a body is not distributed uniformly over its sur-
face. This will be evident from the fact that the different
+ + A B + +
+ < ^ +
Fig. 170. — Distribution of Charge.
portions of a charge repel one another, and that a charge when
placed upon a conductor will so distribute Itself that this ten-
264 ELECTRICITY AND MAGNETISM
dency of the different portions to separate is satisfied as far as
possible under the circumstances. It is easy to see, therefore,
that upon such a body as is represented in AB, Figure 170, a
greater portion of the charge will be distributed over the ends
of the long conductor AB than is to be found on its central
portions. In the same way there will be a heaping up or a
concentration of the charge upon the corners of a square con-
ductor (7, and in the case of an egg-shaped body B, the concen-
tration of charge will be greatest at the point. It is assumed,
of course, that each of the bodies here discussed is free from
the influence of charges on other bodies.
In general there is a concentration of the charge carried by
any body about its angles and corners or those portions of its
surface which are of sharp curvature.
SURFACE DEySITY OF CHARGE
250. The concentration or heaping up of charge at the
corners and angles of an irregular charged body is usually
expressed by sa\-ing that the surface density of the charge on
these portions of the body is great.
The surface density of an electrostatic charge is the quantity of
electricity per unit area of the surface.
THE DISCHARGTSG ACTION OF A POTST
251. As indicated above, the surface density of a charge
upon a conductor is greatest where the curvature of the surface
is greatest. It will be easih' understood, therefore, that any
sharp point upon an electrical conductor will be a region of
great surface density of charge.
The tendency of any charged body to lose its charge, that is to
say, to be discharged by giving up a portion of its charge to
the air ^\hich comes in contact with it, and the particles of
dust, etc., which are carried in the air, depends upon the surface
density of the charge. The action of a point upon a charged
conductor is therefore to facilitate the escape of the charge
which is upon it. In other -words, a point on a charged body
tends to discharge it.
ELECTROSTATICS 265
Problems
1. What is the magnitude and direction of the force acting on a charge
of 15 c. g. s. units ( + ) when placed at a distance of 20 cm. from a charge of
25 c. g. s. units ( — )?
2. The force between two charges is F. What will be the foTce
between them if both charges are doubled?
3. If the force between two charges is F when they are separated a
distance d, what will be the value of the force when the distance is increased
to5d?
4. Three charges, Qi = + 20, Qa = + 30, and Qa = — 40 are placed at the
corners of an equilateral triangle, each side of which measures 20 cm.
What is the magnitude and direction of the resultant force acting on each
charge ?
5. What would be the magnitude and direction of the force acting on
a charge of + 10 placed at the center of the triangle of problem 4 ?
6. A charge is placed at each corner of a hexagon. The charges taken
in order around the figure are, + 10, — 20, + 30, — 40, + 50, and — 60. A
charge of + 25 is placed at the center of the hexagon. What is the
magnitude of the force on this charge ?
7. What is the field intensity at a distance of 20 cm. from a, concen-
trated charge of 500 c. g. s. units?
8. Two charges, + 40 and — 50, are separated by a distance of 30 cm.
What is the field intensity at a point midway between them ?
9. What is the field intensity at the center of the triangle of problem 4
due to the charges at the corners ?
10. What is the field intensity at the center of the hexagon of problem 6
due to the charges at the corners ?
ELE3GTROSTATI0 MACHINES
CHAPTER XXI
THE FRICTION MACHINE
Fig. 171. — The Friction Machine.
252. Various devices are employed for the development of
electrostatic charges rapidly and in large quantities. One
of the earlier forms of electrostatic machine is the friction
macliine. This device, which is represented in Figure 171, is
a machine for developing electrostatic charge by friction. The
essential parts of
the apparatus
are shown in the
figure. -A is a
- disk of glass
which is caused
to revolve on
the axis 0. At
^ is a clamp
fitted with cha-
mois skin pads which presses upon both sides of the glass disk
as it revolves. The friction between A and B develops elec-
trostatic charges on these bodies. The glass plate, revolving
in the direction indicated, passes on from the body B bearing
on both of its faces positive charge, while B is charged nega-
tively. At is placed a metallic comb which presents a num-
ber of sharp points to the face of the disk. The comb is a part
of a large conductor I) as indicated in the figure. It is upon
D that the positive electricity is accumulated. The manner in
which D becomes charged is as follows : When the positive
charge upon the glass plate is brought into the presence of the
comb O the conductor OB becomes charged by induction, posi-
tive charge appearing at the farther end of D, the negative
266
ELECTROSTATIC MACHINES 267
electricity being drawn into the comb. The discharging action
of the points coming into play, this negative electricity is dis-
charged upon the face of the glass plate, to which it is attracted
by the positive charge which that body carries. The positive
charge upon the glass plate is thus neutralized and the plate
passes toward B where it is again charged by frictional contact
with that body. The friction machine is very inefficient.
Most of the work put into the machine is transformed into
heat.
THE ELECTROPHORUS
253. The electrophorus is a device for the rapid accumula-
tion of charge which depends for its action upon the principle
of induction. It consists essentially of a cake of sealing wax
or resinous material A, Figure 172,
and a disk B of conducting mate-
rial, provided with an insulating
handle. The electrophorus is used ' ' "
in the following manner : The
cake of sealing wax A is first rubbed
with cat's fur. In this operation it .
becomes charged negatively. The ^^^ m. - Electrophorus.
disk B is now brought into the
presence of the charge on A and becomes charged by in-
duction, positive charge appearing upon the lower side of the
disk and negative upon the upper side of the disk as indicated
in the figure. If now the disk is " grounded," that is brought
into communication with the earth, the repelled negative charge
upon it will pass off to the earth. The ground connection
being removed, there remains upon the disk B a permanent
positive charge which may be carried upon the disk B and
made use of as desired. This operation may be repeated
again and again without any diminution of the original charge
upon A.
THE TOEPLER-HOLTZ MACHINE
254. The Toepler-Holtz machine, like the electrophorus, de-
pends for its action upon the principle of induction. It consists
essentially of two glass plates, the one stationary and the other
268
ELECTRICITY AND MAGNETISM
arranged to revolve in close proximity to the first plate. These
plates are represented by A and B, Figure 173. Upon the
stationary plate A are placed two "armatures," and D, of
paper and tin foil, represented in outline by dotted lines in the
figure, the stationary plate being behind the moving plate B.
Upon the moving plate B are six conductors (metal buttons)
represented by the circles JE, F, etc., in the diagram. There is
a rod im known as the neutralizing rod, which is provided at
Fig. 173. — The Toepler-Holtz Machine.
each end with a metal brush and sharp points, the brush being
arranged to come momentarily in contact with the buttons as
the plate B is revolved. The terminals of the machine GH
are large metal conductors provided with sharp-pointed metal
combs which stand close in front of the moving plate B in the
positions shown in the diagram.
The action of the machine is as follows: Let it be assumed
that to begin with there is a small positive charge upon the
armature 0. This positive charge, acting inductively upon the
ELECTROSTATIC MACHINES 269
neutralizing rod and tlie two buttons E, F, which are momen-
tarily in contact with its brushes, gives rise to a negative charge
upon the button F and a positive charge upon the button F.
The plate being revolved, the buttons -&and J'go forward, each
carrying a free charge. The next pair of buttons coming into
momentary contact with the brushes of the neutralizing rod
become charged in a similar manner. The rotation being in
the direction indicated by the arrow, it will be seen that t^ie
buttons as they pass across the top of the machine carry free
positive charges toward the left, while those passing across the
bottom of the machine carry free negative charges toward the
right. A button charged positively, coming into the presence
of 0, will share its charge with (7, since it is caused to come in
contact with the brush /which communicates with C. It then
passes on with its residual charge into the presence of the con-
ductor Cr. This conductor Cr now becomes charged by induc-
tion, the positive electricity being repelled and the negative
attracted by the positive charge on the button. The negative
charge streams off from the metal ' comb attached to G and
neutralizes the positive charge upon the button.
Meanwhile, a button charged negatively and passing toward
the right into the pi'esence of the armature D gives up, by
touching momentarily the brush J, a portion of its charge to
that armature. Passing on into the presence of the conductor
-5, it charges that conductor by induction, repelling the nega-
tive and attracting the positive, which flows off the points of
the metal comb and neutralizes the inducing charge on the but-
ton. These two buttons, being now without charge, pass on
into the presence of the neutralizing rod and are again charged
by the action already described. The action of the machine is
therefore continuous and cumulative, the charges upon the
armatures Q and D growing steadily larger, the charges upon
the terminals of the machine Q- and H being likewise steadily
increased. When sufficient quantities of positive and negative
electricities have been accumulated upon Cr and M, a spark will
pass between the knobs L and M. Charges will again be built
up on the terminals until a second spark passes, and so on.
In addition to the charges which are carried by the metal
270 ELECTRICITY AND MAGNETISM
buttons as described above, there is a distributed charge on the
glass plate itself. It will be easily understood that those por-
tions of the glass plate which pass under the metal combs at
the extremities of the neutralizing rod will become charged by
electricity which streams off the points of the comb. This
charge distributed on the glass augments the action of those
charges carried by the buttons. The charging of the conductors
Gr and H is further augmented by the inductive action of the
charges on the armatures and I), as will be apparent upon
inspection of the figure.
THE REVERSIBILITY OF THE TOEPLER-HOLTZ MACHINE
255. A careful analysis of the operation of the Toepler-Holtz
machine will show that while it is generating electricity, work
is being done in dragging apart the metal buttons and the arma-
tures in opposition to the electrostatic force actions which tend
to hold them together. For example, the button E in Figure
173 is assumed to be charged positively, and all buttons coming
to this position receive a positive charge from the comb of the
neutralizing rod. Now the armature D is charged negatively.
There will be at all times, therefore, a force of attraction be-
tween the metal buttons which come to the position E and the
armature D. In order that the plate may be rotated in the
direction indicated by the arrow the force of attraction between
these two charges must be overcome. In other words, they
must be dragged apart in opposition to the electrostatic forces.
The same thing is taking place at the other side of the machine.
The conductors which come into the position F are charged
negatively and are then drawn away from the region of the
armature Q in opposition to the force of attraction between the
charges on Q and F. It would occur to one very naturally that
it might be possible to reverse the Toepler-Holtz machine and
make of it a device for transforming electric energy into me-
chanical energy instead of using it in the manner in which it is
commonly employed, that is, to transform mechanical energy
into the energy of electric charge. This is found to be possible,
and experiment shows that if the terminals of the Toepler-Holtz
machine are connected to a second machine from which it is
ELECTROSTATIC MACHINES 271
allowed to draw a supply of energy in the form of electric
charges, it will tend to revolve as a sort of electric motor,
running backward, that is to say, in the direction opposite to
that in which it must be turned to operate as a generator.
POTENTIAL
256. In moving a charge about in an electrostatic field it is
evident that whenever the motion is parallel to the lines of
force work is being done, since there is present a force action
upon the body being moved, the direction of which is that of
the lines of force. Work is the product of force and the distance
through which the body moves under the influence of that force
in the direction of the force. It therefore follows that if the
charge is caused to move in the direction of the lines of force,
work is done. The work done upon the charge is positive
when the body is moving in opposition to the electrostatic forces
and negative when the body moves in the direction of the elec-
trostatic forces. Referring to Figure 166, Section 246, imagine
a small charge Q to be brought from an infinite distance, or
from a region into which the influence of the charge on A does
not extend, up to the point p. From the foregoing discussion
it is evident a certain amount of work must be done in the
operation. Thus we come to associate with a charge Q, at the
point p, a definite amount of work. If Q is unit positive charge,
the work done is called the potential of the point p. That is,
the potential of a point is the work which must be done upon
unit positive charge to bring it from infinity up to that point.
The difference of potential between two points is the work
which must be done to move unit charge from one of the points
to the other. The potential difference between two points is
unity (c. g. s. unit) if one erg of work is required to carry unit
charge (c. g. s. unit) from one point to the other.
THE POTENTIAL OF A POINT DUE TO A CHARGE ^ AT A
DISTANCE r
257. Consider a point A, Figure 174, at a distance »*j from a
charge Q. Let it be required to find the potential of the point
A due to the charge Q. This potential according to the above
c>
272 ELECTRICITY AND MAGNETISM
definition is the work required to bring unit positive charge
from infinity to
Q . „ ^ tlie point A.
_l h — H A — Let us divide
I this work into
p "" T, -^ [ small parts.
1^ y^ sj Consider first
Fig. 174. — Potential due to a Charge Q at a distance 7-. ^'^^ work re-
quired to carry
the charge from B to A. This is equal to the average force acting
on the charge as it moved from B to J.,multiplied by the dis-
tance BA. The force which acts upon the charge when it is
in the position 5 is —^, (Equation 72). The force acting upon
it when at J. is -^ . Now the average force acting for all points
between B and A is less than ~ and greater than — . Let
'1 '2
it be assumed that A and B are very close together. We may
then write
average force = — ^
T T
since %>^>%
and when r-^ and r^ are nearly equal the average of r^ and r^ is
'•i''2-
We have then : work done in moving unit charge from B
to ^ = average force multiplied by distance AB, i.e.,
This is the difference of potential between the points A and B.
The difference of potential between the point A and infinity is,
(add potential differences A
BtoO, C to B, etc.)
or, W= ^ (75)
Vrj 00 / to iJ,
ELECTROSTATIC MACHINES 273
That is, the potential of the point A, due to the charge Q, is
equal to the charge Q divided by its distance from A.
EQUIPOTENTIAL LINES
258. As indicated in the above discussion on the subject of
potential, work is done in moving a charged body in an electro-
static field when the motion is wholly or partly in the direction
of the field. If the body is moved at right angles to the lines
of force, evidently no work will be done against the electrostatic
forces. To move a charge through an electrostatic field in this
manner would be analogous to sliding a weight along a hori-
zontal plane. In the latter case no work is done against the
gravitational forces. In the former case no work is done against
the electrostatic forces.
A line drawn in an electrostatic field in such manner that it is
everywhere at right angles to the lines of force is called an equi-
potential line. The equipotential lines about the charged body
A^ Figure 164, would evidently be circles concentric with the
sphere A upon which the charge is supposed to be placed. The
equipotential lines in Figure 165 are curves, more or less nearly
circular, drawn about the individual charges A and B. The
lines and curves referred to in the discussions above represent
surfaces called equipotential surfaces whicli surround the actual
charges represented in the figures. Thus the equipotential
surfaces about the charge on an isolated sphere are spherical
surfaces concentric with the charged sphere.
THE WORK DONE IN MOVING A CHARGE FROM ONE EQUI-
POTENTIAL SURFACE TO ANOTHER IS INDEPENDENT OF
THE PATH
259. The work done in moving a charged body from one
equipotential surface to another is independent of the path
along which it is moved, since if the work done were different
along different paths, we might move the charge against the
electrostatic forces along the easier path and allow it to slide
back along the more difficult one. The negative work in this
cycle would therefore exceed the positive work, and for each
274 ELECTRICITY AND MAGNETISM
cycle completed in this direction we could get out of the sys-
tem a little more work than was put in. We would have, in
such an arrangement, a device which would not only operate
automatically, but which would be an inexhaustible source of
energy. This is, of course, absurd. Therefore the work done
in moving a charge from one equipotential surface to another is
independent of the path along which it is moved.
Problems
1. What is the potential at a distance of 20 cm. from a concentrated
charge of 500 c. g. s. units ?
2. What is the potential of a point midway between two charges of
+ 100 separated by a distance 25 cm. ?
3. What would be the potential of a point midway between the charges
of problem 2 if the charges were of opposite sign ?
4. Two charges of + 50 and — 40 are separated by a distance of 20 cm.
At what point between them is the potential zero ?
5. What is the potential at the center of the hexagon of problem 6,
p. 265?
6. How much work would be required to move a charge of + 5 from
the center of the hexagon of problem 5 to each of the corners?
7. How much work would be required to move a charge of — 5 around
the hexagon of problem 5 from the charge + 10 to the charge — 40 ?
8. Does the work done in problem 7 depend upon the path? Is it the
same if the charge is carried straight across ?
9. Sketch roughly the lines of force and equipotential lines about the
charges of problem 2.
10. Sketch the lines of force and equipotential lines about the charges
of problem 4.
ELECTROSTATIC CAPACITY
CHAPTER XXII
GENERAL DEFINITION
260. When an uncharged conductor is brought into contact
with a charged conductor, it receives from the charged con-
ductor a part of its charge. The amount of charge given up by
the charged body to the uncharged body depends upon the
relative capacities of the two bodies in question. In a general
way large conductors have large capacities, small conductors
have small capacities; that is, a relatively large amount of
electricity may be placed upon a large conductor, vi^hereas in
attempting to "place the same amount of electricity upon a small
conductor much greater difficulty is encountered. This will
give a general idea of what is meant by electrostatic capacity.
A more rigid definition of capacity is given below.
THE CONDENSER
261. The capacity of a conductor depends upon the presence
of other conductors in its immediate neighborhood. This is
readily demonstrated in the following manner: Let A, Figure
175, represent a plate of metal supported on a convenient insu-
lating stand. Let this plate be connected by means of a wire
to an electroscope 0. The divergence of the leaves of this
electroscope would measure in a rough way the condition of A
with respect to free charge upon its surface. Let the plate A
be charged, say, by bringing in contact with it the metal disk
of the electrophoi-us. The plate A will acquire in this opera-
tion a strong charge which will result in a large divergence of
the leaves of the electroscope 0. Let the plate A be dis-
charged, and, after bringing near it a second similar plate B,
275
276
ELECTRICITY AND MAGNETISM
E
_l11J1
which communicates with the ground by means of a wire, let
the plate A be once more charged in the same manner as before
by bringing into contact with it the metal disk of the electro-
phorus. It will be found
that under these circum-
stances the divergence of
the leaves of the electro-
scope (7 is much less marked
than in the former case, in-
dicating that while the plate
A has been given the same
electrostatic charge as in
the former case, the charge
does not manifest its pres-
FiG. 175. — The Presence of B increases the ence in the same degree aS
Capacity oi A. • l.^ ^ a. • j. t
m the nrst experiment. In
other words, to charge A to the same degree (apparent) as
before, more electricity must be added. Hence, the capacity of
A is increased by the presence of B.
A more logical statement of the case is the following : The
potential of a point on or near the plate A is diminished by
bringing up the plate B, since _B becomes charged by induc-
tion, and its negative charge operates to lower the potential of
all points in its neighborhood. Hence to raise the potential of
the chosen point to its original value, more charge (-f) must be
added to A.
A combination of two plates (conductors)
separated by a layer of non-conducting
material (insulator) is called a condenser.
The combination shown in Figure 175
constitutes a simple condenser. In its
most common form the condenser is made
of layers of "tin foil" connected alter-
nately, separated by sheets of waxed
paper or mica. Figure 176. Experiment
shows that the capacity of a condenser is proportional to the
size of its plates and inversely proportional to the distance
between them.
^iji;
Fig. 176.
ELECTROSTATIC CAPACITY 277
SPECIFIC INDUCTIVE CAPACITIES
262. The specific inductive capacity of any material is the ratio
of the capacity of a condenser when its plates are separated by
a layer of this material and the capacity of the same condenser
having its plates separated by dry air. Thus, for example, if a
given condenser has the space between its plates filled with
plate glass, its capacity will be found to be six times as great as
if the space between the plates were filled with air. Hence, we
say, the specific inductive capacity of plate glass is six.
The specific inductive capacities of some of the more com-
mon dielectrics are given in the accompanying table :
Specific Inductive Capacities
Glass 3-10
Vulcanite .... 2.5
ParafEne 1.68-2.3
Beeswax 1.86
Mica 4-8
Shellac 2.95-3.6
Turpentine . . . 2.1.5-2.43
Petroleum 2.04-2.42
Equation 72 assumes that the specific inductive capacity of the
medium separating the charges Q and q is unity. In case the
charges are separated by a medium whose specific inductive
capacity is other than unity, the equation for the force action
between two charges is
in which Kis written for the specific inductive capacity of the
medium surrounding the two charges.
THE LEYDElSr JAR
263. The Leyden jar consists essentially of a wide-mouthed
glass bottle coated part way from bottom to top on both inside
and outside with a thin layer of tin foil. For convenience in
estabhshing connection with the inside coating, a metallic knob
is fastened to the stopper of the bottle and made to communi-
cate with the inside coating of the jar by means of a chain.
Thus, the knob on the stopper of the bottle is to be regarded
as one of the terminals of the condenser, the outside coating of
278 ELECTRICITY AND MAGNETISM
the bottle as the other. This form of condenser is used on
electrostatic machines.
SEAT OF THE CHARGE
264. An interesting experiment is performed with the dis-
sectible Leyden jar. This is simply a condenser from which
it is possible to remove one or both plates while the condenser
is charged. If a dissectible condenser is charged and then
carefully insulated and its plates removed one after the other,
they will be found to be almost without charge. If they are
now replaced, the condenser will be found to be charged. The
conclusion to be drawn from this experiment is that the dielectric
is the true seat of the charge.
Experiment shows that the dielectric between two charged
bodies is strained. It has been shown, for example, that the
linear dimensions of a Leyden jar change when it is charged.
The change in size is small but measurable. It is also well
known that if a condenser is too heavily charged, i.e. if the
dielectric is subjected to too great a strain, the dielectric will
be ruptured. If a Leyden jar is charged more and more
strongly, there will come a time when this strain in the glass
separating the plates is so great that the material of the glass
will no longer be able to withstand it and is broken down.
THE RESIDUAL CHARGE
265. If a Leyden jar is heavily charged and then insulated,
and then discharged by bringing the outside and inside coatings
into communication, and then left to stand for a few minutes,
it will be found to have upon it a small charge. Since the two
coatings were brought into metallic contact it is evident that
this residual charge could not have been left upon the coatings.
We must, therefore, in this experiment, as in the experiment
with the dissectible jar, look to the dielectric for the explana-
tion of the charge. The residual charge is explained as follows :
When the jar is first charged, the dielectric is under severe
strain. When the jar is discharged, this strained condition re-
lieves itself largely but not completely. This residue of strain
in the dielectric gradually relieves itself, or better still, distrib-
ELECTROSTATIC CAPACITY 279
utes itself to the surfaces of the dielectric against which are
the coatings. It is the residue of this strain in the dielectric
which accounts for the residual charge in the condenser. The
residual charge is not exhibited by condensers having gaseous
dielectrics. It would seem in such cases that the strain is com-
pletely relieved at the moment of the discharge of the condenser.
THE NUMERICAL SPECIFICATION OF CAPACITY
266. Consider a condenser without charge. Let it be im-
agined that a small positive charge is taken from one of the
plates and transferred to the other. This will bring about a
potential difference between the plates. The greater the charge
transferred, the greater the potential difference thus established.
Experiment shows that the charge Q, which must be trans-
ferred in this manner in order to bring about a potential differ-
ence E, is proportional to E, that is, Q (x E. This may be
expressed as follows :
Q=CE (11)
The factor is the capacity of tlie condenser. Therefore the
capacity of a condenser is that factor which multiplied by the
potential difference between the plates will give the charge on
one of them.
CAPACITY OF AN ISOLATED SPHERE
267. The potential of a point at a distance rj from a charge
'is
E=^ (75 bis)
1
This expression holds for any point outside of the charged
body. It therefore applies to a point near to or upon the sur-
face of the charged body. Therefore the potential of an iso-
lated charged sphere of radius r is
E=^
r
Now from Equation (77) the potential of a charged body of
capacity, 0, is,
Fig. 177. — Condensers in Parallel.
280 ELECTRICITY AND MAGNETISM
Comparing these expressions for E, evidently,
(7=r (78)
That is, the capacity of an isolated charged sphere is equal
to its radius.
THE CAPACITY OF CONDENSERS IN PARALLEL
268. Two condensers are said to be connected in "parallel"
when they are connected side by side with their corresponding
plates in communication with
one another. The conden-
sers (7(7, Figure 177, are con-
nected in parallel.
The capacity of two con-
densers connected in parallel
is the sum of the capacities
of the individual condensers.
If the condensers are- similar in character, it will be evident
that by connecting them in this manner we have, in effect,
simply increased the size of the plates of one condenser by the
size of the plates in .the other.
THE CAPACITY OF CONDENSERS CONNECTED IN SERIES
269. Two condensers are said to be connected in series when
the second plate of the first condenser
is joined to the first plate of the sec- A
ond. Cj and O^, Figure 178, represent
two condensers connected in series.
The capacity of two condensers con-
nected in series is obtained in the fol-
lowing manner. Let the difference of ' q
potential between the points A and B
be represented by E. Let the poten- g
tial difference between the plates Fig. 178.- Condensers in Series.
of C-y be E^ and the potential
difference between the plates of C^ be E^- Then evidently
E=E-y + E^. But_E'i=^ (Equation 77), where Q^ is the
+ I -I-
'^^_~2 c
+ + + +
2
ELECTROSTATIC CAPACITY 281
charge upon one of the plates of Cj and E^=-^. Therefore,
Now there is evidently a condenser of such capacity that if
placed by itself between the points A and B it will completely
take the place of the combination Cj and 0, as represented in
the figure. The capacity of this condenser will be equal to the
capacity of the combination Cj and Cg. Call this capacity O.
If this third condenser is substituted for the combination
shown in the figure, we may write.
Combining these two equations for U,
9x^914-9^
But Q, is equal to Q^ from the following considerations. When
the positive charge Q^ appears upon the upper plate of Op an
equal negative charge appears upon the lower plate of (7^.
According to the electron theory this negative charge on the
lower plate of 0^ has been brought about by the transfer of a
number of electrons from the upper plate of Cg. Therefore the
deficit of electrons on the upper plate of C, is equal to the
excess of electrons on the lower plate of C^. That is, Q^ = Qi.
We have, therefore,
1=^ + 1
in which is the equivalent capacity of the combination of Oj
and 0, connected in series. Equation (47) may be written in
the following form,
i.e. the capacity of two condensers connected in series is equal to
the reciprocal of the sum of the reciprocals of their individual
capacities.
282 ELECTRICITY AND MAGNETISM
THE UNIT JAK
270. The unit jar is a Leyden jar provided with an adjust-
able spark gap. The charge which may be placed upon this
condenser depends upon the distance which separates the knobs
of the spark gap. When such a condenser is connected to the
terminals of the Holtz machine, the potential difference between
the plates and the knobs of the spark gap rises until it is suf-
ficient to break down the air between the knobs. At this
point ' a discharge occurs, and the potential difference again
builds up until a second spark passes, and so on. Evidently
the quantity stored in the condenser at the moment the spark
passes is each time the same so long as the distance between
the knobs remains unchanged.
The unit jar may be used for making a rough comparison of
the capacities of condensers in the following manner : In Fig-
ure 179 let Q rep-
resent a condenser
whose capacity is
to be measured.
It is connected as
shown, in series
with the unit jar
J between the
knobs AB of an
electric machine.
ah are the knobs of
the adjustable
spark gap of the unit jar. AB is a spark gap across which the
condenser C discharges when the potential difference between
its plates rises to a sufficient value. When the machine is oper-
ated, the following action takes place : Positive and negative
charges accumulate upon the plates of the two condensers as
indicated in the figure. When a sufficient quantity, say §', has
accumulated on the condenser J, a spark will pass between the
knobs ah. At this same instant there is a charge q on the
condenser Q also. When the unit jar discharges, it loses its
charge, of course, but the charge q remains upon C. A new
Fig. 179. — Showing how the Unit Jar is Used.
ELECTROSTATIC CAPACITY 283
charge now accumulates upon J until it again reaches the value
g, when a second discharge occurs between knobs ah. The
charge on Q is now 2 q. When the third spark passes between
ah the charge on (7 is 3 g, etc. The process is continued until a
spark passes between AB. Suppose that when this occurs three
sparks have passed between ah. Another condenser (7j is now
substituted for C, all other connections remaining exactly the
same, and the operation is repeated. Suppose that in the case
of this second condenser, four sparks pass between a and h
before the discharge occurs between A and i>, then the capacity
of Q is to the capacity of (7j as 3 is to 4.
THE ENERGY OF A CHAEGE
271. Let it be assumed that a condenser is charged by trans-
ferring a number of unit charges one after another from one
plate of a condenser to the other. Evidently the work done
upon the first unit charge will be zero, since the initial potential
difference is assumed to be zero. A very small amount of
work will have to be done upon the second unit charge, a little
more work upon the third unit charge, and so on. If the final
potential difference is ^, then it is evident that the average
work done upon one of these unit charges is \ E{ = — ^^ ] .
This follows from the definition of potential and from the fact
that the potential difference established is at each instant pro-
portional to the number of unit charges which have been trans-
ferred. But the average work done upon unit charge multiplied
by the number of unit charges brought up will give the total
work done in charging the body. Thus we have,
W= 1 QH (80)
or substituting for E its value in terms of the charge Q and
the capacity of the condenser (Equation 77) we have.
284 ELECTRICITY AND MAGNETISM
THE OSCILLATOEY DISCHARGE
272. If a highly charged condenser is suddenly discharged,
there will, in general, be an oscillation of the condition of elec-
trification between the two plates. It is as if the negative
charge on one plate, rushing over to neutralize the positive
charge on the other, overshot the mark, so that after the first
rush a certain excess of negative charge existed upon the second
plate and an excess of positive upon the first plate. In other
words, the electricity at the moment of discharge behaves as if
it had a sort of inertia. As soon as the first reversal of the
charge has taken place, the condenser will discharge again, and
again the two opposite charges in rushing together will over-
shoot; that is, this inertia effect will again come into play, and
the charge upon each plate of the condenser will be once more
reversed in sign. The successive charges grow rapidly less in
amount and quickly die away to zero value. This sort of dis-
charge is known as the oscillatory discharge.
LIGHTNING
273. A lightning flash is a disruptive electric discharge, some-
times oscillatory in character, which takes place between two
charged clouds or between a cloud and the earth.
The true character of the lightning flash was first proved by
Benjamin Franklin in his classical kite experiment. In this
experiment he " drew " electrostatic charge from passing clouds
by means of a silk kite having a hempen string. The kite was
provided with a sharp point, and the charged cloud in passing
the kite electrified the kite and string by induction, the at-
tracted charge streaming off the sharp point at the kite, the re-
pelled charge appearing at the lower end of the string. In ad-
dition to this experiment, which demonstrated the fact that
clouds, during a thunderstorm, are charged with static elec-
tricity, Franklin proceeded to identify the lightning flash with
the electric discharge by comparing the several different effects
of each. He found them, for example, to be identical in their
heating effects, lighting effects, in the production of sound, in
their mechanical effects and physiological effects.
ELECTROSTATIC CAPACITY 285
THE SOTTRCE OF THE HIGH POTENTIALS OF THUNDERSTORMS
274. The manner in which clouds receive their initial charges
of electricity is not clearly understood. Various attempts have
been made to account for these initial charges, but no entirely
satisfactory theory has yet been evolved. Assuming the pres-
ence of small initial charges, however, it is easy to explain the
development of the enormous potentials which must evidently
be present in order to cause discharges through the great dis-
tances through which lightning is known to "strike." It
would seem that lightning flashes are oftentimes one half or
three quarters of a mile in length, or even more. The poten-
tials represented by these flashes are of course very great. The
development of these high potentials is explained as follows:
Let it be assumed that the cloud is made up of minute par-
ticles of water vapor, each one bearing an infinitesimal charge
of electricity. When condensation sets in and these small
drops coalesce to form larger ones, the charges carried by the
small drops are combined into larger charges upon the larger
drops. Now it has been demonstrated (Section 267) that the
capacity of a sphere is numerically equal to its radius. Any
change in the size of a charged drop of water will therefore be
accompanied by a change in its potential, since, as we have seen,
the potential of a body varies inversely as the capacity, provid-
ing the charge remains the same. Now the volume of a sphere
is proportional to the cube of its radius, therefore, eight small
drops would combine to form one drop of twice the radius of the
smaller drops. Furthermore, this large drop will have upon it
eight times as much charge as the small drop, but since the
capacity of the large drop is but twice as great as the capacity
of one of the small drops, it will be evident that the potential
to which the large drop is charged by these eight small charges
combined upon its surface will be four times as great as the
potential of the small drops when they exist singly. Thus, as
condensation goes forward and the size of the drops increases,
the potential of the cloud rapidly rises. Positive and negative
charges are developed in nearly equal amounts among the clouds,
and the large majority of lightning flashes are therefore from
286
ELECTRICITY AND MAGNETISM
cloud to cloud. The number of flashes which reach the earth
is relatively small,
THE LIGHTNING EOD
275. The manner in which a lightning rod operates to pre-
vent lightning discharge between a cloud and a building is as
follows:
Let A^ Figure 180, represent a charged cloud ; B, a building
provided with a light-
ning rod. The positive
charge on the cloud
charges the surface of
the earth in the neigh-
borhood of the cloud by
induction as indicated
in the figure, so that the
surface of the ground
immediately beneath
the cloud, and the build-
ing B as well, are
strongly electrified.
This negative charge
upon the building B is
discharged by the sharp
pcir.t of the rod, thus
relieving the strain in
the air in the neighbor-
hood of the building.
Evidently, as this action goes forward the potential difference
between A and B grows steadily less and the danger of a dis-
ruptive discharge is diminished. In the absence of the sharp
point with its discharging action the dielectric strain between
A and B might become greater and greater until it was relieved
by a disruptive discharge between A and B^ that is to say,
until the building was " struck by lightning."
PROTECTION AFFORDED BY A LIGHTNING EOD
276. The lightning rod unquestionably protects the building
upon which it is placed providing the dielectric strain which is
Fig. ISO. — Action o( the Lightning Rod.
ELECTROSTATIC CAPACITY
287
developed in the neighborhood of the building is developed
slowly. When the dielectric strain is suddenly developed, a
lightning rod seems to afford little or no protection against a
disruptive discharge. It is customary to distinguish two kinds
of dielectric strain as developed in the thunderstorm. The
first is known as the condition of steady strain. The second
condition is known as the condition of sudden strain.
The case of the steady strain may be illustrated in the
following manner : Let A and B, Figure 181, represent the
terminals of an electric machine.
(7 is a condenser with its plates
connected between the terminals
as indicated. D and U are two
conductors providing a spark gap
for the discharge of the condenser
C. Upon the conductor D is
placed two terminals, one having
a sharp point and the other termi-
nating in a round metal knob. If,
with the connections as repre-
sented, the machine is put into
operation, the potential difference D
between the plates of C will tend Fig. isi.-TheCoDdition of "Steady
, . 1 . , 11-1 Strain."
to increase to higher and higher
values. It will be found, however, that no spark will pass
between the terminals D and U. The reason is that the dis-
charging action of the point comes into play and the dielectric
strain is steadily and continuously relieved. It will be noted
that the strain which comes upon the dielectric between D and
U is gradually developed.
The case of sudden strain is illustrated in Figure 182. A
and B are the terminals of the electric machine. (7j and Cg
are condensers. The first plate of each condenser is connected
to a terminal of the machine. The second plates of (7j and Cg
communicate with one another by means of the conductors I)
and B and the spark gap between them. As the machine is
put into operation, charges accumulate upon the condensers as
indicated in the figure, the separation of electricities upon the
288
ELECTRICITY AND MAGNETISM
second plates of the condensers being made possible by the dis-
charging action of the point at B. As the operation of the
^O
6V
t +
i_JL
c.
Fig. 182. — The Condition of " Sudden Strain.'
machine continues the
charges upon the conden-
sers increase, and the
potential difference be-
tween the plates of the
condensers and between
A and ^ is steadily in-
creased. When this po-
tential difference becomes
suiSciently great, a dis-
ruptive discharge takes
place between A and B.
The charges upon the
first plates of Cj and 0^ are neutralized by this discharge;
the charges upon the second plates of the condensers thus become
suddenly free and rush together, causing a disruptive discharge
between the conductors D and E. This discharge will taiie
place as often between E and the round knob as between
E and the sharp point. In this case the dielectric strain
which comes upon the air between D and H is developed
suddenly.
Comparing the two cases represented by Figures 181 and
182, it will be seen that in the first case the point operates to pro-
tect the rounded metal knob. Wliatever discharge takes place
passes to the point and is prevented from going to the rounded
knob. In the second case the discharge will pass as often to the
rounded knob as to the sharp point. In this case, therefore, the
sharp point affords no protection to the knob.
There are cases on record in which houses provided with
good lightning rods have been struck by lightning, the dis-
charge apparently not passing to the rod at all. It is probable
that these cases are to be explained on the theory of the sud-
den strain. Fortunately the sudden strain condition as pro-
duced in a thunderstorm is, comparatively speaking, of rare
occurrence.
ELECTROSTATIC CAPACITY 289
THE ESSENTIALS OF A GOOD LIGHTNING EOD
277. The essentials of a good lightning rod are :
(a) Continuity.
(5) Sharp points.
(c) Good ground connections.
A lightning rod which is discontinuous or has bad joints in
it is a source of danger, since whatever discharge passes the
rod may be of the nature of a disruptive discharge at the
poor connection. A heating effect is therefore produced which
may result in setting the building on fire. Evidently in order
that the discharging action referred to in Section 275 may take
place readily, it is necessary that the rod be provided with a
sharp point, or points, and that the connection between it and
the moist earth upon which the negative charge is induced be a
good one.
Problems
1. Two spheres of equal size are given charges of +.50 and —30 respec-
tively. They are brought into contact and then separated. How are the
charges altered?
2. How are the potentials altered in problem 1 ?
3. A condenser of capacity 80 and charge 400 is connected by a poor
conductor to earth. When the energy of the charged condenser is reduced
to one sixteenth its initial value, what charge remains on the condenser?
4. How is the potential altered in problem 3 ?
5. A condenser with air between its plates has a capacity of 500. When
glass is substituted for the air, the capacity of the condenser is 8200. What
is the specific inductive capacity of the glass ?
6. The force action between two charged plates separated by a distance
d in air is F. What would be the force if the space between the plates were
filled with a liquid having a specific inductive capacity of 2.4 ?
7. One plate of a condenser is connected to earth. A charge of 500
e.g. s. units on the other plate will raise its potential to 100. What is the
capacity of the condenser ?
8. What energy -would the charged condenser of problem 7 possess?
9. Two condensers in series are charged to a potential difference of
1000 c. g. s. units. The capacities of the condensers are 5 and 15 c. g. s.
units. What energy is stored in the charged condensers ?
10. To what potential difference are the condensers of problem 9 indi-
vidually subjected?
ELECTROKINETICS
CHAPTER XXIII
THE ELECTRIC CURRENT
278. When two conductors between which there is a differ-
ence of potential are connected by means of a wire, a transfer
of electricity takes place. Under these circumstances there is
said to be an electric current in the wire. According to the
electron theory this current consists of a procession of electrons
moving toward the conductor which is most strongly charged
positively. This may be termed the electron current. Under
the theories universally adopted before the advent of the elec-
tron theory, the current is assumed to flow from the body hav-
ing the higher (positive poten-
tial) to that of lower potential.
This is called the positive cur-
rent. For example, referring
to Figure 183, let A represent
, T , , .^-1 J Fig- 183. —The Direction of the Posithe
a body charged positively and Current.
let B represent a negatively
charged body or one having a lower positive potential. If a
wire w is made to form a connection between A and 5, there
will be a positive current in the wire in the direction indicated
by the arrow.
If there is nothing to maintain the potential difference be-
tween A and 5, the current in the wire will exist for a brief
interval only after the wire is attached. If the potential dif-
ference between A and B is maintained, for example, by the
continuous operation of an electrostatic machine, there will be
a steady current in the wire.
THE STRENGTH OF THE CURRENT
279. The strength of the current in a wire is the rate at
which electrostatic charge is transferred. Let it be assumed,
290
ELECTROKINETICS 291
for example, that in the case described in the last section, Q
units of charge pass from A to Bia the time t. Then the ratio
Q^t gives the average strength of the current in the wire dur-
ing that interval, that is,
7 = -2 (82)
in which /stands for the electric current.
If Q in Equation 82 is expressed in c. g. s. electrostatic units
of charge and t in seconds, then / is expressed in terms of the
0. g. s. electrostatic unit of current. This unit is much too
small to be used conveniently in practical measurements. For
this reason a unit equal to 3,000,000,000 = 3(10)9 (,_ g_ g_ gigg.
trostatic units of current has been agreed upon as the practical
unit of current. This unit of current is called the ampere.
ELECTROMOTIVE FORCE
280. The term electromotive force (e. m. f.) is sometimes
used instead of potential difference to specify that which causes
a current to flow in a wire or other conductor. Evidently the
unit of electromotive force is the same as that of potential dif-
ference. The c. g. s. electrostatic unit of electromotive force
or potential difference (Section 256) is too large for convenience
in practical measurements and by universal agreement a unit
equal to 3^(=g-10"^) c. g. s. electrostatic units has been
adopted. This unit of electromotive force or potential differ-
ence is called the volt.
ohm's law
281. It was discovered by Ohm that the ratio of the potential
difference between the terminals of a conductor and the current
which flows in the conductor in response to that potential differ-
ence is constant, so long as the physical state of the conductor
remains the same. That is, calling the potential difference be-
tween the terminals of the conductor U and the current in
the conductor 0, then,
— = i? = a constant (83)
292 ELECTRICITY AND MAGKBTISM
This relation is known as Ohm's Law. The constant R, which
is found to depend upon the dimensions of the conductor and
the material of which it is made, is called the resistance of the
conductor.
Equation (83) is a definition for resistance. The unit of re-
sistance defined by this equation is evidently that resistance
through which unit potential difference will cause unit current
to flow. That is to say, if, when unit potential difference is
maintained between the ends of a conductor, unit current
flows in the conductor, the conductor is said to have unit
resistance.
If the potential difference is measured in volts and the cur-
rent is measured in amperes, the resistance is measured in ohms.
A conductor has a resistance of one ohm when a potential difference
between its ends of one volt will cause a current of one ampere to
flow in it.
FALL OF POTENTIAL
282. In order that water may flow in a pipe there must be a
difference of pressure from point to point along the pipe, the
pressure decreasing in the direction of flow. In order that heat
may flow (be conducted) along a bar of metal there must exist
a difference of temperature from point to point, the temperature
decreasing in the direction in which the heat is flowing. In
the same way there is a difference of potential or " fall of poten-
tial " from point to point along a conductor through which an
electric current is flowing.
The current flows from a region of higher to a region of
lower potential, and the potential difference between the ends
of a conductor carrying current may be regarded as the cause
of the flow of current in the conductor.
A portion of the e. m. f . of a dynamo or electric battery is
therefore required for each part of the circuit through which
it causes current to flow. According to Ohm's law a portion
of the circuit which has high (large) resistance requires a large
part of the total e. m. f., and a portion of the circuit which has
small resistance requires a relatively small part of the e. m. f.
In other words the e. m. f . is distributed throughout the cir-
ELECTROKINETICS
293
cuit as fall of potential according to the relative resistances of
the various parts.
Consider a system like that represented in Figure 184. A
dynamo supplies current to a group of incandescent lamps, the
lamps being connected to the dynamo by the lines (conductors)
ah and cd. If the total current is /and the resistance of the
group of lamps is iJg, the potential difference between the
I
Dynamo
R.
R*
Fig. 184.
lamp terminals is R^I, according to Olim's law. Let R^ be
the resistance of the line ab and R^ the resistance of the line
cd. Then the fall of potential in ab {i.e. the difference of poten-
tial between a and 6) is R^I, and the fall of potential in cd is
RJI. Now the conductors through which the current flows in
the dynamo also have resistance. Call this resistance R-^.
The current flows through this resistance out over the line ah,
through the lamps, back over ed, and again through iZj, and so
on. Therefore a part of the e. m. f. generated by the dynamo
must be apportioned to the resistance Ry This part of the
e. m. f., is of course, ^jJ.
If E is the e. m. f . generated by the dynamo, evidently,
or.
E= R.j:+ R^I+ RJ+ RJ
E=^RI
(84)
that is, the total e. m. f. generated by the dynamo (applied to
the circuit) is equal to the sum of the RI values around the
circuit.
294 ELECTRICITY AND MAGNETISM
Evidently, in order that a large part of the e. m. f. may be
effective at the lamps, the fall of potential in tlie lines must be
small. The fall of potential in the lines may be made small by
making R^ and R^, Figure 184, the resistance of the line, small.
It is for this reason that large copper wires are often used for
such lines. Large copper wires have relatively small re-
sistance.
COMPARISON OP KESISTANCES BY PALL OP POTENTIAL
283. If two conductors having resistances K and X respec-
tively are connected in series (i.e. so connected that the same
current flows through each) and a current I is caused to flow
through them, see Figure 185, the ratio of the resistances may
be found by measuring and comparing the corresponding
potential differences. Call the fall of potential in K, Vy, and
I, V, i Vj * *^;'* "^ ^' ^'a'
; I I ^\'e have, then,
f K X \ (a Ai A
\ j wlience
^ 1^1 x=K-^ (85)
Battery ^i
^"'- '''■ The fall of po-
tential in a conductor may be measured by means of a volt-
meter, which is an instrument for measuring e. m. f. or potential
difference. If, therefore, with such an instrument V^ and V^
are measured, the ratio IK: X may be obtained from the above
relation. If the value of K is known the equation may be
solved for the numerical value of X.
SPECIFIC RESISTANCE
284. As stated above, the resistance of a conductor is found
to depend upon the dimensions of the conductor and upon
the material of which the conductor is made. Assuming
that the physical state (temperature, etc.) remains constant,
ELECTROKINETICS 295
these are the only things upon which the resistance of the
conductor depends. It is found by experiment that the
longer the conductor, other things being equal, the greater is
its resistance. That is to say, the resistance varies directly
with the length. It is also found by experiment that, other
things being equal, the smaller the cross section of the conduc-
tor the greater the resistance ; that is to say, the resistance is
inversely proportional to the cross section of the conductor.
This may be expressed algebraically as follows :
It cc —
9
or, B=Jc-^ (86)
in which L is the length of the conductor and q its cross sec-
tion. The proportionality constant k which appears in Equa-
tion (86) is known as the specific resistance of the material of
which the conductor in question is made. Equation (86) is a
definition for specific resistance. Evidently from tlie equa-
tion, k is the resistance of a conductor of the given material
having a length of one centimeter and a cross section of one
square centimeter.
The specific resistances of a few of the common metals are
given in the following table of specific resistances :
Substance Specific Resistance
IN Ohms at 20* C.
Silver, annealed 1.488 x 10-^
Copper, annealed 1.580 x lO"'
Platinum, annealed .... 8.957 x IQ-^
Iron, annealed 9.611 x 10-»
Mckel, annealed .... 12.320 x 10-»
German silver, pressed 20.76 x 10"^
VARIATION OF RESISTANCE WITH TEMPERATURE
285. In discussing Ohm's Law and specific resistance the
assumption has been made that the temperature of the material
under investigation remained constant. It is found that the
resistance of a conductor depends upon its temperature. The
resistance of metals increases with a rise of temperature. Such
296
ELECTRICITY AND MAGNETISM
substances are said to have positive temperature coefficients.
The resistance of some substances, for example, carbon, de-
creases with a rise of temperature. Such substances are said
to have negative temperature coefficients. It is found that
the lavir of increase in resistance for metals with rise of tem-
perature may be expressed very approximately as follows :
R,= R^Cl + af)
(87)
in which i2j = resistance at t° C. and iZ^ = resistance at 0° C.
a is known as the temperature coefficient of resistance for the
material under investigation. For most pure metals it is
about 0.4 of 1 per cent per Centigrade degree.
The temperature coefficient of resistance of alloys is, in
general, less than that of the metals composing them. Thus,
manganin, an alloy of manganese, copper, and nickel, has a
temperature coefficient which is very nearly zero and is even
negative for certain temperatures.
THE EESISTANCE OF CONDUCTORS IN PARALLEL
286. Let two conductors be connected in parallel between
two points A and B, Figure 186. Under these circumstances
the effective resistance between the points A and B is less
than that of either of the conductors. The actual value of
this effective re-
sistance may be
found as fol-
lows : Let the
resistance of the
upper branch be
R^, that of the
FiQ. 186. — The Resistance ol a Branched Circuit is Less than lower branch
that of Either Branch.
R,,
Call the
total current which divides at A between the two branches L
Let Jj be the current in the upper branch and i^ the current in
the lower branch. Then, evidently.
I=I, + I,
(a)
ELECTROKINETICS 297
Calling the potential difference between the points A and B,
U, it follows at once from Ohm's Law that
and -^2 = -^ C'^)
Now, let it be assumed that there is a third conductor of
resistance It, such that, if placed between the points A and B,
it would in every respect take the place of the branched cir-
cuit. Then, since the same total current I would flow through
this resistance, if it were substituted for M^ and M2, therefore
1=1 w
Substituting the values of Jj, I^ and / from (6), (c), and {d')
in (a), we have ^ _ U E
or dividing through by E,
R R^ R^
or ij=_J__ (88)
-Kj -B2
In other words, the resistance of a branched circuit is equal to
the reciprocal of the sum of the reciprocals of the resistances of
the branches.
THE SHUNT
287. A conductor connected to an instrument in such man-
ner as to form a branched circuit with the instrument is called
a shunt. The effect of a shunt is to switch or " shunt " a part
of the total current past the shunted instrument. For exam-
ple, let A, Figure 187, be an instrument, let us say, an ammeter
(ampere meter), between the terminals of which has been
placed a conductor having a resistance R^. Evidently a cur-
rent / coming to the instrument will divide, a part going
298
ELECTRICITY AND MAGNETISM
Ammeter
through the ammeter and a part going through the shunt.
The current in the ammeter will be but a fractional part of the
total current /. That part of the total current which flows
through the ammeter may be pre-
determined by a proper adjustment
of the resistance of the shunt.
Evidently the ammeter A takes
the place of the resistance R-y in
J the branched circuit of Figure 186.
^^::^ That portion of the total current
which flows through the ammeter
when connected in this manner is
determined as follows: Let £ be
the potential difiierence between the terminals of the branched
circuit as in the discussion of the last section. Then Jj, the
current in the ammeter, equals £ -5- iJj, where M^ is the re-
sistance of the ammeter. But I, the total current, is equal to
E ^ R, where R is the resistance of the branched circuit. We
have, therefore.
Shunt
Fig 187.
^=1=^
R^ R^
(See Equation 88)
Whence,
^ ^ R- i + Rj
R^R^
-^ _ J R^R^
-Kj + R,^
Putting this value ■of JJ in the expression for _Zj, ( -?! = — ) we
ha.vfi 1
have
or,
-Kj R^ + R^
I, = I.
Ro
-Bj -f- R^
(89)
If, for example, i2j = 9 ohms and R^= 1 ohm, then the
fraction, R^ h- (i2j + R^) = 0.1, that is, to say, 0.1 of the total
current would under these circumstances pass through the
ammeter.
ELECTROKINETICS
299
WHEATSTONE S BRIDGE
288. One of the most convenient and at the same time most
accurate methods for measuring resistance is by the use of
Wheatstone's bridge,
tially of a branched
circuit containing four
Wheatstone's bridge consists essen-
B
Fig. 188. — Network of the Wheatstone Bridge.
resistances, two in each
branch, and a cross con-
nection between the
branches containing a
sensitive current de-
tector or galvanometer.
The arrangement of
parts is shown in Figure
188. xtj, R2, Rgi and M^
are the four resistances
forming the branched
portion of the circuit
AC. G is the sensitive galvanometer in the cross connection
BI) as indicated. ^ is supposed to represent a battery which
supplies the current used in the instrument. Let it be as-
sumed that the resistances Up M^, -B3, and R^ have such values
that no current flows in the cross connection BGD. This will be
the case only when the points B and D are at the same potential.
That this condition of affairs is possible will be readily under-
stood by a moment's study of the hydraulic analogue repre-
sented in Figure 189. Let abo and ado be the branches of a
stream of water flowing about the island /. Let it be further
imagined that beginning at the point d, a ditch is dug across
the island. Evidently if this ditch is joined to the upper
branch of the stream at the proper point, there will be no ten-
dency for water to flow in it in either direction. If the b end
of the ditch is connected too far upstream, water will flow in
the ditch g in the direction bgd. If it is connected too far down-
stream, there will be a flow of water in the ditch in the direction
dffb. There is one point, b, therefore at which the ditch may
terminate such that there will be no tendency for the water to
300
ELECTRICITY AND MAGNETISM
flow in either direction in the ditcli g. Evidently that point b is
the one which is at the same level as the point d. Returning to
the discussion of Wheatstone's bridge, if the points B and D
are at the same potential, there will be no flow of current
through the galvanometer (7.
Assuming that this condition has been reached, the following
relation must hold among the four resistances. From Ohm's
t
Fig. 189.^Wheatstoiie Bridge Analogue.
Law we have the potential difference ^4. to _B is I^Ri, and the
potential difference ^ to D is I^R-^- But since B and B are at
the same potential, these two potential differences must be equal.
We may therefore write
In the same manner, we may write
Dividing the first equation by the second, we have,
R^_R2
Ro Ra
or.
R\R i ^^ R^Rz
The Wheatstone bridge is used for measuring an unknown
resistance in the following manner : the resistance to be
measured, call it R^, is connected in series with a known re-
sistance iig, the value of which can be varied. The other side
of the bridge is formed of two known resistances R^ and R^.
ELECTROKINETICS
301
When the connections are complete the resistance of R^ is varied
until no current flows in (?. Then,
B^ = iJg
E^
R.
(90)
EBSISTANCE THERMOMETER
289. Advantage may be taken of the fact that the resistance
of a conductor varies with its temperature in estimating the
variations of temperature to which the conductor is subjected.
That is, it is possible to measure the temperature of a given
region by comparing the resistance of a conductor when placed
in that region with its resistance when kept at some standard
temperature, say 0° C. The form which the resistance ther-
mometer usually takes is that of a Wheatstone bridge with ex-
tended arms as represented in Figure 190. The extended arm
AB is made up of large copper wires joined at their extremi-
D C
B
Fig. 190. — Connections of the Resistance Thermometer.
ties by a coil of fine platinum wire C. It is the variation in
the resistance of C that is made use of in the estimation of tem-
perature by the instrument. The variation in resistance of the
connecting wires A and B, due to change of temperature, is
usually compensated for by placing a similar pair of conductors
DE minus the platinum wire C in an adjoining arm of the
bridge. These conductors are placed alongside of the con-
ductors A and B so that they are subjected to the same tem-
perature variations. The change in resistance of A and B is
thus automatically compensated. The change in the tempera-
ture of the wire C corresponding to a given variation in its
resistance may be determined by means of Equation (87).
302 ELECTRICITY AND MAGNETISM
Problems
1. Upon what four things does the resistance of a wire depend ?
2. When an electromotive force of 110 volts is applied to the terminals
of an incandescent lamp, a current of 0.5 ampere flows through the lamp.
AVhat is the resistance of the lamp ?
3. What potential difference must exist between the ends of a conductor
having a resistance of 20 ohms in order that a current of 5 amperes may
flow in the conductor ?
4. A wire has a length of 10,000 cm. and a diameter of 0.2 cm. Its re-
sistance is 0.5 ohm. What is the specific resistance of the metal of which
the wire is made ?
5. The resistance of a copper wire at 0° C. is 10 ohms. What is its
resistance at 100° C?
6. Two points A and B are connected by two wires in parallel. The
resistances of these wires are 5 and 7 ohms respectively. What is the re-
sistance A to B1
7. What is the resistance between two points when they are joined by
three wires in parallel having resistances of 3, 5, and 7 ohms, respectively?
8. The resistance betTceen two points in a circuit is 10 ohms. What
resistance must be placed in parallel with this to reduce the resistance to
4 ohms?
9. An ammeter has a resistance of 0.27 ohm. What must be the resist-
ance of a shunt for this instrument such that 0.1 of the total current wiU
pass through the ammeter ?
10. The terminals of a wire of 25 ohms resistance are at potentials + 50
and — 50. What current is flowing in the wire ? When the potentials are
-I- 80 and — 20? When the potentials are + 100 and 0?
11. Five hundred coulombs are carried along a wire in 25 sec. What is
the average current in the wire during this interval?
12. A dynamo is connected to a group of 100 incandescent lamps in
parallel. The resistance of the dynamo is 0.2 ohm. The total line resist-
ance is 0.3 ohm. The resistance of each lamp is 200 ohms. If the dynamo
e. m. f. is 100 volts, what current will flow in this circuit?
13. What part of the dynamo e. m. f. is effective at the lamps in prob-
lem 12?
14. What e. m.f. would the dynamo in problem 12 have to generate in
order to supply the lamps with 100 volts at their terminals?
15. What would be the effect on the remaining lamps, problem 12, if
50 lamps were suddenly turned off, the e. m. f . of the dynamo remaining the
same?
MAGNETISM
CHAPTER XXIV
MAGNETS
290. It was discovered by the ancients that a certain iron
ore (now called magnetite) possessed the property of attract-
ing and holding small bits of iron. At the present time this
iron ore is found in Sweden and Spain, in Arkansas and else-
where. Pieces of this ore are called natural magnets and the
property which enables these magnets to attract and hold bits
of iron is called magnetism. The name is derived from Mag-
nesia, a province in Asia Minor in which natural magnets were
first discovered.
In the tenth or twelfth century the discovery was made that
if a magnet is suspended so as to be free to turn in a horizontal
direction, it always sets itself in a definite position with respect
to the points of the compass, a certain part of the magnet tend-
ing always to point toward the north, a certain other part to
point toward the south. These parts of a magnet are called
magnet poles and are distinguished as the north-pointing pole
and the south-pointing pole. The magnetic property of a mag-
net is limited to its poles. If a magnet' is dipped into iron fil-
ings, they will cling to the magnet only at its P9les.
Magnetism is very readily imjia*tedr~t5'"apiece of iron or
steel. A piece of steel possessing the properties of a natural
magnet is sometimes called an artificial magnet. Such magnets
are usually made in the form of straight bars or " horseshoes."
A compass needle is a light bar magnet delicately poised on a
pivot so as to be free to turn in a horizontal plane.
THE FORCE ACTION BETWEEN MAGNET POLES
291. A magnet is found to exert a force action, not only
upon bits of iron and steel as pointed out above, but also upon
303
304 ELECTRICITY AND MAGNETISM
other magnets. It is found by experiment that similar poles
repel one another, while between unlike poles there is a force of
attraction. There is a force of repulsion between two north-
pointing poles or between two south-pointing poles, while a
north-pointing pole attracts a south-pointing pole.
The force action between two magnet poles depends upon
the "strength" of the poles and upon the distance between
the poles. A magnet has great pole strength if the force with
which it acts upon other poles brought near to it is great. A
rough notion of the pole strength of a bar magnet may be ob-
tained by dipping one of the poles into iron filings. If the
pole strength is great, a large mass of filings will adhere to the
pole ; if the pole strength is small, a smaller mass of filings will
adhere. Experiment shows that the force between two magnet
poles is proportional to the strength of each pole and inversely
proportional to the square of the distance between the poles.
This law may be expressed as follows :
in which F is written for the force between the poles, d the
distance which separates them, and m and m' are the strengths
of the magnetic poles.
Evidently Equation (91) is a definition for magnetic pole
strength, and unit pole strength in the c. g. s. system would be
defined as follows : A magnet pole has unit strength if when
placed at a distance of one centimeter from a pole of equal
strength it is acted upon by a force of one dyne.
MAGNETIC FIELD
292. The magnetic field of a magnet is that region of space
into which the influence of the magnet extends. Theoretically,
the field of a magnet is infinite in extent. Practically, for
ordinary forms of magnets the field is quite limited.
The lines of force in a magnetic field are imaginary lines sup-
posed to be drawn through the field in such manner that at
each point they extend in the direction in which a small mag-
net pole would tend to move if placed at that point. Arrow-
MAGNETISM
305
heads are placed on the lines of force to show the direction in
which a north-pointing pole would tend to move in the field.
Evidently a south-point-
ing pole would tend to
move in the opposite
direction. A compass
needle, therefore, always
tends to set itself tangent
to the lines of force of
the magnetic field in
which it is placed.
From the above discus-
sion it can be seen that
magnetic lines of force
emerge from a north- Fm. I91.-Magnetic Field ol a Bar Magnet.
pointing pole and enter a south-pointing pole.
The general character of the magnetic field surrounding a
bar magnet is represented in Figure 191.
The field about two bar magnets placed with the north-point-
ing pole of one opposite the south-pointing pole of the other is
shown in Figure 192.
A convenient way of mapping the magnetic field about a
magnet or system of magnets is to scatter iron filings in the
Fi(i. iy2. — Magnetic Field surrounding two Bar Magnets.
field. The filings as they fall become magnetized and act like
small compass needles, setting themselves tangent to the lines
of force.
MAGNETIC SUBSTANCES
293. There are but few substances to which this property of
magnetism may be imparted in any appreciable degree. These
306 ELECTRICITY AND MAGNETISM
substances are iron (steel), nickel, cobalt, manganese, chromium,
and cerium. From a practical standpoint only the first three
or four of the above mentioned substances are of importance.
These substances are known as magnetic substances. Other
substances, e.g., copper, zinc, aluminum, are called non-magnetic
substances.
MAGNETIZATION
294. A magnetic substance may be magnetized:
1. By contact with a magnet.
2. By means of the electric current.
3. By induction.
(1) Magnetization by contact. If a piece of steel, for ex-
ample, a knitting needle, or a piece of a watch spring, is stroked
with a magnet, it will acquire the property of the magnet or
become magnetized. When a piece of steel has been magnet-
ized in this manner, it may be used for magnetizing other
pieces of steel by the same process, that is, by being rubbed
upon them.
(2) Magnetization by electric current. It is found that if a
wire is wrapped about a bar of magnetic substance and a cur-
rent sent through the wire, the bar will acquire the properties
of a magnet. Let iViS, Figure 193, be a bar of iron, and OD a
wire wrapped spirally about the bar
as shown, and let E represent an
electric battery connected to the wire
CD in such manner as to send a cur-
rent through it. Under these circum-
stances the bar NS becomes a magnet.
(3) Magnetization by induction. If
Fig. 193. — Magnetization by , ~ .• i . • i it
means of the Electric Cur- a bar of magnetic substance is brought
rent. into a magnetic field, it tends to be-
come magnetized. The magnetic substance acts as if it afforded
a better or easier path for the lines of force than the air which it
displaces. Because of this fact there is a tendency for the lines
of force from right and left to bend into and pass through the
bar of magnetic substance. Heuce, the system of lines sur-
rounding it is very much like that surrounding an ordinary bar
magnet and the bar of magnetic substance is found to possess the
MAGNETISM
307
properties of a magnet. It is said to be magnetized by induction.
The soft iron bar ns, Figure 194, brought into the presence
of a magnet NS, becomes a magnet. The magnetic field sur-
rounding the magnet and the soft iron bar is indicated in the
Fig. 191. — Magnetization by Induction.
figure. Under tliese conditions ns exhibits all the properties of
a magnet, and is said to be magnetized by induction. It should
be noticed that the end of the bar ns which is nearest to the in-
ducing magnet NS, is of opposite polarity to that of the induc-
ing pole S.
THE RETENTION OF MAGNETISM
295. If a bar of hard steel is magnetized by any of the
methods above mentioned, it will be found to be more or less
difficult to magnetize it strongly, but once it is magnetized, it
will retain its magnetism for a great length of time. On the
other hand, if a bar of soft iron is magnetized by the same
method, it will be found to yield more readily to the magnetiz-
ing influence brought to bear upon it, but also to lose whatever
magnetism is imparted to it, almost, if not quite completely, as
soon as it is removed from the magnetizing influences. Perma-
nent magnets therefore can be made of steel only. Iron is used
in magnetic devices in which it is desired to quickly change
the magnetic condition of the iron from time to time.
If a magnet is roughly handled, for example, if it is dropped
on the floor or hammered, it is found to lose its magnetism
much more rapidly than would be the case if it were not sub-
jected to such treatment. The effect of heat upon a magnet
with respect to its retention of magnetism is very marked. It
308
ELECTRICITY AND MAGNETISM
is found that if a magnet is strongly heated it loses a large part
of its magnetism, and if it is heated to what is known as the
critical temperature its magnetism disappears.
That property of a substance which enables it to retain its
magnetism is usually referred to as the retentivity of the sub-
stance. The retentivity of steel is great. The retentivity of
soft iron is almost zero. Hard cast iron oftentimes has con-
siderable retentivity.
THE CRITICAL TEMPERATURE
296. Reference was made above to what is known as the
critical temperature. This temperature may be defined as the
temperature at which the magnetic properties of a magnetic
substance disappear. It is found that if any magnetic sub-
stance is sufficiently heated,
it loses its magnetic properties
completely. When this tem-
perature is reached, the sub-
stance is said to be at its
critical temperature. The
manner in which the magnetic
properties of a magnetic sub-
stance change with the rise of
temperature is indicated in
_ ^ Figure 195. The curve ABC
^^'^ is plotted in the following
Fig. 195. -Curve showing the Effect of manner : Distances measured
Temperature upon the Magnetic Prop- horizontally, that is, abscissae,
er y o ron. ^^^ temperatures on the Cen-
tigrade scale. Distances measured vertically (ordinates) rep-
resent the magnetic properties of the magnetic substance at
the different temperatures. The curve indicates that as the
temperature rises, the magnetic properties of the material be-
come more and more pronounced until a certain temperature
known as the critical temperature is reached. At this point
the material becomes almost, if not quite, completely non-mag-
netic. This is indicated by the sudden drop in the curve.
The critical temperature for iron is about 786° C. The critical
H
= 02
IOC
100
n|
80
00
u
60
00
ll
40
00
1
> ( 1
20
00
/ 1
—fy-
_B-
M_
—
—
-—
"
El
zoo°
400 ° 600°
MAGNETISM 309
temperature for nickel is about 350° C. The significance of
critical temperature is not well understood, but it is known that
the sudden disappearance of the magnetic properties of a mag-
netic substance at the critical temperature is accompanied by-
other marked molecular changes. Certain other physical prop-
erties change abruptly at this temperature, and it would seem
as if an entire rearrangement of the molecular parts takes
place.
THE THEORY OP MAGNETISM
297. The theory of magnetism which is most commonly ac-
cepted at the present time is that each molecule of a magnetic
substance is in reality a permanent magnet. In the unmag-
netized body it is supposed that these small molecular magnets
are arranged in heterogeneous fashion so that they neutralize
one another in their effects upon outside bodies. A magnet is
considered to be a body in which these molecular magnets are
turned so that they all face in the same direction. Evidently,
if this were the case, then at one end of the magnet there would
be a number of molecular south poles which, combined, would
constitute the south pole of the magnet, while at the other end
of the body there will be a group of molecular north poles,
constituting the north pole of the magnet.
Evidently, under this theory, the process of magnetization is
simply the process of turning these molecular magnets so as to
cause them all to face in the same direction. A body exhibits
a small amount of magnetism if but a few of these molecular
magnets are so turned. Its magnetic properties become more
marked as larger numbers of these molecular magnets are
turned in the same direction.
The electron theory assumes that the magnetism of the mol-
ecule is due to the revolution of electrons about the positive
part of the atom in the same way that the earth revolves about
the sun. These revolving electrons constitute electric currents
flowing around the molecule. The molecule is, therefore, mag-
netized in much the same way as the iron bar in Figure 193.
310 ELECTRICITY AND MAGNETISM
MAGNETIC FIELD INTENSITY
298. The force which a given magnet pole experiences at a
given point in a magnetic field depends upon the strength of
the pole and the field intensity at the point. A definite notion
of this quantity,
Fig, 196
N I called magnetic field
I intensity, is best ob-
1 tained in the follow-
I
: ing manner: LetP,
Figure 196, repre-
sent a point in the
neighborhood of a
magnet JVS, that is
to say, a point in a
magnetic field. Let
it be imagined that different magnet poles are brought up to
this point P and that the force which each pole experiences
when brought to that point is carefully measured. It will be
found in this experiment that in each case the force I' is pro-
portional to the strength of the magnet pole m placed at P.
That is, -r,
Fxm
or, F=H-m (92)
The constant Hin this equation, which evidently pertains to
the point P, is called the magnetic field intensity at that point.
The field intensity in a given region is sometimes called the
magnetizing force in that region. Evidently,
^ = |' (93)
(Compare Equations 91 and 92)
If the magnetic pole considered in the last paragraph is
thought of as unit magnet pole, then evidently 5"= F. That
is to say, the field intensity at a point is numerically equal to
the force which unit magnet pole will experience if placed at that
point.
MAGNETISM 311
SPECIFICATION OF FIELD INTENSITY BY NTTMBEE OF LINES
OF FOECB
299. In the discussions of the magnetic lines of force which
have thus far been given, the lines have been supposed to
represent the direction of the field only. It is possible to
represent also the field intensity at each point by the lines of
force, by making the number of lines of force per square centi-
meter equal to the field intensity at that point. The field
intensity may then be completely specified by stating the
number of lines of force per square centimeter.
INDUCTION
300. When a magnetic substance is placed in a magnetic
field of given intensity, it becomes magnetized to an extent
which depends upon the magnetic substance itself, and upon
the field intensity or magnetizing force to which it is subjected.
The nnmber of lines of force per square centimeter which thread
through the magnetic substance is called the induction.
PERMEABILITY
301. If different magnetic substances are subjected to the same
magnetizing force or field intensity, the induction will in each
case be different. That is to say, if a piece of iron is subjected
to a given magnetizing force, the induction in the iron will
have a certain value. If a piece of nickel is subjected to the
same magnetizing force, the induction will be quite different.
This is usually expressed by saying that the magnetic permea-
bility is different for these different substances. The induc-
tion in iron, for example, would be in the ordinary case very
high as compared with the induction in nickel or cobalt. We
say, therefore, that iron is more permeable to the lines of force
or has a higher permeability. Permeability is defined as the
ratio of the induction in the substance to the field intensity or
magnetizing force' to which the magnetic substance is subjected.
That is, P
/^ = f (94)
312 ELECTRICITY AXD MAGNETISM
in which the symbol jjl is written for the permeability, B for
the induction, and Hiov the "magnetizing force."
The permeability of any magnetic substance depends upon
the intensity of the field which is acting upon it. The per-
meability of iron increases for a time as the field intensity is
made greater up to a certain point. After this point, known
as the point of saturation, is reached, the permeability grows
less with a further increase of field intensity. This fact is
clearly brought out by the following table, in which are shown
the corresponding values of B, H, and /i for a certain sample
of wrought iron.
B
H
f
41
0.3
128
1460
2.2
670
11540
4.9
2350
14840
10.2
1450
16900
78.0
215
The table shows that for small values of H, B is also small
(relatively) and jj, is small. For larger values of M both B
and /i increase until breaches a value of about 5. For larger
values of H, although B continues to increase, the ratio — , that
is, the permeability, becomes smaller.
The permeability of air is taken arbitrarily as unity. The
permeability of non-magnetic substances, for example, wood, rub-
ber, copper, aluminum, etc., for practical purposes are also equal
to unity.
THE CUEVE OF MAGNETIZATION
302. It will be evi-
dent from the state-
ments made in the last
section that the induc-
tion in a given mag-
netic substance is not
proportional to the
magnetizing force.
The manner in which
the induction changes
Fig. 197. — Magnetization Curve for Wrought Iron, in wrought iron as the
MAGNETISM 313
magnetizing force increases is shown by the curve, Figure
197. This curve is plotted in the following manner : The mag-
netizing force H to which the magnetic substance is subjected
is laid off horizontally. The induction in tl^e substance when
subjected to this magnetizing force is laid off vertically. Thus
the point of the curve indicates that when the magnetic sub-
stance is placed in the field of field intensity H^ the induction
in it is By
THE TORQUE ACTION ON A BAR MAGNET IN A UNIFORM FIELD
303. A uniform magnetic field is one in whicli the lines of
force are straight, parallel, and equidistant. The torque action
upon a bar magnet when placed in such a field is determined
from the following considerations : Let mm. Figure 198, rep-
resent a bar magnet in a uniform magnetic field. Let it be
A
m^
^^^
F
X
^
/^O
^?
^^
/
Mn
Fig. 1
98.
assumed that the strength of this uniform field is H and that
m represents the strength of the magnetic pole. Let it be
further assumed tliat the distance between the poles of the
magnet is L. The north-pointing pole of the magnet will be
acted upon by a force F= mlT (Equation 92) urging it toward
the right, that is, in the positive direction of the lines of force.
This will give rise to a torque action about the center of the
magnet equal to F times OA, in which OA is the perpendic-
ular distance between the center of the magnet and the line
of action of the force F. Calling the angle which the magnet
314 ELECTRICITY AND MAGNETISM
makes with the lines of force (95)
THE TIME OP VIBRATION OF A BAR MAGNET IN A UNIFORM
MAGNETIC FIELD
304. A bar magnet in a uniform field in the position repre-
sented in Figure 198 is acted upon by a torque tending to
turn the bar into a position parallel to the lines of force. If
the bar magnet is assumed to be free to move about the point
0, it will turn as indicated; but because of its inertia it will not
stop when it reaches a position parallel to the lines of force.
Its inertia will carry it beyond this position. It will then be
acted upon by a positive torque which will turn it back. Its
inertia will again carry it beyond the position of equilibrium,
and so on, that is to say, the bar magnet will oscillate to and fro
through the position of equilibrium. The rapidity with which
these oscillations succeed one another is best determined in the
following manner. Referring to Equation (95), it will be seen
that for small values of this expression may be written as
follows: T=M-R.cj>
MAGNETISM 315
since for small angles the sine of the angle is numerically
equal to the angle itself when measured in radians.
Since ilf and H are constants, the magnet satisfies the condi-
tion for simple harmonic motion (Sections 49 to 52). But any
body so conditioned that it will execute simple harmonic mo-
tion of rotation will vibrate in such manner that,
r = - 4 ttVZ'c^ (Equation 11 0)
in which n is the number of vibrations per second, K is the
moment of inertia of the vibrating body, and ^ is the small
angular displacement. Comparing this equation with the one
given above, it will be evident that
4 7rV^=iffl"
Solving for w, the number of vibrations which a bar magnet
will execute in one second under the conditions assumed is
-h<^ (»«)
27r^ if
THE MAGNETISM OP THE EARTH
305. The earth is surrounded by a magnetic field which
varies both in direction and intensity from point to point on the
earth's surface. The earth's magnetic field is such as would
exist if a central portion of the earth were magnetized so as to
have a south-pointing pole in the northern hemisphere and a
north-pointing pole in the southern hemisphere, both poles being
considerably displaced from the axis of the earth and far below
the surface. In Figure 199 is shown the magnetized central
portion the existence of which would, in a general way, account
for the earth's magnetic field. The circle ABCD represents
the earth's surface. 8, iVare the poles of the magnetized cen-
tral portion. The dotted lines show the field in a plane passing
through the magnet poles N, S and the axis of earth.
MAGNETIC DIP
306. A bar magnet suspended in such manner as to be free
to turn in all directions tends to set itself parallel to the mag-
316
ELECTRICITY AND MAGNETISM
Norlrh
netic field in which it is placed. An inspection of Figure 199
will show that at the points B and I) (and all other points on
the great circle pass-
ing through B and I)
and perpendicular to
NS) the lines of force
are horizontal, that is,
parallel to the surface
of the earth. At A
and (7 the lines offeree
are vertical or perpen-
dicular to the surface
of the earth.
The magnetic dip at
any point is the angle
between the magnetic
field at that point and
the horizontal. The
magnetic dip at B and
B, Figure 199, is zero,
at A and (7, 90°, and
for intermediate points its value ranges between these limits.
The point A is on the west coast of Hudson's Bay at about
70° north latitude. At this point a bar magnet tends to stand
on end (parallel to the plumb line).
South
Pole
Fig. 199. —The Magnetic Field of the Earth.
DECLINATION
307. The magnetic declination at a point is the deviation of the
compass needle from the true (geographic) north and south.
The magnetic declination for points near B and D, Figure 199,
on the great circle ABCD is zero. For other points the dec-
lination has a value and direction depending upon the latitude
and longitude of the point. In the United States the declina-
tion ranges from about 17° west in Maine to 23° east in the
state of Washington.
HORIZONTAL INTENSITY
308. Compass needles are mounted on pivots so as to be free
to turn in a horizontal plane only. When a needle is so
MAGNETISM
317
mounted, it responds to the horizontal component of the earth's
field and is unaffected by the vertical component.
In Figure 200 let F represent in magnitude and direction
the earth's magnetic field at a given point. Through the upper
end of the line F draw the horizontal line H and through the
lower end of F the vertical line V.
Then R and V are respectively the
horizontal and vertical components of
the field F.
The horizontal intensity of the
earth's magnetic field is the intensity
of its horizontal component. At B
and D, Figure 199, the vertical com-
ponent of the magnetic field of the
earth is zero, and the total field in-
tensity is effective in directing the
compass. At A and C the horizontal
component is zero, and the compass
needle will stand indifferently in any
position. It is found that over a
large area in the neighborhood of A the compass needle is
very sluggish in its action. That is, in this region the hori-
zontal intensity is so small that the needle is scarcely affected
by it.
The statements made above with respect to the magnetic
elements (dip, declination, and horizontal intensity) of the
points B, D, A, and (7, must be regarded as correct only in the
general sense. As a matter of fact, the magnetic elements of
any point on the earth's surface depend not only upon the geo-
graphical location of the point, but oftentimes upon the pres-
ence near at hand of deposits of iron ore, etc. Furthermore,
the magnetic elements of a point are subject to slight recurrent
variations, both daily and annual, and to slow progressive
changes extending over long periods of time.
Fig. 200.
Problems
1. Two equal magnet poles are placed 10 cm. apart. The force action
between them is 16 dynes. What is the pole strength of each?
318 ELECTRICITY AND MAGNETISM
2. Two north-pointing poles of pole strength 50 and 60 are placed 5 cm.
apart. What is the force of repulsion between them ?
3. The pole strength of the north-pointing pole of a bar magnet is 80
c. g. s. units. What is the field intensity at a distance of 3 cm. from this
pole? Xeglect effect of south-pointing pole.
'. 4. .V bar magnet has a pole strength of 50 c. g. s. units. The distance
between its poles is 10 cm. What is the field intensity at a point 5 cm.
from the nortli-pointing pole and 10 cm. from tlie south-pointing pole of
this magnet? At a point for which the distances are 6 and 6 cm. respec-
tively ? At a point for which the distances are 5 and li cm. ?
5. Sketch roughly the lines of force in the field surrounding three equal
magnet poles placed at the corners of an equilateral triangle. Two of the
poles are north-pointing, the other, south-pointing.
6. What is the field intensity at the center of the triangle of problem 5?
Assume pole strength = 20 and side of triangle = 10 cm.
7. A sample of iron when subjected to a magnetizing force of 5 c. g. s.
units shows an induction of 12,000 lines per square centimeter. What is
the permeability of the iron?
8. A piece of iron has a permeability of 1000. What magnetizing force
will give an induction of 3600 ? Would twice this magnetizing force give
twice the induction? Explain.
9. A compass needle makes 200 vibrations per minute when placed in a
magnetic field having an intensity of .2 c. g. a. units. What would be its
period in a field of which the intensity is .1 ?
10. The horizontal intensity of the earth's magnetic field at a point is
.18 c. g. s. units. The dip is 70°. What is the total intensity of the field at
this point?
ELEOTROMAGNETISM
CHAPTER XXV
OERSTED'S EXPERIMENT
Fig. 201.
309. A wire carrying an electric current is surrounded by a
magnetic field. This fact was first discovered by Oersted in
1819. The experiment which led to Oersted's discovery is as
follows : Let AB, Fig- ^ f^
ure 201, represent a wire
carrying current in the
direction from A to B
as indicated by the ar-
row, and NS represent
a compass needle
mounted just below the
wire. It is supposed that we are looking at the apparatus from
above. Let it be assumed further that the wire AB lies in the
magnetic north and south direction, and therefore that the com-
pass needle stands parallel to the wire before the current is
turned on. When the current flows, the compass needle will
swing into some such position as that represented in the dia-
gram, the north-point-
ing pole having moved
toward the west and the
south-pointing pole
toward the east.
When the current
flows in the opposite
^°' ^^' direction, that is, from
S to A, the wire still remaining above the compass needle,
the deflection of the needle is like that indicated in Figure 202.
That is to say, the north-pointing pole moves toward the east,
the south-pointing pole toward the west.
,S19
320 ELECTRICITY AND MAGNETISM
The angle through which the needle is deflected in this ex-
periment depends apon the strength of the current in the wire
and the intensity of the magnetic field in which the compass is
placed before the current is turned on.
The fact that the compass needle is deflected when the cur-
rent flows in the wire is evidence that the current in the wire
produces a magnetic field. Evidently the poles of the compass
in this, as in all other cases, tend to move in the direction of
the field in which they are placed. Therefore if the compass
needle stands northwest and southeast instead of north and
south we must conclude that the magnetic field in which it is
placed has an east to west component. But the magnetic field
of the earth extends north and south. Therefore a new field
extending east and west must have been introduced by the pas-
sage of the current through the wire, which, combined with
the earth's field in the north and south direction, gives the
resultant field.
THE MAGNETIC FIELD WHICH SUREOUNDS A WIRE CAERYmG
CURRENT
310. The magnetic field surrounding a wire carrying current
is of such nature that the lines of force are concentric circles
having their centers at the axis of the wire and their planes
perpendicular to the wire. The field is most intense close to
the wire and falls off rapidly as the distance from the wire
increases. The general character of the field may be deter-
mined by the following experiment : Let AB, Figure 203,
represent a wire carrying current in the vertical direction AB
as indicated by the arrow. Let it be assumed that a number
of small compasses e, f, g, etc., are arranged about the wire
supported by the cardboard OB. The compass needles will
arrange themselves as indicated in the diagram. If iron filings
are scattered upon the cardboard CD and the cardboard is
gently tapped, they will arrange themselves in concentric cir-
cles as indicated by the dotted lines.
The relation between the direction of the current in the wire
and the direction of the field may be stated as follows : If one
imagines himself in such a position that he can look along the
BLECTROMAGNETISM
321
wire in the direction in which the current is flowing, then the
positive direction of the field is that in which the hands of a clock
move.
~ A
♦ 1 ( (
if))) ^,
>*70
Fig. 203. — The Magnetic Field about a Wire carrying Currettt.
THE FORCE ACTING UPON A WIRE CARRYING CURRENT AND
LYING IN A MAGNETIC FIELD
311. Ampere demon-
strated that a wire car-
rying a current and lying
at right angles to a mag-
net field is acted upon
by a force which tends
to move the wire in a di-
rection perpendicular to
both the field and the
wire. A simple device
for exhibiting this effect
is shown in Figure 204.
AB is a wire suspended fig. 204.
322 ELECTRICITY AND MAGNETISM
from two small cups CD, filled with mercury, which serves to
make good contact between the wire AB and the wires E and F
which convey the current to and from the apparatus. i\r repre-
sents the north-pointing pole of a bar magnet. The lines of
force spread from N in the manner indicated by the dotted
lines. With the arrangement shown in the figure the wire AB
is perpendicular to the lines of force of the field /of the mag-
net. When a current flows in the wire, it is pushed either
toward or from the observer according ■ to the direction of the
current.
ampere's law
312. When a wire carrying current lies at right angles to the
lines of force in a magnetic field, it experiences a force action
which is proportional to the field intensity, to the length of the
wire, and to the current which is flowing in the wire. In other
words, TT TTj^
in which / is the current in the wire, L is the length of the
wire lying in the magnetic field, f is the field intensity which
is supposed to be uniform over the entire length of the wire.
This may be written in the form of an equation by introduc-
ing a proportionality constant, or by what amounts to the same
thing, the choosing of a new unit of current. Thus,
F= ILf (97)
If this relation is written thus in the form of an equation, it
becomes a definition for what is known as c. g. s. electro-
magnetic unit of current. Evidently the c. g. s. electromagnetic
unit of current is that current which flowing through a wire one
centimeter long lying at right angles to a magnetic field of unit
strength experiences a force action of one dyne.
1 c. g. s. electromagnetic unit of current = 10 amperes.
The direction of the force contemplated in Equation (97)
depends upon the direction of the field and the direction of
current in the wire. It is found, by experiment, that the
direction of the force action is always related to the directions
of these two quantities in a simple manner. A good rule for
BLECTROMAGNETISM
323
determining the direction of the force action when the direction
of the field and the direction of the current are known, is the
following left-hand rule. If the thumb and first and second
fingers of the left hand are held in such position as to be at right
angles to one another and the forefinger is made to point in the
direction of the field while the second finger points in the direc-
tion of the current in the wire, then the thumb will indicate the
direction of the force.
THE FORCE ACTION BETWEEN TWO WIEES CARRYING CURRENT
313. Since there is a magnetic field about a wire when it is
carrying current, it is evident that there may be a force action
between two wires which lie near one another when electric
currents are flowing through them.
Experiment shows that parallel wires
carrying current in the same direction
attract one another and parallel wires
carrying current in opposite directions
repel one another. A simple experiment
illustrating the attraction between par-
allel wires carrying currents in the same
direction is that represented in Figure
205. AB represents a spiral of copper
wire supported at its upper end. The
lower end of the wire dips into a cup
of mercury, 0. When a current flows
through the spiral, the adjacent turns
attract one another. The result is that
tlie spiral as a whole contracts, thereby
lifting the lower end of the wire from
the mercury ; but since the circuit is
completed through the mercury cup,
the lifting of the wire from the mercury breaks the circuit.
When the circuit is broken, there is no longer any attraction
between the successive turns of wire, and the spiral falls back.
The circuit is thus once more completed and the operation is
repeated.
Fio. 205.
324
ELECTRICITY AND MAGNETISM
Fig. 206. — Magnetic Field of a Solenoid.
THE MAGNETIC FIELD OF A SOLENOID
314. A solenoid is a spiral of wire, the successive turns oi
which are of the same diameter. Such a coil is obtained by
winding the wire spirally upon a cylinder.
The magnetic field of a solenoid is represented in a general
way in Figure 206. The lines of force which would extend in
concentric cir-
cles about the
individual coils
of the solenoids,
if these coils
were isolated,
unite, forming
continuous lines
of force from
end to end within the solenoid and extending back from end
to end without the solenoid in curves similar to those which
surround a bar magnet. In fact, the external field of a solenoid
is like that of an ordinary bar magnet.
The direction of the lines of force surrounding a solenoid is
easily determined by the following rule : If one imagines him-
self placed at that end of the solenoid from which the electric
current appears to run clockwise about the coils, then he will be
looking in the direction of the lines of force through the solenoid.
Another rule which is often made use of in this connection
is known as the right-handed screw rule. If one imagines a
right-handed screw placed in the axis of a solenoid and turned
in the direction in which the current is flowing about the coils
of a solenoid, then the screw will advance in the positive direc-
tion of the lines of force.
THE ELECTROMAGNET
315. The electromagnet in its simplest form consists of a
solenoid having a core of soft iron. The effect of placing a bar
of soft iron within a solenoid is to increase the number of lines
of force set up by the current in the coils. This will, of course,
increase the intensity of magnetic field at each and every point
ELECTROMAGNETISM
325
in the neighborhood of the solenoid, since all of the lines of force
which thread through the core of the solenoid must return
through the space surrounding it.
The increase in the number of lines of force within the sole-
noid due to the iron is explained by saying that the permea-
bility of the iron is many times as great as that of air, so that
the same magnetizing force is enabled to establish or set up a
larger number of lines of force.
The electromagnet in its most efficient form is so designed
that the magnetic circuit, that is to say, the path through which
the lines of force of the magnet
extend, is as nearly as possible ArmaiUre
all iron. Thus, in the electric
hell, the electromagnet is given
the form shown in Figure 207.
One of the essential parts of a
dynamo or electric motor is a
strong electromagnet. Great
care is taken in the design of
such machinery to limit what is known as the air gap as far as
possible, for the reason that the number of lines of force de-
veloped for a given current in the coils is greatest when the
magnetic circuit is as nearly as possible all of iron.
Fig. 207.
THE MAGNETIC FIELD OF A CIKCULAR LOOP OF WIRE
316. Since the lines of force about a wire carrying current
are concentric circles whose planes are at right angles to the
axis of the wire, it will be evident that the lines of force about
a circular loop of wire carrying current will lie in planes which
pass through the axis of the coil, and that the current in each
part of the circular coil will contribute to the magnetic field
at the center. These statements will be more readily under-
stood by reference to Figure 208, in which A and B repre-
sfejJLthe ends of a circular loop of wire which has been cut by a
plane passing through its axis CD. Let it be imagined that the
current is flowing out at the top of the coil and in at the bottom.
Then the lines of force about the wires in the plane of the
326
ELECTRICITY AND MAGNETISM
paper will be as indi-
cated in the diagram.
From the figure it will
be evident that the cur-
rent in both the A and
B portions of the wire
contributes to the field
at the center of the coil
0. In the same manner
it will be evident that
each and every part of
the loop adds its por-
tion to the field at the
center of the coil. At
the very center of the
coil the lines of
force are parallel and
equidistant, that is to say, the field at the center of the coil is
uniform.
Fig. 208.
FIELD INTENSITY AT THE CENTER OE A OIECULAR LOOP
OF WIRE
317. As de-
fined above (Sec-
tion 298), the
field intensity at
any point is that
constant which
multiplied by
the strength of
a magnetic pole
brought to that
point will give
the force which
acts upon the
pole. Let Fig-
ure 209 repre-
sent a circular Fig. 209.
ELBCTROMAGNETISM 327
loop of wire of radius r, carrying a current I. Let it be im-
agined that there is placed at the center of this coil an isolated
north-pointing magnet pole of strength m. The lines of force
raaiating from this pole will cut the circular loop of wire at right
angles. Furthermore, the magnetic field intensity at the wire
due to the magnet pole m is equal to m-i-r^ (Equation 93).
According to Ampere's Law the circular coil of wire under
these circumstances will experience a force action tending to
move it at- right angles to the radial lines of force, that is, to
lift it perpendicular to the plane of the paper in the figure.
The magnitude of this force action which the coil experiences
is obtained from Ampere's Law. That is,
F=ILf
Since 2 irr is the length of the wire lying in this field and the
field intensity as shown above is — , therefore,
^2
F = • m
r
If the coil has two turns of wire in it, evidently the force action
will be twice that given by the above equation, since each coil
is acted upon by the force F given by the above expression.
If there are n turns of wire in the coil, then,
F=^^I^.m (98)
r
This is the expression for the force acting upon the coil which
tends to lift it vertically, assuming that the coil of Figure 209
lies in a horizontal position. Since action is equal to reaction,
the magnetic pole m must be acted upon by a force of equal
value but oppositely directed. That is to say, m is acted upon
by a force the magnitude of which is given by the above equa-
tion, the direction of which is downward.
Referring again to the definition for field intensity, it will be
evident that the quantity lirnl-i-r is an expression for the
field intensity at the center of the coil due to the current in the
coil. This must be evident from the fact that it is this group
328 ELECTRICITY AND MAGNETISM
of constants which, multiplied by the strength of the pole m,
gives the force which acts upon it.
Problems
1. A wire 20 cm. long lies at right angles to a magnetic field of 50 c. g. s.
units intensity. What is the force acting upon the wire when a current of
30 amperes flows in it?
2. Assume that in the last problem the wire extends in a vertical direc-
tion, the current flowing from top to bottom, and the direction of the
magnetic field is from north to south. What is the direction of the force ?
3. A wire 1 km. in length is stretched horizontally on poles and carries a
current of 100 amperes. The vertical component of the earth's field is
.3 c. g. s. units. What is the total force urging this wire in a horizontal
direction ?
4. A circular coil of 1 turn carries a current of 10 amperes. What is its
radius if the field intensity at its center is unity ?
5. Two circular coils of wire lie in the same plane. One coil consists of
4 turns ; the other of smaller radius has but one turn. What must be the
ratio of their radii in order that the field intensity at their common center
may be zero when they carry the same current in opposite directions ?
6. A circular coil of wire has a radius of 20 cm. There are 50 turns of
wire in the coil and the current flowing is 6 amperes. What is the field
intensity at the center of the coil?
7. Why does a solenoid tend to shorten when a current is passed
through it?
8. A circular coil of wire carrying current is suspended in the earth's
magnetic field. Explain the torque action on the coil when its plane is
vertical and extends north and south. When its plane extends vertically
east and west. When its plane is horizontal.
THE HEATING EFFECT OF THE ELECTRIC
CURRENT
CHAPTER XXVI
JOULE'S LAW
318. The electric current produces a heating effect in any
conductor through which it flows. For example, the filament
of an incandescent lamp is so strongly heated by the current
which flows through it »
as to become incandes-
cent.
An understanding of
this heating effect may
be obtained from the
following considera- p^^ 210
tions. Let AB, Figure
210, represent a portion of a wire of resistance R carrying a
current /. Let it be assumed that the potential difference be-
tween the points A and B is E. The quantity of electricity
which flows down the wire from A to B in t seconds is
^^i^l'^"^ Q = It (Equation 82)
When this quantity of electricity passes from A to B, the po-
tential energy of the system is decreased by an amount which
we may call W. Then W= EQ, since, from the definition of
potential difference, it is evident that this amount of work
must be done upon the charge Q to carry it back from the
point B to the point A and thus restore the original conditions.
Since Q is equal to It, therefore,
W= Elt (a)
But from Ohm's Law, E= IB
Therefore, W= IB ■ It
or, W= PBt (99)
329
330 ELECTRICITY AND MAGNETISM
This apparent loss of potential energ}' in the system appears
in the conductor AB in the form of heat. Therefore, Equa-
tion (99) is an expression for the amount of heat developed by a
current Z in a conductor of resistance R in the time t. This
is known as Joule's Law.
If E is expressed in c. g. s. units of potential difference, that
is, in ergs per c. g. s. unit charge, I is given in c. g. s. units of
current and t in seconds, so that It expresses the charge in
e.g. s. units of charge, then evidentlj' TF (Equation a) is given
in ergs. If £ is expressed in volts and I in amperes, W is given
in joules. This is proven as follow.s : Assume that all quantities
are given in c. g. s. units. In order that U (Equation a), may
be reduced from c. g. s. units to volts it must be multiplied by
300 (Section 280). To reduce /from c. g.s. units to amperes
it must be divided by 3 x 10^ (Section 279). Thus the right-
hand member of the equation is in effect divided by lO-'. In
order that equality may be preserved the left-hand member
must also be divided by 10^. But Win ergs divided by 10'
gives TFin joules, since 1 joule = 10''' ergs (Section 60). Hence
also if, in Equation (99), Zis given in amperes and M in ohms, W
is given in joules.
THE HEAT DEVELOPED BY A CTJEEENT
319. It is oftentimes desirable to express W (Equation 99)
in calories, since the energy appears in the form of heat. To
do this, W as given in Equation (99) must be divided by the
number of joules in one calorie. Now,
one joule = IC ergs (Section 60)
one calorie = 4.187 x 10''' ergs (Section 218)
Therefore
one calorie = 4.187 joules
We have, therefore,
PRt
WCin calories") =
^ ^ 4.187
or W{in calories) =0.24 I^Rt (very approximately).
THE HEATING EFFECT OF THE ELECTRIC CURRENT 331
THE POWER EXPENDED IN HEATING AN ELECTRIC
CONDUCTOR
320. Power is defined as the rate of doing work (Section
86). That is to say, the power expended by any agency is
equal to the total work done by that agency divided by the
time in which the work is accomplished. Dividing Equation
(99) by t, we have tjt-
t
That is to say, the power expended in heating a conductor is
equal to the product of the resistance of the conductor and the
square of the current flowing in that resistance. From Ohm's
Law IR is equal to E where E is the potential difference be-
tween the terminals of the resistance R in which the current
J is flowing. Therefore the equation above may be written,
p = m (100)
Evidently from the above discussion of units, when IE is
expressed in volts and / is expressed in amperes, P will be
expressed in watts (see Section 86).
This discussion leads to the conclusion that in any circuit
which absorbs electric power the total power absorbed is obtained
by multiplying together the current in the absorbing circuit and
the potential difference between the terminals of that circuit.
In the same way the total electric power delivered to a circuit
by an electric generator giving a steady current is obtained by
multiplying together the electromotive force of the generator
and the current which it is supplying.
Examples. If an incandescent lamp when subjected to a
potential difference of 110 volts has a current of J ampere
flowing through it, the power absorbed by the lamp is
Pj = 110 X J = 55 watts.
Again, if a dynamo having an electromotive force of 500 volts
supplies 50 amperes to a circuit to which it is connected, then
the power supplied by the dynamo is,
Pa = 500 X 50 = 25,000 watts
= 25 k.w. (See Section 86.)
332
ELECTRICITY AND MAGNETISM
ELECTEIC HEATING
320. Heating by means of the electric current is accom-
plished by causing current to flow through a suitable resistance,
the value of the resistance being so chosen that when connected
to the given circuit, the proper value of current will flow
through it for developing the amount of heat required. Joule's
law is used in determining the value of the resistance necessary
for such purpose.
ELECTRIC COOKING
322. If it is desired to heat a liquid, a coil of insulated wire
protected by a copper tube and bent in the form of a spiral is
employed, see Figure 211. This spiral cop-
per tube containing the resistance coils is
lowered into the vessel containing the liquid
to be heated. When it is desired to heat
a vessel from below, an electric stove is
employed. This consists of suitable resist-
ance coils arranged immediately beneath an
iron plate, which becomes strongly heated
when the electric current flows in the re-
sistance coils. An electric oven is made of
sheet iron. On the inner walls of the oven
Fig. 211. — Immersion are mounted resistance coils through which
Coil for heating ^j^^ electric current is caused to flow, thus
Liquids by Means of
the Electric Current, heating the OVeu.
ELECTRIC LIGHTING
323. When a body is strongly heated, it becomes incandescent
and gives out light. The incandescent lamp is a device in which
a conductor of high resistance is heated to incandescence by
the electric current, thus becoming a source of light. In the
ordinary form of incandescent lamp the conductor is of carbon.
Since carbon oxidizes readily at high temperature, it is necessary
to inclose a conductor used in this way within a glass bulb
from which the air has been exhausted. Under these circum-
stances, there being no oxygen present, the carbon filament may
THE HEATING EFFECT OF THE ELECTRIC CURRENT 333
be heated to incandescence without danger of oxidation. The
Nernst lamp glower is of magnesium oxide and similar sub-
stances which, being already in an oxidized condition, are stable
chemically in air even when the filament is raised to a very high
temperature. Recently incandescent lamps are being made of
the metals, tantalum and tungsten. The principal advantage
claimed for the incandescent lamp in which tantalum or tung-
sten has been substituted for carbon is that its efficiency is
greater, that is to say, much larger returns in the way of light
are secured for a given input of electric power. The efficiency
of an incandescent lamp is usually specified in terms of the
watts absorbed per candle power of light delivered. Thus in
the carbon lamp the efficiency is roughly 4 watts per candle
power, for the Nernst and tantalum lamps about 2 watts per
candle power, and for the tungsten lamp about one and one
fourth watts per candle power.
The arc lamp is another device made use of in lighting by
electricity. In this device the incandescent body is the tip of
a carbon rod which has been brought into momentary contact
with a second carbon rod and then slightly separated therefrom.
The potential difference employed between the carbon rods tends to
maintain the flow of current between the rods even after they are
separated. The resistance at this point of the circuit being
very high, an intense heating effect is produced. In the pres-
ence of this heat the carbon is vaporized and forms a sort of
bridge of vapor between the ends of the carbon. This vapor
bridge is sufficient to maintain the current, and therefore the
heating action of the current, which keeps the tip of the carbon
white hot. In place of carbon a rod of magnetite is sometimes
employed for the negative terminal of the arc. The magnetite
arc has a higher efficiency than the carbon arc for the reason
that the magnetite, when strongly heated in the vapor state, is
brilliantly luminous, while the carbon vapor under the same
circumstances gives but little light. Another form of high
efficiency arc lamp known as the flaming arc employs carbon
rods impregnated with the salt of some metal of such nature
that the vapor produced is strongly luminous. Calcium fluoride
or calcium borate is usually employed for this purpose.
334
ELECTRICITY AND MAGNETISM
The efficiency of the arc lamp varies from about 2 watts per
3andle power in certain forms of carbon arc to about J of one
watt per candle iu some forms of the flaming arc.
THE ELECTRIC FURNACE
324. The highest temperature known in the laboratory is
that produced in the " crater " of the electric are, carbon rods
being used as electrodes. Certain chemical changes take place
at this high temperature which do not take place at lower tem-
peratures. Thus, it is possible to effect certain chemical com-
binations in the electric furnace which cannot be brought
about by any other means. It is by means of the electric fur-
nace that such compounds as calcium carbide are made.
Calcium carbide is a compound of calcium and carbon. It is
impossible to secure a combination of these two elements except
at the very high temperature of the electric furnace.
ELECTRIC WELDING
325. If two pieces of iron are brought end to end and a
strong current is sent through them, they will become strongly
heated at the point at which they are in contact, because at this
point there is large resistance to the flow of the current. If
the current is sufficiently large, this heating effect is very pro-
nounced, and after the lapse of a short time the ends of the iron
rods may be raised to a welding temperature. When suffi-
ciently heated in this manner, they may be compressed or
■ pounded together, and in this
manner welded.
THE ELECTRIC FORGE
326. The electric forge af-
fords another way of heating
a piece of metal to a high tem-
perature by means of the elec-
tric current. This apparatus
is illustrated in Figure 212.
^
C D#
A
-^
Fig. 212. — The Electric Forge.
AB is a vessel containing a solution of sodium carbonate. Cis
a plate of lead and 2> the bar of iron to be heated. When a
m.
THE HEATING EFFECT OF THE ELECTRIC CURRENT 335
current is sent through the apparatus from the lead to the iron,
a layer of gas forms on the surface of the iron, which, because
of its high resistance, gives rise to a strong heating effect.
With such a forge a piece of iron may be brought to a welding
heat in a few seconds.
FUSES
327. The heating effect of the electric current is taken ad-
vantage of in the use of fuses for protecting circuits against
excessive currents. The fuse in its simplest form is a piece of
wire formed of an alloy of lead and tin. This alloy is chosen
because of its low fusing point. The size of the fuse is so
chosen that it will carry the maximum allowable current with-
out becoming excessively
heated. If a larger current 1 f f
flows, the fuse is "burned" i j^^ ' — ^ ' —
and the circuit opened be-
fore the excessive current
injures the other parts of the Fig. 213.
circuit. The manner in
which fuses are placed in a circuit is illustrated in Figure 213.
ilf represents an electric motor. FF are the fuses. If, from
any cause, an excessive current begins to flow from the line
to the motor M, the fuses will burn out and damage to the
motor will be prevented.
Problems
1. A cuiTent of 10 amperes flows through a resistance of 10 ohms. How
much heat (in joules) is developed per minute?
2. A certain electric oven has a resistance of 9.6 ohms. At what rate
is heat developed in the oven when a current of 12.5 amperes is flowing, —
(a) in watts? (b) incal./sec. ?
3. What power is absorbed by an arc lamp which is supplied with 9
amperes at 50 volts ?
4. A wire having a resistance of 10 ohms is connected to a dynamo hav-
ing an e. m.f. of 50 volts. What power is absorbed by the wire? What
would be the effect as to the power absorbed if the length of the wire
doubled? Halved?
336 ELECTRICITY AND MAGNETISM
5. A certain 16 candle power incandescent lamp having a carbon fila-
ment takes 0.56 ampere at 100 volts. A tantalum lamp of 22 candle power
takes 0.44 ampere at the same voltage. How do the efficiencies of the
lamps compare ?
6. An electric motor requires 15 amperes at 110 volts. The two wires
leading from the dynamo to the motor have a resistance each of 0.3 ohm.
What e. m. f . must the dynamo supply ? How much power is lost in the
line?
7. A current flows through 3 wires of copper, platinum, and silver of
the same length and diameter, connected in series. What are the relative
amounts of heat developed in the 3 wires ?
8. If the 3 wires of problem 7 are connected in parallel, what are the
relative amounts of heat developed in them?
THE CHEMICAL EFFECT OF THE ELECTRIC
CURRENT
CHAPTER XXVII
ELECTROLYSIS
328. When an electric current is caused to flow through a
liquid having a complex molecule, it tends to break up the
complex molecular structure, reducing it to some simpler form.
If, for example, an electric current is passed through acidulated
water, the water molecule is broken up into oxygen and hy-
drogen. This effect of the electric current in bringing about
chemical change is called electrolysis. The liquid acted upon
is called the electrolyte.
IONS
329. Modern theory assumes that in a solution many of the
molecules are normally separated into positively and negatively
charged parts. Under this theory the action of the electric
current when passing through such a solution is largely to
assemble these positively and negatively charged molecular
parts at the points at which the current enters and leaves the
electrolyte, although it probably serves at the same time to
break up other molecules which before the passage of the cur-
rent were more or less stable.
Under the electron theory, the negatively charged part of
the molecule has an excess of electrons, the positive part a
deficit. These charged parts of the molecule tend to move in
the electrolyte in response to the electric forces which are pres-
ent. The positively charged parts tend to move with the (posi-
tive) current (Section 278), the negatively charged parts tend
to move against the (positive) current. These charged parts
of molecules which are supposed to wander about in this man-
ner in the electrolyte are called ions,
z 337
338 ELECTRICITY AND MAGNETISM
It is not believed that the ions move in straight and un-
broken paths through the electrolyte ; but that they are con-
tinually forming combinations and again breaking away from
such combinations to wander for a brief interval, perhaps entirely
free, only to unite a moment later with some other free ion of
opposite sign, to form a complete molecule, and so on. During
all of these changes they are steadily progressing in response
to the electric forces which are urging them forward.
ELECTROLYTIC TRANSFOKMATIONS
330. A general idea of the transformations due to electrol-
ysis may be obtained from a discussion of the following simple
cases. Let AB, Figure 214, repre-
V. » r / ^^'^^ ^ glass vessel containing an
""^^r^ — , I— <^ electrolyte, into which dip terminals
(electrodes) of an electric circuit.
Such an arrangement is called an
electrolytic cell. The electrode by
which the current enters the cell is
called the anode, that by which it
leaves the cell the cathode.
Fig. 214. — Electrolytic Cell. Let it be assumed that the
electrolyte in the cell shown in
the figure is copper sulphate (CuSO^) and the electrodes
metallic copper. When electrolysis takes place, the copper sul-
phate breaks up into Cu ions ( + ) and SO^^ ions ( — ). The
Cu ions move toward the cathode and unite with it, thereby
increasing its weight. The SO^ ions coming into the presence
of the anode combine with it, forming CuSO^, thereby decreas-
ing the weight of the anode. Evidently the average concen-
tration of the electrolyte remains the same.
If the electrolyte, instead of being copper sulphate, is dilute
sulphuric acid and the electrodes are of platinum, the products
of the electrolytic action of the current are gaseous, and layers
of gas gradually accumulate on both electrodes of the elec-
trolytic cell, finally rising in bubbles to the surface of the liquid.
Under these circumstances the hydrogen of the water molecule,
or the HgSO^ molecule, acts like the copper in the CuSO^ solu-
E 1
CHEMICAL EFFECT OF THE ELECTRIC CURRENT 339
tion in tlie cell described above. That is to say, the hydrogen
accumulates at the cathode. Probably in this case it is the
HjSO^ molecule which is broken up ; the SO^ ion coming into
the neighborhood of the anode combines with a molecule of
water, forming HgSO^ and free oxygen.
FARADAY'S LAWS
331. The mass of any ion deposited from an electrolyte by an
electric current is proportional to the quantity of electricity
passed through the electrolyte. That is, M^ Q, or, since Q = It,
we may write, Moz It. Hence,
M=elt (101)
in which e, the proportionality constant, is called the electro-
chemical equivalent of the substance deposited.
If 1= 1 and ( = 1 in Equation (101), e = M. Hence, the
electrochemical equivalent of any substance is the mass of that
substance which is deposited by unit current in unit time. The
law expressed by Equation (101) is known as Faraday's first
law of electrolysis.
If the same quantity of electricity is passed through a number
of electrolytic cells, each containing a different electrolyte, the
mass of each substance deposited is proportional to its chemical
equivalent. Thus the same quantity of electricity will deposit
1 gram of H, 35.46 grams of CI, 107.9 of Ag, etc. This is
called Faraday's second law of electrolysis.
From Faraday's second law it appears that the electro-
chemical equivalents of different substances are widely dif-
ferent. Below is a table which gives the electrochemical
equivalents of a few substances :
Electrochemical equivalents
Electrochemical
sltbstancb equivalent
Silver 0.001118
Copper 0.0003271
Nickel 0.000304
Hydrogen 0.000010
Oxygen 0.000082
Water 0.000093
340 ELECTRICITY AND MAGNETISM
The electrochemical equivalents given in this table are in
grams per ampere-second. It is convenient, when the products
of electrolysis are gaseous, to have the electrochemical equiva^
lent expressed in terms of the volume of gas liberated in the
cell, that is to say, cubic centimeters per ampere-second. Since
the volume of a gas depends upon its temperature and the
pressure to which it is subjected, the electrochemical equiva-
lent must be given in terms of the standard temperature, 0° C,
and the standard pressure, 760 millimeters of mercury. The
electrochemical equivalents of oxygen, hydrogen, and water
specified in this manner are given in the table below :
Electbochemioal
Substance Equivalent
Oxygen 0.0578
Hydrogen 0.1156
Water 0.1734
THE COULOMB METER
332. The electrolytic cell represented in Figure 214 may be
used for the measurement of an electric current. The opera-
tion is as follows : Using, say, copper electrodes in an elec-
trolyte of CuSO^, the cathode is carefully weighed before
placing it in the electrolyte and again after the unknown cur-
rent has been passing through the cell for an observed time t.
The difference between these two weights is the gain in weight
of the cathode, i.e. the amount of copper deposited by the cur-
rent in the time t. This is the mass contemplated in Equa-
tion (101). Since the electrochemical equivalent of copper is
known, we have M, e, and t of Equation (101), which may
therefore be solved for the value of I. Evidently the mass of
copper deposited is a measure of the quantity of electricity
which has passed through the cell rather than the current.
This is evident from the fact that ten amperes in one second
will deposit as much metal as one ampere in ten seconds.
Hence the apparatus, properly speaking, is a quantity meter, or
coulomb meter, the coulomb being the practical unit of quantity
(1 coulomb = 1 ampere-second).
CHEMICAL EFFECT OF THE ELECTRIC CURRENT 341
APPLICATIONS OF ELECTROLYSIS
333. Electrolysis is used extensively in practical operations.
Among the more important applications are the following :
Electrometallurgy. — Very pure copper is obtained by elec-
trolytic refining. For this purpose an electrolyte of CuSO^ is
used. The impure copper is connected as anode, a thin plate
of pure copper serving as cathode. When the current flows,
pure copper is deposited upon the cathode, the " sludge "
(impurities) falling to the bottom of the cell. Metallic alumi-
num is reduced from the oxide of aluminum by subjecting the
oxide in a fused condition to electrolysis.
Electroplating. — Objects made of the baser metals may be
gold or silver plated by causing them to serve as cathodes in
an electrolytic cell containing a suitable electrolyte. The elec-
trolytes used are the double cyanides of potassium and gold or
potassium and silver as the case may be. Surfaces of brass or
steel are often nickel plated to prevent tarnishing or rusting.
For nickel plating a double sulphate of nickel and ammonium
is used as the electrolyte. In order that the density of the
electrolyte may remain constant, the anode used must in each
case be a plate of the same metal as that deposited upon the
cathode.
Electrotyping. — -Most books which are printed in large edi-
tions are printed from copperplate copies of the pages of type
as set up in the ordinary way. A wax impression or mould is
made of the type, and this mould is copper plated by elec-
trolysis. This electrolj'tic copy of the original type is used in
the press.
The electrolytic process is also used in the manufacture of
various chemicals, such as caustic soda and potassium chlorate.
Problems
1- How many coulombs would be required to deposit 10 g. of silver?
2. How long would it take a current of 10 amperes to deposit 1 lb. of
copper?
3. How many grams of copper will be deposited by 5 amperes in 1 hr. ?
342 ELECTRICITY AND MAGNETISM
4. The same current is made to pass through a silver voltmeter and a
copper voltmeter in series. What are the relative amounts of silver and
copper deposited ?
5. In an experiment with the coulombmeter, the weight of the cathode
before it was placed in the electrolyte of CuSO^ was 10 g. After the un-
known current had passed through the cell for 20 min. the cathode was
found to weigh 11 g. What was the average value of the current?
6. How much water would be decomposed by a current of 1 ampere in
Ihr.?
THE VOLTAIC CELL
CHAPTER XXVIII
GALVANI'S EXPERIMENT
334. In 1786 it was discovered by an Italian by the name of
Galvani tiiat some freshly prepared frog legs, which were sus-
pended from a copper hook attached to an iron railing, twitched
when they came into contact with the iron. This effect being
like that produced by an electric discharge, Galvani recognized
it as an electric effect and attempted to explain it by assuming
that an electric discharge was generated in the muscle of the
leg. This explanation was rejected by Volta, a professor in the
University of Padua, who maintained
that the true source of the effect was
the contact of dissimilar metals. Volta
proceeded to verify his theory by de-
vising experiments in which dissimilar
metals were brought into contact
directly and in other cases through the
medium of a suitable liquid. The
most marked effects, that is, the great-
est potential differences, were secured
by an arrangement like that repre-
sented in Figure 215, which is com-
monly known as a Voltaic cell. It
consists essentially of a containing vessel A, an electrolyte B,
and two dissimilar metals and I) dipping into the electrolyte,
in other words, Volta's cell is an electrolytic cell having dissimi-
lar electrodes.
A simple and effective cell is made by using strips of copper
and zinc as electrodes and a dilute solution of sulphuric acid as
an electrolyte.
343
=^— ^
i
—
B
Fig. 215. —The Voltaic Cell.
344 ELECTRICITY AND MAGNETISM
Upon examining such a cell it will be found that there is a
difference of potential between the copper and the zinc so long as
they are allowed to stand in the electrolyte, and furthermore, if
the upper end of the copper and zinc electrodes are joined by a
wire w as indicated in the figure, an electric current will flow in
the direction indicated by the arrow. This current will con-
tinue to flow so long as the conditions represented in the diagram
are maintained. The source of the energy represented by this
current is the chemical transformation which is going forward
within the electrolyte. It is found that the zinc gradually dis-
appears as metallic zinc, being taken up by the acid and con-
verted into zinc sulphate (ZnSO^).
As intimated above any dissimilar metals may be employed
in place of the copper and zinc but the potential difference be-
tween the terminals depends upon the nature of the materials
employed. In the same way it is found that there are many
liquids which will take the place of dilute sulphuric acid as an
electrolyte, but the potential difference between the electrodes
depends upon the nature of the electrolyte.
It is also found upon experiment that with two given metals
the one may be positive and the other negative when dipped in
one electrolyte, while the reverse relation will exist if they are
placed in a second electrolyte. For example, if the metals,
lead and copper, are placed in dilute HCl, the copper is at a
higher potential than the lead. If a solution of potassium
sulphate is used as an electrolyte, the lead is positive with
respect to the copper.
Evidently there is a large choice of materials which might
be employed in the cell both with respect to the electrode and
the electrolyte. The advantages of several of the more impor-
tant combinations will be taken up in detail in the discussion
given below.
THE CHEMICAL ACTION IN THE SIMPLE VOLTAIC CELL
335. When a simple voltaic cell like that represented in Figure
215 is supplying current, that is to say, when its electrodes are
connected by a wire, electrolysis takes place in the cell. Thus
HjSO^ when used as an electrolyte is broken up into hydrogen
THE VOLTAIC CELL 345
on the one hand and SO4 on the other. It is found that the
hydrogen set free in the operation of the cell tends to accumu-
late upon the copper, that is upon that electrode which is at the
the higher potential (i.e. the cathode), and which is commonly
referred to as the positive terminal of the cell. SO^ is found to
accumulate in the neighborhood of the negative terminal of the
cell, and being very active chemically it immediately combines
with the metallic zinc of this terminal, forming zinc sulphate.
The hydrogen does ' not enter into combination with the copper
terminal in the neighborhood of which it appears, but collects in
the form of hydrogen gas on this terminal.
POLARIZATION
336. Evidently, as the chemical action described in the last
section goes forward, the copper terminal or positive terminal
of the Voltaic cell becomes coated with a layer of minute
hydrogen bubbles. The effect of this layer of bubbles is to
prevent contact between the electrolyte and the electrode, and
may result in a very decided diminution of the potential
difference between the electrodes of the cell. This:effect is
known as polarization.
Various means are employed for preventing polarization.
One method used is to give the positive electrode a large
size and a rough surface. This method only serves to post-
pone the effect. It requires, of course, a longer time for
the bubbles to cover a large surface than would be required
for a smaller one. The best method is to take up the hydro-
gen as it is formed, preventing in this way its accumulation
as a layer of minute bubbles against the electrode. If, for
example, the electrode is surrounded by an oxidizing agent
with which the hydrogen combines readily, the formation of
free hydrogen in the neighborhood of the electrode will be
prevented. Chemical agents employed for this purpose are
known as depolarizers.
LOCAL ACTION
337. Another difficulty which is encountered in the opera-
tion of the Voltaic cell in a practical way is that which is
346
ELECTRICITY AND MAGNETISM
known as local action. Local action arises from the fact that
commercial zinc which is so extensively employed in Voltaic cells
contains a certain amount of impurities in the way of iron, car-
bon, and the like. These particles of dissimilar substances em-
bedded in the surface of the zinc constitute with the zinc, when
dipped into an electrolyte, small Voltaic cells. This will be
understood by reference to Figure 216, in which AB represents
a piece of zinc, a bit of carbon em-
bedded in the surface of the zinc.
When the zinc is dipped into dilute
sulphuric acid a difference of potential
exists between and AB. This will
give rise to small electric currents cir-
culating through the zinc and the car-
bon and the electrolyte in the immediate
neighborhood as indicated by the small
arrows. The effect of these local cur-
FiQ. 216. — Illustrating Local rents is to cause the zinc to be gradually
Action. , . . , , ,
converted into zinc sulphate even when
the battery proper is not in operation. The effects of local
action may be largely done away with by what is known as
amalgamation. The zinc rod is amalgamated by dipping it
into dilute sulphuric acid and then into mercury. The mer-
cury forms an amalgam with the zinc. This amalgam spreads
over the surface of the rod, covering up all of the particles of
foreign matter which may be embedded in its surface and thus
preventing contact between them and the electrolyte.
[
A
' -1-
"i
' J
Its
M
Ij
r
y
B
V
THE GRAVITY BATTERY
338 Oiie of the more common forms of modern Voltaic cell
is tlie gravity battery. This has been evolved from the original
form of what was known as the Daniell cell. The Daniell cell in
its original form consisted of copper and zinc electrodes dipping
into zinc and copper sulphate solutions, the copper electrode
dipping into the copper sulphate, the zinc electrode being sur-
rounded by the zinc sulphate, and the two solutions being kept
separate by placing the zinc sulphate in a porous cup, that is,
an unglazed earthenware cup. This cup, while it prevented
THE VOLTAIC CELL 347
the mixing of the solutions, did not separate them electrically.
The copper sulphate in this cell acts as the depolarizer, — the
copper of the copper sulphate, being displaced by the free
hydrogen, is thrown down against the copper terminal. The
SO4 ion combines with metallic zinc at the zinc electrode, thus
increasing the amount of zinc sulphate present in the cell.
In the gravity battery the copper electrode and solution of
copper sulphate are placed in the bottom of a vessel, the zinc
electrode and zinc sulphate being placed in the top of the
vessel. The two solutions are kept separate by gravity, the
copper sulphate being more dense than the zinc sulphate. This
battery is also called the crowfoot battery because of the form
which is commonly given to the electrodes.
THE GROVE CELL
339. In the Grove cell the electrodes are of platinum and
zinc. The platinum in the form of a thin strip is placed in a
small porous cup and surrounded by strong nitric acid. This
porous cup is then placed in a solution of sulphuric acid into
which the zinc electrode is also placed. Evidently in this cell
the strong nitric acid is the depolarizer. The free hydrogen
coming into the presence of the nitric acid combines with it in
such manner as to form nitric oxide (NjOj) and water (HgO).
The nitric oxide escaping from the cell takes up additional
molecules of oxygen when it comes into contact with the air,
forming nitrogen tetroxide. One of the serious drawbacks
to the use of this cell is the formation of these noxious fumes.
Another objectionable feature from a commercial standpoint is
the high cost of the platinum electrode.
THE BUNSEN CELL
340. To overcome the objectionable features of the Grove
cell, Bunsen substituted carbon for the platinum and potassium
bichromate (KgCrgO^) for the nitric acid. It is found that
when potassium chromate combines with a small amount
of sulphuric acid a very strong oxidizing agent, chromium
trioxide (CrOg), is formed.
Bunsen also discovered that when potassium bichromate is
348
ELECTRICITY AND MAGNETISM
used as a depolarizing agent, the porous cup may be dispensed
with, the two electrodes dipping into the same solution. A
cell made up in this manner is usually called a bichromate cell.
THE LECLANCHE CELL
341. In this cell the positive electrode is of carbon and is
surrounded by or mixed with manganese dioxide (MnO^), which
acts as a depolarizer. The electrolyte is ammonium chloride
(NH^Cl). The negative electrode of the cell is zinc.
THE DRY BATTERY
342. The batteries described in the foregoing sections all
possess the disadvantage of containing liquids which evaporate
when used for any appreciable length of time and which are
apt to spill as the cells are carried about. The dry battery is
largely free from these defects. As usually constructed it con-
sists of a zinc vessel which serves as the negative electrode and
at the same time as the containing vessel for the electrolyte.
Against the walls of this vessel are placed several layers of
damp blotting paper. Within this
is a moist mixture of manganese
dioxide, graphite, plaster of paris,
and calcium chloride, in the midst
of which is placed the carbon or
positive electrode. The mangan-
ese dioxide serves as a depolarizer.
The calcium chloride is used be-
cause of its property of taking up
moisture from the air. Evidently
the contents of the cell must be
-Hg£So4 Paste kept moist in order that it may
. Hg. continue in operation.
-Zn
...Stopper
-!• 1 Zn So4 Sol.
.ZnSo4 Crys.
Fig. 217.
THE STANDARD CLARK CELL
343. The Clark cell is used as a
standard of potential difference
and is usually made up in small sizes, oftentimes conveniently
in a test tube. The positive electrode is mercury. The
THE VOLTAIC CELL
349
depolarizer is mercurous sulphate (HgjSO^). The electro-
lyte is a saturated solution of zinc sulphate, and the negative
electrode is pure metallic zinc. One of the more simple forms
of the Standard Clark cell is shown in Figure 217, in which
the arrangement of the various parts is shown. The e.m. f. of
the Standard Clark cell is very constant, changing only slightly
with the temperature. At 15° C. its e. in. f. is 1.434 volts.
E. M. F.'S OP COMMON BATTERIES
344. The electromotive forces of the cells described above
are given in the following table :
Name of Cell E. M. F.
Gravity Battery ... 1.08 volts
Grove Battery . . . .1.9 volts
Bichromate Cell ... 2.1 volts
Leolanch^ Cell . 1.5 volts
Dry Battery ... . . 1.35 volts
Clark Cell ... 1.434 volts
B
Fig. 218.
THE REVERSIBILITY OP THE VOLTAIC CELL
345. In certain forms of Voltaic cell it is possible to com-
pletely reverse the chemical transformations which take place
in the cell when
it is supplying
current by forc-
ing a current
through it in
the reversed di-
rection. Let it
be imagined, for
example, that
three gravity
batteries are
connected in se-
ries as shown in
Figure 218. A
and B are joined
in the same sense, that is to say, in such manner that the poten-
tial difference between the zinc and the copper in the cell A is
1
/
V
/
^~ >
\
--A Zn :
-
; — — =^- -
:znh-
~4 Zn :
/
\
N
^
N
'
Cu 1
CU 1
1 cu !
350 ELECTRICITY AND MAGNETISM
added to the potential difference between the zinc and the
copper in the cell B. The cell C is connected in the opposite
sense. Cells A and B work together to send a current around
the circuit in the direction indicated by the arrows. The cell
is so connected that it tends to oppose this flow of current, that
is, it tends to cause current to flow in the opposite direction.
Since A and B are working together, they will overcome the
effect of the cell Q and will force current through the cell
C in the reverse direction. Under these circumstances the
chemical transformations which take place in the cell C are the
reverse of those which take place in the cells A and B. In A
and B the hydrogen of the H^SO^ molecule appearing in the
neighborhood of the copper electrode displaces the copper of
the copper sulphate, thus freeing the metallic copper, which is
deposited on the copper plate. The SO^ ion going into the
presence of the zinc electrode combines with metallic zinc, form-
ing zinc sulphate. In the cell the hydrogen of the HgSO^
molecule, which is broken up by the current, goes into the
presence of the zinc plate, displaces the zinc in the zinc sul-
phate, thus freeing the zinc, which is deposited on the zinc
electrode. The SO^ ion coming into the presence of the copper
plate combines with the metallic copper, forming copper sul-
phate. That is to say, in the cells A and B the weight of the
zinc decreases and zinc sulphate is formed. In the cell (7 the
weight of the zinc increases and zinc sulphate grows less in
amount. In the cells A and B the copper plate increases in
weight and the copper sulphate grows less in amount, while
in the cell Q the copper plate decreases in weight and more
copper sulphate is formed.
The thought here suggests itself that it might be possible to
restore a gravity battery which is pretty well worn out by con-
necting it to other batteries in the manner in which is con-
nected in Figure 218, and sending through it a current in the
reverse direction. This is possible. Evidently when the cell
has again exhausted itself, that is to say, when it becomes dis-
charged, it may again be charged in the manner indicated. A
cell used in this manner is called a storage battery.
THE VOLTAIC CELL 351
THE STORAGE BATTERY
346. Evidently the essential features of a storage battery are
that the chemical reactions which take place when the cell is
used as a source of energy shall be completely reversible. In
addition to this the nature of the electrodes and electrolyte
must be such that there is no local action of any sort when the
cell is standing idle.
THE LEAD STORAGE CELL
347. The simplest form of storage battery copsists of lead
electrodes dipping into dilute sulphuric acid. When a current
is sent through a cell of this character, the anode becomes
strongly oxidized, its surface being coated with a dark brown
layer of peroxide of lead. The other plate is gradually formed
into spongy lead. If such a cell, after being charged in this
manner, is connected to a receiving circuit, it will be found
capable of furnishing electric current. The potential difference
between its electrodes when charged is about 2.1 volts. As
the cell discharges the oxidized condition of the positive plate
gradually disappears, the oxide being reduced to a lower oxide
and this finally to metallic lead. When the oxide entirely dis-
appears, the action of the cell may be restored by again charg-
ing it. In the practical form of the storage battery it is
customary to make the plates in the form of grids upon which
is placed a paste of lead oxide. This construction simplifies
the process of charging the cell and makes the cell more efS-
cient in that larger surface is exposed to the action of the current
than would be the case if the plate were made up in the solid
form.
The chemical reactions in the cell are as follows: Assume
the cell to be charged, then one plate is spongy lead Pb, and
the other lead peroxide PbOg. During discharge the Hj of the
electrolyte (HgSO^) goes to the cathode, the SO4 going to the
anode ; we have, therefore,
PbOa -h H2 -F H2SO4 = 2 H2O -1- PbSOi
t
Pb -f-SO^ =PbS04
352 ELECTRICITY AND MAGNETISM
The arrow shows the direction of the current in the cell.
The products of the reaction are lead sulphate (on the plates)
and water. Evidently during discharge the concentration of the
electrolyte decreases.
For the reactions during charge we have,
Pb SO4 + SO4 + 2 HjO = PbOj + 2 H2SO4
Pb SO^ + H^ = Pb + H^SO^
During the charging operation the plates P and N return to
their original chemical form, and the concentration of the elec-
trolyte increases.
ELECTRICAL MEASURING INSTRUMENTS
CHAPTER XXIX
GALVANOMETERS
348. A galvanometer is an instrument for measuring electric
current. There are several kinds of galvanometers, the more
important of which are described in the following paragraphs.
THE TANGENT GALVANOMETER
349. The tangent galvanometer consists essentially of a cir-
cular loop of wire of one or more turns standing in a vertical
position and having suspended at its center a small bar magnet
or compass needle. In the use of the instrument the coil is
carefully adjusted to stand in a magnetic north and south direc-
tion. Evidently the compass needle at its center will, under
these circumstances, be parallel to the plane of the coil. When
current is sent through the coil, an east and vi^est magnetic
field is set up at the center of the coil, as pointed out in the
discussion of the magnetic field at the center of a circular
loop of wire (Section 316). This
east and west field due to the
current in the coil combines with
the earth's magnetic field, form-
ing a resultant field in the direc-
tion of which the compass needle
tends to set itself. Let H, Figure
219, represent the horizontal com-
ponent of the earth's field. Let ^^^ 219 "
/ represent the field due to the
current in the coil. The resultant, -B, of these two fields is in-
dicated in magnitude and direction by the diagonal of the
2 a 353
354
ELECTRICITY AND MAGNETISM
rectangle in the figure. Let 6 be the angle between the re-
sultant field R and H. Then evidently,
tan 9 =
H
but /, the field due to the current in the coil, is 2 irnl-i- r
(Section 317). Substituting this value of/ in the expression
for tan 6, we have,
tan 6 =
Solving this equation for /,
1=
2-7rnI
rH
rH
■ t&nO
I Trn
r and n are constants depending only on the dimensions of the
instrument ; and H, the horizontal component of the earth's
magnetic field, may also be taken as a constant for any one
point. It will be seen, therefore, that the coefficient of tangent
6 in the above equation is a constant. We may therefore write,
7= ir- tan (102)
in which K has been written for rH-i- 2 ttm. That is to say,
the tangent of the angle 6 through which the compass needle
is deflected is proportional to, and therefore a measure of, the
current which flows in the coil of the instrument.
^is known as the constant of the instrument. It is best
determined by sending through the instrument a known cur-
rent and observing the corresponding
deflection. Knowing / and tan 6, the
constant K may be calculated from the
equation.
THE THOMSON GALVANOMETBK
350. The equation of the tangent gal-
vanometer indicates that a given current
will produce the largest effect in the
instrument when the number of turns
of wire is large and the radius of the
coil is small. It will be evident, there-
galvanometer, that is, one which will
Fig. 220.
fore, that a " sensitive
respond to very small currents, should be made up in this way,
ELECTRICAL MEASURING INSTRUMENTS
355
An instrument having a small coil of many turns of fine wire,
with a small magnet suspended at its center, is called a Thomson
galvanometer. With an instrument of this type currents of
less than one ten-billionth of an ampere may be measured.
Figure 220 represents a Thomson galvanometer in its sim-
plest form. The magnet is supported by a slender fiber of silk
or quartz, and its deflections are observed by means of a small
attached mirror. The law of the tangeat galvanometer cannot
be applied to the Thomson instrument for the reason that the
field in which the needle turns is not uniform.
THE d'ARSONVAL GALVANOMETEK
351. The D'Arsonval galvanometer is an instrument in
which the arrangement of parts is the reverse of that found in
the Thomson galvanometer, that is, in this instrument it is the
coil carrying the current which moves, the magnet producing
the field in which the coil lies remaining stationary. A com-
mon form of the D'Arsonval
galvanometer is represented
in Figure 221.
N and S are the poles of
a strong horseshoe magnet
mounted in an upright posi-
tion. Between these poles
is suspended a vertical coil
of fine wire, one end of the
wire serving as a support
for the coil and at the same
time as one of its terminals,
the other end of the wire ex-
tending downward and serv-
ing as the other terminal of
the suspended coil as repre-
sented in the diagram. Let
it be imagined that the cur-
rent passes clockwise around the coil as indicated. Applying
the left-hand rule, it will be evident that the left-hand side
of the suspended coil will tend to move toward the observer
5
I
y
N
Fig. 221. — Esseutial Parts of a D'Arsonval
Galvanometer.
356
ELECTRICITY AND MAGNETISM
and the right-hand side from the observer. This tendency of
the suspended coil to turn is opposed by the twist in the sup-
porting wires. Evidently the larger the current which is flow-
ing, the larger the force action on each side of the coil, and the
larger the angle through which the coil will turn. This in-
strument is capable of being made very sensitive. It possesses
the distinct advantage over the Thomson galvanometer that
it is but little affected by external magnetic influences.
THE PLITN6EE IKSTEUMENT
352. Another type of electrical measuring instrument which
depends for its indications upon the magnetic action of the
current is known as the plunger type. The essential parts of
this instrument are shown in Figure 222.
It is seen to consist essentially of a sole-
noid AB, through which the current to be
measured is passed, and a soft iron plunger
'. O, which becomes magnetized in such way
that the plunger is drawn down into the
solenoid. For example, if the current in
the solenoid flows clockwise in the coils as
seen from above, the positive direction of
the lines of force is that indicated in the
diagram. The lower end of the soft iron
plunger therefore becomes a north-pointing
pole and the upper end a south-pointing
pole. The north-pointing pole tends to
move in the positive direction of the lines
of force, that is, toward the center of the
solenoid, while the south-pointing pole
tends to move in the opposite direction.
The field intensity, however, is greater in
the neighborhood of the north-pointing pole or lower end of the
plunger than it is in the neighborhood of the upper end of
the plunger. Hence the tendency of the north-pointing pole
to move in a downward direction is greater than that of the
south-pointing pole to move in the opposite direction. The
plunger as a whole moves downward. This tendency to move
Fig. 222.
ELECTRICAL MEASURING INSTRUMENTS
357
downward is opposed by the spiral spring. The extent to
which the plunger moves downward is indicated by the pointer
attached to the plunger which plays over the scale EF. Evi-
dently the stronger the current in the solenoid the greater the
intensity of the magnetic field within it, and hence the greater
the distance to which the plunger is drawn into the coil.
THE ELECTRODYNAMOMETER
353. In the electrodynamometer a movable coil mounted
like the coil of a D'Arsonval galvanometer is placed at the
center of a coil (large) like that used in the tangent galva-
nometer. The two coils are
arranged to stand perpendicular
to each other. When a current
is passed through the two coils,
the small coil tends to turn
about its suspending wires.
This tendency to turn
will be understood from _L
the following considera-
tions. Let A, Figure /i^
223, represent the larger coil
and D the smaller coil. When
a current flows in the coil A, a
magnetic field is established in the neighborhood of D, the
direction of which is perpendicular to the plane of coil A. If
a current is caused to flow through the coil D, it will tend to
turn in this magnetic field, being acted upon by forces propor-
tional to the current in the coil I) and the field in the neigh-
borhood of D due to the current in the larger coil. But the
field at D due to the current in A is proportional to the cur-
rent in A (Section 317). Therefore the torque on D is propor-
tional to the product of the currents in D and A, that is,
Tccli or T=KIi. If the same current / flows through
both coils, then T = K I"^
If there were nothing to oppose this tendenc}'^ to turn, the coil
Ji would set itself parallel to the coil A. In turning, however,
Fig. 223.
358 ELECTRICITY AND MAGNETISM
it twists the vsuspending wires. The opposing torque intro-
duced in this manner balances the torque due to the action of
the current in the coil. The coil D will therefore turn through
a definite angle for each value of the current which flows
through it. The deflections of the coil D may be observed by
means of a mirror il!f attached to the suspending wire near the
coil.
THE HOT WIRE INSTRUMENT
354. In this instrument the heating effect of the electric
current is taken advantage of. Consider a wire AB, Figure
224, supported at its
V\ /^ ends and attached at its
" center to a spring c. If
a current is passed
p through AB, it will heat
the wire, causing it to
•p increase in length (Sec-
tion 158); the "slack"
will be taken up by c.
Fig. 224.
The pointer SP pivoted
at is attached at P to the end of the spring and moves with
it. Hence as AB lengthens, the pointer moves over the scale
»S' according to the current effect in AB.
AMMETERS, VOLTMETERS, AND "WATTMETERS
355. An ammeter is a low resistance galvanometer provided
with a scale so marked as to indicate directly the current which
passes through the instrument.
A voltmeter is a high resistance galvanometer provided with a
scale so marked as to indicate directly the electromotive force
applied to its terminals.
The method of connecting ammeters and voltmeters is shown
in Figure 225. A represents an ammeter and V& voltmeter;
2/ is a group of lamps being supplied with current from the
dynamo D. With this arrangement the total current going to
the lamps passes through and is measured by the ammeter A.
The e. m. f. which is applied to the lamps acts also upon the volt-
ELECTRICAL MEASURING INSTRUMENTS
359
meter V, sending through it a current i = - . Hence the indica-
R
tions of V are proportional to E, the e. m. f . acting upon the lamps.
Obviously, in order that A and Fmay absorb but little power,
Fig. 225. — Diagram ot Circuit showing Connections of Ammeter and Voltmeter.
A, which carries the whole current, must have small resistance,
and V must be of large resistance in order that the current
which flows through it may be small.
A Wattmeter is an electrodynamometer having one coil (the
current coil) of low resistance and one coil (the pressure coil) of
high resistance, and provided with a scale so marked as to indi-
cate directly the power absorbed by the circuit to which it is
connected.
The manner in which a wattmeter is connected to a circuit
is shown in Figure 226. TTis the wattmeter which is to meas-
ure the power sup-
plied to the lamps L
by the dynamo J).
The circle represents
the current coil and
the ellipse the pres-
sure coil of the watt-
meter. With the
connections indicated
in the figure the total Fig. 226.
current / which goes
to the lamps passes through the current coil of the wattmeter.
The pressure coil of the wattmeter is connected as a shunt to the
lamps. Let H be the e. m. f . acting on the lamps. Then the
current which will flow through the pressure coil is «' = — , in
- Diagram of Circuit showing Connections
of Wattmeter.
since » = -=, •■•
II
t.{^)m.
i.e.
T=Ki:i
360 ELECTRICITY AND MAGNETISM
which R is the resistance of the coil. It has already been
pointed out (Section 353) that the torque acting on the sus-
pended coil of an electrodynamometer is proportional to the
product of the currents in the coils, therefore T = K^ ■ I ■ i ox.
E
EI= a const, x EI.
(103)
But EI is the power absorbed by the lamps L (Section 320) ;
therefore the torque acting on the movable coil of the wattmeter
is porportional to the power (watts) absorbed by the circuit to
which the instrument is connected.
Problems
1. When a certain tangent galvanometer is in adjustment, it is found
that a current of 5.5 amperes in the coil will deflect the needle 45°. What
is the constant of the galvanometer ?
2. A current of I amperes deflects the needle of a tangent galvanometer
60°. A current of i amperes gives a deflection of 30°. What is the ratio -"i
I
3. What current vfill deflect the needle of the galvanometer of problem
1 50°?
4. The coil of a tangent galvanometer has a mean radius of 20 cm. and
consists of 40 turns of wire. If used where the horizontal intensity of the
earth's field is .2 c. g. s. units, what is the constant of the instrument ?
ELECTROMAGNETIC INDUCTION
CHAPTER XXX
INDUCED ELECTROMOTIVE FORCE
356. It was discovered by Faraday in 1831 that whenever a
current is started or stopped in an electric circuit, there is a
momentary current in any other closed circuit in its immediate
neighborhood. The circuit in which the current is started or
stopped is called the primary circuit. The circuit in which the
momentary current circulates at the moment of starting or
stopping the current in the primary is called the secondary
circuit. The temporary currents which circulate in the sec-
ondary at the moment of starting or stopping the current in the
primary are called induced currents. The e. m. f.'s in the sec-
ondary in response to which the induced currents flow are
called induced electromotive forces.
'^(^
■VVS-W
©
Secondary
t)\C_
R
Primary
K
BI=
Fig. 227. — Illustrating Electromagnetic Induction.
The statement made in the foregoing paragraph will be more
readily understood by reference to Figure 227, in which ABO
represents an electric circuit containing a battery £ joined by
a wire AC to a resistance i2 and a key K. DUGr represents a
second circuit consisting of a galvanometer, the terminals of-
which are connected by a wire DJE. It is supposed that DS
361
362 ELECTRICITY AND MAGNETISM
and A C are parallel portions of the two circuits which lie close
to one another. With the arrangement shown in the diagram
it will be observed that, at the moment of closing the key K, and
thus starting the current in the circuit J.5 (7, the galvanometer
Cr will indicate the presence of a current in the circuit DEO.
Again, upon stopping the current in the primary circuit ABO,
the galvanometer G- will indicate a momentary current in the
secondary circuit BEGr, in the opposite direction to that which
was produced when the current was started in the primary.
A careful study of the conditions represented in the diagram
will discover that a momentary current is produced in the
secondary, not only upon starting or stopping the current in the
primary, but whenever any change is made in the primary,
either with respect to the magnitude of the current flowing in the
primary or with respect to its position relative to the secondary.
In other words, an induced current is present in the secondary:
(a) when the current is started in the primary ; (5) when the
current is stopped in the primary ; (c) when the value of the
current in the primary is changed (either increased or di-
minished) ; (<£) when the primary is moved nearer to or farther
from the secondary, the current in the primary remaining the
same.
A more convenient arrangement of apparatus for studying
these effects is that shown in Figure 228. ^ is a coil of wire
connected to a battery E, a resistance R and a key K. This is
the primary circuit. 5 is a coil of wire connected to a sensitive
galvanometer Cr. This constitutes the secondary circuit. In-
duced currents are present in the coil B whenever the current
in A is started, stopped, or changed in magnitude, or whenever
the coil A, with a steady current flowing in it, is caused to ap-
proach or recede from the coil B.
Since the coil A, Figure 228, is surrounded by a magnetic
field which changes with respect to the coil B when the current
in the coil A is changed or when the position of the coil A is
changed, we very naturally conclude that the induced electro-
motive forces in the coil B are in some way associated with the
changing magnetic field due to the coil A.
If it is the changing magnetic field in the neighborhood of
ELECTROMAGNETIC INDUCTION
363
the coil B which gives rise to the induced electromotive force
in B, it ought to be possible to demonstrate the presence of
such electromotive forces in B whenever the magnetic field in
its neighborhood changes from other causes. It ought, for ex-
ample, to be possible to demonstrate the presence of induced
electromotive forces in the coil B when a bar magnet is brought
up into its presence or removed from its neighborhood, since
under thqse circumstances there would be a changing field in
the neighborhood of the coil B. That there are induced elec-
tromotive forces in the coil B under these circumstances may
be shown by means of the apparatus represented in Figure 229.
Primary
Secondary
Fig. 228.
NS is a permanent bar magnet which is supposed to have been
thrust into the coil B from the left to its present position.
Upon making the experiment, it is found that there is an in-
duced electromotive force in the coil B while the magnet is
moving up to its present position. If the magnet is now with-
drawn, there will again be an induced electromotive force in the
coil B, but in the opposite sense.
Again, if it is the changing field in the neighborhood of B
which is responsible for the induced electromotive force, it
ought to be possible to induce electromotive forces in such a
coil by turning it over in a magnetic field. For example, in
Figure 230, let B represent a coil lying in a horizontal position.
364
ELECTRICITY AND MAGNETISM
Under these circumstances a certain number of lines of force
due to the earth's magne-
tism are threading through
the coil in the direction in-
dicated. If the coil is re-
versed, eYidently, when it
comes into an edgewise po-
sition with respect to the
direction of the lines of
force, there will be no lines
of force threading through
it. And, finally, when it is
turned into the reverse posi-
tion, this same number of
lines of force will be pass-
ing through the coil in the opposite direction. Under these cir-
cumstances it is found that when the coil is reversed there is an
induced electromotive force in the coil.
Fig. 229.
THE INDUCED ELECTROMOTIVE FORCE DEPENDS UPON THE
RATE AT WHICH THE MAGNETIC FIELD CHANGES
357. In all of the experiments outlined in the preceding
paragraphs it can be very readily determined that the induced
electromotive force depends
upon the rate at which the
magnetic field in the neigh-
borhood of the coil B is
changing. Thus if, in the
arrangement of apparatus
represented in Figure 229,
the north pole of the bar
magnet is very suddenly
thrust into the coil B, the
induced electromotive force
in the coil is correspond-
^ . Fig. 230.
ingly great. If the pole is
inserted slowly, the induced electromotive force is correspond-
ingly small. Again, if in the experiment represented in
ELECTROMAGNETIC INDUCTION 365
Figure 230, the coil is very quickly reversed, the induced elec-
tromotive force is greater than when the coil is slowly revolved
in the magnetic field.
THE LAW OF INDUCED ELECTROMOTIVE FORCES
358. Thus it may be demonstrated by experiment that
there is an induced e. m. f. in any circuit through which the
magnetic flux (i.e. total number of lines of force) is changing,
and that the magnitude of the induced e. m. f. depends upon
the rate at which the flux is caused to change. There are two
statements of the law of induced e. m. f.'s, both of which are
useful in the discussion of the various applications of the prin-
ciple of electromagnetic induction.
(a) In terms of changing flux:
There is an induced electromotive force in any coil when the
number of lines of force threading through it is changing, and
the value of the electromotive force is equal to the rate at which
the number of lines of force through the coil is decreasing.
(6) In terms of cutting lines of force :
There is an induced electromotive force in any conductor
which is cutting {i.e., moving across) lines of force, and the
value of the induced electromotive force is equal to the rate at
which lines of force are cut by the conductor.
MAGNITUDE OF THE INDUCED ELECTROMOTIVE FORCE
359. Let it be imagined that iV lines of force are withdrawn
from a coil of wire in t seconds. The induced electromotive
force in the coil under these circumstances is given by
e=^ (104)
This then is the algebraic expression of statement (a). (See
above.)
If a conductor moves through a magnetic field in such man-
ner that it cuts i\riines of force in t seconds, the induced electro-
motive force is again given by the above equation. Hence,
this is also the algebraic expression of statement (6). In other
366 ELECTRICITY AND MAGNETISM
words, Equation (104) is the general law of induced electromo-
tive force.
It should be observed that the above expression is really a
defining equation for electromotive force. That is, a new unit
of e. m. f. is here contemplated such that if one line of force is
cut per second the induced e. m. f. is unitj*. It can be shown
that one volt is equal to 10,000,000 such units. Therefore, if e
is to be expressed in volts, we have, —
e (m volts ) =
^ -^ t 10'
DIRECTION OF THE INDUCED ELECTEOMOTIVE FORCE
360. Experiment shows that the induced current in any circuit
is in such direction as to oppose that change of conditions which
gives rise to the induced current. This is known as Lenz'sLaw.
As examples of the application of Lenz's Law consider the fol-
lowing. In the experiment illustrated in Figure 229 it is the
approach of the magnetic pole which gives rise to the induced
electromotive force in the coil B. According to Lenz's Law the
induced current in the coil B will be in such direction as to op-
pose the approach of this north-pointing pole. In other words,
the induced current in the coil B under the assumed conditions
will be in such direction as to establish a magnetic field within
the coil of such nature that the lines of force pass through the
coil from right to left. As the north-pointing pole is pushed
into the coil it is therefore being carried forward in opposition to
this magnetic field.
Again, considering the experiment illustrated in Figure 228,
we have seen that when a current is flowing in the coil A, an
induced current is present in the coil B if it is caused to ap-
proach nearer to the coil A. Applying Lenz's Law to this case,
it will be understood that the induced electromotive force in the
coil B, under these circumstances, is in such direction as to op-
pose by its magnetic reaction on the coil A this approach or
coming together of the two coils; that is, the direction of the
induced current in the coil B will be opposite to that in the coil
A, since, as has been demonstrated, parallel currents flowing in
ELECTROMAGNETIC INDUCTION
367
opposite directions repel one another. If we consider the in-
duced electromotive force in the coil B when it is drawn back
from the coil A, we can see that, according to Lenz's Law, the
induced current in B would be in the same direction as the
current in A, since parallel currents flowing in the same direc-
tion attract one another; and this force of attraction would
therefore constitute a resistance to the separation of the coils,
or in other words, would tend to oppose that motion which sep-
arates them.
In determining the direction of the induced electromotive
force in a conductor, the right-hand rule may be used. This
rule is as follows : Hold- c
ing the thumb and first x;
and second fingers of the
right hand in such manner
that they are at right
angles to one another, if
the first finger points in
the direction of the field
or lines of force, and the
thumb points in the direc-
tion in which the conduc-
tor is moving, the second
finger gives the direction
of the induced electromo-
tive force in the conductor.
Figure 231 is designed to
illustrate the right-hand rule and to show the relative direc-
tions of these three quantities. For determining the direction
of an induced electromotive force in a coil the following rule is
applicable: Let it be imagined that one is looking through the
coil in the direction in which the lines of foi-ce extend. Then
a decrease in the number of lines of force in the coil will give
rise to an induced current which will flow clockwise in the coil.
An increase in the number of lines of force will give rise to an
mduced electromotive force which will flow counterclockwise
m the coil. See Figure 232. The curved arrows represent
the direction of the induced electi'omotive force in the coil
Held
Fig. 231
368
ELECTRICITY AND MAGNETISM
Fig. 232.
when the number of
lines of force threading
through the coil from
left to right is decreas-
ing.
The effect of decreas-
ing the number of lines
of force threading
through a coil from left
to right is the same as
that produced by
m-
creasing the number of lines of force threading through the
coil from right to left.
EDDY CURRENTS
361. "Eddy currents "are currents which eddy or circulate
locally in masses of metal in the neighborhood of which the mag-
netic field is changing. Consider, for example, a disk of copper
A, Figure 233, which is rotating between the poles of a horse-
FiG. 233. — Eddy Currents.
shoe magnet. Consider any radial element of the disk, for
example, that one lying horizontally to the right and between
the poles of the magnet. According to statement (6) of the
law of induced electromotive force, an electromotive force will
be induced in this element as it cuts the lines of force due to
the magnet, the direction of which is from the right toward the
ELECTROMAGNETIC INDUCTION
369
left as may be determined by the right-hand rule. In response
to this induced electromotive force induced currents will flow-
along this element of the disk from the circumference toward
the center and back through the adjacent parts of the disk as
represented by the dotted lines C. These currents circulating
in small closed paths within the copper are known as eddy
currents. They are true electric currents, being characterized
by the effects of the electric current. Thus, they produce
heating effects and magnetic effects. Since their circulation in
the copper disk means the expenditure of energy, it will be
evident that work must be done upon the disk to supply this
energy. That is to say, it will require greater expenditure of
energy to rotate the disk when these eddy currents are present
than would be required in their absence. Furthermore, the disk
will become heated if it is caused to rotate between the poles
of a strong horseshoe magnet as contemplated in the discussion.
ARAGO S EXPERIMENT
362. An interesting example of the generation of eddy
currents is that afforded by Arago's experiment. Figure 234
represents a thick copper
disk rotating about its
center in the direction
of the arrows OD. Let
NS represent a small bar
magnet pivoted at and
standing just in front of
the copper disk. Under
these conditions it will be
found that the magnet
tends to follow the cop-
per disk in its rotations,
although every precau-
tion is taken to shield it
from any other than the
magnetic influence of the
eddy currents in the disk AB. The explanation of this rota-
tion of the magnet NS is as follows: Consider that radius of
2b
Fig. 234. — Arago's Rotations
370 ELECTRICITY AND MAGNETISM
the disk which is just passing beneath the north-pointing pole
of the bar magnet. Some of the lines of force which radiate
from N pass directly through the copper disk. These lines of
force are cut by each radius of the revolving disk. Applying
the right-hand rule, it is easily determined that the induced
electromotive force on that radial element of the disk which is
just passing beneath the north pole of the bar magnet is from
circumference to center along the radial element as indicated.
The eddy currents which are set up in the copper will circulate
as indicated by the curved dotted lines. If now we consider
the reaction of these currents upon the magnetic field of the bar
magnet, it will become at once apparent that the north pole
of the bar magnet is urged in the direction in which the disk
is moving. To determine the magnetic reaction between the
eddy current and the north-pointing pole of the bar magnet,
apply the left-hand rule (see Section 312). The application
of this rule shows that the eddy current in this portion of the
disk is urged in a direction opposite to that of the rotation; but
since reaction is equal to action and oppositely directed, there-
fore the force acting on the north pointing pole of the magnet is
in the direction in which the disk is rotating. If we consider
the radial element of the disk which is passing under the
south-pointing pole of the bar magnet, it is evident that the
radial eddy current in this part of the disc is from center to
circumference, the direction of the field and the direction of
motion being both reversed. Hence the force acting upon the
south-pointing pole of the bar magnet under these circum-
stances urges it in the direction in which the disc is rotating.
THE PEBVENTTON OF EDDY CURRENTS
363. Since power is required to maintain eddy currents and
through their agency energy is transformed into heat, it is de-
sirable to eliminate the eddy current effect as far as possible
from commercial electrical apparatus. All massive metal parts
which are subject to fluctuating magnetic fields and all iron
parts which are repeatedly magnetized and demagnetized will
have eddy currents generated in them except in those cases in
which the circulation of such currents is prevented.
ELECTROMAGNETIC INDUCTION
371
Fig. 235.
The method of preventing the flow of eddy currents most
commonly employed is to laminate (cut in thin sheets) the metal
parallel to the magnetic
field and perpendicular to
the plane in which the
eddy currents tend to cir-
culate. For example, let
A, Figure 235, represent a
mass of iron which is being
magnetized in the direc-
tion indicated by the ar-
rows B. Since the outer
portions of this mass of
iron A constitute a closed conductor about these lines of force,
it will be evident that induced currents will tend to flow in
these outer portions as indicated by the dotted lines. The
arrowheads indicate the direction in which the induced current
will flow in the mass of iron when the number of lines of force
threading through the mass A from the side B is increasing.
Now the mass of iron A may be laminated in any plane parallel
to tlie lines of force B without breaking its magnetic continu-
ity in the direction BA ; but such lamination will interrupt the
continuous path C in which the eddy currents tend to circulate.
The metallic path for the eddy currents being broken up, the
eddy currents are largely prevented, especially so if the lamina-
tions are insulated from one another. Evidently in the case
represented in Figure 235, the same effect might be secured by
using a bundle of small iron wires.
SBLF-INDTJCTION
364. A careful consideration of tlie law of induced electro-
motive force leads to the conclusion that induced electromotive
forces are present in any coil in which current is being started or
stopped or changed in magnitude. Consider, for example, coil
A, Figure 228. When the key /fis closed and the current be-
gins to flow in the coil,,^, a magnetic field is set up about the coil.
In other words, the starting of the current in the coil amounts
to a threading of a number of lines of force through the coil.
372 ELECTRICITY AND MAGNETISM
But according to statement (a) of the law of induced electro-
motive force, tliis threading of a number of lines of force through
the coil will result in an induced electromotive force. In the
same way, if after the current is established in the coil A, and
the magnetic field surrounding it becomes constant at each and
every point, the circuit is opened and the current stopped, there
will again be an induced electromotive force in the coil A, since
to stop the current is in effect to withdraw those lines of force
which are threading through the coil. Thus we see that induced
electromotive forces are present in a coil in which a current is
started or stopped, and in the same way in a coil in which the
current is varied in magnitude. These electromotive forces are
termed self -induced electromotive forces or electromotive forces
of self-induction since they are developed in the coil in which
the current which produces the changing magnetic field about
the coil is flowing.
To determine the direction of this e. m. f. of self-induction we
have only to apply Lenz's Law of induced currents. For example,
imagine the current to be increasing in the coil. This means
an increasing number of lines of force threading through the
coil in response to the increasing current. But Lenz's Law
states that the induced current will oppose that which gives
rise to the induced current, namely, the increasing number of
lines of force threading through the coil. In other words, the
induced current under these circumstances will be opposed in
direction to the current supplied by the battery, since an oppos-
ing current would tend to establish a magnetic field in the
opposite direction, or what amounts to the same thing, to oppose
the introduction of this increasing number of lines of force.
Again, considering the moment of opening the circuit and
stopping the current from the battery, the decreasing current,
under these circumstances, means a decreasing number of lines
of force threading through the coil. Therefore, according to
Lenz's statement, the induced current will be in the direction of
the current from the battery, since a current in this direction
will tend to prevent that which gives rise to the induced current,
namely, the decreasing number of lines of force threading through
the coil.
ELECTROMAGNETIC INDUCTION
373
Stated briefly, the self- induced electromotive force is in such
direction as to oppose the current from the battery when the current
is increasing in magnitude or when the current is being started,
and in such direction as to tend to maintain the current from the
battery when the current is decreasing or is being stopped.
The presence of the self-induced e. m. f. in a circuit may be
demonstrated by means of tlie apparatus represented in Figure
236. MM is an electromagnet
having a laminated core of soft
iron. This electromagnet is
connected to a battery B and a
key K. L is an incandescent
lamp connected as a shunt across
the terminals of MM. Let it
he assumed that the key is
closed and a current is flowing
in the direction indicated by
the arrow Q. There will also
be a current i in the lamp as
indicated. At the moment of
opening the circuit at K there will be a self-induced e. m. f. in
MM in such direction that it tends to prevent the decrease in the
current in MM. This self-induced e. m. f. will send a reverse
current through L. If MM and B are properly chosen, the
lamp L will glow brightly for an instant under the induced
current in MM even when the battery e. m. f. is too small to
"light" the lamp directly.
Fig. 236. — Arrangement of Apparatus
for showing Self-induced E. M. F.
THE COEFFICIENT OP SELF-INDUCTION
365. Experiment shows that the magnetic flux (i.e. the total
number of lines of force) which is established by a current in a
coil is proportional to the current. Tliat is, iVocJ, in which
i^T represents the magnetic flux and /the current. Hence, we
may write
N=LI (105)
in which i is a constant for the coil under consideration and is
called the coeflficient of self-induction of the coil.
374
ELECTRICITY AND MAGNETISM
Equation (105) holds rigidly only for coils surrounded by air or
some other medium of constant permeability (Section 301). If
a coil has an iron core, the magnetic flux through the coil is
not proportional to the current (Section 302), i.e. the relation
Nxl does not hold. The coefficient of self-induction of such
a coil depends upon the value of the current flowing in it.
A non-inductive circuit is one in which the conductors are so
disposed with respect to one another that their individual
magnetic effects are neutralized. For example, two wires lying
closely side by side and carrying the same current in opposite
directions form a practically non-inductive system. Similarly
a coil of wire consisting of two layers wound in opposite direc-
tions is non-inductive.
THE INDUCTION COIL
366. The induction coil is a device which is used for develop-
ing high electromotive forces by taking advantage of the principle
Fig. 237. — The Induction Coil.
of electromagnetic induction. Its essential features are a core
or bundle of iron wires, a few turns of heavy insulated copper
wire which is known as the primary coil and a secondary coil
consisting of a very great number of turns of fine insulated
ELECTROMAGNETIC INDUCTION 375
copper wire. The arrangement of parts and the connections
for the apparatus are shown in Figure 237. PP is the primary
and SS the secondary coil. The primary coil is connected to a
battery B and a key K by means of which the primary current
is started or stopped. When the key K is closed, the current
flows from the battery through the primary circuit and mag-
netizes the core CC. This means that a large number of lines
of force are threaded through each turn of wire wound upon
the core. This again means that induced electromotive forces
are present in each turn of wire while the number of lines of
force is changing. According to statement (a) of the law of
induced electromotive force, the induced e. m. f. is numerically
equal to the rate at which the number of lines of force is chang-
ing in the coil. Considering the secondary circuit, it will be
evident that the induced electromotive forces in the successive
turns are in the same direction and that they act together to
produce a large electromotive force between the terminals SS.
Since the value of this electromotive force depends simply
upon the rate at which the lines of force are being threaded
through each turn, and the number of turns, evidently by using
a large number of turns and causing the magnetic flux through
each turn to change rapidly, large values of electromotive force
may be established between the terminals SS.
When the circuit is opened and the current stops in the
primary coil, this bundle of lines of force is in effect suddenly
withdrawn from each and every turn of wire in the secondary
circuit. Therefore at this moment there will again be an in-
duced electromotive force in each of the turns and between the
terminals of the secondary.
To secure a rapid interruption of the primary current a
spring vibrator F", carrying a piece of soft iron H, is included
in the circuit, as shown in the figure.
From the foregoing statements the effectiveness of the coil is
greatest when the current in the primary is very quickly started
or very quickly stopped. Now it is found in practice that, be-
cause of the effects of self-induction in the primary circuit, a
large spark will be produced at the point at which the primary
circuit is opened. This spark is in effect an arc which enables
376 ELECTRICITY AND MAGNETISM
the current to continue for a certain interval of time after tlie
circuit is actually broken. The effect of this is to allow the
current in the primary circuit to die away slowly instead of
stopping suddenly as it should do in order to secure the max-
imum effect in the secondary. In order to obviate this diffi-
culty of sparking at F'in the primary circuit a condenser (? is
shunted around the spark gap at V. With this arrangement the
self -induced current in the primary, instead of causing a spark
at the key K, tends to charge the condenser, which after a very
brief interval discharges again in the reverse direction through
the battery and the primary circuit. This device not only does
away in a large measure with the effect of the spark at the key
K, but makes it possible to reduce the magnetic flux in the
core from its maximum value to zero in a very brief interval of
time. Without the condenser the core retains some of its mag-
netism after the primary circuit is broken. When the con-
denser is used, the reverse current from the discharging con-
denser just after the primary circuit is broken demagnetizes the
core, thus doing away with any residual magnetism.
Induction coils may be constructed in this manner to give
very high voltages between the terminals of the secondary,
voltages of such magnitude as to cause a discharge through air
of several inches (several feet even) and give effects analogous
to those produced by the discharge from the electrostatic
machine.
As was stated above, the core of an induction coil is made of
a bundle of small iron wires instead of one large mass of iron
such as is sometimes employed in an electromagnet. The
object in using a bundle of iron wires instead of one large mass
of iron is to prevent eddy currents in the iron (Section 363).
THE TESLA COIL
367. The effects secured by means of the Tesla coil illustrate
in a very striking manner the principle of induced electro-
motive forces. This apparatus is essentially an induction coil
without an iron core. In other words, it consists of two coils,
one of a few turns of coarse wire which is called the primary,
ELECTROMAGNETIC INDUCTION
377
and the other of many turns of fine wire called the secondary.
The secondary is placed within the primary.
When a rapidly varying current is passed through the pri-
mary, e. m. f.'s are induced in the secondary. In order that
the apparatus may be effective, a very rapidly changing current
must be present in the primary. A convenient way of securing
such a primary current is to make of the primary a discharging
circuit for a Leyden jar. Under suitable conditions the dis-
charge of a Leyden jar is oscillatory, that is, at each discharge
of the jar the discharging current surges to and fro in the dis-
charging circuit (Section 272). These surgings succeed each
other very rapidly, often at the rate of a million per second.
The connections for a Tesla coil are given in Figure 238.
is a Leyden jar connected to the terminals PP of the primary of
the Tesla coil, (r- is a
spark gap across which ^^
the discharge of the jar
takes place. A and B
are connections extend-
ing to some suitable
source for charging the
jar, e.g. the secondary of
an induction coil. When
the apparatus is in opera-
tion and a rapid success-
ion of discharges passes
the spark gap (?, very
high e. m. f.'s are in-
duced in the secondary
SS. These e. m. f.'s are
of very high frequency, i.e. they change rapidly in magnitude
and direction. The discharge from such a coil has peculiar
properties, among which is that of passing through or over the
surface of the human body with but little sensible effect.
Fig,
Diagram of Connections of a Tesla
Coll.
THE DYNAMO
368. The dynamo is a device for transforming mechanical
energy into electrical energy. It depends for its action upon
378
ELECTRICITY AND MAGNETISM
Fig. 239. — Simple Dynamo.
the principle of electromagnetic induction. It consists essen-
tially of a powerful electromagnet and a series of conductors
which are made to move rapidly
through the magnetic tield, due to
the electromagnet. One of the sim-
plest forms of dynamo is represented
in Figure 239, in which NS are the
poles of an electromagnet NABS.
When a current is flowing in the coils
A£, there is established a strong
magnetic field between iVand S from
left to right as indicated by the hori-
zontal lines. Let it be imagined that
in the space between iV and S there
are a number of conductors extending in a direction perpendicu-
lar to the plane of the paper in the diagram, and all revolving
about the point C, which is the center of the magnetic field be-
tween iV and S. It will be evident that each of these conduc-
tors cuts all of the lines of force extending across from Nto S
twice for each revolution which it makes about the point 0.
It is evident, therefore, that in each and all of these moving
conductors, electromotive forces will be induced. By properly
connecting these conductors the individual induced electromotive
forces may be added together so as to get one large electromotive
force acting through the entire series. In the practical form
of the dynamo the revolving conductors are mounted upon a
soft iron cylinder having its axis at 0. This diminishes the air
space through which the lines of force must flow, and very
materially increases the magnetic flux of the electromagnet.
Furthermore, it gives a solid support for the revolving con-
ductors. This revolving part of the dynamo is called the ar-
mature. The iron core of the armature is laminated in order to
prevent the eddy currents which would otherwise be developed
in the iron core as it revolves in the magnetic field. The lam-
inations extend at right angles to the axis of rotation. In other
words, the core is made up of a series of thin disks.
ELECTROMAGNETIC INDUCTION
379
Fig. 240.
INDUCED ELECTROMOTIVE FORCE IN A COIL REVOLVING IN
A MAGNETIC FIELD
369. The general character of the induced electromotive force
in a coil which is caused to revolve in a magnetic field will be
understood from a dis-
cussion of the following
simple case : Let AB^
Figure 240, represent a
rectangular coil of wire
of one turn arranged to
rotate on the axis CD.
Let it be imagined that
this coil is lying in a uni-
form magnetic field ex-
tending from left to right
as indicated in the figure,
and that the coil is caused to rotate counterclockwise as seen
from C. Evidently when the coil is in the position represented,
there will be induced electromotive forces in the sides A and
-B, in the directions indicated by the dotted arrows. That is
to say, the electromotive
force in the side A is from
the observer, while that in
the side B is toward the ob-
server. Therefore, if the
coil forms a closed circuit,
the induced current will cir-
culate in a counterclockwise
direction about the coil as
seen from above.
When the coil has rotated
through 90 degrees and has
come into the position represented in Figure 241, there are no in-
duced electromotive forces present. In this position the greatest
number of lines of force are threaded through the coil, but it
will be remembered that the induced electromotive force depends,
not upon the total magnetic flux present, but upon the rate at
Fig. 241.
380
ELECTRICITY AND MAGNETISM
Fig. 242.
which it i.s changing. It is evident that in this position of the
coil the sides A and B are moving parallel to the lines of force,
hence the flux through the coil is momentarily constant.
When the coil comes into the position represented in Figure
242, the sides A and B are again cutting lines of force and there-
fore have induced elec-
D/ tromotive forces in
them. It v^ill be no-
ticed, however, that the
directions of the induced
electromotive forces in
the sides A and B are
opposite to those in the
first position as repre-
sented in Figure 240.
That is, the induced
electromotive force in
the side A is now toward the observer, and that in the side B
from the observer. The induced e. m. f.'s are, as before, in a
counterclockwise direction as seen from above, but the coil has
been reversed so that the induced current flows in each part of
the coil in a direction opposite to that in which it was flowing
in the first position.
When the coil has made another quarter revolution and is
again in a vertical position, the induced e. m. f.'s will again be
zero. It will be evident that it is only when the coil is exactly
vertical that the induced e. m. f.'s are zero, since when it has
turned only slightly from a vertical position the sides A and B
will begin to cut lines of force. The sides A and B have in-
duced e. m. f.'s in them throughout the entire revolution of the
coil except for the brief instant during which the coil is in a
vertical position. The rate at which the sides A and B cut the
lines of force steadily increases from zero up to a maximum
when the coil lies in a horizontal position, and then steadily
decreases as the coil turns once more into the vertical position.
ALTERNATING AND DIRECT CUEEENTS
370. Evidently during one revolution of the coil described
in the last section the induced e. m. f. (or current), rises to a
ELECTROMAGNETIC INDUCTION
381
maximum value twice, and twice during the revolution is equal
to zero. Furthermore, the two maximum values attai'ned in
each revolution are in opposite directions in the coil. Such a
current is called an alternating current. That is, an alternat-
ing current is one which begins to flow in one direction, rises to a
maximum value, and then falls off to zero, then begins to flow
in the opposite direction, rises to a maximum value, and then
falls to zero, and so on repeatedly.
A direct current is one which flows continuously in the same
direction.
THE ALTERNATING CURRENT GENERATOR
371. The alternating current generator is a dynamo so con-
structed that the alternating currents which are developed in
its rotating coils are trans- _^^^
mitted to the outside circuit, w D/
with which it is connected,
as alternating currents. The
principle of the method em-
ployed for accomplishing this
will be understood by refer-
ence to the simple case repre-
sented in Figure 243. AB
represents a rotating coil like
that described in the last sec-
tion. 8 and S' are "slip
rings," that is to say, insu-
lated metallic rings to which
the terminals of the coil A
and B are connected. The
side A of the coil is con-
nected to the ring S, and the side B is connected to the ring ;S"
as indicated. These rings are attached to the axis upon which
the coil AB is mounted, and hence rotate with the coil. If
two strips of metal E and F, connected to an outside circuit,
for example an incandescent lamp circuit, L, are pressed against
the slip rings S and S' while the coil AB is rotating, evidently
the induced alternating current in the coil AB will circulate
as an alternating current through the lamp circuit, L.
-Simple Alternating Current Gen-
erator.
382
ELECTRICITY AND MAGNETISM
THE TRANSFOEMER
372. Since induced electromotive forces depend for their
existence on a varying condition of magnetism, it will be evi-
dent that induced electromotive forces must always accompany
alternating currents. It is possible by means of the alternating
current to transfer electric energy from one circuit to another
with which it has no metallic connection by utilizing the effect
of electromagnetic induction. This is done by a means of a
device called a transformer.
The transformer depends for its action upon the principle of
electromagnetic induction. Its action will be understood from
the following discussion : Let
AB, Figure 244, represent a
frame of soft iron upon which
are wound two coils of wire P
and /S, in the manner indicated
in the diagram. Such an ar-
rangement is called a trans-
former. Let it be imagined
that an alternating current is
-Illustrating the Principle of flowing J^ the Coil P. This
the rransiormer. °
alternating current will give
rise to an alternating magnetic condition of the iron frame AB.
The lines of force which extend through the iron when it is
magnetized by current in the coil P will pass around the frame
in the direction of the dotted lines. Since they pass through
the coil S, evidently induced electromotive forces will be pres-
ent in the coil S whenever these lines are being threaded
through or withdrawn from that coil. If the terminals of the
coil ^S* are connected through any circuit, induced currents will
flow in this coil in response to these induced electromotive
forces. Thus energy is transmitted from the coil P to the coil
S by means of the fluctuating magnetism in the iron frame.
Transformers are used for raising or lowering the electro-
motive force of an electric system. For example, electric current
is distributed over the city at an electromotive force of 1000
volts. It would be dangerous in many ways to use such volt-
FiG. 244.
ELECTROMAGNETIC INDUCTION
383
age in dwellings. It is, therefore, necessary to lower or
" step-down " the voltage of such a system before the current
is carried into the houses. This is done by means of trans-
formers. If 100 volts are desired, a "ten to one" transformer
is used, i.e. one having ten times as many turns in its primary
as in its secondary coil. The secondary e. m. f. will then be
100 volts.
THE DIRECT CURRENT GE^'ERATOR
373. The direct current generator is a dynamo so constructed
that the alternating currents which are developed in its rotat-
ing coils are transmitted to
the outside circuit, with
which it is connected, as a
direct current. This is ac-
complished as follows : Let
AB., Figure 2-15, represent a
rotating coil like that de-
scribed in Section 369. The
terminals of this coil are
connected respectively to
and C", the two parts of a
metal cylinder which has
been divided lengthwise as
indicated in the figure.
and C" are insulated from
one another and mounted
upon the axis upon which Fig. 245. — Simple JJU-ect Current Generator.
the coil AB rotates. If two
strips of metal (brushes) E and F, connected to an outside cir-
cuit, for example an incandescent lamp circuit, L, are caused to
make contact with Cand C at the extremities of the horizontal
diameter of the cylinder GC, then as the coil AB rotates, a direct
current will flow through the lamp. This will be evident from
the following considerations : Consider the moment at which
the coil AB reaches its vertical position. At this instant the
induced e. m. f . in the coil is zero. As the coil passes the verti-
cal position, the direction of the induced e. m. f .'s in it is changed
in direction, but at this instant the brush IE passes from the
384
ELECTRICITY AND MAGNETISM
C to the C" part of the cylinder and I' passes from C" to C
That is to say, at the moment in which the e. m. f . in the coil
AB changes direction, the connections with the lamp circuit L
are reversed. It follows, therefore, that the direction in which
the current flows in the circuit ELF remains unchanged as in-
dicated by the arrow. The divided cylinder CQ' is called a
commutator.
THE ELECTEIC MOTOR
374. The electric motor is a device for transforming electrical
into mechanical energy. It is essentially a dynamo, which,
being supplied with electrical energy from some outside source,
becomes a source of mechanical energy. Most dynamos are
reversible, that is to say, they may be used either as generators
of electricity, in which case they are supplied with mechanical
energy, or they may be used as electric motors by supplying
them with electric energy.
The motor action of a
dynamo will be understood
from a consideration of the
following simple case. AB
(Figure 246) represents a
coil like that discussed in
the preceding section. It is
provided with a commutator
OG' and is connected by
means of " brushes " EF to
a battery as shown. Let it
be assumed that to begin
with the coil is stationary in
the position shown. The
battery current will flow
through ^5 in the direction
indicated by the arrows.
Evidently the sides A and B
are acted upon by forces tending to move them at right angles
to the magnetic field in which they are lying. Applying the
left-hand rule (Section 312), it is evident that A is acted upon
by a downward force and B by an upward force so that the coil
Fig. 246. — Simple Direct Current Motor.
ELECTROMAGNETIC INDUCTION 385
as a whole tends to rotate clockwise as seen from the commutator.
The forces acting upon A and B continue to be more or less
effective in producing rotation of the coil until as the coil turns
the vertical position is reached.
As soon as the coil has passed the vertical position, which the
inertia of the moving coil will enable it to do, the current from
the battery will be reversed in the coil; but the coil having been
inverted with respect to the magnetic field, evidently the forces
acting upon A and B will tend to continue the clockwise rota-
tion. Hence a steady current flowing from the battery to the
moving coil will maintain a continuous rotation.
The rotating coil described in the last paragraph may be kept
in continuous rotation by supplying it with an alternating current
instead of a continuous current such as is obtained from a battery,
provided a definite relation between the speed at which the coil
rotates and the alternations of the alternating current exists.
The relation referred to is as follows: The coil must revolve
through 180° while the current is making one alternation. It
will be understood from the description of the direct current
motor above, that the function of the commutator is to convert
the steady current from the battery V into an alternating current
in the coil A, since the commutator reverses the current in the coil
A every half revolution. It follows, therefore, that if an alter-
nating current from some outside source is conducted by means
of slip rings to the coil AB, the condition for continuous rotation
will be secured as before, provided, as stated above, that the coil
revolves through 180° for each alternation of the current, and also
that the coil is in its horizontal position when the alternating
current is changing from positive to negative, or vice versa. This,
of course, means that in order to cause such a coil to rotate by means
of an alternating current it must be set in rotation first and made
to revolve at a definite speed before the alternating current is
turned on. Alternating current motors which operate on this
principle are called synchronous motors.
There is another form of alternating current motor, called an
induction motor, which depends for its operation upon a rotating
magnetic field. One of the simplest ways of developing a rotat-
ing magnetic field by means of alternating currents is as follows:
2c
386
ELECTRICITY AND MAGNETISM
Fig. 247.
liSt AA'BB', Yigure 247, represent 4 coils of wire placed upon
a soft iron ring MR'. If the coils AA' are supplied with current,
the iron ring will be
magnetized in such man-
ner as to have its mag-
net poles at B and B'.
When current is caused
to flow in the coils BB',
the current in A A' being
zero, the ring will be
magnetized in such a
manner as to liave its
magnet poles at A and
A'. When current is
flowing in both sets of
coils, the magnet poles
will lie between the
coils. For example, if
the current in the coils
AA' is in such direction that it tends to produce a north-point-
ing pole at B, and the current in the coils BB' is in such direc-
tion that it tends to produce a north-pointing pole at A', then
the combined effect of the currents in both coils will produce
a north pole at B', and the
south pole at R. Evi-
dently it is possible to
cause the condition of
magnetism in the ring
BR' to shift or rotate by
properly switching the
currents on to the pairs of
coils AA' and BB'. The
eight successive stages of
this shifting magnetic
field, together with the
corresponding directions
of the currents in the coils A and B which produce them, are
Fig. 248.
shown in
Figure
248. In order that this rotating field may
E-LECTROMAGNETIC INDUCTION 387
exist, it is only necessary to supply the coils AA' and BB' with
alternating currents which are "out of step" in their alternations.
Under these circumstances the magnetic effect produced by the
A coils will reach its maximum value before the B coil effect
reaches its maximum value. Hence, the condition for rotating
magnetic field is secured. Alternating currents which are
" out of step " in this way are usuallj^ said to differ in phase,
and such an alternating current system is known as a two-phase
system.
If now within the iron ring MR' there is placed a copper
cylinder so mounted as to be free to turn, it will tend to rotate
with the rotating magnetic field which passes through it, be-
cause of the eddy current effect wliich at once arises in the
metal cylinder. This apparatus as described constitutes an
induction motor. In the practical form of the apparatus the
rotating part is filled with laminated iron.
Problems
1. A wire is moved across a uniform magnetic field cutting 10,000 lines
of force in 1 sec. What is the induced e. m. f . in the wire ?
2. A wire 1 m. long is moved through a uniform magnetic field having
an intensity of 1000 lines per square centimeter. The wire moves perpen-
dicular to its own length and at right angles to the field. If the velocity of
the wire is 20 cm. /sec, what is the induced e. m. f. in the wire?
3. If the direction of motion of the wire in problem 2 makes an angle
of 60° with the field, what is the induced e. m. f. ?
4. A circular coil of wire of 50 turns having a mean radius of 20 cm.
lies on a table, i^ = 0.18, dip = 70°. The resistance of the coil is 0.01 ohm.
The coil is picked up and turnsd over in one second. What is the average
induced current in the coil?
5. What is the total quantity of electricity set in motion in turning the
coil of problem 4 ? Would it make any difference in the quantity if the
coil were turned slowly or quickly?
6. A coil of 50 turns having an area of 4 sq. cm. is " snapped " from
between the poles of an electromagnet where the field intensity is 8000
lines per square centimeter to a point where the field intensity is negligibly
small in 0.01 see. What is the average induced e. m. f. in the coil ?
7. A coil is rotated uniformly about a horizontal north and south axis.
The average induced e. m. f. is 60,000 c. g. s. units. When the same coil is
rotated about a vertical axis at the same speed, the average induced e. m. f.
is 40,000. What is the dip of the earth's field V
388 ELECTRICITY AND MAGNETISM
8. A rectangular loop of wire 20 x 30 cm. is rotated in a uniform mag-
netic field of 5000 lines per square centimeter, at a speed of 1800 R. P. M.
The axis of the coil is at right angles to the field. What is the average
induced e. m.f. ?
9. Would it mate any difference in problem 8 whether the axis is par-
allel to the long side or short side of the coil ? Whether it is at the center
or the edge of the coil ? If it extended along one of the diagonals ?
10. If the rotating coil of problem 8 had 5 turns iijstead of 1 and formed
1 closed circuit (5 turns in series) of 0.5 ohm resistance, at what rate would
heat be generated in the coil ?
11. What average torque would be required to rotate the coil of prob-
lem 10 ? In what positions of the coil would the torque have its maximum
and minimum values ?
TELEGRAPHY AND TELEPHONY
CHAPTER XXXI
THE ELECTRIC TELEGRAPH
375. The electric telegraph is a device for the transmission
of intelligence in which advantage is taken of the magnetic
action of the current. An impulse of current sent over a line is
made to magnetize a distant electromagnet. This electro-
magnet attracts a small armature or piece of soft iron with suf-
ficient force to produce an audible click. The electromagnet
arranged to be used in this way is called a sounder. The
sounder gives an audible click for every current impulse that
goes over the line. Hence by a system of prearranged signals
Ground Ground
Fig. 249. —Simple Telegraph Circuit.
it is possible to transmit intelligence by means of this device.
The simplest form of telegraph circuit is that shown in Figure
249. JT is a key, B a battery, and S a sounder. A short-cir-
cuiting switch (i.e. a switch which closes the gap left by the
open key^, is provided for each key. If the short-circuit switch
at either end of the line is opened and the key tapped, both
390
ELECTRICITY AND MAGNETISM
sounders will respond. In this manner signals may be sent to
the farther end of the line. When not in use the key is short-
circuited so as to complete the circuit for incoming signals.
As indicated in the figure, a single wire extends from one
station to the other. Both ends of the wire connections are
" grounded," and the circuit is completed through the earth.
THE RELAY
376. For long distance telegraphy it is found that the line
currents are too feeble to operate a sounder. Under such cir-
cumstances a " relay " is used. The relay is a sensitive electro-
magnet which, responding to the feeble line currents, opens and
closes a local circuit containing a battery and sounder. The
arrangement of apparatus will be understood by reference to
Figure 250. R is the relay which is connected to the line L
Ground
Relay Circuit.
and the ground Gr as indicated. AD is a very light, soft iron
lever held to the left by a slender spiral spring. When R is
energized, this lever AD is attracted and makes contact with
the stop on the right, thus closing the local circuit ABSD.
The armature or lever CD of the sounder S will evidently
repeat the movements of the lever A. The current in the local
circuit may, of course, be made sufficiently strong to render
audible the click of the lever CD. Evidently this arrangement
of relay, local circuit, and sounder may be used in place of the
ordinary sounder in any circuit in which the line current is
found to be too feeble to operate the ordinary sounder.
TELEGRAPHY AND TELEPHONY
391
DUPLEX TELEGRAPHY
377. In duplex telegraphy the apparatus is so arranged that
signals may be sent in opposite directions over the same line at
the same time. This may be accomplished by the use of the
differential magnet, upon which the coils are so placed that out-
going currents do not magnetize the core while incoming cur-
rents produce the usual effect, and hence record signals from
the distant station.
The arrangement of apparatus is shown in Figure 251. S
and *S'j are the differential magnets. When the key ^is closed,
a current flows over the line from the battery B. It does not.
Line
Fig 251. — Circuit for Duplex Telegraphy.
however, magnetize the core of the electromagnet »S', since the
current is caused to divide as it enters the coil of this magnet,
one half of the current flowing in a clockwise direction through
one half of the coil, the other half of the current flowing in a
counterclockwise direction through the other half of the coil.
In the practical form of the apparatus one half of the coil is
superimposed upon the other. Thus the magnetizing effect of
one half of the coil is neutralized by that of the other. That
half of the current which traverses the line to the distant station
will energize the magnet *S" and cause its armature to respond
to the key K. In the same way, when the key K^ is closed, the
current from the battery Bj^ divides in the magnet S-^, producing
no effect upon L', but the part of the current which flows to S
392
ELECTRICITY AND MAGNETISM
energizes that magnet, causing its armature L to respond to
the motions of the key Ky In order that the differential
magnet may be used successfully in this way, it will be evident
that the current from the home battery must be divided equally
between the two halves of the coil of the differential magnet,
since if the currents in the two halves of the coil are unequal,
their magnetic effects will be unequal, and they will not com-
pletely neutralize one another. This equal division of the cur-
rent between the two halves of the coil is secured by adjusting
the resistance R which is placed in series with that half of the
coil which is grounded at the home station. When R is prop-
erly adjusted the current divides equally between the two
halves of the coil. Because of the effects of electrostatic
capacity in the line it is found necessary in practice to employ
a condenser in the ground connection SR. It is connected as
a shunt to the resistance R.
Local
Circuib
THE POLARIZED EELAY
378. The polarized relay is a relay which responds to a re-
versal of the current in the circuit in which it is placed, but
does not respond to a change in the
strength of the current. This mech-
anism will be understood from the
sketch given in Figure 252. NS is
a C-shaped bar of soft iron upon
which is wound a magnetizing coil
G. ns is a permanent steel magnet
pivoted at s so as to be free to move
back and forth between the poles NS
of the electromagnet. Let it be as-
sumed that a current is flowing in
the electromagnet in a direction in-
dicated by the arrows. Then the
north and south poles of the electro-
magnet will be as indicated ia the
figure. Now the north-pointing pole
of the steel magnet ns will be attracted by the south-pointing
pole *S' of the electromagnet, and repelled by the north-pointing
Fig. 252. —Polarized Relay.
TELEGRAPHY AND TELEPHONY 393
pole N. So long as the direction in which the current is flowing
in the coil C remains unchanged, the steel magnet ns will remain
in the position shown, even though the strength of the current in
C is caused to vary between wide limits. A change in the
direction of the current in will be followed at once by a
change in the position of ns, since a change in the direction
of the current means a reversal of the magnetism in the iron.
When the current is reversed, the right-hand pole of- the elec-
tromagnet becomes south-pointing, and the left-hand pole,
north-pointing, and the small steel magnet then moves to the
right. Thus the polarized relay is a device which is unaffected
by changes in the strength of the current but which responds at
once to a reversal of its direction.
DIPLEX TELEGRAPHY
379. In diplex telegraphy arrangements are made for sending
two messages simultaneously in the same direction over the same
wire. One of the simple arrangements by means of which this
is accomplished is the following : In Figure 253, P and /
represent the two keys at the home station, which are used
simultaneously, for sending messages over the line L. I is an
ordinary key connected in shunt with the resistance R. Evi-
dently the effect of closing the key I is to short-circuit the
resistance R, thereby diminishing the resistance of the line and
hence increasing the current which flows from the battery B to
the distant station. The key P is called a "pole changer" for
the reason that when it is manipulated it reverses the connec-
tions of the battery B^. When the key P is at rest, it is held
down by the spring, so that it makes contact with the lever A
and the positive pole of the battery. The line at the same
time is in communication with the negative pole of the battery
through the lever B. Now if the right-hand end of the key P
394 ELECTRICITY AND MAGNETISM
is caused to rise, it makes contact with the lever B and the
negative pole of the battery, while the line is placed in commu-
nication with the positive pole of the battery through A.
When, therefore, P is operated, the current in the line is re-
versed in direction.
At the distant station are two relays represented by iHf and
P' in the same figure. M is an ordinary relay which is so ad-
justed as to respond to relatively strong currents without
regard to the direction in which the current is flowing in the
line. P' is a polarized relay which responds only when the
current in the line is reversed. It will be evident that with
this arrangement of apparatus M will respond to every motion
of I, while P' will respond to the motions of P. Therefore,
M will record all messages transmitted by /, P' will record all
messages transmitted by P, and each system of transmitter
and recorder will be independent of the other.
QtJADRUPLEX TELEGKAPHY
380. By combining the principles of duplex and diplex
telegraphy it is possible to send simultaneously, two messages
in each direction over one and the same wire. This is called
quadruplex telegraphy.
THE TELEPHONE
381. By taking advantage of the principle of electromagnetic
induction it is possible to transmit speech electromagnetically.
L
Fig. 254. — Simple Telephone Circuit.
The telephone is the apparatus by means of which this is ac-
complished. Figure 254 represents the simplest form of electro-
magnetic telephone. The apparatus at each end of the line
TELEGRAPHY AND TELEPHONY 395
consists of a permanent magnet N'S, a coil of fine wire (7 placed
over one end of this magnet, and a soft iron diaphragm D placed
near the end of the bar magnet which is surrounded by the
coil. The two coils are connected in series by the lines LL as
indicated in the figure. The operation of this telephone will
be understood from the following considerations. Let it be
imagined that a person stands at the instrument, represented
by DO, Figure 254, and speaks to the diaphragm D. The
sound waves of the voice cause the diaphragm D to vibrate, and
each vibration of the diaphragm causes a redistribution of the
lines of force spreading from the adjacent pole of the bar magnet.
The moving lines of force give rise to induced electromotive
forces in the coil 0. Induced currents will therefore flow from
this instrument through the lines LL to the instrument at the
other end of the circuit, and passing through the coil O in the
distant instrument will alter the magnetism of the magnet N'S',
But any alteration in the magnetic field due to N'S' will cause
the diaphragm L' to change its position, since, if the field is in-
creased, the diaphragm vs^ill be more strongly attracted ; while if
the field is weakened, the diaphragm will be less strongly at-
tracted. Evidently, therefore, the diaphragm I)' will respond
to every motion of the diaphragm D. But the vibrations of
B' give rise to sound waves in the air in its neighborhood.
Hence, sound waves falling upon I> are reproduced by L'.
LONG DISTANCE TELEPHONE
382. The apparatus described in the last paragraph is not
adapted to the transmission of speech to any great distance,
since the currents developed in the manner indicated are not
sufficiently strong. The arrangement of apparatus which is
employed in modern long distance telephone systems is rep-
resented in Figure 255. It consists essentially of a transformer
PS wound upon a bundle of straight iron wires. The primary
of this transformer is connected to a battery B and the trans-
mitter T. The essential parts of the transmitter are two plates
between which there is placed a small quantity of granular
carbon. This granular carbon forms a conductor of rather high
resistance between the plates. A motion of either plate in the
396
ELECTRICITY AND MAGNETISM
transmitter will cause a variation in the resistance of the granular
carbon conductor between the plates and therefore a variation in
the current which is circulating about the coil P from the battery
B. The mouthpiece of the transmitter is placed in front of
one of the plates just mentioned. Evidently, sound waves
Fig. 255. — Long Distance Tele
phone Talking Circuit. v
falling upon this plate will produce the effect mentioned above.
Therefore, when a person speaks into the transmitter, the
current in the primary circuit PBT fluctuates in accordance
with the sound waves which fall upon the plate of the trans-
mitter. This varying current in the coil P causes a variation
in the magnetism of the core of the transformer PS and gives
rise to induced currents in the coil S which is connected by
way of the receiver to the lines running to the distant station.
These induced currents traversing the line give rise to sound
waves in the distant receiver as explained in the paragraph
above. The apparatus at the distant end of the line is the exact
duplicate of that installed at the near end.
In the arrangement shown in Figure 255 the apparatus em-
ployed for calling or attracting the attention of the person at
the farther end of the line is not included. In the practical
form of the instrument an automatic switch called the receiver
hook is so arranged that when the receiver is hung up the circuit
represented in Figure 255 is open and the line is connected to
the call bell, and the calling device or magneto, which is, in fact,
a small dynamo that may be employed for sending current over
TELEGRAPHY AND TELEPHONY
397
the line to the distant bell. When the receivers at both ends
of the line are off the hooks, the connections are as shown in
Figure 255.
CENTRAL ENERGY SYSTEMS
383. In the arrangement of apparatus described in the last
section, a battery is required at each telephone. There must
also be provided a magneto, that is, a small hand dynamo for
sending a signal to the distant station in " calling up." The
central epergy system is now quite commonly employed. It
possesses, among others, the following advantages : (a) the bat-
teries used for operating the telephones are all located at the
central station and hence are more conveniently cared for; and
(b) the subscriber to " call up " Central has only to lift the
( p
5
1
^
N.
L
L
, r
^
^
1^^'
M
iU4
Ob
V V ™
H
T
^
_J
u
H
f
T
-^
k
R'zi=~
J
b
"c
V
Fig. '256. — Central Energy Circuit.
receiver from the hook. In Figure 256, one of the more simple
central energy circuits is shown. The subscriber's station, con-
taining a transmitter T, a hook switch H, a transformer PS^ a
receiver i2, and a bell 5, is represented at the left. The central
station, containing a battery B\ a magneto M^ a relay i?', and an
incandescent lamp J, is represented at the right. The two
stations are connected by the lines LIj.
The operation of this system is as follows : When the receiver
R is hanging upon the hook switch jff, the main circuit from
the transmitter T through the primary of the transformer P is
broken at so that no current from the battery B can flow
through this circuit. C is a condenser placed in series with
the bell B, which, of course, prevents the flow of current
through the bell circuit. When Central wishes to call the sta-
398 ELECTRICITY AND MAGNETISM
tion represented, an alternating current is sent over the lines
LL from the magneto M or other suitable source. This alter-
nating current, surging into and out of the condenser C, will
ring the bell B. If it is desired to call Central from the sta-
tion represented, it is only necessary to lift the receiver R from
the hook H. When this is done, a spring raises the hook and
closes the circuit at 0. This allows current to flow from the
battery at the station over the line L through the transmitter
T and by way of the hook switch H and the primary coil P of
the transformer, thence back over the line L to the central sta-
tion. The current which flows in this circuit energizes the
relay B,', which closes the circuit of the incandescent lamp I.
The lamp is then lighted and constitutes the signal to Central.
When the subscriber talks into the transmitter T, the current
through the primary P of the transformer fluctuates as de-
scribed above, and this gives rise to induced currents in the
secondary S, which travel over the lines LL to the central sta-
tion or beyond, according to connections. To connect this sta-
tion with that of any other subscriber. Central makes connections
at the points J-T. From this arrangement of apparatus, it is
evident that the "talking current" and the primary current
flow over the same lines. It is found in practice that this
does not interfere with the transmission.
ELECTROMAGNETIC WAVES
CHAPTER XXXII
MAXWELL'S THEORY
384. Reference has already been made in the study of heat
and electrostatics to the universal medium called the ether
which is supposed to fill all space. We have seen that the
ether in the neighborhood of a charged body is in a state of
strain. It is also assumed that the ether in a magnetic field is
in a strained condition, the strain under these circumstances
being of a different nature from that produced by the electro-
static charge. It follows that upon the sudden discharge of a
charged body or the sudden demagnetization of a magnet, a dis-
turbance of the ether will take place. This is simply another
way of saying that the ether tends to relieve itself of the strain
to which it is subjected while in the presence of a charged body
or the magnet producing the magnetic field. One can imagine
that this disturbance spreads through all space very much as a
disturbance spreads in all directions over the surface of a pond
of still water when a stone is dropped into it. Such a disturb-
ance of the ether is known as an electromagnetic wave, since both
the magnetic and the electrostatic effects are present.
In 1864 Maxwell explained on a purely mathematical basis
that it ought to be possible to establish electric waves in the
ether. In 1888 Hertz succeeded in carrying out Maxwell's sug-
gestion, and not only produced electromagnetic waves by appa-
ratus designed by himself, but succeeded in detecting their
presence at some considerable distance from the apparatus
from which they were caused to spread.
hertz's apparatus
385. The apparatus used by Hertz in his investigations con-
sisted of an oscillator and a resonator. The oscillator is repre-
399
400
ELECTRICITY AND MAGNETISM
Fig. 257. — Hertz Oscillator.
sented in Figure 257. It consists of two plates of metal A
and £ to which are attached small rods terminating in knobs
as indicated in the figure. In the use of this apparatus the rods
are connected to the
secondary terminals
of an induction coil
as indicated in the
diagram. When the
coil is set in operation,
the plates A and £
become charged, and
a difference of poten-
tial is established be-
tween the knobs
■which, when a certain
value is reached, causes a spark discharge between the knobs.
If this were a simple discharge, the disturbance of the ether
resulting would consist of a single pulse which would move for-
ward through space as a single ripple moves across the surface
of a still pond. It is found, however, that with the arrange-
ment of apparatus indicated in the figure, the discharge is oscilla-
tory in character, so that, instead of a single ether pulse, there
will be a succession of pulses constituting a train of waves some-
what like the series of waves which travels over the surface of
a still pond when a stick is
moved up and down in the water
several times in succession.
The resonator Employed by
Hertz is shown in Figure 258.
It consists of a circular loop of
wire AB terminating in small
polished knobs 0. In the actual
form of the apparatus, arrange-
ment is made for adjusting the
distance between the knobs C by
means of a micrometer screw.
Hertz found that when this resonator was placed in certain
positions opposite the oscillator, a discharge would take place
Fig. 258. — Simple Resonator.
ELECTROMAGNETIC WAVES 401
between the knobs C whenever a discharge occurred in the
oscillator. That is to say, ether waves set up by the oscillator,
falUng upon the resonator AB, were able to impart to the
resonator a certain amount of energy which appeared at the
knobs in the form of the electric discharge.
Hertz found in his experiments that the position of the res-
onator with respect to the oscillator had much to do with the
magnitude of the effect produced in the resonator. He found
that it was necessary to place the resonator in such position
that the knobs of the resonator would become charged by elec-
trostatic induction from the charges on the plates of the oscil-
lator, or in such position that the lines of force set up by the
discharging current between the plates of the oscillator would
cut the conductor of the resonator in such way as to produce a
difference of potential between its knobs. The maximum
effect obtained was when the resonator was so placed that these
two effects were combined.
Hertz also showed the resonator might be made up in the
form of the oscillator shown in Figure 257. When this form
of resonator is employed, it is placed with its axis AB parallel
to that of the oscillator.
THE OSCILLATORY DISCHAEGE OP THE OSCILLATOE
386. The largest effects in the transmission of energy by
means of the electromagnetic waves are secured when the dis-
charge of the oscillator is oscillatory in character. The reason
for this is at once apparent when it is understood that for each
oscillation a wave or pulse passes. out through the ether, each
producing its own effect. Now if the receiver or resonator upon
which the waves fall is of such character that it vibrates elec-
trically at the same rate that the oscillator does, then the effect
of each wave falling upon the resonator will be added to that of
all of the others, so that by a succession or train of waves an
effect may be secured which is many times as great as that
which would be given under the same conditions by a single
wave or pulse. The rate at which the oscillations take place in
an oscillator is found to depend upon the electrostatic capacity
and the self-induction of the oscillator. By adjusting the values
2d
402
ELECTRICITY AND MAGNETISM
of these quantities the oscillations may be giveu- any period
desired.
EESOifANCB
387. As pointed out in the last paragraph, the effect upon the
resonator is greatest when the dimensions of the resonator are
such as to cause the charge to vibrate upon it at the same rate
that the charge vibrates upon the oscillator. When this condi-
tion is secured, the resonator is said to be in the condition of
resonance. The resonator may be "tuned" (brought into the
condition of resonance) by altering its dimensions, that is to
say, by changing either its inductance or its capacity, since, as
in the case of the oscillator, the rate at which the charge oscil-
lates is dependent upon these two quantities.
lodge's experiment
388. The effect of resonance may be very convincingly shown
by the following experiment, which is due to Lodge. Let ABCD,
Figure 259, represent a rectangular circuit containing a con-
denser J'and a spark
gap S. Let it be
assumed that the
terminals of the
condenser J" are con-
nected to the sec-
ondary terminals of
an induction coil I
as indicated in the
figure. When the
jar is sufficiently
charged by the
e. m. f. of the induc-
tion coil, a spark
will pass at S. This
discharge of the jar through the circuit ABCD will be oscilla-
tory, and electromagnetic waves will spread out into space
from this discharge circuit. Let EFGrR be a second circuit
of approximately the same form and dimensions, having its
plane parallel to that of the circuit ABCD. The side I'Gr of
Fig. 259.
ELECTROMAGNETIC WAVES
403
this circuit is supposed to be free to slide upon the conductors
EF and SGr so that the size of the rectangle EFGrH may be
varied at will. It is found that when FG- is in a certain
position so as to make the rectangle EFGrK of approximately
the same size »s>ABCD, sparks will pass across the spark gap
of the circuit EFCrH when the discharge occurs at S. In
other words, under these circumstances, ABCD is serving as
an oscillator, ^i^6r-ff as a resonator. If, now, the size of the
resonator rectangle is altered by moving the side FGi, all else
remaining the same, the discharge of this circuit will cease,
indicating that the resonator is no longer responding to the
electric waves which fall upon it. A very slight motion of
the side FCr is sufficient to throw the resonator out of "tune."
WIRELESS TELEGRAPHY
389. In wireless telegraphy a practical application is made
of the principles enunciated in the foregoing paragraphs.
Electromagnetic waves, generated by an oscil- i i i i
lator, may be detected at great distances, pro- \ I / /
viding a resonator is used which is tuned to \ I / /
resonance with the oscillator at the sending \ \ / /
station. If, therefore, a succession of signals is \ 1 I /
sent out according to some prearranged code, \ 1 / /
messages may be transmitted by means of these \ I / /
electromagnetic waves. The form of oscillator \\ 1/
commonly employed in wireless telegraphy is \\ 1/
that of a more or less nearly vertical wire, or M //
group of wires, attached to suitable supports. yl
This system of wires is called the aerial. For J
long distance work the aerial is sometimes 150 I
One form of aerial is that
•G
or 200 feet in height.
shown in Figure 260. It consists of a fan-shaped
group of wires A, supported on a mast or tower
(not shown in the figure), communicating with
one knob of the spark gap G-. The other knob
is connected to the earth as indicated. The terminals of the
induction coil are connected above and below the spark gap.
The discbarge which occurs between the knobs of the spark
Ground
Fig. 260,
404
ELECTRICITY AND MAGNETISM
gap is oscillatory in character, and gives rise to a train of elec-
tromagnetic waves.
The receiving apparatus may consist of a similar aerial upon
which the electric waves are allowed to impinge. But in order
that the receiving apparatus may be operated at great distances
from the sending station, it is necessary to substitute for the
spark gap some device which is more sensitive in its indications.
Various devices have been used for this purpose. One of these
is known as the coherer.
DETECTORS
390. The coherer consists essentially of a small glass tube
AC£, Figure 261, and two metal rods which reach into the
tube, leaving a small opening C at the center which is filled
A, , B
with metal filings. The resistance of this device between the
points A and B is found to be quite high, owing to the loose
contact between the filings at O. When the coherer is con-
nected in a receiving circuit upon which
electric waves are falling, the discharge
which takes place through the metal fil-
ings tends to cause them to cohere in such
manner that the resistance of the device
from A to B is very much decreased.
A simple arrangement of apparatus il-
lustrating the use of the coherer is repre-
sented in Figure 262. AO is the coherer
which takes the place of the spark gap in
the aerial. A local circuit, containing a
sounder S, or some equivalent device, and
a battery B, is completed through the
coherer AC a,s indicated. Under ordinary
conditions no appreciable current flows in
this local circuit, because of the high re-
sistance of the coherer. When electromagnetic waves fall upon
Ground
Fig. ■2&2.
ELECTROMAGNETIC WAVES
405
the aerial, and oscillations take place in the system of conduc-
tors connected to the coherer, its resistance is decreased, as
pointed out above, and a current of sufficient strength to op-
erate the sounder S flows in the local circuit. The armature
of the sounder is arranged to vibrate when current is flowing
in the local circuit, and is placed close to the coherer so that
it strikes the coherer when vibrating. In this manner the filings
of the coherer are caused to " decohere," and the apparatus is
made ready for the succeeding signals.
The magnetic detector of Marconi is a detector which has
recently come into extensive use in wireless telegraphy. The
principle upon which it depends for its action is as follows.
Let Figure 263 represent the aerial of a receiving station. In
place of the usual spark gap, it con-
tains at (? a coil of wire of compara-
tively few turns. Over this is a
second coil connected to an ordinary
telephone receiver T. WW is a fine
iron wire supported by suitable clock-
work by which it is made to travel
slowly in the direction of its own
length through the two coils. A
magnet NS is fixed permanently near
the moving wire WW, as indicated
in the figure. This magnet tends to
magnetize the iron wire, which retains
a certain amount of this magnetism
as it passes into the coil Cr. So
long as this magnetic condition in
the wire TFTFis constant, no effect is produced by the moving
magnetized wire upon the coil connected to the receiver ; but
any change in the magnetic condition of the wire is followed
at once by induced electromotive forces in the coil referred to,
and results in sounds in the receiver. When electric oscilla-
tions take place in the aerial, the magnetic condition of the
wire WW is altered by the surging currents in the aerial.
Hence oscillations in the aerial are always accompanied by
sounds in the receiver, and by these sounds the oscillation may
be recognized.
Ground
Fig. 263.
ELECTRIC DISCHARGE
CHAPTER XXXIII
POINT DISCHARGE
391. The simplest kind of electric discharge is that which
takes place from a sharp point ; this is called point discharge.
It is found that under ordinary atmospheric conditions a
charge is retained by a conductor for a limited time only, even
though the conductor is carefully insulated. The discharge
which takes place under these circumstances was formerly attrib-
uted to the presence of moisture and dust particles in the air.
Careful experiment has shown, however, that this discharge
takes place when moisture and dust both have been carefully
removed from the air.
The modern theory attributes this discharge to what is
known as the ionization of the air.
IONIZATION
392. A gas is said to be ionized when it contains free electrons
or free positive atoms. The simplest way of ionizing a gas is
by heating it. Thus the gases in and about a flame or near an
incandescent solid are ionized. It has been shown that a very
hot body loses its charge more quickly than a cold one. The
explanation is that in the more violent vibratory motion of the
molecules, corresponding to the higher temperatures (Section
198), the electrons become separated from the atoms.
When electrons or free positive atoms are caused to move
with high velocity through a gas, they collide violently from
time to time with molecules of the gas. As a result of such
collisions the gas may become more completely ionized.
406
ELECTRIC DISCHARGE 407
THEORY OF THE POINT DISCHAEGB
393. Let it be imagined that in the neighborhood of a nega-
tively charged body there are a few free positive atoms. These
atoms, being attracted by the negative charge, move toward the
charged body. The nearer they opproach the more rapid their
motion and hence the more violent the collisions when they
occur near the charged body. The effect of these collisions is to
ionize the air in the immediate neighborhood of the charged
body. The electrons and positive atoms thus set free start into
rapid motion, the electrons are repelled, and the positive atoms
attracted to the charged body. The positive atoms thus falling
upon the body gradually neutralize the charge.
A point facilitates this process since, as we have seen (Sec-
tion 250), the surface density of charge on a point is greater
than at any other part of the conductor, and ionization will
therefore take place more rapidly in that vicinity.
A positively charged body is discharged in a similar manner.
The electrons in this case are the attracted bodies, which by
their rapid motion and collisions with the neutral portions of
the air effect its ionization.
THE BRUSH DISCHARGE
394. The point discharge is accompanied by a faint bluish
glow which extends to a short distance from the point. When
the difference of potential between the charged body and its
surroundings is very great a "brush " is formed near the point.
This consists of a large number of faint sparks or streamers
which radiate from the point to a distance, it may be, of several
inches. This discharge is very beautiful, but is only faintly
luminous and can only be seen in the dark. It is to be thought
of as a modified point discharge taking place in essentially the
same manner.
THE DISRUPTIVE DISCHARGE
395. If the terminals of an electric machine in operation are
brought sufficiently near together, sparks will be observed to
pass between them. This is known as the disruptive discharge.
This discharge is accompanied by the development of heat,
408 ELECTRICITY AND MAGNETISM
light, and sound. In general the energy of the charge is con-
verted into these three forms of energy. The sound and the
light developed are at once apparent. The heating effect is
readily proven by placing in the path of the discharge some
inflammable material, for example, ether, or by causing the
discharge to take place along a very fine wire, in which case,
providing the charge is sufficiently large, the wire will be fused
in consequence of the heating action of the discharge.
THE EFFECT OF PEESSUKE UPOX THE DISCHARGE
396. We are already familiar with the characteristic features
of the electric discharge when it takes place in air at ordinary
pressures. We have seen that the " spark " is narrow and,
except when the terminals between which the spark occurs are
very close together, its path is zigzag and oftentimes forked.
This form of discharge is accompanied by heat, light, and
sound.
If the electric discharge is caused to take place in a region
in which the pressure is somewhat less than atmospheric pres-
sure, very marked changes take place in the character of the
discharge. When the pressure of the air in which the dis-
charge takes place has been reduced to about one thousandth
of an atmosphere, the discharge is known as the Geissler dis-
charge.
THE GEISSLER EFFECT
397. The changes in the character of the electric discharge
in regions of low pressure were studied quite extensively by
Geissler. In his investigations he employed a tube like that
represented in Figure 264. £0 is a, glass tube sealed at both
ends and having platinum wires A and K sealed through the
walls of the glass. By connecting the tube to an air pump by
means of a side connection U the air may be gradually ex-
hausted from the tube and the corresponding changes in the
character of the discharge observed. Such a tube is called a
Geissler tube. When the air has been exhausted until the
pressure within the tube is about one thousandth that of the
atmosphere, the effect within the tube is known as the Geissler
ELECTRIC DISCHARGE 409
effect. The characteristics of the discharge under these con-
ditions may be briefly described as follows; Let A represent
the wire by which the discharge enters the tube, i.e. the anode,
and K the
cathode. When °
the pressure
conditions are
as indicated
above, the cath-
ode K is sur-
roiindpfl bv a ^'°' ^^*' — Vacuum Tube for showing the Effect of Pressure
■^ upon the Character of the Electric Discharge.
luminous layer
known as the negative glow, which extends from the sur-
face of the cathode to a comparatively short distance. Next
to this negative glow is a dark space which is apparently de-
void of any luminous discharge. Beyond this and reaching
from the dark space to the anode is the positive column. This
consists of a peach blossom colored luminosity which appar-
ently fills the entire tube.
If the pressure of the air within the Geissler tube is still fur-
ther diminished, it will be observed that the dark space grad-
ually increases in length until it occupies the entire length of
the tube. At this point in the exhaustion a new set of phe-
nomena appear.
THE CROOKES EFFECT
398. When the exhaustion of the Geissler tube has been
carried forward as indicated in the last paragraph until the
dark space occupies the entire length of the tube which takes
place at a pressure of about one millionth of an atmosphere, the
walls of the tube become brilliantly fluorescent. The luminous
effect at this stage of the exhaustion seems to be limited almost
entirely to the walls of the tube. The luminous discharge in
the gas itself which is so prominent in the Geissler effect is
at this stage of the exhaustion almost, if not quite, entirely
absent. This effect is known as the Crookes effect, and a tube
which has been exhausted to this degree is known as a Crookes
tube.
410
ELECTRICITY AND MAGNETISM
Fig. 2{i.')
- Gciasler Tube.
CATHODE RAYS
399. The fluorescence of the walls of a Crookes tube is caused
by the bombardment of the walls by matter which is projected
from the cathode. It can be shown experimentally that there is
such a projection of material par-
ticles from the cathode and that
they are projected at enormous
velocities ; further, that they
move in straight lines. The fol-
lowing experiment is designed to
demonstrate the rectilinear propa-
gation of cathode rays. Let Fig-
ure 265 represent a Geissler tube
more or less nearly spherical in
form. Let ^ be a cathode having
a concave surface. Let it be as-
sumed that the terminal A is used
as the anode. It will then be
observed that the luminous discharge within the spherical bulb
tends to bend in a curved path between A and IC as represented
by the dotted line. When B is used as an anode, the dis-
charge curves between B and IC;
and when is used as an anode,
the discharge curves as indicated
bv the dotted line running from
CtoK.
If now the exhaustion is carried
forward until a Crookes vacuum
is reached, the character of tlie
discharge will be that shown in
Figure 266, in which the cathode
rays passing from the terminal K
are concentrated at the center of
curvature of the concave cathode.
Beyond this point they again di-
verge and fall upon a portion of the opposite wall of the tube
2>, which is rendered highly fluorescent by their impact. The
Fig. 2(iG. —Crookes Tube.
ELECTRIC DISCHARGE
411
path of the cathode rays as represented in Figure 266 is inde-
pendent of the terminal which is used at the anode. It will
thus be seen that the cathode rays travel in straight lines and
that their direction of motion is not at all determined by the
position of the anode.
PROPERTIES OF THE CATHODE RAYS
400. The cathode rays are characterized by three effects, the
mechanical effect, the heating effect, and the production of X-rays.
The mechanical effect
of the cathode rays is
very readily shown by
the apparatus represented
in Figure 267. Let A
represent the anode and
K the cathode of a
W
Fig. 267. — Mechanical Efl'ect ol Catliode Rays.
Crookes tube of the form shown in the diagram. The cathode
rays will then stream across from ^toward A as iiidicated by
the arrows. If a wheel having light vanes is mounted as indi-
cated in the diagram so that the cathode rays impinge upon
these vanes, the wheel will be set in rapid rotation, thus
demonstrating the mechanical effect of the cathode rays.
The heating effect of the cathode rays is shown by means of
a tube like that represented in Figure 268. A is the anode and
K a concave cathode. P is
conveniently a piece of plati-
num supported at the point at
which the cathode rays are
concentrated. When such a
tube is connected to a power-
ful induction coil, the piece of
platinum P becomes strongly
heated. It may be even melted
by the action of the cathode
rays.
The production of X-rays
is demonstrated conveniently by the use of a tube like that
shown in Figure 269, which is known as a focus tube. The
Fia. 268. -
•Heating Effect of Cathode
Rays.
412 ELECTRICITY AND MAGNETISM
bulb of this tube is nearly spherical, having extensions on
opposite sides through which the anode and cathode are in-
troduced. In the tube represented the anode A is supposed to
terminate at the center of the bulb in a small sheet of platinum
which is placed at an angle of 45° to the axis of the tube.
/ / / I I 1 \ \ \
X
Fig. 269. — X-ray Tube.
The cathode rays are concentrated upon this piece of platinum
when the discharge is passing. With this arrangement and a
Crookes vacuum in the tube, it is found that, when the dis-
charge passes, P becomes the source of what is known as the
X-rays or Roentgen rays. The X-rays under the assumed
conditions are given off in greatest abundance in the directions
indicated by the arrows X.
OTHER CHARACTERISTICS OF THE CATHODE RATS
401. In addition to the characteristics mentioned above it has
been demonstrated that the velocity of the cathode rays is
about one tenth the velocity of light (2.8 x 10^ centimeters per
second). It has also been demonstrated that the mass of the
charged particles which are assumed to make up the cathode
rays is very small as compared even with the atom of hydrogen,
being equal in mass to about one-thousandth that of the
hydrogen atom. The demonstration of this fact has led to the
ELECTRIC DISCHARGE 413
belief that the atom, which hitherto had been regarded as the
smallest portion of matter, is, in reality, made up of a great
many small particles. These smaller parts of the atom are
called corpuscles or electrons.
A stream of cathode rays may be deflected by a magnet. If
the tube represented in Figure 267 is placed in a strong mag-
netic field of such direction that the lines of force pass perpen-
dicular to the paper in the diagram, the cathode rays, instead of
passing in straight lines from ^toward A, will be curved toward
the top of the tube or toward the bottom of the tube, depending
upon the positive direction of the magnetic field.
CANAL RAYS
402. If a perforated cathode is used in a highly exhausted
tube, luminous streams may be observed to emerge from the
perforations, and pass in a direction opposite to that of the
cathode rays. These rays have been called canal rays. They
are capable of producing phosphorescence, and may be deflected
by a magnetic or an electric field. They have been shown to
consist of positively charged particles, their masses being of
the same order of magnitude as that of the hydrogen atom.
X-RAYS
403. The principal effects which characterize the X-rays are
the following : They do not seem to be regularly reflected or
refracted. They cannot be focused by a lens. In these respects
they seem to be different from ordinary light. They cannot
be deflected by a magnet. In this respect X-rays differ from
cathode rays. They will affect a photographic plate. In this
respect they are like cathode rays and ordinary light. They
are capable of penetrating considerable thicknesses of solid
matter, opaque to ordinary light. Thus the X-rays will pass
through a piece of wood several centimeters in thickness.
They will also pass through comparatively thick layers of
aluminum and ebonite. They will also pass through thin
layers of such metals as zinc and iron and lead, layers which
are quite opaque to ordinary light waves. They will pass
414 ELECTRICITY AND MAGNETISM
through flesh and bone. They give rise in certain bodies to
very strong fluorescence. In this particular they are also like
the cathode rays. X-rays also produce strong ionizing effects.
Tlie real nature of X-rays is not clearly understood, but most
of the phenomena to which they give rise may be explained on
the assumption that they consist of irregular pulses in the ether.
THE SKIAGRAPH
404. If a layer of material which fluoresces strongly under
the action of the X-rays is placed in some such position as X,
Figure 269, and the hand is placed between the focus tube and
the layer of fluorescing material, a " shadow picture " of the
hand will be produced upon this fluorescent layer, since the
flesh of the hand will intercept, in part at least, the X-rays
which would otherwise fall upon the fluorescent layer. The
bones of the hand, being more opaque than the flesh, will cast
deeper shadows and may be seen clearly outlined upon the
fluorescent screen.
A substance which lends itself very readily to use as a fluo-
rescent material is the chemical compound known as the platino-
cyanide of barium. The tungstate of calcium is also employed
for this purpose.
If a sensitive plate such as is used in photography be used in
place of the fluorescing screen, a permanent shadow picture or
" skiagraph " may be obtained by developing the plate iu the
usual manner, after an exposure before the X-ray tube.
RADIOACTIVITY
CHAPTER XXXIV
BECQUEREL'S DISCOVERY
405. In 1896 Becquerel discovered that a certain compound
of uranium emitted a radiation which produced an eifect upon
a photographic plate similar to that produced by X-rays. Fur-
ther study showed that this property was possessed by other
uranium compounds and by the element itself. It was also
found that the action upon the plate was independent of the
nature of the uranium compound employed, and was determined
solely by tlie quantity of uranium present. Becquerel also
showed that this radiation was capable of discharging electri-
fied bodies.
It was later shown by Rutherford that this discharging of
electrified bodies was due to the ionizing action of the radia-
tion. The following experiment may be performed for showing
the ionization due to the
radiation from uranium : Let
-8, Figure 270, represent a
battery, one terminal of which
is connected to a metal plate ^__
a, the other terminal being
connected to ground. An- -^ n
other metal plate b is placed ~
a few centimeters above a
and is connected to ground
through an electrometer or
'sensitive instrument for de-
tecting electrostatic charge. Under ordinary conditions, h will
not acquire a permanent charge, since it is insulated from a
and the connections in J57 are such as to insulate it from the
415
xxxxyxxx
Fig. 270.
416 ELECTRICITY AND MAGNETISM
ground. If a small quantity of uranium compound is sprinkled
upon a, b will acquire a charge of the same sign as that on a.
That is, if a is connected to the positive pole of the battery, b
will acquire a positive charge. If a is connected to the nega-
tive pole of the battery, b becomes negatively charged. This
is exactly that which would be expected if the air between the
plates were ionized.
RADIOACTIVE STTBSTAXCES
406. The study of uranium compounds led to the belief that
this radiating property is characteristic of uranium, and that
its radiation is not due to any outside cause but is emitted
spontaneously. This property of emitting radiations of the
character described above is called radioactivity. A substance
capable of emitting such radiations is called a radioactive
substance.
The discovery of the radioactivity of uranium led investiga-
tors to exalnine other substances for the same property. It
was found that thorium and its compounds also possess this
property, though perhaps in smaller degree. Later it was
found by M. and Mme. Curie that certain specimens of pitch-
blende showed a higher degree of radioactivitj' than uranium
itself. This led to a more careful study of pitchblende, as a
result of which two new substances, polonium and radium, were
discovered. Polonium was found to differ from uranium in
this important respect, that while the radioactivity of uranium
is constant, that of polonium gradually becomes less as time
passes.
Radium, the most active of the known radioactive substances,
is an extremely rare element, although found in minute quan-
tities in various minerals from different parts of the world.
The chief source of radium is the pitchblende of Bohemia.
Several tons are required even of this substance to yield a
small fraction of a gram of radium. It is usually prepared in
the form of radium bromide. In this compound it is highly
active. It is phosphorescent and causes other substances Hke
calcium tungstate, platino-barium cyanide, and willemite to
phosphoresce.
RADIOACTIVITY
417
Another radioactive substance, actinium, was discovered by
Debierne. Pitchblende is the source of tliis substance also. It
is similar to thorium in its chemical nature, but is much more
active.
Radium is by far the most active of all the radioactive
substances. Its radioactivity is at least a million times as
great as that of uranium. A few milligrams of radium bromide
will produce powerful photographic and ionization effects and
will render a fluorescent screen brilliantly luminous. Its radio-
activity is so great that it is really dangerous to handle because
of its painful effects upon the skin when the exposure lasts for
an appreciable length of time.
If a slender stream of
THREE KINDS OF RADIATION
407. The radiation emitted by radium is complex and may
be separated into three distinct parts. These parts are called
the a-rays, the /3-rays, and the 7-rays.
radiation from radium is passed through
a strong electric field, a portion will pass
on unchanged in direction (7-rays),
two other portions will be deflected
(a and /S-rays), the one being deflected
in a direction exactly opposite to that
of the other. It was this fact that led
to the separation of the a- and /3-rays
and their identifioation. The arrange-
ment of the apparatus for this experi-
ment is shown in Figure 271. A
quantity of radioactive material R is
placed beneath a heavy plate of lead
having a small hole E at its center.
By this means a slender stream of rays
passing vertically upward above the
plate is secured. Metal plates P and
N are arranged as shown. P is
strongly charged positively and N negatively, pp is a photo-
graphic plate arranged to receive the rays and show their sepa-
ration. When the experiment is performed, the developed
2e
Fig. 271.
418
ELECTRICITY AND MAGNETISM
plate shows that a portioa of the rays are undeviated. This
portion consists of 7-rays. Another portion is strongly deviated
towards the positive plate P- This portion consists of /8-rays.
A third part is slightly deviated towards the negative plate N.
This part is composed of a-rays. The rays may also be sepa-
rated by a strong magnetic field. This may be effected by an
arrangement of apparatus like that of Figure 272, which is simi-
lar to that shown in Figure 271 except that strong magnet poles
iVand ^S" are substituted for the charged plates. In this experi-
ment the deflection of the rays is at right angles to the mag-
netic field, that is, perpendicular to the plane of the paper,
p p The a-rays are deflected away
from the /3-rays toward the reader.
THE a-KAYS
i
^A
^
R
Fig. 272.
408. The a-rays consist of posi-
tively charged atoms (a-particles)
of matter. They are projected
from the radioactive substance at
a velocity of something like 20,000
miles per second. Their principle
characteristics are their powerful
ionizing action and their inability
to penetrate (that is, carry their
ionizing action) to a distance of
more than two or three inches in
air. They are completely absorbed by a sheet of aluminum
0.01 centimeter thick. The mass of the a-particle is about
twice that of the hydrogen atom.
THE /3-EATS
409. The /S-rays consist of electrons (yS-particles) and carry,
of course, negative charges. They leave the radioactive sub-
stance with a velocity nearly as great as that of light (186,000
miles per second). They produce ionizing effects, but are very
much less powerful in this respect than the a-particles. Their
penetration is much greater than that of the a-particles, their
ionizing action being lost only after they have travelled several
RADIOACTIVITY 419
feet in the air. A layer of aluminum 0.5 centimeter thick will
ahsorb most of the /3-particles. The mass of the /3-particle is
about -^ that of the hydrogen atom.
THE 7-RAYS
410. The 7-rays consist in all probability of very abrupt
waves in the ether. They seem to be identical in character
with Roentgen rays. They are characterized by the fact that
they cannot be deviated by an electric or magnetic field and do
not carry charge. They have the property of ionizing a gas
and a penetration exceeding tliat of the most penetrating
X-rays.
If an X-ray tube is operated at low pressure (high vacuum)
it gives very penetrating X-rays, sometimes called " hard "
X-rays. These hard X-rays and /3-rays are much alike. It is
known that X-rays are produced by the bombardment of a solid
by cathode rays, that is, by moving electrons. In other words.
X-rays are produced by the sudden stopping of moving electrons.
It seems reasonable to suppose that they would also be produced
by the sudden starting of electrons. Now in a radioactive sub-
stance atomic explosions are supposed to be continually taking
place, as a result of which a- and /8-particles are projected from
the substance. The resulting ether disturbances may be thouglit
of as being somewhat like those produced by suddenly bringing
moving electrons to rest, and the 7-rays as the resulting wave
disturbance in the ether. This view is strengthened by the
fact that 7-rays always occur in conjunction with /3-rays.
MASS AND VELOCITY OP a- AND /3-PAETICLES
411. The method used in determining the mass and velocity of
an a- or /3-particle will be understood from the following anal-
ogy: Consider the case of a cannon ball fired horizontally.
The conditions are, an initial velocity « in a horizontal direction
and a constant force (the weight of the ball) acting at right
angles to v. The ball describes a curve AB, Figure 273. That
is, while traveling a distance D horizontally it falls a distance
d. Let t be the time of flight from A to £. Then J) = vt and
420
ELECTRICITY AND MAGNETISM
d = ^gt^. Combining these expressions and eliminating t, we
have
(106)
v^ = S^
2d
Fig. 273.
from which v may be obtained, if corresponding values of B
and c? are known.
Now consider an a- or /3-particle projected at right angles to
an electric field. Let P and i\/', Figure 274, represent two metal
+
+
+
+
N
plates charged respectively with positive
and negative electricity. Let the dotted
line represent the direction of the ve-
locity V with which the charged particle
enters the electric field between P and
JV. The particle will be acted upon by
a constant force F at right angles to
the field, which is equal to the product
of the charge q carried by the particle
and e, the intensity of the electric field.
Lnder these circumstances the particle
will describe a curve similar to that described by the cannon
ball in the case above considered.
In a given time t the particle will travel a vertical distance
D, such that D = vt. In the same interval it will travel a dis-
tance d horizontally, such that d = ^ at\ in which a is the uni-
form acceleration in the direction of the electric field produced
by the action of the field upon the charged body. From these
relations, we obtain, as above.
aJfi
'Id
(106 bis)
RADIOACTIVITY
421
Now the force acting on the charged particle is F = qe, as
pointed out above. Hence (since F= ma), we have
a =
m
Substituting in the expression for v^, vf& obtain,
2 dm
(107)
from which we may find the value of v, if q, e, and m are
known and I) and d are observed.
This relation may also be written in the form
m
em
2dv^
(108)
m
from which we may find the value of — when e and v are
known and d and J) are observed. "
In the case above considered, the charged particle describes
a parabola. This form of path is due to the fact that the
deflecting force is constant in magnitude and direction. If the
charged particle is deflected by a magnetic field, it describes a
circular path, since in this case the deflecting force is constant
in magnitude but continually changing in direction. A moving
charge is equivalent to an electric current,
and the deflecting force is perpendicular to
the field and to the path of the moving
charge. Let the broken line. Figure 275,
represent the direction in which the charged
particle is moving as it enters the magnetic
field. Assume that the magnetic field is
perpendicular to the paper, the lines of force
being represented by the dots in the figure.
The path of the moving charge is then circu-
lar, the force F being constant in direction
and always perpendicular to the path. The value of F is qvf,
in which q is the charge, v its velocity, and / the intensity of
the magnetic field. The acceleration of the particle in the
■D
•
1 l'^"'^
• •
A
Fig. 275.
422 ELECTRICITY AND MAGNETISM
direction J'is, therefore, '^- But in uniform circular motion
m
the radial acceleration is — • Hence,
r
qvf _ v^
m r
2d'
But
since AB is a circular arc, of which d is the sagitta and D the
half -chord.
Therefore, qrf^2d>^
m D^
whence, m^^
q 2dv
Now let it be imagined that a moving charged particle is
acted upon by an electric field and a magnetic field at the same
time. Let it be assumed that the fields are at right angles to
one another and so related that the deflecting force of the mag-
netic field is equal to that of the electric field and opposite in
direction. Under these conditions the charged particle wiU
pass through the electric and magnetic fields without deviation.
For this case, we have
qvf=qe
or, v = ^- (110)
That is, the velocity of the moving particle is given by the
ratio of the intensities of the electric and magnetic fields, when
these intensities are so related that they produce no deflection
of the moving particle.
It has been found possible to carry out an experiment of this
kind, and by this means to determine v, the velocity of the
moving particle. When v has been determined, its value may
be used in Equations (108) or (109), and the value of — maybe
1
obtained. Finally, there is reason to believe that the charge q
carried by an electron is the same as that carried by an hydro-
gen ion in electrolysis, and this charge is known.
RADIOACTIVITY 423
It, therefore, becomes possible to determine from — the value
1
of m, the mass of the charged particle. The mass and velocity
of the a- and /3-particles given above have been determined in
in this manner.
URANIUM X AND THORIUM X
412. It has been shown by Crookes that by a chemical pro-
cess there can be separated from uranium a substance which is
much more active photographically than the uranium from
which it is derived, and that when the separation is made it
leaves the uranium without photographic activity. The active
substance separated from uranium in this manner is called
uranium X.
Uranium is active photographically because of the /8-rays
which it gives oS.. The uranium left behind when uranium X
is separated is active electrically but inactive photographically,
because it gives off a-rays only.
Now it is found that after the separation is made the uranium
X gradually loses its activity, while that of the uranium is
gradually regained. Further, the rate of decay in uranium X
exactly equals that of recovery in the remaining uranium. In
all discussions of decay or recovery in radioactive substince,
the time required for one half the change to be completed is
called the period. Thus the period of decay for uranium X is
equal to that of recovery for the remaining uranium. This
fact leads to the conclusion that they are intimately associated.
It points, in fact, to the probability that uranium X is continu-
ally being formed in uranium, the constancy of the radiation of
the unseparated substance being accounted for by the fact that
the quantity of uranium X present is such that its rate of de-
cay just equals its rate of formation. If this view is correct, it
ought to be possible after the remaining uranium has stood for
a time to again separate from it uranium X. This is found to
be the case.
Other experimenters have succeeded in separating from
thorium a substance similar to uranium X which has been
called thorium X. The period of thorium X is about 4 days.
424 ELECTRICITY AND MAGNETISM
while that of uranium X is about 22 days. Thus within
uranium and thorium a process of transformation is continually
going on accompanied by the formation of uranium X or
thorium X. Similar processes have been shown to be continu-
ally taking place in radium and actinium.
EMANATION
413. Radium, thorium, and actinium have been found to
give off a kind of gas which is called emanation. It is not
affected by an electric field, hence does not consist of charged
particles. It differs from an ionized gas in that it does not
lose its ions in those processes which remove the ions from an
ordinary gas. Its gaseous nature, however, is proved by the
fact that it can be condensed. It condenses at a temperature
of -150° C.
When emanation is separated from thorium, the activity of
the emanation dies away, while that of the thorium rises, much
as in the case of uranium and uranium X. The period of
thorium emanation is about 54 seconds. That of radium ema-
nation is about 3.7 days.
KADIOACTIVE TRANSFORMATIONS
414. Many facts developed by experiment, of which those
mentioned in the discussions of uranium X, thorium X, and
emanation are illustrations, have led to the formulation of the
theory of successive changes. This theory states that radio-
active substances are gradually undergoing a process of trans-
formation by which they are changing from one product to
another. These changes are spontaneous and independent of
all outside influences.
A large number of radioactive products are now known.
The transformations of these products have been studied and
their relations definitely established. Below is given a table of
the radioactive products of uranium, together with their trans-
formation periods and the rays emitted by them. Similar
series have been developed for thorium and actinium.
RADIOACTIVITY
425
Radioactive Products
Transformattokt Period
Nature of Rays Emitted
Uranium
5 X 10" years
a
Uranium X
22 days
P and y
Ionium
?
a
Kadium
2000 years
a
Emanation
3.7 days
a
Radium A
3 minutes
a
Radium B
26 minutes
a.
Radium C
19 minutes
a, /3, and y
Radium D
40 years
No rays
Radium E
6 days
No rays
Radium F
45 days
/3 and y
Polonium
140 days
a
The list of substances given in the first column of the table
are to be understood as radioactive products of uranium.
Thus uranium changes to uranium X, uranium X to ionium,
ionium to radium, etc. Radium and polonium are thus seen
to be evolved from uranium and appear at certain definite
stages of its radioactive transformation. Actinium and
thorium, with their transformation products, are apparently
to be regarded as distinct families of radioactive substances.
Recent experiments have shown, however, that there is some
relation between actinium and uranium, although neither
actinium nor any of its radioactive products appear in the
direct line of descent set forth in the above table. Actinium
and its products may perhaps be regarded as an offshoot of the
uranium family. No such connection has thus far been estab-
lished for thorium, so that from the standpoint of our present
knowledge, this element with its products must be regarded
as a distinct and independent radioactive group.
PART IT
SOUND
SOUND
CHAPTER XXXV
WAVES
415. The general topics of sound and light, which are dis-
cussed in the following pages, have to deal with phenomena of
wave motion. Sound in the general sense consists of air waves
or waves in some other material substance. In the same way,
light is a wave phenomenon, the medium in which the waves are
formed in this case being the ether. In view of these facts, it
is convenient before taking iip these subjects in detail to make
a general study of wave phenomena and particularly of the
laws governing the production and propagation of waves.
Such a study is useful, not only because of its bearing on the
subjects of sound and light, but also because of the knowledge
it gives of other phenomena which partake of the nature of
wave motion.
WATER WAVES
416. Every one knows that the surface of a body of water
exposed to the wind is covered with waves. It is also gen-
erally known that such waves are sometimes large and some-
times small, and that they travel over the surface of the water
at varying speeds. Water waves may also be produced by
dropping a stone into a body of still water, or by moving an
object rhythmically up and down at the water surface.
However they may be formed, we observe that certain effects
always accompany water waves. We know, for example, that
at the crest of the wave a certain mass of water is lifted above
the normal level, while at a trough the surface is lower than
normal. This means that energy is involved in such wave
motion, and the important fact suggested is that wave motion
affords a means of transferring energy.
429
430
SOUND
A system of waves involves not only the potential energy
suggested above, but kinetic energy as well. Consider the
water waves represented in Figure 276. The vertical displace-
ments at the crests Q and the troughs T constitute the potential
Fig. 276. — Water Waves.
energy effects. Now, in front of each crest, at such points as
a, the water is rising. Behind the crest, at such points as 5,
the water is falling. Furthermore, at crest and trough there
is horizontal motion. It wiU be evident, therefore, that all
these portions possess kinetic energy.
In such waves the disturbance consists in the elevation of
certain portions of the surface water above the normal level and
the depression of certain other portions. Gravity tends to re-
store equilibrium (normal level). But the rising and falling
masses, because of tlieir inertia, pass and repass their equilibrium
positions, and thus the wave motion is maintained. Neverthe-
less, as the motion continues, energy is absorbed by the water,
due to friction (viscosity) effects, and, if the supply of energy
is withdrawn, the wave motion gradually ceases. The gradual
dying away of waves under such circumstances is called damp-
ing, and the gradually decreasing waves are said to be damped.
ESEEGY OF SOUND AXD LIGHT WAVES
417. As in the case of water waves, light waves and sound
waves involve energy and afford a means of energy transfer.
Sound waves consist of alternate compressions and rarefactioiLS
in the transmitting medium. These compressed and stretched
portions of the medium have potential energy much like that
possessed by a compressed or stretched spring. Those parts of
the medium which lie between the compressions and rarefactions
are in motion, and therefore have kinetic energy. Lightwaves
also represent energy. A light wave, as it travels through the
WAVE MOTION
431
ether, carries with it a store of energj^ in much the same manner
as a water wave.
THE FOEM OF A WATEE WAVE
418. The motion of the surface in deep water waves is such
that each particle moves in a circular orbit. Let the straight
line, Figure 277, represent the undisturbed surface of a body of
water. When a train of waves is passing over this surface, the
surface particles describe circles as indicated. This explanation
assumes that particles slightly separated differ in their motion
in point of time only, that is, they differ in phase. Two par-
FiG. 277.
tides are said to differ in phase when they arrive at the highest
point of the circular orbit (or other convenient reference point)
at different times. The few widely separated particles repre-
sented in the above figure differ in phase by 45°. That is,
when one particle is at the highest point of its orbit, the next
is 45° from the corresponding point in its orbit, etc.
In shallow water the particles move in ellipses which become
flatter and flatter as one approaches the bottom.
THE EELATION BETWEEN VELOCITY, WAVE LENGTH AND
FEBQUENCY
419. A wave is conveniently represented by a curved
line like that shown in Figure 278. A and b are wave crests;
! X j
432 SOUND
c and d are troughs. Evidently, a complete wave contains a
crest and a trough. The wave length of a wave is the distance
from any point on a wave to the corresponding point on the next
wave. For example, it is the distance from a to b, from c to d,
or from e to /in the figure. The amplitude of a wave is the
distance between the crest or trough and the mean position of
the wave. Oa is the amplitude of the wave shown in Figure
278.
Let it be imagined that the wave represented in Figure 276
is traveling toward the right. A given particle which rises
upon the successive crests and sinks into the successive troughs,
as the system of waves passes it, evidently makes a complete to
and fro excursion for every complete wave that passes the par-
ticle. Suppose that w complete waves pass the given particle in
one second. The frequency of the disturbance is then said to
be n. Evidently, the time required for one wave to pass the
particle in question will be the nth part of a second. This is
called the period of the disturbance. Let T represent this
period. Then,
The distance through which the disturbance (wave) moves
toward the right in the time T is evidently the length of one
wave. Call the wave length \. It follows at once that
T
or, V = nX (111)
RIPPLES
420. Water waves having a length of a few millimeters only
are called ripples. Surface tension has a great deal to do with
the formation and propagation of ripples. A moment's con-
sideration will show that surface tension acts in much the same
waj^ that gravity does to restore equilibrium when the surface
of a liquid is broken into waves. For long waves, the gravity
effect predominates, while for ripples, surface tension produces
WAVE MOTION 433
the larger effect. The propagation of waves a few centimeters
in length depends partly on gravity and partly on surface
tension.
THE VELOCITY OP WATER WAVES
421. The velocity of propagation of a water wave can be
shown to be ,-— t^ — ts
. = V|^+2^ (112)
where g is the acceleration of gravity, X the wave length, T the
surface tension, and p the density of the liquid. Evidently,
for very long waves, the second term under the radical becomes
negligible in comparison with the first. This equation is based
upon the assumption that the depth of the water is great as
compared with \. In shallow water the velocity of a water
wave is less than in deep water.
STATIONARY WAVES
422. An important effect is produced by two waves of the
same wave length and amplitude which are traveling in opposite
a,
a;
a,
/'~\
t
/'
-"v
\
/
/
\
1
\
/
\
/
1
\
A /
\
\
\
A'
/
/
f
l^''*^^
1
\
.^ t
\
W-s
/
•\7
\ /
\
V~^
\ /\
\
•/
\
^'' \
\/ \
\
-_k-
Va__.
A'b ^
\t-—
Vb
)(-—4-
\
r,
i\
"t
/
1 \
/
/\ /
/'. /
\ i\
/
1 \
/
/ ^ /
/ \ /
\ 1 \
/
LJ
/
o/
V\
/
\
\
\
\
1
$
i
\
1
— -^
\
\
\
\
\
\
\
\
1
1%
\
/
\
\
/
\
/
\
/
\
*,
a.
\
. '
a.
\
a.
\
Fig. 279.
directions through the same region. The elTect of such a com-
bination of waves will be understood by reference to Figure
279. Let A represent a wave traveling toward the right, B a
2f
434 SOUND
similar wave traveling toward the left. Consider the combined
efEect of these two waves upon the particle a. Evidently this
particle under the conditions represented in the figure is rising
on the crest of each wave, that is to say, it is being lifted by
the crest of the wave A^ which is approaching it, and it tends at
the same time to rise on the crest B, which is approaching from
the opposite direction. Since, supposedly, these waves are
traveling with the same velocity, the crests A and JB will reach
the particle a at the same instant. Since the particle a responds
to each wave, independent of the pregence of the other, it will
be elevated to a greater height than it would be if moving
under the influence of either wave alone. At the instant,
therefore, that the crests A and B reach the particle a it will
occupy some such position as a^ Again, when two troughs
reach the particle a it will sink to some such point as flj-
Considering now the particle h, it will be evident that at the
moment in which the conditions represented in the diagram are
supposed to exist, the particle h will remain in its mean position.
That is to say, it will be displaced neither up nor down, since
the effect of the wave A would be to raise it onto a crest, while
the effect of the wave B is to sink it into a trough. Under the
combined influence of the two waves, the particle b therefore
remains at rest. A moment's consideration will show that this
particle is at all times at rest, since at any instant it is ele-
vated by the one wave to exactly the same distance that it is
lowered by the other. In other words, the particles 6 remain
stationary at all times, while the particles a are displaced through
amplitudes much greater than those of the individual waves.
Particles lying between a and b vibrate to and fro through am-
plitudes which are small for those particles lying near b, and
large for those particles lying near a. The dotted lines in the
figure represent the stationary wave which results from the com-
bination of the waves A and B. The stationary points h are
called nodes, and the regions midway between the points h are
called loops.
The conditions necessary for the production of stationary
waves are :
1. The component waves must be of the same wave length.
WAVE MOTION 435
2. They must have the same amplitude.
3. They must be traveling in opposite directions.
A simple illustration of stationary waves is the following.
Let AB, Figure 280, represent a rubber tube or a slender spiral
spring, the end A being fas- y fork when it is caused to vibrate
y oyer an air column of suitable di-
mensions. The simplest way of ob-
taining the effect is as follows. Let F, Figure
300, represent a vibrating tuning fork and AB
an air column inclosed by a cylindrical jar. In
the general case the sound of the tuning fork is
not reenforced by bringing the fork into the pres-
ence of the air column. But if the air column, which
is capable of being set into vibration and giving
off sound waves, is of such length that its frequency
or proper period of vibration is the same as that of
the fork, then it will be found that upon the approach of the
tuning fork the air column will start into sympathetic vibra-
tion and will reenf orce the sound given off by the tuning fork.
B
Fig. 300.
462 SOUND ^
Problems
1. What is the velocity of sound in diy air at 30° C. ?
2. What is the velocity of sound in air at — 20° C. ?
3. A bell makes 100 vibrations (sends out 100 waves) per second.
What is the wave length of the disturbance in air at 30° C. ? At — 20° C?
4. A tuning fork is found to give off waves 130 cm. in length in dry
air at 0° C. How many vibrations does the fork make per second ?
5. A man is observed chopping wood. He m.akes 25 strokes per min-
ute. The sound of each stroke reaches an observer as the ax strikes the
wood in the following stroke. Temperature of the air is —10° C. What is
the distance of the chopper from the observer ?
6. A flash of lightning is seen and 5 sec. later the first sound of the
thunder is heard. What is the approximate distance of the nearest point
of the discharge? Why is it impossible to determine the distance accu-
rately by this means?
7. A man fires a rifle at a target 1000 ft. away. Velocity of the bullet
= 1200 ft. / sec. At what point must an observer stand on a line drawn
through the target perpendicular to the line joining gun and target, in
order that the sound of the rifle and the impact of the bullet may reach
him at the same instant? Assume velocity of sound = 1120 ft./sec.
8. A siren has a disk containing 16 holes. What is the pitch of the
tone it gives when it makes 50 revolutions per second ?
9. The pitch of the whistle of a locomotive drops a half tone in passing
an observer. If the velocity of sound is 1100 ft./sec, what is the speed
of the locomotive in miles per hour ?
10. Draw a diagram of the wave system accompanying a body moving
over the surface of a pond of water with a velocity J^"- that of the waves gen-
erated by the moving body?
THE MUSICAL SCALE
CHAPTER XXXVII
MUSICAL INTERVALS
441. The musical interval between two tones is the ratio of
their frequencies, the frequency of the higher tone being taken
as the numerator of the fraction. Thus two musical tones are
said to be in unison when their frequencies are as 1:1, that is,
when the ratio of their frequencies is unity. Tlie interval
between two musical tones is called an octave when the ratio of
their frequencies is 2. The principal intervals employed in
music are given in the following table :
Name of Interval
Ratio of Frequencies
Unison
i
Octave
i
Fifth
3
5
Fourth
1
Third
1
Minor Third
, f
Minor Sixth
f
The value of a musical interval does not depend upon the
absolute pitch of its components. For example, the interval
between two tones whose frequencies are 60 and 120 is the octave.
The interval between two tones whose frequencies are 256 and
512 is also the octave. That is, a musical interval depends only
upon the ratio of the frequencies of the tones which bound it
and is independent of absolute pitch.
CONSONANCE AND DISSONANCE
442. Two tones sounded together produce a pleasing effect
when the ratio of their frequencies can be expressed by small
numbers. Thus, aside from unison, the most pleasing interval
463
464 SOUND
is the octave, the next the fifth, etc. The pleasing effect of
tones sounded together is called consonance.
When two tones are sounded together the ratio of whose fre-
quencies can be expressed only in large numbers, an unpleasant
effect is produced. This effect is known as dissonance.
ADDITION AND SUBTRACTION OF MUSICAL INTERVALS
443. From the definition for the musical interval between
two tones it is evident that the sum of two intervals is found
by the process of multiplication rather than by the process of
addition. Consider, for example, the musicaV interval a to c.
Assume that this interval is made up of the two intervals a to 6
and b to c. Since the musical interval is defined as the ratio
of the number of vibrations which the upper note makes to
that made by the lower note, evidently the interval a to J is -.
a"
The interval between b and a determined in the same manner
f COO
is -. Now the product of - and - is - which is the musical
b baa
interval a to a. Thus the sum of the two intervals is obtained
by multiplying their values together.
In the same way the difference between two musical in-
tervals is obtained by dividing the one by the other. Let
it be required, for example, to find the difference between
the octave and the fifth. Dividing the octave interval (f)
by the fifth (|), we have
1 4
Octave — fifth = — i— = — = fourth.
2 "
That is, the difference between an octave and a fifth is a
fourth, or the sum of a fifth and a fourth is an octave.
THE MAJOR CHORD
444. The combination of three tones whose frequencies
are to each other as 4, 5, and 6 produces an effect upon the
ear which is especially pleasing. The combination tone is
rich, full, and satisfying. Such a combination is known as
the major chord. The major chord is of importance in the
THE MUSICAL SCALE 465
present study, since upon it is based what is known as the
major scale.
THE MAJOR SCALE
445. Let three tones c', e', and g' be taken, whose fre-
quencies are to each other as 4, 5, and 6. Let a fourth tone
c" he taken, which is the octave of c'. This major chord
forms the skeleton of the major scale, c' is known as the
tonic of tlie scale and g' as the dominant. In order that
we may have a specific example before us, let it be assumed
that the frequency of c' is 256. Then the frequency of e'
is I X 256 = 320, and the frequency of g' is f x 256 or 384,
and the frequency of c" is 2 x 256 or 512, since the frequency
of the octave of any tone is twice the frequency of the tone.
Let now g' be taken as a basis for the formation of a new
major chord. To form a major chord upon g', it is only neces-
sary to place in combination with it two tones whose frequencies
are to the frequency of g' as 5 : 4 and as 6 : 4 respectively.
That is to say, in any major chord the interval between the
first and second tones is the major third. The interval be-
tween the first and third tone is the major fifth. Let b' repre-
sent the second tone of the new chord formed upon g'. Since
the frequency of b' must be to the frequency of g' as 5 : 4,
therefore the ratio of the frequency of b' to that of c' must be
I X I = -1^, since the interval c' to g' is |. The absolute
frequency of b' is therefore f X 384 or 480. In the same
manner the frequency of the third tone, which, combined with
b' and g', will form a major chord, is determined. Call this
third tone d". Then the frequency of d" must be to the
frequency of g' as 8 : 2, and therefore the ratio of the frequency
of d" to that of c' = I X I = f, since the ratio of the fre-
quency of g' to that of c' is equal to f. The absolute
frequency of d" in this scale is given by | x 384 = 576. If
now a third major chord is formed upon f as fundamental, the
major scale will be complete. Comparing the frequencies of
c" and f, we find the interval to be a fifth, hence c" is the
third tone on a major chord on f. To determine a', the
additional tone necessary, we have simply to remember that
the frequencies of f and a' must be such that a' : f as 5 : 4.
2h
466 SOUND
Thus, in the scale above mentioned, the frequency of f is
1^ X 256 = 341 (approximately), and the absolute frequency
of a' = I X 341 = 427 (approximately). The several tones
whose frequencies have been determined in this manner may
now be written as follows:
c' d' e' f g' a' b' c" f (1"\
256 288 320 341 384 427 480 512 [ruGJ
The numbers written below the letters indicate the absolute
frequencies of the corresponding tones. The frequency of d'
is obtained from that of d", which is its octave, therefore the
frequency of d' is 288.
There are several different ways of representing the suc-
cessive notes of the major scale, as follows:
y 1 1
/t , 1 1 1 1 ,! ^
-^B — ^—
n
— a —
— =i—
— «- —
-=>
1
' li' -J-
25l
1
\ 1
\ 1
\ 1
1 1
V
\ /
/ s
\ /
X
/ \
\ /
/ \
/ \
* /
/ *
/ \
\ /
/ \
' \
\ /
);
/ \
i 1
1 1
/ \
>N
/ \
\ ;
/ \
1
\ 1
/ \
•J
V
/ \
/ \
/ \
( )
V
/ \
1 1
\
1 1
/ \
1 ;
' 1
\ 1
t /
t /
! 1
i
V
V
\ /
\ /
V
V
a b c
Fig. 303. — Overtones of a Closed Pipe.
modate shorter waves. The only conditions which the stationary
waves must satisfy, in order that they may be accommodated by
the closed pipe, that is to say, in order that they may exist in
SONOROUS BODIES
475
the closed pipe, are that in every case there must be a node at
the closed end of the pipe and an antinode at the open end.
Thus, Figure 303 shows a closed pipe of length L with some of
the longer wave lengths which it will accommodate. The wave
length Xj present in (a) is 4 L. The wave length X^ present in
(J) is evidently ^L. When this wave length is present in the
pipe, there are two nodes, one at the closed end of the pipe and
the other at one third of the distance from the top. The wave
T
I
A
X
X
X
Fig. 304. — Overtones of an Open Pipe.
length Xg present in (c) is ^L. The wave length X^ present in
(d') is equal to ^ L. That is to say, the wave lengths of the
various waves which may be accommodated by a closed pipe
are to each other as 4, |-, -I, ^, etc. Since the frequency of a
tone varies inversely as its wave length, the corresponding fre-
quencies are |, |, ^, ^, or, in other words, the several successive
frequencies are in the order 1, 3, 5, 7, etc.
In Figure 304 is shown an open pipe with some of the longer
476
SOUND
waves which it will accommodate, that is to say, which may
exist as stationary waves within the pipe. Call the length of
the pipe L. The length of the wave Xj present in the case
represented in (a) is evidently 2 L. The length of the wave
Xj present in the case represented in (h) is evidently L. Xg
in the case represented in (c) is equal to |- L, and in the case
represented in (i) X^ is equal to | L. That is to say, the several
wave lengths are to each other as the numbers, 2, 1, |, l. The
relative frequencies corresponding are given by the numbers 1,
2, 3, 4, etc.
It is thus seen that the frequencies of the various tones which
a closed pipe can give are to each other as the odd numbers, 1,
3, 5, 7, while those which the open pipe can give are to each
other as the numbers 1, 2, 3, 4, a series containing both the even
and the odd numbers. The tones corresponding to the waves
present in cases 5, c, d, etc., are called the overtones. Generally
speaking, when an air column is thrown into
vibration, it vibrates in several different ways at
once, for example, as a whole, giving its funda-
mental note ; in halves, giving the octave of
this note ; in thirds, giving the fifth above this
octave ; and so on.
MANNER IN WHICH AN AIR COLUMN IS SET
IN VIBRATION
453. Let Figure 305 represent an open
organ pipe. In addition to the box contain-
ing the air column proper, there is a device at
the bottom whereby a stream of air is directed
_.^^^^^ against a sharp edge in one of the walls of
j^^^^H the pipe. The direction in which the air cur-
^^^ ^5 rent moves is indicated by the arrow. When
I I the stream of air first starts, a disturbance is
'-f created in the lower part of the column which
travels to the top of the column, is there re-
flected and returns to the point 0, and reacting upon another
disturbance travels up the tube in the same manner. Evi-
dently the number of impulses which the stream of air at
SONOROUS BODIES
477
receives from the vibrating column is determined by the
length of the column. Thus, the stream of air is started flut-
tering or vibrating to and fro at a frequency which is deter-
mined by the dimensions of the pipe. The vibrating column
imparts its motion to the surrounding air, and thus a sound
wave of a frequency corresponding to the dimensions of the air
column is established. Since the pipe is open at (7, evidently
an antinode is present at this point. An antinode is present
also at the upper end of the pipe. There is, therefore, a node
at a point midway between these points. The wave given off
by such a pipe is therefore twice as long as the pipe itself.
Evidently, from the consideration shown above, if the pipe is
closed at the top the fundamental wave length will be twice as
great. The corresponding frequency will be -| as great, that is,
one octave below that corresponding to the open pipe.
APPLICATION OF THE LAWS OF VIBRATING AIR COLUMNS
454. An interesting application of the law of vibrating air
columns is the following : Let it be required to determine
the pitch of a tuning fork. This may
be done by finding the length of air
column which will vibrate in unison
with the fork. Thence by measuring the length
of the column the wave length of the sound wave
given off by the fork is at once known and from
this the pitch may be calculated. Let Figure 306
represent a jar partly filled with water as indi-
cated, above which is placed the vibrating tuning
fork. Let it be imagined that water is gradually
turned into the jar, thus shortening the air column
between the surface of the water and the mouth
of the jar until the resonance effect (Section 440)
is secured. Let L represent the distance between
the mouth of the jar and the surface of the water. When the
condition of resonance is reached, a quarter wave length is
present in this air column. That is to say, the sound wave
given off by the tuning fork has a length such that X. = 4 i.
But X = D -^n (Equation 111) in which v is the velocity of
Fig. 306.
478
SOUND
eound in air and n the frequency or pitch of the tuning fork.
We have, therefore,
vjn = \li
.-. n =
4:L
(118)
KTJXDT S EXPERniEXT
455. The longitudinal vibrations of a rod of metal or of glass
are like those of an air column. Imagine, for example, a glass
rod clamped at the center and stroked endwise with a damp
cloth. The rod will be set vibrating in very much the same
manner that an open air column vibrates. The ends of the rod
correspond to anti-nodes or vibrating segments. The center
of the rod constitutes, of course, a node. In such a vibrating
rod the same relation exists between the length of the rod and
the length of the stationary wave which is present in it as that
which holds for the air column inclosed by an open pipe, that
is to say, the wave length of the stationary wave present is
twice that of the rod.
HI'
D
Fig. 307. — Kundt's Apparatus.
Application has been made of this and the foregoing prin-
ciples to the determination of the relative velocities of sound in
various solids and gases. The apparatus employed is that rep-
resented in Figure 307. AB' is a glass tube, the bottom of
which is sprinkled with a small quantity of cork dust or lyco-
podium powder.
BO is a. rod, let us say, of brass, clamped at its center, as
indicated at J). When the brass rod is stroked, it is thrown
into longitudinal vibration as indicated above. These vibra-
tions are communicated to the air column in AB by means of
a disk of paper or wood attached to the end of the rod. The
disk at A is moved backward and forward until the distance
SONOROUS BODIES 479
AB is such as to accommodate the stationary wave train thus
set up in the air column AB. Wlien this adjustment is reached,
it will be observed that the lycopodium is violently agitated at
certain points within the tube, these being the points of greatest
disturbance, of course. At other points the Ij'copodium remains
at rest. These points are evidently nodes in the wave train.
They are distinctly marked by the heaping up of the powder.
The distance between two of these nodes is, as we have seen,
A, -H 2, in which A, is the wave length in air of the sound pro-
ceeding from the rod. Thus the half wave length of the sound
wave in air is determined. The corresponding half wave length
in brass is, of course, BC. It follows, therefore, that the velocity
of sound in brass is to the velocity of sound in air as BO is to
UF. If the tube is now filled with another gas, the length of
the wave will be found to be different, since the velocity of
sound is different in different gases. But evidently the velocity
of sound in the new gas is to the velocity of sound in air with
which the tube was first filled as the corresponding distances
between the nodes of the stationary wave trains.
TRANSVERSE VIBRATION OP STRINGS
456. Consider a string tightly stretched between two points
as represented by AB, Figure 308. Imagine it to be displaced
(drawn aside) and re-
leased. The disturb- J^ ?
ance (distortion of the .^^^. - _ ^ ^ _ ^ -^ ^ ^^
string) will be propa-
gated to both A and
r> Ml 1 n n Fig. 308.
is, Will be reflected at
those points. These reflected disturbances will return to
the center of the string, pass one another, and go on to the
ends to be once more reflected, and so on. The actual vibrations
of the string are therefore made up of two waves traveling to
and fro along the string in opposite directions with equal ve-
locities. These waves combine to form stationary waves in
the string, and these stationary waves constitute the visible and
effective vibrations of the string. Evidently the string makes
one complete vibration (assuming that but two nodes are pres-
480 SOUND
ent, namely, those at A and 5) while one of the component dis-
turbances travels from A to £ and back again, a distance 2 L.
Call the frequency of the vibrations n, and the velocity of the
component disturbances up and down the string v. We have,
then,
V = nX
in which X is the wave length of the component disturbances.
But since the disturbance travels twice the length of the string
for each vibration, therefore the wave length, \= 2L. Hence,
V = u ■ 2 L
Now, experiment shows that the velocity of the component dis-
turbances is given by the square root of the tension in the
string, divided by the mass of the string per unit length.
That is.
Wf,
.1/
in which T is the tension and M the mass per unit length.
Combining these two expressions for v, we have,
2L = -f^
31
n = ^VJ (119)
which is an expression for the frequency of the vibrating string
in terms of its length and the tension and mass per unit
length of the string.
This is known as the Law of Vibrating Strings. By means
of this relation, the frequency with which a string will vibrate
or the pitch of the sound which it will give off as it vibrates
may be predicted. The law is exemplified in the stringing of
a violin. The strings of a violin are all of the same length, so
that the relative pitches of the strings are determined by the
tensions to which they are subjected and their masses per unit
length. The E string, the string of highest pitch on the violin,
is given a high pitch by making it thin and liglit and by sub-
jecting it to relatively high tension. The G string, or lowest
SONOROUS BODIES
481
string, is given a low pitch by subjecting it to a relatively low
tension, and by giving it a large mass per unit length. This is
done by winding it with wire. The A and D strings, or strings
of intermediate pitch, are thicker than the E string and are
subjected to a higher tension than the G string. The law of
vibrating strings is readily verified by means of the apparatus
represented in Figure 309. Let AB represent that portion of
Fig. 309. — Sonometer.
the string which is caused to vibrate. TFis a weight which is
attached to the end of the string by a cord passing over the
pulley P. By changing the value of this weight the tension
in the string may be varied. Thus the dependence of pitch
upon the tension may be investigated. By changing the posi-
tion of the support B the length of the vibrating segment of
the string may be varied at will and in this manner the depend-
ence of the pitch upon the length of the string, other things
remaining the same, may be determined. Finally, by taking
strings of different weight or mass per unit length, subjecting
them to the same tension, and making use of the same length of
vibrating segment AB, the dependence of the frequency upon
the mass per unit length may be determined.
TRANSVERSE VIBRATION OF RODS
457. It may be shown experimentally that the rate at which
a given rod will vibrate transversely is given by the following
expression :
Ad fW
n =
i^'V
(120)
2i
482 SOUND
in which U is the modulus of elasticity, p is the density, L is
the length of the rod, d is its thickness in the direction of vibra-
tion, and ^ is a constant depending upon the manner in which
the rod is fastened and upon the number of nodes present in the
vibrating rod. The manner in which the rod vibrates is indi-
cated in Figure 310, in which the heavy line represents the rod
a b
Fig. 310.
supported at two points a and h. If a rod supported in this
manner is struck at its center with a mallet, it will be bent into
the position shown by the dotted line. From this position it
will vibrate to the reverse position as shown by the second-
dotted line, and so on, the points of support in this case con-
stituting the nodes. By supporting rods of different length
and thickness, the above law may be verified. For example,
according to the above law, if two rods of the same material
and the same length but having thicknesses in the ratio of 2 to
1 are employed, the thicker rod will give the octave of the note
given by the thinner rod, and so on.
It should be noted in this connection that the width of the
rod does not enter. In other words, the frequency is inde-
pendent of the width of the rod.
LONGITUDINAL VIBRATION OF EODS AND STRINGS
458. Consider an elastic rod attached at one end to a rigid
wall, Figure 311 (a). Let it be assumed that the rod is compressed
by the application of a force at the outer end and then released.
Under these conditions the rod will vibrate longitudinally and,
if the vibrations are of proper frequency, will give a musical
tone. The general character of the vibratory motion of the rod is
shown at (a), (J), (c), etc.
As soon as the compressing force is removed, the compression
in the rod begins to relieve itself. The first layer of particles
at the free end of the rod moves to the right. This is followed
by similar motion of the second, third, and other successive
SONOROUS BODIES 483
layers. Each moving layer continues in motion until all layers
have been relieved of their compression and until it receives
a pull to the left due to the stopping of the adjacent layer. The
condition of the rod shortly after the outward motion of the
successive layers has begun is shown in (5). When the com-
pression has been relieved throughout the rod, the condition is
that represented in (c). If the time for one complete vibration
of the rod is t, then ^ t is the time required for the rod to change
from the condition (a) to the condition (c).
The condition represented in (c) exists for an instant only,
since the layers adjacent to the wall are stretched by the out-
ward motion of the rod. The stretch in the rod near the wall
spreads outward and the various layers are successively brought
to rest. When the last (outward) layer has been brought to rest
the rod is in the condition shown at (e). The entire rod is now
stretched and stationary. The time which has elapsed since the
beginning of the vibration is ^ t. The condition of the rod for
a time intermediate between ^ t and | t is shown at d.
The stretch in the rod (e) now begins to relieve itself. The
end layers at the right are set in motion toward the left and the
strain in the successive layers disappears. The condition
shortly after the stretch begins to relieve itself is represented
(/)• When the stretch has been entirely relieved, all parts of
the rod are moving to the left (^). The time elapsed is | t.
The motion of the rod is now stopped by a compression which
sets in at the wall end of the rod. After this compression has
started the conditions are like those shown at (Ji). When all
layers have been brought to rest the rod is in its initial condition
(i) and (a). The time elapsed is (t).- The rod will now begin
a second vibration like that outlined above. These vibrations
will continue until the energy of the vibrating rod has all been
frittered away in friction effects or spread through the surround-
ing medium in wave motion.
It is interesting to note the energy transformation in the
vibrating rod. At (a), (e) and (z) the energy of the rod is poten-
tial, like that of a stretched or compressed spring. At (c) and
(g^ the energy of the rod is kinetic. At intermediate conditions
the energy is partly potential and partly kinetic.
484
ca)
(b)
SOUND
Compressed
Compressed
Stationary
t = o
Stationary
Compressed Unstrained.
■.-..•^f:: .'^j.- ;.■■ •:■■■■,■> ;r.-.ov.u
Stationary Moving
Unstrained
Stretched
Moving
Unstrained
(e)
(f)
' Stationary Moving
Stretched
i^^^^_*i^i^^^^^bri
^^^^^^^^^^^^^^^^^^^
t =
it
StatTonary
Stretched
Unstrained
I I im 1^L^^^^»J^^
Stationary
Unstrained
ivioving
»-::*^s^^s&^^«;
= fT
Moving
Unstrained
Compressed Unstrained ^^
(1)
Stationary IVloving
• Compressed
j'-'.^'.'jj'.'.yj'-.l!
>stykiJ,''4VA*i
y.^^.^>j-i.:>t..-..--^...-:..;y:-.-^-..-'~
t=r
Stationary
Fig. 311.
SONOROUS BODIES
485
A rod clamped at its center, or a string stretched between two
supports, may be set into vibratory motion analogous to that of
the rod in Figure 311. The motion of a rod clamped at its
center is such that its ends move in opposite directions, each
half having a motion like that described above.
THE TUNING FORK
459. When a straight rod is caused to vibrate transversely,
the nodal points a and h, P^igure 312, are at a distance from
the ends of about \ the total length of the rod. If the rod is
curved as shown at
B, Figure 312, it
will be found that
the nodal points
are nearer to-
gether. If the rod
is more sharply
curved as at Cand
B, the nodal points
will be found to
be still closer to-
gether. Finally, if
the rod is bent into
the form shown at
JS, the vibrating segment at the center is quite short and
especially if the rod is made thick in this region. A tun-
ing fork consists of a steel rod bent into the shape represented
at E and supplied with a stem at the center. The principal
object in giving the tuning fork this form is to exclude the
secondary or harmonic vibrations which are present in the
straight bar. These secondary tones being excluded, the
tuning fork gives a very pure musical tone.
Fig. 312.
THE VIBRATION OF PLATES
460. The vibration of a plate may be examined b}^ regard-
ing it as a bar from each of two adjacent edges, and by con-
sidering the manner in which the vibrations in these two
486
SOUND
directions are combined. For example, in Figure 313, let
AH CD represent a vibrating plate. Let it be assumed that,
as regarded from the edge AB, it is vibrating like the bar rep-
resented in Figure 310. Evidently there will be two nodal
C DC d'
+ I - +
+ / - \ +
A
\ — ^ f
B A'
B'
Fig. 313.
lines across the plate as represented by the dotted lines in the
figure. Let it be assumed that at a given instant the edges of
the plate are rising while the center of the plate is falling.
This may be represented by the + and — signs as indicated
in the figure. Let it be assumed that the plate regarded from
the edge AQ \s, also vibrating like the bar represented in Fig-
ure 310. Let it be further assumed that at the instant under
consideration the plate in response to this vibratory motion is
rising at the edges and falling in the center as indicated at
A!B' CD'. Now it will be evident that the actual motion of
any part of the plate is the sum of the motions due to these
two vibratory motions. For example, the center of the plate
is fa;lling under both vibratory motions, the four corners of the
plate are rising under the influence of each of the two vibratory
motions. Evidently there is a point at the center of each edge
which is rising in response to one of the vibrations and falling
in response to the other. These points will, therefore, remain
stationary. Upon examination it is found that under these
circumstances the nodal lines upon the plate are as shown by
the dotted lines in the square at the right of Figure 313.
If it is assumed that the vibratory motion of the plate at the
moment under consideration is the opposite of that shown at
A'B' Q' B', then the result obtained is that represented in Fig-
SONOROUS BODIES
487
ure 314. Under these circumstances the central part of the
plate remains stationary, and the corners of the plate remain
stationary, while the center points of the edges rise and fall
under the influence of both of the component vibrations.
The nodal lines represented in Figures 313 and 311 are readily
obtained by placing a vibrating plate in a horizontal position
and sprinkling the surface with sand. The sand is driven
away from the vibrating segments and collects along the nodal
lines, outlining them in a very definite manner. The vibrations
- 1 +
\
_
/
\
^^_
-'
\
+ +
>::'
++
,'
^
/
V
\
/
—
Fig. 314.
considered in the above discussions represent the more simple
cases. Ordinarily a plate breaks up into a' great many more
vibrating segments than are indicated in these discussions.
The vibrations represented in Figures 313 and 311 correspond
to the lowest pitches which the vibrating or sounding body is
capable of giving.
Problems
1. What is the pitch of the fundamental tone of an open pipe 100 cm.
long? Temperature = 0° C.
2. What is the pitch of the first overtone of the pipe of problem 1 ? Of
the second overtone?
3. What are the pitches of the fundamentals and first and second over-
tones of a closed pipe 100 cm. long ? Temperature = 0° C.
4. An open pipe is giving its fundamental tone. A hole at the middle
of its length is suddenly opened. What effect is produced?
5. If the pipe of problem 4 is giving its first overtone and the hole is
opened, what effect will be produced?
6. What are the wave lengths of the sound waves given off by the pipe
of problem 3 ?
488 SOUND
7. A tuning fork making 256 vibrations per second is fixed in front of
a, tube the lengtli of which is adjusted to resonance at 0°C. What change
in the length of the tube will be necessary to secure resonance at 20° C. ?
8. A closed tube is adjusted in length to give resonance with a tuning
fork making 64 vibrations per second. Give the frequencies of three other
forks of higher pitch to which this tube will also respond.
9. Three closed pipes containing air, oxygen, and hydrogen at 2U°C. aie
of such length as to respond when giving their fundamental tones to a fork
having a frequency of 1000. What are their lengths ?
10. An organ pipe containing air at 20° C. gives a tone having a pitch
of 500. What will be the pitch when the pipe is filled with hydrogen at
the same temperature ?
PART y
LIGHT
LIGHT
CHAPTER XXXIX
THE NATURE OF LIGHT
461. It was pointed out in the discussion of heat waves that
a hot body gives off waves of different wave lengths. Those
which produce heating effects are called heat waves, those which
affect the optic nerve are called light waves. Aside from the
question of wave length, however, these two kinds of waves
are identical. They travel in the same medium (the ether)
with the same velocity, and obey the same laws of reflection,
refraction, etc. In the subject at hand, we shall confine our
attention to a discussion of those wave lengths which are capa-
ble of affecting the optic nerve.
Various theories have been advanced in attempts to explain
the manner in which a luminous body is capable of affecting
the eye. One of the theories which held its ground for a long
time was known as the corpuscular theory. This theory as-
sumes that a luminous body is continually throwing off small
particles of matter. These particles are repelled from the
luminous body at very high velocities. Falling upon the sur-
face of other bodies, they are reflected, thus rendering these
bodies luminous. Falling directly upon the retina of the eye,
they produce the sensation of light.
Although this theory was for a time popular, it was eventu-
ally displaced by the wave theory, which is the one at present
imiversally accepted. This theory assumes that the disturb-
ance known as light consists of a wave motion in the medium
known as the ether. That the ether is the medium of propa-
gation of these light waves is evidenced by various facts and
phenomena ; for example, the ordinary incandescent lamp bulb
is exhausted of air and other gases, the vacuum being made
491
492 LIGHT
as nearly perfect as is possible with the best of air pumps. ■
But the incandescent filament of such a lamp is capable of
illuminating its surroundings in spite of the fact that the dis-
turbance which travels outward from the filament must pass
through a vacuum. The fact that the light of the sun reaches
the earth is perhaps sufficient proof that light does not require
for its transmission a material medium. This is evident from
the fact that practically the entire space between the luminous
surface of the sun and the earth is devoid of matter, at least
in the ordinary sense, that is to say, it is a vacuum.
The real nature of this wave motion will be better under-
stood after we have discussed some of the various phenomena
which serve to prove that light consists of a wave motion. For
the present it will suffice to state that this wave motion is a
transverse wave motion, and that it travels through the medium
of propagation, the ether, with a finite velocity. The manner
in which the velocity of the disturbance is determined and
the proofs for the statement that the waves are transverse are
described below.
THE TELOCITY OF LIGHT
462. The velocity of light is so great that for a long time
it was considered to be infinite. That light does require a
definite length of time for traveling a given distance was first
determined by Roem.er in 1675. Roemer came upon this dis-
covery in an accidental way while making astronomical observa-
tions upon one of the satellites of the planet Jupiter. He was
endeavoring to determine the period of the satellite, that is to
say, the time required by the satellite to make one complete
revolution about the planet Jupiter. His observations ex-
tended over a period of many months. In comparing the
results of his observations he found that, beginning at that
time of year at which the earth was situated directly between
Jupiter and the sun, for a period of half a year the succes-
sive observations taken upon Jupiter's satellite showed that
the period was apparently increasing. Continuing his observa-
tions for another six months he determined that the period of
the planet during this interval was apparently decreasing so
THE NATURE OF LIGHT
493
that at the end of a year the period of the planet was once
more the same as that observed at the beginning. Roemer's
explanation of this apparent variation in the period of Jupiter's
satellite was as follows : Referring to Figure 315, let ^S' repre-
sent the sun, UW the orbit of the earth about the sun, JJ' the
Fig. 315.— Illustrating Roemer's Method.
orbit of the planet Jupiter. The small circles drawn about J
and J' represent the orbit of the satellite T. The curved
arrows indicate the direction in which the various bodies are
supposed to be moving in their orbits. When the earth is in
the position E and Jupiter is in the position J", the distance
between the earth and the planet is given by the diifereuce
between the radii of their orbits. When the earth is in the
position _E', six months later, the planet Jupiter will have passed
to some such position as J' (it requires about 12 years for
Jupiter to pass once around its orbit). Under these circum-
stances the distance between the earth and the planet is equal
to the sum of the radii of their orbits. Thus, the distance
between the earth and the planet when the earth is at W is
greater than the distance between the earth and the planet
when the earth is at E by the distance EE' , that is, by the
diameter of the earth's orbit. Therefore a light signal passing
from the planet Jupiter in its J' position would have to travel
494
LIGHT
farther to reach the earth by the distance EE' than it would
have to travel when the planet is in the J position. If light
travels with finite velocity, a definite interval would be required
for the light to travel this extra distance. Roemer found in his
observations that the light signal came apparently 1000 seconds
too late when the earth is at E' . The mean diameter of the
earth's orbit is 186,000,000 miles. Therefore, if this explana-
tion of the observed facts is correct, it required 1000 seconds
for light to travel 186,000,000 miles. In one second, there-
fore, it would travel the thousandth part of this distance or
186,000 miles. Roemer's explanation of the facts observed by
him, while undoubtedly the true one, was not well received
and in a short time was forgotten. Fifty years later a noted
English astronomer, by the name of Bradley, determined the
velocity of light by an entirely different astronomical method,
obtaining practically the same result as that obtained by Roemer.
This served to direct the attention of the scientific world to
the work of Roemer, who was then given due credit for the
discovery he had made.
BRADLEY S DETERMINATION OF THE VELOCITY OP LIGHT
463. Bradley's determination
of the velocity of light was based
upon the principle of aberration.
This principle, briefly stated, is
as follows : The apparent velocity
of one body as seen from a second
body is given by the vector differ-
ence of the two individual veloci-
5 ties. Consider, for example, the
case of falling raindrops as
viewed from a moving train.
Let it be assumed that the rain
is falling vertically. Let the
magnitude and direction of this
velocity be represented by the
arrow a6, Figure 316. Let the
arrow cd represent the velocity
\
-^— _
3
^
\
\
1
/-0-.
1
\
\
^
L
1;''
e'*^-
b
\
c ^d
Fig. 316. — Apparent Motion of a Fall-
ing Raindrop as seen from a Mot-
ing Car.
THE NATURE OF LIGHT
495
of the car. Then, according to the principle given above, the
apparent velocity of the raindrops is given by ae, the vector
sum of ab and — cd, that is to say, the vector difference be-
tween ah and cd (Section 21). Let AB represent the frame of
a car window. Then to a person within the car a raindrop ap-
pearing at a will travel across
the window in the direction
ae. This effect of the appar-
ent change in the direction
of motion of the one body
(the raindrop) due to the
motion of the body from
which the observation is taken
(the car) is called aberration.
The angle is called the
angle of aberration.
Just as the apparent motion
of the raindrops is different
from their real motion, so the
apparent direction of a wave
motion is altered by the
motion of the observer. Let
the dotted circle. Figure 317,
represent the orbit of the
earth, the curved arrows in-
dicating the direction in which
the earth moves in its orbit.
When the earth is at A, it is
moving toward the right;
when at £, it is moving
toward the left. Let it be imagined that from the earth as it
travels about its orbit observations are being made upon a star
located, for example, at iS. It will be evident from the discus-
sion given above that when the observation is taken from A
the star will be apparently displaced to some such position as
»S". When the observation is taken from B, the apparent dis-
placement of the star will be in the opposite direction, the star
appearing in some such position as S" . When _ the earth
Fig 317. — Illustrating Bradley's Method.
496 LIGHT
reaches some such position as C or B, the star will be seen in its
true position. By taking observations upon the star, while the
earth makes one complete revolution in its orbit, the double
displacement S' S" of the star is readily determined. This
double displacement is determined as an angle and is evidently
equal to twice the angle of aberration. Call the velocity of
light V. Let the velocity of the earth in its orbit be repre-
sented by M, we have, then,
tan ^ = -
V
or, ^ = r^ (121)
tan
If u is known and the tangent of (^ determined by measure-
ment, we have at once the means of calculating the velocity of
light.
poucault's method
464, In 1849 Fizeau devised a method for determining the
velocity of light bj^ measuring the time required for light to
travel over a comparatively short distance on the surface of the
earth. About one year later Foucault developed what might be
called a laboratory method. The essential parts of Foucault's
apparatus are shown in Figure 318. S is a, source of light, M
Fig. 318. —Diagram of Foucault's Apparatus.
is a plane mirror revolving at high velocity about an axis in its
own plane and perpendicular to the paper as shown in the dia-
gram, the direction of rotation being indicated by the curved
arrow. The beam of light proceeding from the source tS* falls
THE NATURE OF LIGHT 497
upon the mirror M, and is thence reflected to a second mirror
M', which is stationary. It falls upon this second mirror perpen-
dicular to its surface and therefore returns along the same path
to the revolving mirror M. If now M were stationary, the light
would also retrace the path SM. Since, however, the disturb-
ance requires an appreciable time to pass from M to M' and
back again, the mirror M will have turned through a small
angle before the disturbance again reaches it. Therefore, in-
stead of being reflected back to S, it will be reflected to some
such point as *S". The angle 2 a is twice the angle through
which the mirror has turned in the interval during which the
disturbance travels from ilf to M' and back again. Let it be
assumed that the mirror makes n revolutions per second. It
will therefore require - second for it to make one revolution.
n
Evidently if it requires - second to turn through the angle 2 tt,
n
the time required for it to turn through the angle a is given by
the following relation : ^
n
/y
or, t=-
'1 Trn
where t is the time required for the mirror to turn through the
angle a. This is also the time required for light to travel a
distance 2 MM'. We have, therefore,
^ _ 2 MM'
Zi —• ■
V
Equating these two values of t, we obtain
^^i^n.MM' (^22)
a
Hence, knowing the speed of the mirror M, the distance MM',
and the angle «, the velocity of light is readily calculated from
this formula. 2 a is given by the ratio of the distance *S'*S" to
SM.
The best determinations of the velocity of light have been
made with this form of apparatus.
2k
498 LIGHT
The most recent determinations of the velocity of light indi-
cate that its value is not far from
z> = 3 X 10^" centimeters per second
= 186000, miles per second (approximately).
THE INDEX OF EEFRACTION
465. One of the most important applications of the labora-
tory method for the determination of the velocity of light has
been in the investigation of the value of the velocity of light
in different media. A study of this kind is readily carried out,
for example, with Foucault's apparatus by placing between the
mirrors M and W a long tube containing the medium under
investigation. The tube is provided with glass ends and is so
placed that the light in traversing the distance MW and back
again must travel lengthwise through the tube.
The ratio of the velocity of light in air to its velocity in a
second medium is called the index of refraction of the second
medium as referred to air. This ratio is found to be greater
than unity for all substances whose densities are greater than
that of air. This fact, when discovered, had a decided effect
upon the displacement of the old corpuscular theory by the
new wave theory, since under the old theory it was tiecessary to
assume that light traveled more rapidly in a dense medium
than in a rare medium.
THE RECTILINEAR PROPAGATION OF LIGHT
466. Light travels in straight lines in all directions from its
source. This is evidenced by the fact that the light proceed-
ing. from any luminous body is capable of forming an image of
that body. The formation of an image in this manner will be
understood by reference to Figure 319. Let AB represent a
box having a single small opening at A. Imagine a luminous
object CD to be standing in front of this box. Each point on
this luminous object may be thought of as a distinct and sepa-
rate source of light. Consider the light which is being emitted
by the point C at the upper extremity of this object. This
disturbance travels in all directions. A very small part of this
THE NATURE OP LIGHT
499
disturbance will find its way through the opening A and fall
upon some such point as B on the opposite wall of the box.
Thus B is a luminous point, in effect an image, of the point C.
In the same way the point D will produce an image of itself at
Fig. 319.
E and the light proceeding from points intermediate between
Cand I) will fall upon corresponding points between B and B.
Thus, for every point on the object CD there is a corresponding
point in the image BE. The fact that QD is thus able to form
an image of itself at BB is proof of the fact that light travels
in straight lines.
WAVE FRONT AND BAY
467. The wave front of a wave is defined as a line drawn
through all points on the wave which are in the same condition
as regards displacement and direction of motion. For example,
a line drawn along the crest of a water wave would be a wave
front. A line drawn along the trough of a water wave or a
line drawn along the side of a water wave joining particles
which are equally displaced from their positions of rest, would
also be a wave front.
The term ray of light, as it will be used in the following dis-
cussions, is intended to indicate a line drawn perpendicular to the
wave front.
A wave front always moves perpendicular to itself ; hence a
ray is a line drawn through a wave front to indicate the direc-
2k
500
LIGHT
tion in which the disturbance is traveling. For example, the
lines GB and DH, Figure 319, are ra^s. These rays are not to
be thought of as limiting in any sense the direction in which
the disturbance is spreading, since we know that the disturb-
ance proceeding from each point in the luminous object CB is
radiated in all directions.
HUYGHENS PRINCIPLE
468. In determining the position of a reflected or refracted
wave the application of what is known as Huyghens' principle
is found to be of service. This principle is as follows : Each
point on a wave may be regarded as a separate disturbance, and
the combination of the secondary wavelets proceeding from these
individual sources determines the position of the advancing wave.
The principle is made
clear by the example
represented in Figure
320. Let S represent a
source of light and W
a part of one of the
spherical waves proceed-
^1 ing from it. Let it be
required to find the po-
sition of this wave after
the lapse of the time t. Let WW represent the distance over
which the disturbance travels in the given time. With the
successive points A, B, C. D, etc., on the wave WW as centers,
draw circles each having a radius WW. Then the envelope
of these secondary wavelets will be the wave front WW re-
quired. In this envelope the secondary wavelets conspire to
produce a maximum disturbance. At other points they inter-
fere in such manner that practically the entire disturbance
WW is handed on, as indicated, to W'W'.
Fig. 320.
HUYGHENS CONSTRUCTION FOR A REFLECTED WAVE
469. The application of Huyghens' principle will be under-
stood by considering the following cases : Let it be required,
for example, to determine the position of a plane wave after it
THE NATURE OP LIGHT 501
had been reflected by a plane mirror. The application of Huy-
ghens' principle to this case, and in fact to all cases of reflection,
consists in considering each point of the approaching wave as it
reaches the mirror a secondary source of disturbance. Then,
by combining the secondary wavelets which proceed from these
several sources, the total reflected wave is found. In Figure
321 let WW represent a plane wave approaching the plane
mirror MM. Let it be required to find the position of this
W
W
Fig, 321.
wave after reflection. The first step in the application of Huy-
ghens' principle is to draw what is known as the " dotted posi-
tion " of the wave W W". This is the position to which the
wave would pass, in the length of time required for the dis-
turbance to travel from Wio W, if the mirror were absent.
In the presence of the mirror, however, the entire disturbance
is turned back, so that after reflection it is traveling on the
same side of the mirror MM. Consider that point of the wave
WW which is in contact with the mirror at M. This part of
the disturbance begins, at the instant corresponding to the posi-
tion of the wave shown in the figure, to travel back in the
medium above MM. Evidently it will travel as far above MM
as it would have traveled below in the same length of time,
that is, WW. If, therefore, a circle is drawn about the point
M with a radius equal to WW', it will be evident that the re-
flected disturbance corresponding to the first point of the wave
to come in contact with the mirror will have reached some
502
LIGHT
point on this circle above MM, when the other end of the wave
has reached the mirror. In like manner, when the center point
of the wave has reached the mirror at B, it is reflected and be-
gins to travel back from itOf and at the moment that TF comes
to W" the A disturbance will evidently have traveled from the
mirror a distance equal to BC, that is, equal to the distance
which it would have traveled forward in the same length of
time had its course been unobstructed by the mirror. If, there-
fore, a circle is described about the point 5 as a center and
having a radius equal to CB, it is evident the A disturbance
will lie somewhere on this are above B. Thus, the disturbance
which travels away from the mirror from each point on its sur-
face is determined by drawing circles about the several points as
centers tangent to the dotted position. Evidently the reflected
wave is the envelope of these circles on the
opposite side from the dotted position.
V^" Thus in Figure 321 W"AW is the reflected
wave.
A PLANE WAVE REFLECTED FROM A
CONCAVE MIRROR
470. The position of a plane wave after
reflection at a curved surface is obtained in
a" similar manner. In Fig-
ure 322 let P represent a
plane wave approaching the
concave mirror MM. Let WW represent
the dotted position of the plane wave, that
is, the position to which the plane wave
would have passed in the absence of the
mirror. Taking the successive points on
the mirror MM as centers, draw circles
which are tangent to the dotted position.
Then W"W", the envelope of the small cir-
cles, is the reflected wave, its direction of
motion being indicated by the small arrows.
The distances MW and MW" are equal, each being measured
perpendicular to the corresponding wave front ; but these two
Fig. 322.
THE NATURE OP LIGHT
503
wave fronts are parallel for points near 0, and for such points
the distances MW and MW" would be measured perpendicular
to the line WW. The distance OP, the greatest distance
between an arc and its chord, is called the sagitta of the arc.
The above statement with reference to the distances MW and
MW" is therefore equivalent to saying that the sagitta of the
reflected wave is twice that of the mirror, provided but a small
portion of the wave in the neighborhood of O is considered. This
being the case, it is an easy matter to show the relation which
exists between the radius of
curvature of the reflected wave
and the radius of curvature of
the mirror.
Tlie relation between the
sagitta of an arc and its radius
is obtained as follows. Let
MM, Figure 323, be the arc
of a circle having its center
at 0. Let the straight line
MM be the chord of this arc.
Call the distance MO, that is,
the radius of the arc, It. Call the sagitta h. Call the chord
d. Then evidently the distance from the chord to the center
of the circle is R—h. Therefore from the right-angle triangle,
we have, „„ , „ ^
4
That is, W = R^~2Rh + h^ + ^
Fig 323
Whence,
HR
(123)
providing h is so small that its square may be neglected, which
is usually the case in the application of this formula to curved
mirrors. In the example given in the last paragraph, let it be
assumed that the sagitta of the mirror is h. Call the sagitta of
the reflected wave K. From the relation given in Equation
(123), we therefore have
A=-^ iind ir=2^ = -^
SR Sb
504
LIGHT
in which R is the radius of curvature of the mirror and h is the
radius of curvature of the reflected wave. Combining these
equations, we obtain,
In other words, the radius of curvature of the reflected wave
under the assumed conditions would be one half the radius of
curvature of the mirror.
CONVEX AND CONCAVE WAVES
471. A wave front is said to be convex if its rays diverge. A
concave wave front is one whose rays converge. Thus WW,
Convex
Wave-front"
Convcave
Wave-front
Fig. 324.
Figure 324, is a convex wave front,
front.
TF' TP is a concave wave
A CONVEX WAVE REFLECTED BY A CONCAVE MIRROR
472. The case of a convex wave reflected from a concave
surface is of particular importance. In Figure 325, let MM
represent the spherical mirror, WW the dotted position of
the incident wave, W"W" the position of the reflected wave at
the moment the last point on the wave leaves the mirror. By
THE NATURE OF LIGHT
505
and
Then evidently,
and
But
construction cl is equal to cd. Call the sagitta of the mirror
A, the sagitta of the approaching wave h\ and the sagitta of the
reflected wave h" . That is,
ac = h
ad = h'
ab = h"
cd= h' — h
bd=2(h' -h)
ab = ad — bd
.-. h" = h' -2(ih' -K)
or, A" = 2 A - A'
If we call the radius of curvature of
the mirror R, the
_ radius of curvature
of the approaching
wave a, and the radius of curvature of
the reflected wave b, we have from the
last section
d^
d^
A' = -^, A = ^,and A" =
8a HR 8b
Substituting these values in the above
equation, we obtain,
85 SR 8a
Fig. 325.
or finally,
a b R
(124)
THE FORMATION OF IMAGES BY MIREOES
473. A real image of an object point is a point at which the
light, proceeding from the object point, is concentrated or focused
after being reflected by the mirror. A virtual image of an object
point is a point from which the light proceeding from the object
point appears to come after being reflected by the mirror.
The case discussed in the last section is again represented in
Figure 326, in which MM is the curved mirror. The incident
and reflected disturbances are represented by rays. is sup-
506 LIGHT
posed to represent a source of light. The spherical waves
proceeding from toward the mirror are reflected and focused
at I. OP = a, the radius of the incident waves ; IP = 6, the
radius of the reflected waves ; CP = H, the radius of the
mirror MM. I, the point at which the disturbance proceeding
from is focused after reflection from the mirror M, is called
Fig. 326.^Coniugate Points.
the conjugate of 0, and and I are called conjugate points.
I is also called a real image of the point 0. Evidently in this
case the effect of the mirror has been to change the form of
the wave front from convex to concave. In the discussion of
Section 472 it was assumed that the curvature of the incident
wave was in the same direction as that of the mirror, that is,
that the center of curvature of the mirror and the incident
wave lay on the same side of the mirror. Under this heading
there are certain special cases that it is worth while to note in
particular.
Case I. R = oo . If the mirror is plane, then i? = oo. We
have, therefore, from Equation (124),
1 + 1=0
a
whence, h = — a
That is to say, the curvature of a wave reflected from a plane
mirror is the same as the curvature of the incident wave but
opposite in direction. Hence a wave reflected from a plane
mirror appears to come from a point as far behind the mirror
THE NATURE OF LIGHT
507
as the real object, or source of the wave, is in front of the
mirror. See Figure 327.
Case II. a =00. If the incident wave is plane, then a = qo.
This case has already
Evidently if a = — , then
I
Therefore from Equation 124, 6 = —.
been fully discussed in Section 470
b = co. That is to
say, a spherical wave
originating at a point
in front of a concave
spherical mirror half
way from the mirror , ', ' '
to its center of cur-
vature, will be plane
after reflection. It
will also be evident
that a spherical wave
having its origin at
a point less than the
distance — from the
2
concave mirror, will after reflection have a curvature opposite
to that of mirror. This is indicated in the formula by the
negative sign which appears before the expression for b.
Case III. a = R. It a = R, then from Equation (124) we
have b = Ii= a. That is, a disturbance originating at the
center of curvature of the mirror will return after reflection to
the point from which it started. In other words, under these
circumstances, the image and object coincide.
A CONVEX WAVE REFLECTED BY A CONVEX MIRROR
474. When the curvature of an incident wave is opposite in
direction to that of the mirror, Equation (124) becomes
1-1 = 1
b a R
This will be evident from the following considerations: Let
MM, Figure 328, represent a convex mirror of radius R, hav-
508
LIGHT
ing its center of curvature at the left. Let W W be the
dotted position of an incident wave having its center of curva-
ture at the right. Then W" W"
is the reflected wa,ve, as deter-
mined by Huyghens' construc-
tion. In this case, evidently,
Id =103.
.-. ab = '2cd — ad
cd = ac + ad
. . ah = 2(ac + ad') — ad
ab = 2 ae + ad
h"=2h + h'
1 _ 1 _ 2^ (See equa-
b a~ B tion 123)
From the figure it is
evident that h" will always be
greater than A and the direction
of curvature of the reflected wave
the same as that of the mirror.
This explains the fact that a con-
vex mirror always gives a virtual
image.
The case discussed in the last
paragraph is represented again in
Figure 329, in which the incident
and reflected disturbances are
represented by rays. MM is the
convex mirror having its center of curvature at 0. is the
object or point from which light passes to the mirror. / is
the virtual image of 0, that is the center of curvature of the
reflected wave or point from which the reflected disturbances
apparently proceed.
Remark. One important assumption has been made in the
derivation of the above expressions ; namely, that the reflected
wave is spherical. This assumption is justified provided a lim-
ited portion of the wave only is considered.. In other words
these laws will be found to hold only in those cases in which the
Fig. 328.
THE NATURE OF LIGHT 509
Fig. 329.
width of the mirror is small as compared to its radius of curva-
ture. In Figures 326 and 328 the sagittas of the mirrors and
the waves are greatly exaggerated for the sake of clearness.
Problems
1. In an arrangement of Foucault's apparatus the distance between
mirrors is 5 km. What is the angular velocity of the rotating mirror
that gives a = 3'^?
2. Draw Huyghens' diagram for a plane wave reflected from a curved
mirror.
3. Show by diagram that the image formed by a plane mirror appears
to be as far behind the mirror as the object is in front.
4. If a wave after reflection at a plane mirror is to converge to a point,
what must be its form before reflection ?
5. Two mirrors are placed at an angle of 90^. A candle is placed
between them. Locate the images.
6. Plane waves falling upon a concave mirror are focused at a point
15 in. from the mirror. What is the radius of curvature of the mirror?
7. A luminous object stands 30 in. in front of a, concave mirror having
a radius of curvature of 35 in. What is the distance of the image from the
mirror?
510 LIGHT
8. A luminous object stands 20 cm. in front of a concave mirror. The
radius of curvature of the mirror is 8 cm. Determine by diagram the posi-
tion and size of the image.
9. Assume the curvature of the mirror in problem 8 to be reversed.
Determine position and size of the image.
10. A concave mirror has a radius of curvature of 50 cm. Determine
two pairs of conjugate foci.
REFRACTION
CHAPTER XL
THE BENDING OF A BEAM OF LIGHT
475. When a wave front passes from one medium into another
of different density, it is said to be refracted. The refraction of
a wave front is usually accompanied by a change in its direc-
tion. The simplest case of refraction is that of a plane wave
passing from one medium to another, the surface separating
the media being also plane. Consider the case represented in
Figure 330, in which MM' represents the interface or surface
"■-'W
Fig. 330.
separating the two media, let us say, air and water. MW
represents a plane wave approaching the surface as indicated.
As in the case of a reflected wave the first step in the applica-
tion of Huyghens' principle is to draw the dotted position of the
wave M'CW. This dotted position, it will be remembered, is
the position to which the wave would have gone had the second
511
512 LIGHT
medium, the water, been absent. In the presence of the water
the disturbance at M will not travel as far in the time under
consideration as it would have traveled in the same length of
time in air. Since the velocity of light in water is about three
fourths the velocity of light in air, evidently a disturbance
entering the water at M will have traveled only three fourths
as far as it would have traveled in air in the same length of
time. If, therefore, a circle is described about the point M
as a center with a radius equal to three fourths the distance
MW, it is evident that the arc will mark the distance to which
the disturbance has traveled in water while that part of the
wave which is at W travels to M' . In the same way, if the
center point B on the wave is considered, it is evident that
instead of traveling the distance BC, which it would have
traveled in air, it will travel but three fourths of the distance
BQ in water. Describing a circle about the point B with a
radius equal to three fourths BC, evidently this arc will mark
the distance to which the disturbance has traveled in the water
when the disturbance W comes to M'. In like manner the
disturbance in the water corresponding to any point on the
approaching wave is determined. The refracted wave is then
obtained by drawing the envelope of these several arcs.
It will be evident that the direction of motion of the wave
front has been changed by its passing into the second medium
as indicated by the large broken arrow. The amount of this
change in direction is determined as follows : Call the angle
between the approaching wave and the surface of the water,
that is, the angle WMM\ i. Call the angle between the re-
fracted wave front and the surface of the water, that is, the
angle MM' W", r. Then evidently,
WM'
sin I =
MM'
, . MW"
and sm r = ,^,^,
MM'
Dividing the first expression by the second, we obtain,
sin t ^ WM'
sin r MW"
KEFRACTION 513
WM
But — — ~7 is the ratio of the distance which light travels in air
MW"
to the distance which it travels in the same length of time in
water. Hence this ratio is equal to tlie ratio of the velocities
of light in the two media. In other words, this ratio is the
"index of refraction " of water as referred to air (Section 465).
The symbol /x is commonly used to represent the index of re-
fraction. We have, therefore, finally,
?HL1 = y, (125)
sin r
The angle i is usually known as the "angle of incidence."'
The angle r is called the "angle of refraction." It should be
noticed that when the disturbance is traveling, as in the above
case, from the rarer to the denser medium, that the wave front
becomes more nearly parallel to the interface after refraction.
It is evident that if the wave were traveling in the other direc-
tion so that WW" represented the wave approaching the inter-
face, then ilfTF would be the refracted wave. In this case the
approaching wave is more nearly parallel to the interface than
is the refracted wave.
THE SHALLOWING EFFECT IN WATER
476. An interesting, result of refraction is the shallowing
effect observed in water. Let MM^ Figure 331, represent the
surface of a shallow pond. Let represent a luminous object,
for example, a bright pebble lying upon the bottom. To an
eye placed above MM this luminous object appears nearer the
surface than it really is. This is known as the shallowing
effect and is explained in the following manner : Consider the
spherical waves which are proceeding towards the surface of
the water from the point 0. Let WW be the dotted position
of one of the wave fronts, that is to say, the position to which
the wave would have gone had it been traveling all the time in
water. Since, however, the central portion of the wave has
been traveling for a certain length of time in air, it will have
traveled to a position beyond the dotted position. The actual
position of the central portion of the wave when the edge
2i,
514
LIGHT
portions reach the surface is determined by a process analogous
to that used in the last section. If OB represents the distance
which the wave front would have traveled in water in a given
length of time, then |- of OB will be the distance which it has
traveled in air in the same length of time, since it travels in
air with | the
velocity with
which it travels
in water. If,
therefore, with
C as a center
a circle is de-
scribed having a
radius equal to
I OB, this arc
will measure the
distance to
which the dis-
turbance has ac-
tually traveled
in air. In the
same manner other arcs may be drawn about different points
on the line MM as centers, each having a radius equal to
I the perpendicular distance from that point to the dotted
position. The envelope of these several arcs determines the
position of the refracted wave WW. Evidently the curva-
ture of this wave has been increased by refraction. The direc-
tions in which the different parts of the refracted wave are
proceeding are repi-esented by the small arrows. This wave
therefore appears to come from some point 0' above 0. In other
words, the luminous object appears nearer the surface than it
really is. From the discussion of the relation between the
sagitta and the radius of curvature of an arc it will be evident
that O'Cis equal to | of 00. That is to say, since the sagitta
of the wave front has been increased in the ratio of 4 to 3, its
radius at the point B has been decreased in the ratio of 3 to i-
For points on the wave near M the change in curvature is
greater and the shallowing effect more marked.
Fig. 331.— The Shallowiag Effect in Water.
REFRACTION 515
TOTAL KEFLECTION
477. In general, when a wave front passes the surface sepa-
rating media of different density a part of the disturbance is
refracted and another part reflected. For example, in Fig-
ure 332, let a represent a ray of light falling upon the interface
MM separating media of different density. A part J of the
Fig. 332.
disturbance represented by the ray a is turned back into the
same medium. A second part c is refracted into the second
medium. Under certain conditions the refracted portion c is
absent, that is to say, the disturbance is totally reflected. The
conditions necessary to secure this effect are : First, the wave
must approach the interface from the side of the denser medium.
Second, the angle of incidence must be greater than a certain
angle known as the "critical angle." The value of this angle
depends upon the nature of the two media involved. In
Figure 383, let represent a luminous object located, let us
say, in water. Consider the rays proceeding from to the
surface of the water MM. Evidently the ray a which falls
vertically upon the surface MM will be transmitted without
change of direction to a'. The ray h will be refracted upon
entering the rarer medium and bent farther away from the
vertical. The ray o falling at a still greater angle upon the
interface MM will be bent still farther from the vertical. It
will be evident that a certain ray proceeding still farther to
the right, for example /, will fall upon the interface at such an
angle that the refracted ray will coincide with the surface
MM. The angle a. which the ray / makes with the perpen-
516
LIGHT
dicular to the surfiice MM is called the critical angle (evi-
dently this angle is the angle of incidence for that portion of
the wave front which travels in the direction Of). Any ray,
Fig. 333. —The Critical Angle.
for example ^, for which the angle of incidence is greater than
a will be totally reflected. It will be evident from the con-
struction used in this figure that total reflection is possible only
when the ray is proceeding from the denser
toward the rarer medium.
THE LUMINOUS FOUNTAIK
478. The luminous fountain affords an
interesting example of the effect of total
reflection. Figure 334 shows a simple
form of luminous fountain. Let AB
represent a water-tight tank, the
walls of which are opaque, ex-
cepting a small portion at i,
which is of glass. Let it
be assumed that water escapes ^ from this tank through
an orifice in a horizontal jet as \ indicated in the figure.
Let it be assumed that a powerful light is placed at the
point L beyond which is a concave mirror MM. This mirror
is so placed that the light which falls upon the mirror from
A
g^
----- ;33
Fig. 334. —The Luminous Fountain
REFRACTION
517
the source L is focused at the orifice 0. The rays of light
which enter the stream of water fall upon the surface of the jet
at angles which are greater than the critical angle. They are
therefore reflected back and forth within the jet, being unable
to escape into the rarer medium which surrounds it. Owing
to the small air bubbles and particles of foreign substance
with which the stream is filled, a certain amount of this light
is scattered (diffused), thus rendering the stream luminous.
That part of the light which is not scattered in this manner is
reflected back and forth until it reaches that portion of the jet
which breaks into drops.
THE CONVEX LENS
479. A simple convex lens is a piece of glass, one or both faces
of which are spherical, the lens being thicker at the center than
at the edges. Such a lens produces a modification in a wave
front which passes through it. The simplest case is that of
the plane wave. The effect of the simple convex lens upon a
plane wave is shown in Figure 335. LL is the lens and WW
w W
W
FiQ. 335.
the approaching plane wave front. Consider the wave at the
moment it reaches the position W W' . When in this position
the central portion of the wave is just entering the glass. The
edge portions are still traveling in air. It will be evident that
518 LIGHT
the edge portions will run ahead of the central portion, since the
velocity of light in glass is less than the velocity of light in air.
Consider the wave just as the central portion is emerging on
the opposite side from the glass. Call the thickness of the lens
at the center s and let the index of refraction of the glass as
referred to air be ijl. Then evidently the edge portions of the
wave will have traveled to WW" such that W W" is equal to
fjLS. Evidently the edge portions will be ahead of the central
portion by the distance yits — s = s(/i— 1). This distance is
the sagitta of the modified wave, but since the radius of
curvature of a circular arc is given by
in which i2 is the radius of the arc, d the chord, and h the
sagitta, we have, therefore,
d^
^ 8 s(^ - 1)
in which p is the radius of the modified wave.
Let it be assumed that the radius of curvature of one side of
the lens is R^ and the radius of curvature of the other side of
the lens B„. Then,
d^
' 8i?i
d^
and, §2 = g-^
2
in which Sj and s^ are the sagittas of the arcs forming the two
sides of the lens. But evidently,
S = Sj + Sj
^JL + JL
8 Jxj 8 M2
=iY— +—
8 \-Z2j -B2
Substituting this value of s in the above expression for p, we
obtain 1
p- fr—T ^''^^
i?i R.
REFRACTION
519
Evidently f represents the distance from the lens to the
point at which the wave W"W" is concentrated. This dis-
tance is known as the principal focal length of the lens. The
point at which the plane wave is concentrated after passing the
lens is known as the principal focus.
If one face of the lens is plane, its radius of curvature is
infinite, and — for this face is zero. The principal focal length
R
of such a lens is given by.
R
^ = (.-1)
(put R^= (X) in Equation 126).
(127)
THE CONCAVE LENS
480. The simple concave lens consists of a piece of glass one or
both faces of which are spherical, the center of the lens being
thinner than the edges. The effect of a concave lens upon a
plane wave front is shown in
Figure 336. Let WW represent
the approaching wave, and LL
the lens. Let it be assumed, for
the sake of simplicity, that the
thickness of the lens
at the center is 0. ,-''_.
Evidently as the
wave passes such a
lens its center portion
will get ahead of the edge por-
tions. The distance by which
the center portion is ahead of the
edge portions after the wave
passes the lens is given as before
by the expression s(/it — 1) where s represents the thickness
of the lens at the edge. The wave as it leaves the lens will
be convex, the curvature of the wave being given by the same
expression as that derived in the last section. This wave, after
passing the lens, appears to be proceeding from a point which
520 LIGHT
is the center of curvature of the modified wave. The distance of
the point from the center of the lens is called the principal
focal length of the lens and the point is known as the prin-
cipal focus.
CONJUGATE FOCI
481. Spherical waves proceeding from a point on the one side
of a convex lens are in general brought to a focus at a point on
the opposite side of the lens. The second point is known as the
conjugate of the first. The distance of the conjugate point
from the lens is determined as follows. In Figure 337 let LL
represent the convex lens, the source of light from which
L
0^-''
.■&''
the spherical waves proceed toward the lens LL, and / the con-
jugate of the point 0. Call the radius of the approaching
wave as it falls upon the lens, a, and the radius of the modified
wave front just as it leaves the lens, b. Call the sagitta of the
approaching wave h and the sagitta of the receding wave h.
Evidently the sum of these sagittas, that is, h + k, is equal to
the distance which the edge portions of the wave have gained
over the central portion in passing the lens. We have, there-
fore, A -h ^ = s(m - 1)
where s is the thickness of the lens at the center. Substituting
the value of h and k in terms of a and h and of s in terms of R^
and i?2 ^s above, we have,
ii + ii=(^-i)r-^ + ^l
REFRACTION
521
That is,
a b
\R^ R,
But the expression on the right is evidently equal to - (Equa-
P
tion 126). We have, therefore, finally,
a p
(128)
THE IMAGE FORMED BY A CONVEX LENS
482. The conjugate points and I, Figure 337, being on the
axis of the lens, their positions are best determined by the dis-
cussion given above. Points in the neighborhood of the point
0, but not on the axis, are found to have conjugate points in
the neighborhood of / on the opposite side of the axis. Con-
jugate points which are not on the axis are most readily
determined by the construction shown in Figure 338. Let L
represent a convex lens of which F and F' are the principal
foci, that is to say, F' is the point at which plane waves from
the left would be focused ; F is the point at which plane waves
from the right would be focused by the lens. Let 00' be a
luminous object. Consider the rays proceeding from the point
0. The directions in which three of these rays are traveling
after being refracted by the lens are readily determined. First,
that ray a which passes from toward the lens L parallel to
the axis, passes, after refraction, through the point F' . Second,
522 LIGHT
that ray b which passes directly towards the center of the lens
continues unchanged in direction after passing the lens. Third,
that ray c which passes through F will, after refraction, pass
parallel to the axis. It is seen that these three rays a, b, c are
brought together at the point /'. Experiment shows that all
other rays proceeding from the point to the lens L are brought
to a focus at the point I' . Thus, /' is the conjugate of 0. In
other words, an image of the point is formed at the point I'.
In like manner it may be shown that I is the image of 0'.
Also that each point lying between the points and 0' has a
corresponding conjugate or image-point lying between /and J'.
Thus, at II' is formed a complete image of the object 00'. It
should be noted that this image is inverted and real.
From the construction of Figure 338, it will be evident that
the size of the image is to the size of the object as the distance
of the image from the center of the lens is to the distance of
the object from the center of the lens. This is evident from the
fact that the triangles O'CO and /' Clare similar by construction.
Note. The construction used in Figure 838 assumes that
the lens is very thin, so that the rays a and c extend without
change in direction to the middle of the lens, and the rays a' and
c' extend without change in direction from the middle of the
lens to the point I. Also that the ray h passes through the
center of the lens without change in direction. The conditions
assumed are approximately realized in thin lenses.
THE IMAGE FORMED BY A CONCAVE LENS
483. The principle applied in the foregoing paragraph may
be used for determining the position and size of the image
formed by a concave lens as follows: Let L, Figure 339, be a
concave lens and F and F' its principal foci. Consider the rays
from a luminous object 00'. As in the preceding case, the
rays a, b, e, proceeding from the luminous point 0, may be deter-
mined in their refracted positions by remembering that the ray
a which passes in the direction parallel to the axis of the lens,
appears after passing the lens to come from the point F; that
the ray b which passes toward the center of the lens, continues
unchanged in direction after passing the lens; and the ray c which
REFRACTION
523
passes in the direction OF will be parallel to the axis of the lens
after refraction. The three rays a', b', and o' appear to come
from the same point I. This is the image of the point 0. In
the same manner it may be shown that all rays proceeding from
0' to the lens will, after passing the lens, appear to come from
the point I'. Points intermediate between and 0' have their
conjugate points between / and /'. Thus II' is an image of
the object 0'. It should be observed that this image II' is a
virtual image, that is to say, unlike the image formed by the
convex lens, it exists only in the sense that the image formed
by a plane mirror exists. It should be observed, further, that
it is an erect image and smaller than the object.
Problems
1. I{ the velocity of light is altered in passing from one medium to an-
other, does the frequency change? Does the wave length change?
2. The water in a certain vessel is 12 in. deep. What is its apparent
depth to an eye looking vertically down upon its surface ?
3. A lens has two convex spherical faces, the radius of one is 15 cm.,
that of the other, 20 cm. The index of refraction of the glass is 1.5. What
is the principal focal length of the lens ?
4. A luminous object is placed in front of the lens of problem 3, at a
distance of 100 cm. What is the position and relative size of the image?
524 LIGHT
5. What is the velocity of light in water ? /i = f .
6. A glass cube is placed on the bottom of a vessel filled with water.
The angle of incidence of a beam of light on the water is 60°. What is its
direction in the glass? Index of refraction of water = |, of glass = |.
7. Under what conditions does light travel in a curved line? Explain
how the sun is visible after it has passed below the horizon.
8. Show by diagram the path of a beam of light passing through a
glass prism submerged in CSj. Index of refraction of glass, f ; of CSj, 1.63.
9. The curved surface of a plano-convex lens has a radius of curvature
of 10 cm. j«. = |. What is its principal focal length when submerged in
water ? /t = |.
10. The radii of curvature of a biconvex lens are 20 and 30 cm. Its
principal focal length is 24 cm." What is the index of refraction of the
glass?
OPTICAL INSTRUMENTS
CHAPTER XLI
THE SIMPLE MICROSCOPE
484. In the discussion of the formation of an image by a
convex lens given in Section 482, it was assumed that the dis-
tance between the object and the lens was greater than the
principal focal length of the lens. If an object is placed between
a convex lens and its principal focus, the image of the object is
virtual and magnified by an amount which depends upon the
principal focal length of the lens as shown in the following dis-
cussion. In Figure 340 let L represent a simple convex lens
I
I
Fig. 340.
of which F and F' are the principal foci. Let the small arrow
represent a luminous object placed between the lens and its
principal focus F. Applying the principles employed in the
construction of Figure 338, the image / of the arrow point is
found to lie on the same side of the lens and at greater distance
from the lens than the point on the luminous object. Thus IF
is the magnified image of the luminous object. An eye placed
at A receives the rays a'ft' as if they were proceeding from /.
525
526 LIGHT
Thus the object appears magnified. A lens used in this way
is called a simple microscope. The magnification secured by
the use of the simple microscope is determined in the following
manner: The magnification may be defined as the ratio of the
apparent size of the image to the size of the object when placed
at the same distance from the eye. The normal eye sees ob-
jects most distinctly when at a distance of about 25 cm. In
using the simple microscope one unconsciously adjusts the
position of the lens with respect to the object until the image
is apparently at a distance of about 25 cm. from the eye. The
ratio of the size of the image under these circumstances to the
actual size of the object is the magnifying power of the instru-
ment. Referring to Figure 340, it will be evident from similar
triangles that the magnifiying power m is given by
size of image h
m = — — -=- = -
size of object a
where h is the distance of the image from the center of the lens
and a is the distance of the object from the center of the lens.
Now Equation 128 may be made applicable to this case provid-
ing h is regarded as a negative quantity, since, under the con-
ditions assumed in the development of this equation, b and a
were oppositely directed. We have, therefore, for the case of
the simple microscope,
1_1^1
a h p
- = -+1 = m
or multiplying through by b,
h^b_
a p
since - is the magnifying power as given above. But for distinct
a
vision, 6 =: 25 cm. We have therefore for m, the magnifying
power of the simple microscope,
m = ~+l (129)
P
OPTICAL INSTRUMENTS
527
THE COMPOUi^^D MICKOSCOPB
485. It is found impracticable to secure by means of the
simple microscope a magnification of more than about 100
diameters. In order to secure higher magnifying powers the
compound microscope is used. This instrument contains two
lenses. The one, called the object glass, represented by A in
Figure 341, forms a real and inverted image of the object 00' at
1 -'.--
■---—, F
Fig. 341.
IT . The other, 5, called the eyepiece, plays the part of a
simple microscope, giving a magnified image /"7"' of the image
produced by the first lens. The magnifying power of the com-
pound microscope is evidently equal to the magnifying power
of the eyepiece multiplied by the ratio of the size of IT to that
of 00'.
ITJ)_
00' a
But,
We may, therefore, write for the magnifying power of the com-
pound microscope
i»f=^(|+l) (130)
THE TELESCOPE
486. The telescope in its simplest form contains two lenses
arranged somewhat like those of the compound microscope. In
the telescope, however, the object glass is designed to give a
real image of small size of a distant object, wliereas in the mi-
croscope the object glass is designed to give a magnified image
528
LIGHT
of a near object. In other words, the object glass of the tele-
scope is a lens of long focal length, while the object glass of the
compound microscope is a lens of short focal length. The mag-
nifying power of a telescope is given by the ratio of the focal
length of the object glass to the focal length of the eyepiece.
This may be demonstrated as follows: In Figure 342, let A
Fig. 312.
represent the object glass and B the eyepiece of a telescope.
Let //' be the image formed by the object glass of the distant
object which it is supposed is being viewed through the tele-
scope. Let I"I"' be the magnified image of II', formed by the
eyepiece. The angle subtended by //' at the center of A is
evidently the angle which the distant object subtends at the
object glass of the telescope, or since the length of the telescope
may be neglected in comparison with the distance of the object,
this may be taken as the angle which the distant object subtends
at the ej'e. Call this angle /8. yS is thus the angular diameter
of the distant object as viewed by the naked eye. The angle
subtended by /"/"' at the center of B is the apparent angular
diameter of the object as viewed through the telescope. Call
this angle a. The magnification secured by the use of the in-
strument is therefore given by
m =
/8
Now if the object is at a great distance then II' is at a distance
from A which is practically equal to the principal focal length
OPTICAL INSTRUMENTS
529
of the object glass; and if the eye is adjusted for nearly parallel
rays, then II' lies for practical purposes at the principal focus
of B. But since the angular diameter of an object as seen
from a given point is inversely as its distance from that point,
therefore,
/3 p
where P is the principal focal length of the object glass and p
is the principal focal length of the eyepiece. Therefore,
P
V
m ■■
(131)
THE PROJECTION LANTERN
487. The projection lantern consists essentially of a source
of light and tvv^o lenses. One lens, called the condenser, is used
to secure a uniform and intense illumination of the object.
Fig. 343.
The second lens, called the projecting lens, is used to form a
magnified image of the brightly illuminated object. The
arrangement of the different parts is shown in Figure 343.
A is the condenser, which usually consists of two convex lenses,
as indicated in the diagram ; B is the projecting lens. 00 is
2m
530
LIGHT
the object, a magnified image of which is " projected " upon the
distant screen II'. If it is desired to increase or decrease the
distance of the image from the lantern, the position of the lens
B is changed. This changes the distance a of the object from
the lens, which of course results in a change of J, the distance
of the image from the lens, i is a brilliant source of light,
for example, an arc lamp.
THE PHOTOGRAPHIC CAMERA
488. In the photographic camera a lens is employed for
forming a sharp image of more or less distant objects upon a
photographic plate. The camera proper consists of a light
tight box, at one side of which is fixed the photographic plate.
Opposite this is the lens. Evidently, if the distance between
the lens and the photographic plate is fixed, the object of which
it is desired to form an image on the plate would necessarily
have to be at a fixed distance in front of the lens. It is con-
venient to be able to secure upon the plate sharp images of
objects at different distances. This is effected by attaching
the lens to a bellows, which makes it possible to alter the dis-
tance between the lens and the photographic plate, and at the
same time to prevent the entrance of extraneous light. The
essential parts of the camera are shown in Figure 344. L is
°r--^~.
O'L"'-
Fig. 344.
the lens, PP the photographic plate at the back of the camera.
The bellows enables the lens to be pushed forward or drawn
back, so as to secure upon the plate a sharp image of the object
to be photographed. This adjustment of the lens for securing
a sharp image upon the photographic plate is called focusing.
OPTICAL INSTRUMENTS 531
If a camera lens is of short focal length the distance between
lens and plate changes but slightly in focusing upon objects
at widely different distances. If such a camera is focused
upon an object at a distance say of 50 feet, it will give fairly
sharp images of all objects in the field of view whose distances
may vary from a few feet to infinity. A camera having a short
focus lens mounted at a fixed distance from the plate, is called
a universal focus camera.
THE EYE
489. Optically the eye is very much like the photographic
camera, the retina performing the same function in the eye that
the photographic plate does in the camera. That is to say, the
retina receives the image formed by the lens of the eye. Just
as in the camera it is found necessary to be able to change the
distance LP, Figure 344, in order that sharp images may be
secured of objects at various distances, so in the eye a similar
adjustment is necessary. The eye is focused, not by clianging
the distance between the lens and the retina, but by chang-
ing the thickness of the ej^e lens. When the eye is directed to
near-by objects, in which case, if the curvature of the lens were
constant, the image would tend to be formed too far from the
lens, the curvature of the lens is involuntarily increased ; that
is to say, its thickness at the center is made greater, so as to
bring upon the retina a sharp image of the near-by object.
When the eye is directed toward a distant object, the lens is
flattened so that the image of the distant object falls properly
upon the retina. This involuntary adjustment of the curvature
of the lens of the eye is called
accommodation.
Sometimes the accommoda-
tion of the eye is limited to
such an extent that it is im-
possible for the eye lens to form
a sharp image upon the ret ina.
(Til .. , . -LI • Fig. 345. — Nearsighted Eye.
ihus it may be impossible in
a given eye to flatten the lens sufficiently to make distant ob-
jects distinctly visible. An eye so affected is called a near-
Fig. 34fi. — Farsighted Eye.
532 LIGHT
sighted eye. In a nearsighted eye, the image of a distant object
is formed in front of the retina, as shown in Figure 3i5.
Nearsightedness may be corrected by the use of a concave lens.
Such a lens, placed in front of the eye, has the effect of in-
creasing the principal focal
length of the eye lens.
In another eye it may be
impossible to thicken the
lens sufficiently to make near-
by objects distinctly visible.
Such an eye is called far-
sighted. In a farsighted eye,
the image of a near-by object tends to form behind the retina,
as shown in Figure 346. Farsightedness may be corrected by
the use of a convex lens. Such a lens placed in front of the
eye has the effect of decreasing the principal focal length of the
eye lens.
Problems
1. What is the magnifying power of a simple lens used as a microscope,
the principal focal length of the lens being 2.5 cm. ?
2. Two lenses are used in combination as a compound microscope. The
focal length of the objective is 0.5 cm., that of the eyepiece 5 cm. If the
instrument is focused on an object 0.52 cm. from the objective, what is the
magnifying power of the microscope ?
3. What is the magnifying power of a telescope, the focal length of its
objective being 200 cm. and that of its eyepiece 5 cm., when focused on a
very distant object? Assume observer's eye to be adjusted for parallel rays.
4. The focal length of the objective of a projecting lantern is 15 in.
The lantern slide is 3 x 4 in. and the distance between the slide and the
screen upon which the image is to be formed is 40 ft. What is the size of
the image ? How far is the projecting lens from the slide ?
5. The focal length of a photographic lens is 10 in. If the camera is
focused sharply upon an object 6 ft. away, how far will the lens have to be
moved to give a clear image of an object whose distance is 100 ft. ?
DEFECTS OF MIRRORS AND LENSES
CHAPTER XLII
CHROMATIC ABERRATION
490. In discussing the effect of the simple lens in changing
the form of a light wave it has been assumed for the sake of
simplicity that all parts of the disturbance are equally affected
in passing through the lens. Now, white light is complex in
its nature, in that it consists of a large number of light waves
of different wave lengths, and experiment shows that these
component parts are not equally affected in passing through a
lens.
The effect of a lens in changing the curvature of a wave
front depends, as we have seen, upon the index of refraction of
the glass of which the lens is made. It is found by experiment
that the index of refraction of a given kind of glass is different
for light of different wave lengths, being greater for short waves
than for long ones. The shortest waves in white light are violet
waves. The red waves are the longest.
It follows, therefore, that when white light passes through a
convex lens, its component wave lengths tend to become
separated, the curvature of the shorter waves being more
affected than that of the longer waves. This effect is known
as chromatic aberration.
The effect described above is shown in Figure 347. Let A
be a convex lens receiving white light from a source on the
left. Because of the dispersive action of the lens A^ the violet
rays will be focused at some such point as Fi while the red
rays will be focused a point farther away such as H. If,
therefore, a screen is placed at V as represented by the dotted
line, there will be formed upon this screen a violet image of the
source surrounded by a red fringe. If the screen is placed at
533
534
LIGHT
R, there will be formed upon the screen a red image surrounded
by a violet fringe, the violet rays after passing the point V
having diverged so as to surround the point H.
A
Fig. 347. — Chromatic Aberration in a Convex Lens.
The effect of a concave lens upon converging rays is just the
reverse of that of a converging lens. For example, let £,
Figure 348, represent a concave lens. The converging rays a,
b are rendered less convergent upon passing the lens B, and as
in the case above, the effect of the lens upon the violet waves is
greater than upon the red waves, so that the red waves after
Fig. 348. — Chromatic Aberration in a Concave Lens.
passing the lens will be focused at some such point as R, the
violet waves at a point farther from the lens, the waves of the
intermediate colors falling between R and V.
It is possible b}"^ combining a convex lens with a concave
lens to bring the violet and the red waves to a common focus.
Lenses combined in this manner are made of different kinds of
glass. For example, one of the lenses may be made of crown
glass and the other of flint glass. A pair of lenses which brings
the violet and the red waves to the same focus, does not
DEFECTS OF MIRRORS AND LENSES
535
altogether prevent the dispersion of the other colors. Generally
speaking, however, it is possible by placing two lenses together
in this manner to render the combination sufficiently free from
dispersive action for most practical purposes.
SPHERICAL ABERRATION
491. A light wave after passing through a simple lens is not
quite spherical. Evidently such a wave will not be sharply
focused. In other words, a simple lens does not form a perfect
image of a luminous object. If such an image is examined, it
will be found to be " fuzzy " or blurred. This is due to the
fact that those rays of light which pass through the edges of the
lens are focused at a point nearer the lens than those which pass
Fig. 349. — Spherical Aberration.
through the center of the lens. This effect is distinct from and
independent of chromatic aberration. If a simple lens is used for
forming an image of an object from which light of one color
only is proceeding, for example, red light, evidently the effects
of chromatic aberration will be absent. Nevertheless, the
image will be defective, since those light waves which pass
through the edges of the lens are focused uearer the lens than
those which pass through the center. The effect is illustrated
in Figure 349. Let be a luminous object sending out waves
of light of one color only, let us say red. Those portions of a
wave which pass through the extreme edges of the lens will be
brought to a focus at some such point as A. Those which pass
through the lens near its center will be focused at a point
farther away from the lens, for examnle B. This is known as
spherical aberration. ^B
Various means are employed for reoucing spherical aberra-
536
LIGHT
tion. In photographic cameras " stops " are sometimes used
which cover the edges of the lens and confine the beam of
light to the central
portions. The spheri-
cal aberration of such
lenses is also reduced
by presenting the sur-
face of greater curva-
ture to the incident
light. It is also pos-
sible to grind the sur-
faces of a lens so that
they differ slightly
from the spherical
form and thereby, for
a given pair of conju-
gate focal distances,
eliminate spherical
aberration. A lens
corrected for spherical
aberration is called an
aplanatic lens.
Spherical aberra-
tion is also noticeably
present in a spherical mirror if the width of the mirror is com-
parable to its radius of curva-
ture. Figure 350 shows the rays
reflected from a wide spherical
mirror. The incident rays are
assumed to be parallel. It will
be seen from the figure that if
a limited portion, say CD, is
used, the reflected rays will all
be concentrated at F. Such a
mirror is contemplated in the
above discussions of concave
and convex mirrors. K^en a
larger portion of the mirror Fiq, 351. - Parabolic Mirror.
/f
/
'/
c/V
l%\
"^^
^> ^
^^^
\)//
dVv
\
^
Fig. 350. — Caustic Curve formed by a Spherical
Mirror.
DEFECTS OF MIRRORS AND LENSES 537
surface is used, the reflected rays are not all focused at F.
The majority of them fall behind F, as shown in the figure.
The effect of this crossing of the reflected rays is to form a
region of intensified brightness called a caustic. The caustic
curve is a curve drawn tangent to these crossing rays. It has
a cusp at F.
If the section of the mirror is a parabola instead of a circle,
the reflected rays are focused at one point. A parallel beam
of light falling upon such a mirror is concentrated at the
focus of the parabola. Conversely, light rays proceeding from
a source placed at the focus of a parabolic mirror are rendered
parallel after reflection. Figure 351.
ASTIGMATISM
492. A lens is said to possess astigmatism or to be astigmatic
when it is incapable of giving an image in which all lines that
pass through the center of the image are equally in focus. For
example, the horizontal lines of an image, formed by a lens
possessing this defect, may be sharply focused, while the vertical
lines of the image are blurred, or vice versa. This defect is
present in a lens the faces of which are not truly spherical, but
have different curvatures in different directions. Astigmatism
due to this cause is often found in the lens of the e3'e. The
simple lens does not possess this defect to any extent except
for rays entering the lens obliquely, and at a considerable angle
to the axis.
Astigmatism may be corrected by employing two lenses in
which the astigmatic effects are opposite. Astigmatism of the
eye is corrected by the use of a cylindrical spectacle lens so
placed before the eye that its convexity is, as it were, added
to that of the eye in the direction of its least curvature. A lens
corrected for astigmatism is called an anastigmatic lens.
DISTORTION
493. The image formed by a simple lens, of an object made
up of straight lines, is imperfect in that straight lines in the
object are reproduced as curved lines in the image. Let it be
imagined, for example, that the lens is used to form the image
538
LIGHT
of an object like that represented at A, Figure 352. Then the
image may be like the figure shown at B or at 0. This defect,
known as distortion, is due to the fact that the magnifying
power of a simple lens depends upon the angle at which the rays
Fig. 352.
A C
- Barrel and Pincushion Distortion.
enter the lens. When the magnifying power is less for the
oblique rays than for the direct rays, the distortion shown at JB,
called " barrel distortion," is the result. When the magnifying
power for the oblique rays is greater than for the direct rays,
the effect shown at O, called "pincushion distortion," results.
This defect of the simple lens is obviated by the use of two
lenses placed a short distance apart and on opposite sides of a
screen having an opening at its center. A lens corrected for
distortion is called a rectilinear lens.
CURVATUEE OF FlELD
494. When a simple lens is used to form an image of a plane
object, the plane of the object being perpendicular to the axis of
the lens, it is found that the images of the different points on
A B
Fig. 353. — Curvature of Field.
the object are not exactly in the same plane, the image of edge
points in the object being formed nearer the lens than the
images of central points on the object. This effect is illustrated
DEFECTS OF MIRRORS AND LENSES 539
in Figure 353, in which A is an object all points of which lie
in a plane perpendicular to the axis of the lens. L is a simple
lens which forms an image of the object at £. The image B
is curved in the manner shown. This effect is known as curva-
ture of field. This defect in a simple lens may be corrected by
combining it with a suitably proportioned concave lens, the
principal focal lengths of the two lenses of the combination
being so chosen that they give, when acting together, the desired
focal length. With such a combination it is possible to produce
a " flat field." Evidently this correction in a photographic lens
for reproducing drawings, etc., is of the highest importance.
DISPERSION
CHAPTER XLIII
THE PRISM
495. A prism is a piece of glass or other transparent medium
having a triangular cross section. When a beam of parallel
rays of light falls upon a prism, two effects are observed. First,
the beam as a whole is changed in direction. This is known as
deviation. Second, the rays after passing the prism diverge to
a certain extent and exhibit color. This effect is known as dis-
persion. The general results obtained in this experiment are
shown diagrammatically in Figure 354, in which ABC is sup-
FiG. 354. — Effect of a Prism upon a Beam of White Light.
posed to represent a prism of glass. A beam of parallel rays of
white light falls upon the face of the prism AB, as shown. It
will be found under these circumstances that the rays marked
B, 0, Y, Gr, B, I, V, are in color, red, orange, yellow, green,
blue, indigo, and violet, respectively. The experiment shows
among other things that white light is really a compound of a
number of different colors. One of the effects of the prism is
540
DISPERSION
541
to disperse the light waves corresponding to these different
colors. It will be observed that the violet rays are most
strongly deviated, while the red suffer the least deviation.
The angle marked a is the angle of deviation for the red rays ;
the angle ^ is the angle of dispersion for the red and violet
rays.
The deviation of a ray in passing through a prism is effected
in precisely the same manner as that of a ray passing through
the edge of a lens. The deviation of any given ray is deter-
FiG. 354 a.
mined by the index of refraction of the glass of which the prism
is made. But the deviation of the red ray is different from
that of the violet, since the index of refraction of the glass is
different for light waves of different colors. It has been
pointed out that light travels more slowly in glass than in air.
It is because of this fact that the direction in which the wave
is traveling changes as the wave passes from the air into the
glass. Evidently if the velocity of light in glass is a great
deal less than the velocity of light in air, the change in the
direction of propagation will be correspondingly great. It
thus appears that violet light travels less rapidly in glass than
red light does. In other words, the index of refraction of glass
is greater for violet than for red light.
The change in direction of a wave front as it enters the
prism, and again as it emerges from the prism, is shown in
Figure 354 a. (Compare Figure 283.)
542
LIGHT
THE SPECTRUM
496. The band of light with the succession of colors, red,
orange, yellow, etc., which appear in the above experiment
upon a screen held at iJFi constitutes what is known as a
spectrum. The red shades gradually into the orange, and the
orange gradually into the yellow, and so on, so that it is impos-
sible to distinguish where one color leaves off and another be-
FiQ. 355.
gins. In other words, there is between R and F'an infinite
number of colors. We may, however, distinguish the groups
red, orange, yellow, etc. enumerated above, and use their names
in referring to the different parts of the spectrum.
That the spectrum is due to the composite nature of white
light rather than to any transformation occurring in light
which passes the prism may be shown by recombining the
colors of the spectrum. When this is done, it is found that
the combination of the separate colors produces white. The
separate colors of the spectrum may be combined by means
of a lens as shown in Figure 355. P is a prism upon which
falls a beam of white light W. This beam is dispersed, form-
ing a spectrum at L. A
lens L placed as shown will
recombine the various colors
in the colored beam, form-
ing white light at W.
Another way of recombin-
ing the colors of the colored
beam is to use a second
prism, so placed with respect to the prism P that it tends to
deviate the beam in the opposite direction. This arrangement
Fig. 356.
DISPERSION 543
is shown in Figure 356. The white light TF which is dispersed
by the prism P is recombined by the prism P' . Under these
conditions W will be white light, and the direction in which it
is traveling will be parallel to that of W. It is here assumed
that P and P' are prisms of the same form and the same kind
of glass.
DEVIATION WITHOUT DISPERSION
497. The deviation produced by a prism, i.e. the angle a,
Figure 354, depends upon (a) the angle of incidence f, Figure
354 : (6) the refracting angle A, i.e. the prism angle opposite the
base of the prism ; and (c) the index of refraction of the glass of
which the prism is made. The dispersion produced by a prism
depends upon these same things, but the dispersion produced
by a given prism is not proportional to the deviation. It is,
therefore, possible by using two prisms made of different kinds
of glass and arranged as shown in Figure 356, to neutralize by
means of the second prism the dispersive action of the first
prism without entirely correcting its deviation. That is, with
such a combination of prisms it is possible to deviate a beam
of white light without dispersing it. Such a combination of
prisms is known as an achromatic combination. This is the
principle employed in correcting for chromatic aberration in
lenses (Section 490).
DISPERSION WITHOUT DEVIATION
498. Evidently from the statements made in the preceding
paragraph, it is equally possible to combine two prisms of dif-
ferent kinds of glass in
the manner indicated
in Figure 356 so as to
neutralize, by means of
the second prism, the
deviation produced by a ~ r"
the first without en- Fig. 357. — Dispersion Without Deviation.
tirely correcting for its
dispersion. A beam of white light passing through a combina-
tion of prisms like that shown in Figure 357 will be dispersed
544
LIGHT
without, as a -whole, suffering any deviation. A and G are
made of the same kind of glass. B is made of a different
glass, so chosen that it completely neutralizes the effects of A
and C so far as deviation is concerned.
CONDITIONS NECESSAET FOR THE PRODUCTION OP A PURE
SPECTRUM
499. As pointed out above, when a beam of parallel rays of
white light passes through a prism, a dispersion of the rays
of different wave lengths takes place. If the beam, after
passing the prism, is allowed to fall upon a screen, a spectrum
is formed. A spectrum formed in this manner is, in general,
impure, because of the overlapping of certain colors. This
Fig. 358.
effect will be understood from a consideration of Figure 858.
C is a source of light, F a prism; A and B are rays from
the upper and lower parts of the source. B, and T^are the
red and violet of the ray A and i?', and F"' the red and violet
of B. As indicated in the figure, V and IS fall together
upon the screen; in other words, there is an overlapping of
the spectra from A and B. It is evident, therefore, that for
producing a pure spectrum a narrow source of light must be
used.
THE SPECTROSCOPE
500. For the purpose of facilitating the study of spectra
the spectroscope is employed. The essential parts of the
spectroscope are shown in Figure 359. A narrow, adjustable
slit S, which is strongly illuminated by the light to be ex-
DISPERSION
545
amined, is used as the source. This slit is located at one
end of a closed tube AB. At the opposite end is a con-
verging lens. The length of the tube AB is equal to the
principal focal length of the lens B. Evidently, with this
FiQ. 359. —The Spectroscope.
arrangement, the rays of light which fall upon the prism P
will be parallel. The tube AB is called the collimator.
The rays, after passing the prism, are received upon the
the converging lens C, and form a pure spectrum at RV.
This spectrum is in reality formed of a series of images of
the slit S. Evidently at R there is a red image of the slit,
at J^ a violet image, and images in the other spectrum colors at
points intermediate. A simple microscope U is used for view-
ing this spectrum. The converging lens and the simple
microscope _Z?are contained in the same tube CB, which is called
the telescope.
When the spectroscope is provided with an arrangement for
measuring the angle of deviation it is called a spectrometer.
THE DIFFERENT KINDS OF SPECTEA
501. The Continuous Spectrum. — If a white-hot, solid body
is placed before the slit of the spectroscope, the spectrum
formed in the instrument will be a continuous one; that is
to say, the succession of images between R and V, Figure 359,
will be so complete that a continuous band of light is formed.
A continuous spectrum is also given by an incandescent liquid
or by an incandescent gas under very high pressure.
The Bright Line Spectrum. — The light given off by a gas
under low pressure, when heated to incandescence, forms in
2n
646 LIGHT.
the spectroscope a bright line spectrum; that is to say, if
such light is caused to illuminate the slit of the spectroscope,
the spectrum formed at i2Fi instead of being one continuous
band shading imperceptibly from the red through the orange
and the yellow, etc., on to the violet, will be found to be
discontinuous, there being in general but comparatively few
colored images of the slit present. For example, if the light
from incandescent sodium vapor is used, there will be but a
single line or image of the slit in the spectrum. This line
is in the orange-yellow. If the light used is that which is
given off by lithium vapor, there will be but two colored images
of the slit. Other incandescent gases have greater numbers
of lines in their spectra. It is found, however, that each
substance, when heated to incandescence in the vapor state, gives
a bright line spectrum which is characteristic of that substance.
This fact is taken advantage of in spectrum analysis, and con-
stitutes a very sensitive test for the presence or absence of a
substance in a given compound.
The Dark Line Spectrum. — If the slit of the spectroscope is
illuminated with sunlight, we obtain a,t RV, Figure 359, what
is known as the solar spectrum. It will be observed that the
solar spectrum is crossed by a great many dark lines. It is
as if the colors corresponding to the positions of these dark
lines were not present in the sunlight. This being the ex-
planation commonly accepted for the dark lines, it becomes
necessary to account for the absence of the particular wave
lengths corresponding. Undoubtedly the central portion of
the sun sends off light waves of all lengths, ranging from the
extreme red to the extreme violet. Since some of those wave
lengths do not reach the earth, we must conclude that they
have been lost or absorbed on the way. The central or
hot portion of the sun is surrounded by an atmosphere of
various gases at high temperatures. The light proceeding
from the central portion of the sun must pass through this
atmosphere on its way to the earth. In its passage through
the sun's atmosphere, as well as through the earth's atmos-
phere, the sun's light loses some of its wave lengths. It is
found that the light waves absorbed in this manner are those
DISPERSION
547
light waves which would be given off by the elements of the
sun's atmosphere and the earth's atmosphere as bright line
spectra if they were heated to incandescence.
That this absorption effect is sufficient to account for the
dark lines observed in the solar spectrum is easily demonstrated
by placing an incandescent solid before the slit of the spectro-
scope and securing all adjustments for a continuous spectrum.
If then incandescent sodium vapor is interposed between the
source and the slit, a dark line in the orange is immediately
observed. Finally, if the slit is screened from the source and
only the light from the sodium flame is allowed to fall upon it,
a bright orange-colored line will be found in the position
formerly occupied by the dark line.
fraunhofeb's lines
502. The dark lines in the solar spectrum were first observed
by Wollaston in 1802. They were studied by Fraunhofer in
1814. Fraunhofer called a number of the principal dark lines
RED
ORANGE
B
D
YELLOW GREEN BLUE
VIOLET
H
1 1 If 1 1 1 1 I 1 1 1 11 1 1 1 1 1 ? I I 1 1 1 1 1 1 1 11 M I 1 1 M'l I i r
•7 -6 .5 .-f
Wave Length in Thousandths of a Millimeter
Fig. 359 a, — Fraunhof er's Lines.
of the solar spectrum A, B, 0, D, etc., and his classification is
in use at the present day. The principal Fraunhofer lines are
given in Figure 359 a, together with the corresponding wave
length of light measured in thousandths of a millimeter.
Problems
1. A narrow beam of white light passes from air to water, the angle of
incidence being 60°. Is there a tendency for the components to become
separated ? Explain.
2. The index of refraction of CS2 for red (C) is 1.6336, for yellow (D)
is 1.6433, for blue (F) is 1.6688. What are the yelocities of red, yellow,
and blue light in CS2?
548 LIGHT
3. The index of refraction of water for red (C) is 1.3318, for blue (F)
is 1.3377. What is the difference in cm. /sec. between the velocities of red
and blue light in water ?
4. The index of refraction of a certain kind of glass for red (C) is
1..5826, for yellow (B) is 1..5867, and for blue (F) is 1.5967. A plano-con-
vex lens is made of this glass. iJ = 50 cm. What is the principle focal
length of this lens for red? for yellow? for blue?
• INTERFERENCE
CHAPTER XLIV
CONDITIONS UNDER WHICH LIGHT WAVES INTERFERE
503. In discussing the subject of interference of wave trains
(Section 432) it was pointed out that two sound waves may be
so related as to completely destroy one another so that their
combined effect would be silence, and that two water waves
passing over the same surface may be so related as to leave the
surface under their combined influence undisturbed. If light
is really of the nature of a wave disturbance, it ought to be
possible by combining two light waves, properly related, to
produce darkness. This is found to be the case. In order that
this effect may be brought about, it is only necessary to have
two light waves of the same wave length traveling in the same
direction, the one wave being half a wave length behind the
other so that the crests of one wave train will fall opposite the
troughs of the other. Attempts to secure interference between
waves of light from two different sources are unsuccessful, the
explanation being that the phase of the disturbance proceeding
from any source is continually changing, so that if at a given
instant the phase relation existing between the two trains is
such as to cause interference, a very short interval of time
later they may be so related as to add their effects. It is
therefore necessary, in making a study of this phenomenon, to
secure two beams of light from the same source. One of the
simplest devices for securing this is that due to Fresnel.
frbsnel's bipeism
504. For the purpose of securing interference effects Fresnel
employed what is known as a biprism. This is a double prism,
the two refracting angles of which, A and B, Figure 360, which
549
550 LIGHT
are small, are turned away from one another. Imagine a source
of light, for example a slit, placed at S and brightly illuminated
by monochromatic light, tliat is, by light of but one wave length
like that given off by incandescent sodium vapor. The light
will reach the point P on the right of the prism along two
paths as indicated. The one ray passing toward the side A is
refracted by the prism A. The other ray is refracted by the
P"
P'
P
P'
B
Fig. 360. — Interference produced by Fresnel's Biprism.
prism £. Since the two paths SP are equal, it will be evident
that the waves coming to the point P along the two paths will
be in the same phase, that is to say, crest will correspond with
crest and trough with trough so that the two disturbances will
be added at this point. Consider, however, a point such as P'
a short distance above or below the point P- Rays of light
will reach this point P' from the source S along two paths.
It will be evident, however, from the construction of the figure
that the two paths SP' are unequal in length. Let it be assumed
that the point J" is so situated that the difference in the
lengths of these two paths is half a wave length of the light
proceeding from S. Then the wave train proceeding to the
P' along the shorter path will get ahead of that which travels
along the longer path by half a wave length, so that of the two
wave trains arriving at P' the crests of one will fall opposite
the troughs of the other. The result is that the two wave
trains are in condition to interfere at the point P', and the
effect of one of the wave trains is destroyed by that of the
other. Hence there is darkness at the point P'. Again, con-
INTERFERENCE 551
sider a point P" a short distance farther from the central point
P such that the distances SP" measured along the two paths
as before differ by one whole wave length. For this point,
since the one wave train gains over the other a complete wave
length, evidently the waves will be in a condition to add their
effects together at this point and P" will be a point of maximum
illumination. If P'" is so located that the two paths SP'" dif-
3 X
fer in length by -— (\ = the wave length) then P'" will be a
Jt
region of interference and darkness. If P'^' is so located that
the two paths AP'^ differ in length by -— , P'^ will be a region
of brightness.
The general statement covering all points on the screen CD
is as follows : Let the difference in length of the two paths be
d, and put ->
d=n-'^ (132)
where \ is the wave length of the light under consideration and
n is any wliole number odd or even. When n is odd, interfer-
ence effects will be present. When n is an even number, inter-
ference effects will be absent.
THE COLORS OF THIN PLATES
505. Brilliant color effects are observed in very thin plates
or layers of transparent media ; for example, in films of oil on
water, in thin layers of oxide on polished metal, in soap bubbles,
etc. These colors are the result of the interference of light
waves. The brilliant colors sent to the eye from the soap bub-
ble, for example, are white light minus one or more of its colors
which have been destroyed by the interference effect in the
film. The manner in which this interference effect takes place
will be understood from the following discussion: Let AB,
Figure 361, represent a thin film upon which a beam of white
light a is falling at the angle indicated. This beam of light is
broken up at m, one part b being reflected, a second part c being
refracted and passed on to the point n at the opposite face of the
film. At this point the ray e is broken into two parts, the one
being reflected to / and the other d passing into the surround-
552
LIGHT
Fig. 361.
-Interference in a Thin
Plate.
ing medium below. That component which is reflected back
to the point /is again divided, a part e being refracted, a sec-
ond part being reflected, and so on.
The rays e and b are parallel, and
if the film AB is very thin, they
proceed practically from the same
point. Evidently the ray e has
Q traveled over a longer path than
the ray b and if in traversing this
greater distance it has fallen behind
an odd number of half wave lengths
it will be in a condition to interfere
with the ray b.
The fact that the ray e falls be-
hind the ray b is sufficient to
account for the interference colors
observed in thin plates, but various
accompanying phenomena indicate that this explanation is
not complete. For example, if with tliis explanation in mind
we imagine that the plate or film AB is made extremely
thin, then the interference effect should disappear since under
these circumstances the ray e would not fall appreciably behind
the ray b. The fact is, however, that the interference effect is
very marked for an extremely thin film. Such a film appears
black by reflected light. It will therefore be evident that in
the mere process of reflection there is a loss of half a wave length
by one of these trains of waves. This is explained in the fol-
lowing manner : It will be noted that the ray b has been re-
flected in the rarer medium, while the ray e has been reflected in
the denser. Now it is not a difficult matter to show that when
a wave is reflected at an interface on the side of the rarer
medium, it suffers a change of phase of half a wave length,
while if it is reflected on the side of the denser medium, no
such change of phase is brought about. Thus the ray b loses
(or gains) half a wave length in the process of reflection at m.
Therefore in order that interference effects may take place, e
must lose in virtue of its greater path an even number of half
wave lengths.
INTERFERENCE
553
Let it be assumed that the film AB is of such thickness that
the conditions for interference of violet waves are present.
Then the eye placed at E will receive by reflection from the
film white light minus violet light. The film at this point will
therefore appear to be brilliantly colored.
A film slightly thicker than the one discussed above would
cut out the blue light by interference, a still thicker film would
extinguish the green, and so on.
DIFPEACTION GRATING
506. An important experiment demonstrating the wave
nature of light is the formation of spectra by what is known
as a diffraction grating. A diffraction grating consists of a
Fig. 362. — Diffraction Grating.
large number of very narrow parallel openings placed close
together. One of the simplest ways in which a grating may
be formed is by ruling lines with a diamond point upon a glass
plate. The scratches produced in this manner may be regarded
as opaque. The spaces left between are the portions which
transmit the light. Let the broken line AB, Figure 362, rep-
554 LIGHT
resent such a grating. The dashes, let us say, correspond to
the opaque portions, while the spaces between the dashes rep-
resent the portions which transmit light. Let it be imagined
that plane waves parallel to the grating are falling upon it
from the left as indicated in the figure. Each of the openings
a, h, c, d, etc., will constitute a separate source of disturbance
so far as the medium on the right of the grating is concerned.
Let it be imagined then that trains of secondary wavelets have
been for some time proceeding toward the right from these
several secondary sources. Consider the condition of affairs
at the moment a crest of the wave is just passing the opening
a. Evidently there is also a crest of a wave at the point h and
a second crest at a distance \ from the point b. Let it be
imagined that a line is drawn from the center point of a, tan-
gent to the crest which is at a distance X from h. This line
will also be tangent to a crest which has spread from c to a
distance 2 X and to a crest which has proceeded from d to a,
distance 3 \, and so on. In other words, it will be evident that
the secondary wavelets proceeding from the sources a, 6, c, S'j, is caused to rotate rapidly, the eye at B
is enabled to compare the illumination of the screen 8 by
the source B, and that of »S'j by the source A. When these
screens are unequally illuminated, the light which reaches the
eye will appear to flicker. When the illuminations are equal,
the flicker disappears, and in this way it is known when the
the adjustment is reached.
It is found that the flicker photometer is especially useful
PHOTOMETRY 565
in comparing the candle powers of two sources of different
color. It is impossible to make accurate settings on either
the Lummer-Brodhun or the Bunsen photometer when the
color of the light falling upon one side of the screen is
different from that falling upon the other. In the use of
the flicker photometer it is found that the color flicker is
distinct from the intensity flicker, and that by properly choos-
ing the speed of the disk the color flicker may be caused to
disappear while the intensity flicker still remains. When this
speed has been secured, the screen may be moved to and fro
to secure a balance of intensities, exactly as in the use of the
Bunsen or Lummer-Brodhun screen. The effect upon the eye
of the color difference is thus avoided.
ILLUMINATION
516. The intensitj^ of illumination of any surface is defined
as the ratio of the light received by the surface to the area of
the surface upon which the light falls. A unit of intensity which
is oftentimes employed is known as the foot candle, and is de-
fined as the intensity of illumination which would be present upon
a screen placed at a distance of one foot from a standard candle.
The meter candle is a unit of intensity which is employed to
some extent.
The table below gives a number of values of illumination
such as are commonly observed, the intensity of illumination
being expressed in foot candles.
Suitable for drafting table .
. 5 to 10
Suitable for library table
3 to 4
Suitable for reading table
1 to 2
Required for street lighting .
. 0.0.5 to 0.60
IVTonnliclit ('full nioon^
. 0.025 to 0.03
JjH.\JKJ^11^Uv ixu.11 iLiyjyjLii . .
Probleir
B
1. What is the intensity of illumination at a distance of 4 ft. from a 16
candle power lamp ?
2. At what distance from a 32 candle power lamp is the intensity of
illumination 1 foot candle ?
566 LIGHT
3. A photometer is in adjustment with a standard 16 candle power lamp
at a distance of 1 m. and an unknown source at 80 cm. from the screen.
What is the candle power of the unknown source ?
4. Two sources of light of 16 and 48 candle power respectively are
placed 16 ft. apart. At what point will the illuminations produced by them
be equal ?
5. What intensity of illumination is produced by each source at the
point determined in problem 4?
6. At what distance from an arc lamp of 3000 candle power is the intens-
ity of illumination 3 foot candles? ,
COLOR
CHAPTER XL VI
THE ORIGIN OF COLOR
517. We have seen that colors may be produced by refrac-
tion, as exhibited in the prism, or by interference, as exempli-
fied by the soap film. In nature many color effects are pro-
duced in this manner. There are also certain bodies to be
found in nature which exhibit marked color, whose colors are
not to be explained as due to either of these causes. Such
bodies appear colored, because of absorption effects which take
place in them. That is to say, the surfaces of some bodies ap-
pear to possess the property of reflecting certain colors readily,
while other colors are more or less completely absorbed. This
property is known as selective absorption. We may therefore
say that, in general, there are three principal modes of color
production ; namely, by refraction, by interference, and by
selective absorption.
THE COLOR OF TRANSPARENT BODIES
518. If a transparent body transmits with equal readiness all
of the various colored components of white light, the body is
colorless. If, however, it transmits any part of the spectrum
more readily than another, the transparent body will appear
colored when seen by transmitted light. The color which it
exhibits is a mixture of those colors which it transmits. This
color is evidently white light minus those colors which have
been absorbed. Thus, a piece of red glass, when held before
the eye, appears red, not because the glass changes the light
which passes through it in any way, but because it has sifted
out of the white light which falls upon it certain of its com-
ponent wave lengths, and has allowed only the red light to pass
567
568 LIGHT
freely through it. A piece of blue glass appears blue because
it absorbs the red and the yellow, and allows only the blue to
pass, or, at least, allows the blue to pass most readily.
The color exhibited by two transparent objects when so
placed that a beam of white light is allowed to traverse both of
them in succession is evidently determined by those colors
or wave lengths which pass through both bodies. If, for ex-
ample, a solution of copper sulphate in a narrow vessel is placed
before the slit of a spectroscope, and white light is used, the
spectrum will be observed to consist of green together with
some of the more refrangible colors, that is to say, colors of
shorter wave length, the red and yellow having been completely
absorbed. If a similar vessel filled with a solution of potassium
bichromate is employed, the spectrum observed will consist of
the green together with the longer wave lengths, yellow and
orange, while the shorter wave lengths, blue and violet, will be
entirely absent. If, now, both solutions are placed before the
slit of the spectroscope, then, evidently, the only color found in
the spectrum will be green, since green is the only color which
is transmitted by both solutions. This will be evident from the
following table, in which the colors absorbed by each solution
are indicated by underscoring the corresponding letters :
Copper Sulphate Solution . . . R Y a B V
Potassium Bichromate Solution . R Y Gr B V
In this example, therefore, the color of these two solutions, as
exhibited by transmitted light, is green, while the color of the
copper sulphate solution alone is blue, and that of the potassium
bichromate solution is yellow. The effect is evidently a differ-
ential one.
THE COLOR OF OPAQUE BODIES
519. When white light falls upon the surface of an opaque
body it is, generally speaking, diffusely reflected. If the sur-
face of the body is of such nature that it reflects with equal
facility all of the various wave lengths which enter into the
composition of white light, the body will appear in this reflected
light, white in color. A sheet of white paper or a white screen
reflects equally well all of the various colors of the spectrum.
COLOR 569
and hence, when placed in white light, it appears white. If
the surface of the opaque body is of such nature that it reflects
the waves corresponding to one part of the spectrum more read-
ily than those corresponding to the other parts of the spectrum,
its color will be something other than white. Suppose, for ex-
ample, that the surface is of such nature that everything but
the red light is reflected. Then the color of the surface, under
these circumstances, will be white minus red. It is the color
that would be obtained by combining all of the colors of the
spectrum excepting red. Such a color is said to be comple-
mentary to red. Complementary colors are colors which, com-
bined, will give the effect of white.
Most objects absorb from the white light which falls upon
them certain wave lengths in larger proportion than others.
Such objects exhibit what is known as body color. Body color
is due to the same effect which gives rise to color of transparent
bodies, or colors by transmitted light, since it is found that the
white light incident upon such bodies penetrates to a certain
depth into the surface layers, is then irregularly reflected and
again traverses the surface layer. In thus passing twice through
the superficial layers of the body, the same absorption effect
upon the white light takes place as that which accompanies the
transmission of light through transparent bodies. Thus, when
a building is painted red, its surface is covered with a pigment
which possesses the property of reflecting red light and absorb-
ing in large measure the other colors of the spectrum. An
opaque body which appears green is one which reflects the green
light most readily. Such a body probably absorbs all of the
red and the violet which falls upon it.
MIXING PIGMENTS
520. Since the body color of an opaque object is determined
by this absorption effect, it is not difficult to predict the effect
of spreading two pigments of different color upon the same
surface. If, for example, the colors of the pigments chosen are
yellow and blue, the resultant will be green, since the experiment
would be essentially the same as that of placing the blue and
yellow solutions before the slit of the spectroscope (Section 518).
570
LIGHT
MIXING COLORED LIGHTS
521. It has been seen that in the mixing of pigments a color
effect is obtained which is differential and is determined by those
wave lengths which are transmitted by both pigments. In the
mixing of colored lights the result is very different, since in
this, case the effect is a summation of the effects due to the
colored lights individually.
There are various ways in which colored lights may be mixed
for making a study of this kind. One method employed for
this purpose is to form a spectrum of the light given off by a
white-hot body, thus securing a continuous spectrum. The
colors of this spectrum may be recombined, as pointed out
Fig. 370. — Apparatus for mixing Colored Lights.
above (Section 496), to form white light. By the use of suit-
able screens placed in the plane in which the spectrum is formed,
it will be possible to remove from the beam of light such colors
as are not desired in the experiment, leaving only those which
it is desired to place in combination. These may be recombined
by the use of a suitable lens. For example, in Figure 370, let
P represent the prism of a spectroscope, L the lens upon which
the light falls after passing the prism, R V the position of the
spectrum formed by the instrument. L^ is a lens by means of
which the colors of the spectrum may be recombined forming a
bright spot of light on the screen S. If, now, an opaque ob-
stacle A is placed so as to receive a part of the spectrum, the
wave lengths corresponding will be obstructed in their passage
toward the right. Therefore, a portion of the spectrum only
will pass to the lens L^ and be combined at S. The result is a
COLOR 571
colored spot of light at S which is the sum of all those colors
which reach the screen. If, for example, the obstacle is of such
width and placed in such position that it obstructs all but the
red and the violet, the resultant color effect on the screen S
will be that due to the addition of red and violet (purple).
Another method for studying the effects of combining various
colored lights is by the use of colored disks of paper. As em-
ployed for this purpose the disks are slit radially so that they
may be placed together and the amount of each disk exposed
varied at will. When such a combination of two disks is caused
to rotate rapidly while illuminated with white light, the effect
upon the eye is the same as that secured by mixing two beams
of light, the colors of which correspond to the colors of the
disks. Let it be imagined, for example, that blue and yellow
disks are employed, the adjustment being such that one half of
each disk is exposed. When this combination of two disks is
rotated rapidly in white light, the result (white) is the same as
that secured by combining, by the process described above,
beams of blue and yellow light. This result is due to what is
known as persistence of vision, i.e. the retention of an impression
by the retina of the eye for a certain length of time after the
stimulus (light) has been removed. Thus, as the disks rotate,
alternate flashes of blue and yellow light reach the eye. The
corresponding impressions persist, and in effect the blue and
yellow are added.
PRIMARY COLORS
522. It is customary to call the colors violet, indigo, blue,
green, yellow, orange, and red, primary colors, since all parts of
the spectrum are thereby included and are described in terms
with which we are all familiar. The term " primary," however,
as used in this connection has but little significance. It is sup-
posed that the presence of each primary color is necessary to the
production of white light. But it is very easily demonstrated
that white light may be secured by the combination of three or
even but two of the spectrum colors.
Because of certain phenomena which manifest themselves in
the study of color vision it is thought that there are three
primary color sensations, namely, green, blue, and red. For
572 LIGHT
this reason these colors are often spoken of as primary colors.
The study of pigments has led to the conclusion that the three
primary colors proper are red and yellow and blue, since a pig-
ment of any one of these colors is found to absorb all of the
light transmitted by the other two. Again, it is found possible
to match any color by combining any three spectrum colors
providing they are somewhat separated in the spectrum. In
this sense, therefore, there is a number of groups of primary
colors.
COMPLEMENTARY COLORS
523. If, with the arrangement of apparatus represented in
Figure 370, an opaque obstacle is placed at i2 F in such position
as to intercept the red light only, the resultant color upon the
screen ;S' will be complementary to red, that is to say, it is that
color which combined with red will produce white light. It is
thus apparent that it is possible to combine two colors and secure
as a result white light. The experiment also shows that com-
plementary colors are not necessarily simple colors, that is to say,
they do not necessarily consist of a single wave length only, but
each of the two complementary colors may be a compound of
several wave lengths. Foi> example, if the opaque obstacle in
the experiment referred to is placed so as to intercept the yellow
and all of the longer wave lengths, a certain color will result
at *S', which is of course a compound of violet, blue, and green.
If, now, the obstacle is shifted in position so as to intercept the
green and all of the shorter wave lengths, the color intercepted
upon the screen will be complementary to that obtained in the
first experiment, and will be a mixture of yellow, orange, and
red.
THE CHARACTERISTICS OF A COLOR
524. For the complete description of any color three things
must be stated:
1. Hue.
2. Saturation.
8. Luminosity.
The hue of a color is a specification of the wave length of the
color, for example, red, orange, blue, etc.
COLOR
573
The saturation of a color is a specification of the amount of
white light it contains. If a beam of red light is allowed to
fall upon a screen which is already illuminated with white
light, the red which appears upon the screen is non-saturated,
that is to say, it is red plus a certain amount of white. A color
is said to be saturated when it is free from the admixture of
white light. The pure spectrum colors are examples of saturated
colors.
The luminosity of a color is a specification of its brightness.
If the spectrum formed by a prism is allowed to fall upon a
printed page, it will be observed that the portion illuminated
by yellow is much more easily read than the other portions.
/
V B G Y R
Fig. 371. — Luminosity Curve.
This is expressed by saying that the yellow is the most lumi-
nous of all the colors of the spectrum. In the same sense
violet is the least luminous. The ordinates of the curve shown
in Figure 371 represent the relative luminosities of the corre-
sponding spectrum colors.
NON-SPECTEAL COLORS
525. Among the more prominent colors aside from those
found in the spectrum of white light are the following : purple,
which consists of a mixture of violet and red ; magenta, which
consists of a mixture of blue and red; and brown, which is a red
or a yellow of low luminosity. These three colors are saturated
colors. As examples of non-saturated colors might be men-
tioned pink, lavender, etc.
574
LIGHT
maxwell's color diagram
526. Maxwell's color diagram affords a convenient means of
specifying a given color in terms of its components and its
saturation. This diagram is represented in Figure 372. It
consists of a tri-
angle, the corners
of which are sup-
posed to represent
the colors, red,
blue, and green,
as indicated by the
letters. Since in
the spectrum or-
ange and yellow
are found between
red and green,
these two colors
will be represented
by points on the
line RQ-. Since
Y YG
Fig. 372. —Maxwell's Color Triangle.
the green in the spectrum gradually shades into blue, a point
midway between B and Q- might be called blue-green. Violet
is to be found on the line BR, but close to B. Purple and
magenta are found near the middle of this line BR since they
consist of combinations of red and violet, and red and blue,
as pointed out above. The point W corresponds to white, the
various colors of the spectrum being arranged symmetrically
about it.
A saturated red is represented, of course, by the corner of
the triangle. A non-saturated red, that is to say, a red having
an admixture of white, would be represented by a point between
i2 and TFi If close to R, the red is nearly saturated. If close
to W, it is nearly white.
The complementary colors of this diagram are obtained by
drawing lines through W. For example, yellow and blue are
complementary, yellow-green and purple, orange and green-blue,
red and blue-green.
COLOR 575
THE DEPENDENCE OP BODY COLOE UPON THE CHARACTER
OP THE INCIDENT LIGHT
527. We have seen that the body color of aily object is clue
to the fact that it reflects certain wave lengths readily, while
others are more or less completely absorbed. It is therefore
evident that an object can show its true body color only when the
light in which it is viewed contains those wave lengths which it
most readily reflects. Suppose, for example, that the body color
of an object is red, and that it absorbs from white light all of
the various wave lengths except red. Such an object will ap-
pear black when illuminated by any of the spectrum colors except
red. In white light it will show its true body color, since white
light contains red. It will be seen, therefore, that in white light
all objects exhibit their true body colors, and white light is the
only kind of illumination of which this is true.
This is of great importance in comparing the various sources
of light used in artificial illumination. For example, the gas
flame and in some cases the incandescent lamp give a light
which is decidedly yellowish. The mercury vapor lamp gives
a greenish light. The flaming arc gives a reddish light, and so
on. Evidently from the principle stated above neither of these
forms of light is capable of exhibiting all objects in their true
body color.
YOUNG-HELMHOLTZ THEORY
528. Various theories have been advanced to account for
color perception and the various characteristic phenomena
related thereto. The theory which is most commonly accepted
at the present day is known as the Young-Helmholtz theory.
This theory may be briefly outlined as follows : It is assumed
that there exists in the retina of the eye three sets of nerve
terminals. One of these sets is particularly sensitive to red
light, and the corresponding nerve terminals are usually re-
ferred to as the red nerve terminals. The second set is par-
ticularly sensitive to blue light, and the nerve terminals of this
set are called the blue nerve terminals. The third set, consist-
ing of the green nerve terminals, are especially sensitive to
green light. It has been determined that each of the three
576
LIGHT
nerve terminals is affected by any color of the spectrum. The
name " blue nerve terminal " is not to be understood as mean-
ing that the corresponding nerve terminal is sensitive to blue
alone, but that it is more sensitive to blue than to any other
color. It is fairly sensitive to those colors of the spectrum
which lie in the neighborhood of the blue, for example, the
green and the violet. The blue nerve terminal is least sensi-
/
/red
--f
GRE
EN
rii:
BLUE
R
Y
G
Fig. 373.
V
five to those colors which are the farthest removed from the
blue in the spectrum, that is to say, the red. The curves
shown in Figure 373 are drawn to indicate the sensitiveness of
each of the three nerve terminals to the various colors of the
spectrum. For example, the curves show that orange light
affects the red terminals very strongly and produces consider-
able effect upon the green terminals, while upon the blue termi-
nals its effect is almost negligible.
SUBJECTIVE COLORS
529. The Young-Helmholtz theory enables us to explain
satisfactorily most of the phenomena of color vision. One of
the most important of these is the development of color by
COLOR 577
what is called after effect, or the phenomenon of subjective colors.
This effect will be understood from the following simple experi-
ment: If a brilliantly colored object, let us say, a blue card, is
placed against a gray background and strongly illuminated
with white light, and the eyes are directed toward the card for
a few (about 20) seconds and then turned aside to a gray wall,
there will immediately appear in the field of vision an image of
the card colored yellow. If a bluish-green card is employed,
the after image will be red in color, that is to say, the colors of
the object and the after image are complementary. The expla-
nation of the phenomenon under the Young-Helmholtz theory
is as follows : When-the eye is directed steadily toward a blue
object, the blue nerve terminals gradually become fatigued. If,
after this effect has set in, the eye is turned to a gray object,
which in the normal condition of the eye would affect all three
nerve terminals equally, the effect upon the red and green
nerve terminals will predominate. In other words, the eye
will perceive gray (or white) minus the blue, that is to say, the
color which is complementary to the blue, namely, yellow.
COLOE BLINDNESS
530. In the normal eye, it is possible, as we have seen, to
produce any color sensation by combining the effects of what
we might call the three primary sensations, namely, red, green
and blue. For the normal eye it is therefore possible to match
any color by combining red, green, and blue. In matching
colors in this way it would of course be necessary to have the
luminosity of each of the primary colors under control.
For certain eyes it is found possible to match every color per-
ceived by combining green and blue. Such an eye is said to be
red-blind. In other cases it is found possible to match all
colors for a given eye by combining red and blue. Such an
eye is said to be green-blind. Under the Young-Helmholtz
theory color blindness is explained by assuming that in the color-
blind eye one of the three sets of nerve terminals described above
is either entirely wanting or much less sensitive than the others.
A study of Figure 373 will enable us to determine in a gen-
eral way how various colors would appeal to the red-blind eye.
2p
578 LIGHT
For such an eye the upper curve would be lacking, and evi-
dently if the green and the blue are the only sensations possi-
ble, then when these two sets of terminals are equally excited,
the result will be white or gray just as the normal eye receives
the impression of white when all three nerve terminals are
affected equally. It will therefore be evident that for the red-
blind eye that portion of the spectrum which lies about midway
between the green and the blue will appear white, since the
ordinates of the green and blue curves are equal in this region.
Colors lying near this region will evidently be very pale, since
they have in effect a large amount of white mixed with them.
The peculiarities of color vision for the green-blind eye may be
determined in the same way.
A TEST FOE COLOR BLINDNESS
531. Since all of the various systems of signaling, both on
railways and at sea, require the use of colored lights, it is evi-
dently of the greatest importance that those who are supposed
to interpret such signals be able to distinguish colors in their
proper values. Railway companies subject their employees
to a test for color blindness. A test which is quite commonly
employed for this purpose is known as the Holmgren test. For
making this test a number of samples of colored worsteds are
employed. The samples used consist largely of worsteds of
green, blue, purple, and brown in various degrees of saturation,
and a number of skeins of neutral tint. In addition there are
three samples known as the confusion samples, one of which is
a very pale green, the second a brilliant red, and the third a
magenta which is not very near saturation. The test is made
in the following manner. The group of colors is placed before
the individual whose color vision is to be tested, and one at a
time the confusion samples are placed before him and he is
asked to select those colors from the general group which match
the confusion sample.
To the red-blind eye the magenta confusion sample appears
blue. For such an eye, therefore, it will be found that the
blues will be placed with the magenta. The browns will also
be placed with the grays. To the green-blind eye a green is
COLOR 579
a gray, as we have seen above. Therefore, a person possessing
this defect in color vision will place the grays with the pale
green confusion sample. In this manner, by observing the
selections made to the different confusion samples, it is an easy
matter to detect color blindness when it exists, and to deter-
mine which of the three sets of nerve terminals is defective.
POLARIZATION
CHAPTER XLVII
LIGHT WAVES ARE TRANSVERSE WAVES
532. As we have seen, there are various phenomena which
lead us to believe that the disturbance we call light is of the
nature of a wave motion. We would not, however, be able to
determine from any of the phenomena thus far discussed
whether light waves are transverse or longitudinal. There
are certain phenomena which afford conclusive proof that light
waves are transverse waves ; that is to say, that the ether par-
ticles which transmit light vibrate at right angles to the direc-
tion in which the disturbance is being propagated. The
experiment described in the following paragraph affords evi-
dence of this kind.
THE EXPERIMENT WITH CROSSED TOURMALINES
533. By taking advantage of the natural cleavage of the
mineral it is possible to separate tourmaline into crystals of the
form shown in Figure 374. These
y.™^ crystals are quite transparent, and if
/ ^\s. one of them is held in the path of a
' ^^ narrow beam of light as indicated in
the figure, a large percentage of the
=^=~^ light will be transmitted. So far as
the unaided eye is able to discover
V ^ the transmitted beam is in no way
^y y^' different from the incident beam.
P Upon careful examination, however,
it is found that the transmitted beam
Fig. 374. — Tourmaline Plate. ,. . ^. .
possesses peculiar properties. It the
transmitted beam is allowed to fall upon a second tourmaline,
it will be transmitted, providing the second tourmaline is placed
580
POLARIZATION
581
Parallel Tourmalines.
with its greatest length parallel to the corresponding dimension
of the first tourmaline. That is, if the two tourmalines are
arranged as shown in
Figure 375, the beam
which is transmitted by
the first tourmaline AB
will pass almost undimin-
ished in intensity through
the second tourmaline
A'B' . If, however, the
tourmalines are " crossed,"
that is to say, arranged
as shown in Figure 376,
the beam which is trans-
mitted by the tourmaline AB will be completely intercepted by
the tourmaline AB\
Referring again to the experiment represented in Figure 374,
let it be imagined that the single tourmaline AB is rotated
about the beam of
light as an axis.
Under these circum-
stances, no change in
the intensity of the
light transmitted will
be observed. If, how-
ever, two tourmalines
are employed, a rota-
tion of the second tour-
maline is accompanied
by a change in the intensity of the transmitted beam. When
the tourmalines are parallel. Figure 375, the maximum amount
of light is transmitted. When they are crossed. Figure 376,
the minimum amount of light is transmitted. For positions
intermediate between these two, the amount of light varies
depending upon the angle between the tourmalines. Since no
change in the intensity of the transmitted beam was observed in
the first case, that is, when a single tourmaline was used, it will
be evident that the light in passing the tourmaline AB acquires a
Crossed Tourmalines.
582
LIGHT
property which it before did not possess. It evidently has dif-
ferent properties as seen from different sides, since the position
of the tourmaline A'B' determines the amount of light which is
transmitted. Such a beam of light is said to be polarized.
The phenomena of polarization are best understood by con-
sidering the following mechanical analogy. Imagine that a
long flexible rubber tube AB, Figure 377, has one end fastened
to the wall and the otlier is held in the hand. By moving the
end of the tube which is held in the hand to and fro it is pos-
sible to cause transverse waves to travel down the length of
the tube. If a block of wood with a slot cut in it is placed
Fig. 377. — Mechanical Analogue of Polarizing Apparatus.
over the tube, it will be evident that the motion of the tube
will not be interfered with so long as the slit is parallel to the
direction of motion. If, however, we attempt to cause the
tube to vibrate at right angles to the slot, evidently the vibra-
tory motion will not be able to pass the block of wood. Let it
be further imagined that the end of the tube which is held in
the hand is caused to vibrate in a number of different direc-
tions, horizontally, vertically, and at various angles to the
horizontal. Let it be assumed at the same time that the slot is
in a vertical position. Then of all these vibratory motions
which are imparted to the tube only those which are in a vertical
direction will be transmitted or passed beyond the block. If
now a second slotted block is placed over the tube, those vibra-
tions which pass the first slot will pass the second providing
the second slot is parallel to the first. If, however, the second
slot is placed at right angles to the first, no vibrations will pass.
POLARIZATION 583
The inference is that ordinary light consists of a transverse
wave motion, the vibrations talking place in many different
directions. When such a beam is caused to pass through a
tourmaline crystal, only certain vibrations, that is to say, vibra-
tions in a certain direction, are allowed to pass so that the
transmitted beam differs from ordinary light in that the vibra-
tory motion of the ether particles are all in the same plane.
This being the case, it is very evident that this beam of light
can pass a second tourmaline only when it is parallel to the
first.
The beam which passes the first tourmaline plate is said to
be plane-polarized. Evidently in the experiment the second
tourmaline acts as a sort of detector of the polarized condition.
It is customary, therefore, to refer to the first tourmaline as the
polarizer and the second tourmaline as the analyzer.
THE PLANE OF POLARIZATION
534. It is assumed that in the beam of light which passes
the polarizer, the vibratory motion takes place parallel to the
length of the plate. The plane which extends through the
beam of light at right angles to this vibratory motion is called
the plane of polarization, that is, the ether particles are supposed
to vibrate at right angles to the plane of polarization.
POLARIZATION BY REFLECTION
535. Light may be polarized by reflection from a non-metallic
surface. Thus it is found that when light falls upon a glass
plate the reflected beam is more or less completely polarized
depending upon the angle of incidence. It has been determined
by experiment that for each substance there is a definite angle
of incidence for which the polarization of the reflected beam is
most complete. This angle is known as the polarizing angle.
For example, the incident beam. Figure 378, after reflection at
the mirror M is found to be plane-polarized, the plane of polari-
zation being the same as that of incidence, that is to say, it con-
tains the incident ray and the perpendicular to the mirror M.
Since, as we have seen above, an analyzer is not essentially dif-
584
LIGHT
Fig. 378.
-Apparatus for demonstrating Polarization
by Reflection.
ferent from a polarizer, evidently a second mirror might be
employed as an analyzer. It is found that when a second
mirror M' is placed in the position
shown in the figure that the polarized
beam of light which falls upon it is
reflected exactly as an ordinary beam
of light is reflected from such a
mirror. If, however, the mirror M'
is turned through
an angle of 90°
about the line M'M
as an axis, no part
of the polarized
beam will be re-
flected from the
second mirror. If
M' is turned in
the same direction
through another 90°, the polarized beam will once more be
reflected in its full value, and so on. In other words, it is
possible to make up a complete polarizing apparatus of two
plates of glass. In the .
use of glass plates for
this purpose the angle of
incidence <^ should be
made equal to the polar-
izing angle. The polar-
izing angle for glass is
between 57° and 58°.
BREWSTER S LAW"
536. Brewster found
that polarization by re-
flection is most complete
when the angle between
Fig. 379.
the reflected ray and the refracted ray is 90°. From this circum-
stance it follows that the polarizing angle for any medium may
be very simply expressed in terms of its index of refraction.
POLARIZATION 585
Let A, Figure 379, represent a ray of light in air incident upon
the surface of a denser medium. Let i be the angle of inci-
dence, It the angle of reflection, and r the angle of refraction.
When the angle between the reflected ray £ and the refracted
ray is 90° as shown, then i is the polarizing angle. For these
conditions , + 900 + ^^=180°
.-. r + i2 = 90°
i.e.., R = i= the complement of r (a)
(Section 475)
(from a)
Now
— 1 1
P
siu r
but
sin r = cos i
. sin i
■ ■ :=/*
COS I
or
tan z = /u
(136)
That is to say, the tangent of the polarizing angle of any
medium is equal to its index of refraction.
DOUBLE REFRACTION
537. In considering the various phenomena of refraction we
have assumed that the media considered were isotropic, that is,
that they liad the same physical properties in all directions.
There are certain substances in nature which are transparent
and which at the same time have decidedly different physical
properties in different directions. When a beam of light is
refracted into a crystal of such a substance, certain phenomena
are observed which are not present when refraction takes place
in an isotropic medium.
If, for example, a harrow beam of light is caused to pass
through a crystal of Iceland spar (crystalline calcium carbo-
nate) it will be found that in general the beam becomes divided
into two beams. A study of these two beams of light will
show that one of them obeys the ordinary laws of refraction.
This beam is called the ordinary ray. The other beam, known
as the extraordinary ray, does not obey the ordinary laws of
refraction. A further examination will show that both the
ordinary and extraordinary rays are polarized, their planes of
586
LIGHT
::e
:V V
Fig. 380. — The Ordinary and Extraordi-
nary Rays.
polarization being at right angles to one another. Figure 380
represents a beam of ordinary light a falling in perpendicular
direction upon one face of a
crystal of Iceland spar AB.
A part of the transmitted
ray passes unchanged in di-
rection to 0. This is known
as the ordinary ra3\ An-
other part of the disturbance
is refracted and passes out of
the crystal in the direction
ee' . This is known as the
extraordinary ray. The ordi-
nary ray is plane -polarized,
the plane of polarization being
that of the paper. The ex-
traordinary ray is plane-po-
larized at right angles to this plane. The short cross lines on
ee' are drawn to indicate the direction in which the ether parti-
cles are supposed to be vibrating. The ether particles in the
beam oo' are vibrating in a plane perpendicular to the page.
There are many crystalline substances which exhibit the
phenomena of doable refraction of which we may mention the
following :
Iceland Spar Selenite
Tourmaliue Mica
Quartz Sugar
As was pointed out above, when a beam of light passes into a
double refracting medium, it is usually separated into two beams.
It is found, however, that there are certain directions in double
refracting media along which bifurcation does not take place.
These directions are known as the optic axes of the crystals.
A crystal of Iceland spar is in the form of a rhombohedron, two
opposite solid angles of which are bounded by three obtuse angles.
The optic axis of this crystal is parallel to a line drawn through
one of these solid angles equally inclined to all three faces. Let
ABOD, Figure 381, represent a crystal of Iceland spar. Let
B and D represent the solid angles which are bound by three
POLARIZATION
587
obtuse angles. Then
a line drawn in the
crystal through B
or D making equal
angles with the three
planes containing
these points is the
optic axis of the
crystal. When the
edges of the crystal
are all equal, the
Fig. 381. — Rhomb of Iceland Spar.
optic axis coincides with the diagonal BD.
THE DOUBLE IMAGE PEISM
538. When an ordinary beam of light is passed through a
crystal of Iceland spar, the opposite faces of which are parallel,
the ordinary and extraordinary rays which
emerge from the crystal are parallel. If the
opposite faces of the crystal are not parallel,
the ordinary and extraordinary rays diverge
and become more and more separated the
farther they pass from the crystal. Such a
crystal is of great service in the study of
double refraction and polarization. Such a
crystal of Iceland spar would act as an ordi-
nary prism, giving rise to dispersion. To
obviate this difficulty, it is customary to place
against the Iceland spar a prism of glass with
its refracting edge turned in the opposite
direction from that of the prism of Iceland
spar. The refracting angle of this glass prism
is so chosen as to make of the two prisms an achromatic com-
bination. This combination is represented in Figure 382, in
which / is the prism of Iceland spar and G the prism of glass.
Fia. 382. — Double
Image Prism.
NICOL S PRISM
539. One of the best means of obtaining a beam of plane-
polarized light is by the use of what is known as a Nicol's prism.
588 LIGHT
Referring to Figure 381, let it be imagined that the crystal of
Iceland spar, ABCD, is cut by a plane passing through the
line BB parallel to the diagonal FG-. Let it be further imagined
that the two faces which are thus formed are carefully ground
and polished and cemented together by means of a thin layer of
Canada balsam. Let ABQD, Figure 383, represent such a
crystal, the diagonal -4. (7 representing the plane along which the
crystal was cut. That is to say, the crystal as represented in
the figure is so placed that the plane in which it has been cut is
perpendicular to the page. Let it be imagined that a beam of
ordinary light a is caused to enter this crystal from the left as
0'
Fig. 383. — Nicol Prism.
indicated in the figure. The crystal being double refracting,
this beam will be broken into two beams, the ordinary beam
passing off in the direction ho and the extraordinary ray trav-
eling in the direction he. Now the index of refraction of Can-
ada balsam is less than that of Iceland spar for the ordinary
ray and greater for the extraordinary ray. The values of these
indices are given below :
Canada balsam 1.55
Iceland spar ordinary ray 1.658
Iceland spar extraordinary ray . . . 1.468
It will therefore be evident that if the angle at which the
ordinary ray falls upon the interface ^<7 is greater than the
critical angle (Section 477), this ray will be totally reflected,
since for this ray Canada balsam is optically less dense than the
Iceland spar. Thus the ordinary ray will be turned aside.
The extraordinary ray, however, will pass practically without
change of direction across the interface AQ and emerge as
indicated in the figure. Thus the nicol prism separates
POLARIZATION
589
ordinary light into the ordinary ray, which is suppressed as
indicated above, and the extraordinary ray, which is transmitted.
The transmitted ray is polarized in a plane perpendicular to the
page as indicated by the short cross lines, which are placed to
represent the direction in which the ether particles vibrate.
THE POLARISCOPE
540. A polariscope consists essentially of a device for polar-
izing light and a second device used for analyzing the polarized
beam. Evidently a polariscope might be made of two plates of
tourmaline, two mirrors, or two Nicol prisms, or of combinations
of these various devices.
THIN PLATE OF A DOUBLE EEFRACTING SUBSTANCE IN
POLARIZED LIGHT
541. An instructive experiment is the following: Let it be
assumed that in a polariscope the analyzer is so turned as to
completely suppress the plane-polarized light which proceeds to
it from the polarizer. Under such conditions the analyzer is
said to be "crossed." If, now, a thin plate of a double re-
fracting substance, for example mica, is placed between the
polarizer and the analyzer, it will be found that in general
light will pass the analyzer. Furthermore, it will be observed
that the light which passes the analyzer under these circum-
stances is more or less brilliantly colored.
Fig. 384. — Thin Plate of Double Refrafting Medium between Crossed Analyzer and
Polarizer.
These phenomena are explained as follows : Let P, Fig-
ure 384, represent the polarizer of the polariscope used in the
experiment and A the crossed analyzer. Let it be assumed
that the polarized light which passes P is polarized in the plane
590 LIGHT
perpendicular to the paper so that the vibratory motion of the
ether particles will be represented by the double-headed arrow
p. The doable-headed arrows a represent the vibratory char-
acter of the light before it falls upon the polariscope. It is
assumed that A is so placed as to entirely suppress the polarized
beam which passes P- Let D represent the thin plate of
double refracting substance. Let it be assumed that this plate
is in such position that, if ordinary light were passing, it would
be broken into two beams whose vibrations are at 45° to
the vertical as shown at q. Under these circumstances it is
evident that the polarized beam p will pass the plate i), being
separated into two beams as represented at q. These two
beams, whose vibrations are at right angles to one another, will
each be broken into two beams upon passing A, but A is sup-
posed to be so placed that it will allow nothing but horizontal
vibrations to pass. The result is, the horizontal components of
both vibrations shown at q will pass the analyzer. It will thus
be understood how the presence of D enables the polarized
beam p to pass the analyzer A.
The color effects observed are to be explained in the follow-
ing manner : The two components q into which the plane-
polarized beam p is separated by the plate D pass through the
plate D with different velocities, since, as has been pointed out
above, the index of refraction of a double refracting medium is
different for the ordinary and extraordinary rays. Thus, one
of the components q falls behind the other component in pass-
ing the plate D. If the amount by which the one component is
retarded is equal to an even number of half wave lengths of the
light under consideration, the horizontal parts of these components
will be in condition to interfere. Hence they will neutralize one
another. Under these circumstances no light would pass the
analyzer A^ providing light of one wave length only were em-
ployed. If white light is made use of, it may happen that the
above effect will take place for some one wave length, for
example red, which will therefore be extinguished. The other
wave lengths will pass the analyzer, since the retardation for
these is not such as to bring them into proper relation for com-
plete interference. Hence the light which passes the analyzer
POLARIZATION 591
will be colored. It will be white light minus those wave
lengths which have been extinguished by interference.
If, now, the analyzer A is rotated about the beam of light as
an axis through 90°, evidently that wave length which was ex-
tinguished by interference in the first position of the analyzer
will now pass the analyzer, since the vertical components of q
act together. The vertical parts of the components q, Figure
38-1, extend in the same direction, while the horizontal parts of
these components are opposed. With the analyzer in this posi-
tion those wave lengths will be extinguished for which the
retardation in the plate D is an odd number of half wave
lengths. It is, therefore, evident that such wave lengths as
are quenched by the analyzer in the first position will predomi-
nate in the transmitted beam when the analyzer is in the second
position, and vice versa. We thus obtain complementary colors
in the two positions of the analyzer.
If in the experiment described above the plate D is rotated
about the beam of light as an axis, evidently there will be cer-
tain positions in which the above described phenomena will not
take place. For example, if the plate J) is rotated 45° from the
position which it is supposed to occupy in the above discussion,
the polarized beam p will be transmitted by the plate D without
alteration. It will, therefore, not be able to pass the analyzer.
There is another position for the plate D 90° from this one for
which the same statement is true.
EOTATIOK
542. If a thin plate of quartz cut perpendicular to the optic
axis is placed between the polarizer and crossed analyzer of a
polariscope, light will pass the analyzer. This effect is distinct
from that described in the last section, since it is found that a
rotation of the quartz plate about the beam of light as an axis
will produce no effect upon the intensity of the transmitted beam.
Evidently, therefore, the passage of light through the analyzer
under these circumstances is to be explained in some other way.
The beam of light which enters the quartz plate under these
circumstances is broken up into two polarized beams, but the
character of the vibratory motion in each of these beams is
592 LIGHT
different from that which has been considered in the preceding
sections.
These beams of light within the quartz are said to be circu-
larly polarized, that is, the ether particles instead of moving to
and fro in straight lines are thought of as whirling in circles
whose planes are perpendicular to the ray of light. These cir-
cular components pass through the quartz with different veloci-
ties. They recombine upon emerging into the air to form a
beam of plane-polarized light, but since one of these circularly
polarized beams has been retarded in passing the plate, when
they recombine it will be in a plane different from that in
which the beam p is vibrating. Thus, in effect, the quartz plate
rotates the plane of polarization of the. beam p. The beam
which passes the quartz plate is in no respect different from
that which is incident upon it, except that its plane of polariza-
tion is different. By rotating the analyzer a position is found
for which the beam is completely quenched.
Double refracting substances which are capable of producing
this effect are said to be "optically active." Certain solutions
are found to be optically active. The rotation produced by a
solution of an optically active substance is proportional to the
mass of the substance contained in the solution and the thick-
ness of the solution in the direction in which the light is passing.
If the thickness of the solution is kept constant it thus becomes
possible to estimate the amount of the optically active substance
dissolved, by measuring the angle through which a beam of
plane-polarized light is rotated in passing the solution. This
method is employed in a practical way for the determination of
the percentage strengths of sugar solutions.
INDEX
Numbers refer to pages.
a and (3 particles, mass and velocity of, 419
Aberration, 494
chromatic, 533
spherical, 535
Absolute humidity, 205
Absolute temperature, 172
Absorption, ol heat, 226
selective, 567
Acceleration, 26
angular, 39
Acceleration of gravity, 37
Accommodation, 531
Achromatic lens, 534
prism, 543
Action and reaction, 43
Activity, optical, 592
radio-, 415 et seq.
Addition of musical intervals, 464
Adhesion, 150
Adiabatic process, 240
Aeroplane, problem of, 19
After effect, 577
Air columns, fundamental tones of, 473
overtones of, 474
vibrations of, 471, 476
Air, compressed, 135
Air pumps, 131
Air thermometer, 164, 173
Alternating current, 380
generator, 381
motors, 385
Amalgamation, 346
Ammeter, 358
Ampere, definition of, 291
Ampere's law, 322
Amplitude ol a wave, 432
Amylacetate lamp, 556
Analyzer, 583.
Anastigmatic lens, 537
Angle, critical, 515
measurement of, 40
of deviation, 541
of dispersion, 541
of incidence, 437, 513
of reflection, 437
of refraction, 513
polarizing, 583
2q
refracting, 543
unit of, 40
Angular velocities, addition of, 61
Anode, 338
Aplanatic lens, 536
Arago's experiment, 369
Arc, flaming, 333
magnetite, 333
Arc lamp, 333
Archimedes' principle, 112, 114, 115, 120
Area, unit of, 4
Armature of dyDamo, 378
" Artificial ice," 237
Astigmatism, 537
Atmospheric electricity. 284
Atmospheric pressure, 121
measurement of, 121
value of, 123
Audition, limits of, 45()
Auditory nerve, 439
Avogadro's principle, 211
Axis of a crystal, 586
Axis of precession, 62
Axis of spin, 62
Axis of torque, 62
Balanced forces, 41
Balanced torques, 41
Ball and jet, 146
Ball bearing, 83
Ball nozzle, 14;')
Ballistic pendulum, 105
Balloon, 114
Barometer, simple, 122
siphon, 122
Baseball, curves of, 146
Battery, crow-foot, 347
gravity, 346
storage, 351
Beats, 447
Beaume''s hydrometer, 119
Becquerel's discovery, 415
Biprism, Fresnel's, 549
Blindness, color, 577
Holmgren test for, 678
Block and tackle, 87
Body color, 569, 575
593
594
INDEX
Boiling point, 187
at high altitudes, 188
effect of pressure on, 187
Bouguer's principle, 560
Boyle's law, 125
Bradley's method for velocity of light, 494
Brake, Prony, 98
Brakes, 97
Branched circuit, resistance of, 296
Brewster's law, 534
Bridge, A'S'heatstone's, 299
Bright line spectrum, o-io
British standard candle, 566
British thermal unit, 178, 232
Brush discharge, 407
Bulk modulus, 102
Bunsen photometer, 561
Bursting flywheel, 51
Calcium carbide, manufacture of, 334
Caliper, micrometer, 7
vernier, 5
Calorie, 178, 232
Calorimeter, 182
ice, 184
steam, 185
Calorimetry, 178 et seq.
Camera, photographic, 530
pinhole, 499
Canal rays, 413
Candle, British standard, 556
international, 556
Candle power, 556
Capacity, electrostatic, 275, 279
specific inductive, 277
thermal, 179
Capacity, of condensers in parallel, 280
of condensers in series, 280
of anisolated sphere, 279
Capillarity, 153
Carbon dioxide experiment, 195
Carcel lamp, 556
Card, indicator, 243
Carnot's cycle, 240
Cartesian diver, 114
Cathode, 338
Cathode rays, 410
properties of, 411, 412
Caustic, 537
Cell, Bunsen, 347
Clark, standard, 348
Daniell, 346
dry, 348
Grove, 347
lead storage, 351
Leclancho, 348
Centigrade thermometer, 163
Centimeter, 3
Central energy telephone system, 397
Central force in uniform circular motion,
50
Center of gravity, 92
Centrifugal drier, 52
Centrifugal force, 51
C. g. s. unit, of current, electrostatic, 291
of current, electromagnetic, 322
of e. m. f., electrostatic, 291
of force, 36
of length, 3
of mass, 4
of power, 95
of time, 4
of work, 70
Characteristics, of a color, 572
of a musical sound, 452
Charge, distribution of, 263
energy of, 283
residual, 278
seat of, 278
surface density of, 264
unit of, 2(50
Charges, electrostatic, force between,
259
Charging, by friction, 247
by induction, 254
Charles, law, 170
Chemical change, effect of heat on, 161
Chemical effect of the electric current,
337 ei seq.
Chemical equivalent, electro-, 339
Chemical hygrometer, 205
Chord, major, 464
Chromatic aberration, 533
Circular loop, magnetic field of, 325
Circular motion, uniform, 47 et seq.
central force required in , 50
radial acceleration in , 4!)
Circular polarization, 592
Clark cell, standard of, e. m. f., 348
Clock, 10
Closed pipe, 473
Closed vector polygon, 16
Clouds, formation of, 208
Coefficient, of cubical expansion, 170
of linear expansion, l(i5
Coefficient of simple rigidity, 102
Cohesion, 149
Coil, induction, 374
Tesla, 376
Collimator, 545
Color, 567 et seq.
body, 569, 575
chara(!teristics of a, 572
of opaque bodies, 568
of transparent bodies, 567
origin of, 567
INDEX
595
Color blindness, B77
test for, 578
Color diagram, Maxwell's, 574
Color flicker, 565
Colored lights, mixing, 570
Colors, complementary, 5B9, 572
nonspectral, 573
of the spectrum, 540
of thin plates, 551
of thin plates in polarized light, 589
primary, 571
subjective, 570
Colza oil, 556
Comma, 467
Commutator of dynamo, 384
Complementary olors, fiOi), 572
by polarized light, .Wl
Compound microscope, 527
magnifying power of, 527
Compressibility of a gas, 125
Compression, in sound waves, 444
Concave lens, 519
image found by, 522
Concave mirror, 505, 506
Concave wave, 504
Condensation of vapor, 192, 193
Condenser, 275
optical, 529
Conduction of heat, 216, 219
Conductivity, thermal, 220
measurement of, 221
Conductors, electric, 251
of heat, 219
Confusion sample, 578
Conjugate foci, 520
points, 506
Consonance, 463
Contact difference of potential, 343
Continuous spectrum, 545
Convection, 216
Conversion of work into heat, 230
Convex lens, .517
image formed by, 521
Convex mirror, 508
Convex wave, 504
('ooking, electric, 332
Cooling effect of vaporization, 194 et
seq.
Corpuscles, 413
Corpuscular theory of light, 491
Coulomb, 340
Coulomb meter, 340
Couple, thermo, 176
Crane problem, 17
Cream separator, 52
Critical angle, 515, 516
Critical temperature, 196, 308
Crookes effect, 409
Crookes tube, 409
Crossed tourmalines, 581
Crovi'foot battery, 347
Cryophorous, 194
Cubical expansion, 170
Regnault's method for, 175
Curie, M. and Mme., 416
Current, electric, 290
alternating and direct, 381
c. g.s. electromagnetic unit of, 322
c.g.s. electrostatic and practical units
of, 291
chemical effect of. 328 et seq.
heating effect of, o2\f
induced, 361
magnetic effect of, 319
strength of, 291
Currents, eddy, .3i;S
Curvature, radius of, 503
Curvature of field, 538
Curves, distribution, 557
Cycle, Carnot's, 240
reversible, 243
Dalton's law, 212
Damping of waves, 430
Daniell's cell, :346
Dark heat Maves, 228
Dark line spectrum, 546
D'Arsonval galvanometer, 355
Debierne, 417
Declination, 316
Defects of mirrors and lenses, 533 ei
seq.
Defining equation, 39
Degree, 40
Density, 109
maximum, of water, 175
measurement of, 115
Depolarizer, 345
Detectors, wireless telegraphy, 404
Deviation. 540
angle nf. 541
without dispersion, 543
Dew, 208
Dew point, 206
Dew point hygrometer, 206
Diaphra;,'ms in optical instruments, 536,
538
Dielectric, 277
Dielectric theory, 249
Diesis, 41)7
Diffraction, 553
Diffraction grating, 553
Dimension formulse, 44
Diminution of pressure, 144
Dip, magnetic, 315
Diplex telegraphy, 393
596
INDEX
Direct current, 381
generator, 383
motor, '■'•Hi
Direction of induced e. m. f., 366
Discharge, electric, 406 et seq.
brusli, 407
disruptive, 407
effect of pressure on, 408
oscillatory, 284
point, 406, 407
Discharging action of a point, 264
Dispersion, 540 et seq.
angle of, 541
without deviation, 543
Disruptive discharge, 407
Dissonance, 463
Distortion, 537
Distribution curves, 557
Distribution of charge, 263
Diver, Cartesian, 114
Dominant, 465
Doppler's principle, 456
Dotted position, Huyghen's construction,
501
Double image prism, 5S7
Double refracting substances, 586
in polarized light, 58!)
Double refraction, 585
Drier, centrifugal, 52
Dry battery, 348
Ductility, 108
Duplex telegraphy, 391
Dynamo, the, 377
Dynamometer, 37
Dyne, 36
Earth, magnetism of the, 315
magnetic field of the, 315, 316
Ebullition, 187
Echo, 449
Eddy currents, 368
prevention of, 370
Effect, Crookes, 409
Geissler, 408
Effects of heat, 160
Efficiency of a simple machine, 85
Efficiency of heat ennines, 242
Efflux, 138
from air-tight spaces, 140
Elastic bodies, 100
Elastic limit, 102
Elasticity, 100 p( seg.
modulus of, 102
Electric batteries, e. m. l.'s of common,
349
Electric conductors, 251
Electric cooking, 332
forge, 334
furnace, 334
heating, 332
lamps, 332
lighting, 332
welding, 334
Electric current, c. g. s. electromagnetic
unit of, 322
c. g. s. electrostatic unit of, 291
chemical effect of, 337 et seq.
heating effect of, 329 et seq.
induced, 361
magnetic effect of, 319
strength of, 291
Electric discbarge, 406 et seq.
Electric motors, 384
Electrical measuring instruments, 353
et seq.
Electricity, positive and negative, 247
Electrification, 247
Electrochemical equivalent, 339
Electrodes, 338
Electrodynamometer, 3J7
Electrokinetics, 290 et seq.
Electrolysis, 337
applications of, 341
Electrolyte, 337
Electrolytic transformations, 338
Electromagnet, 324
Electromagnetic induction, 361 et seq.
Electromagnetic unit of current, 322
Electromagnetic waves, 399 et seq.
Electromagnetism, 319 et seq.
Electrometallurgy, 341
Electromotive force, 291
u. g. s. electrostatic and practical units
of, 291
induced, 361
self-induced, 371
Electrons, 2.30, 413
Electron theory, 250, 231, 254
Electrophorus, 267
Electroplating, 341
Electroscopes, 251
Electrostatic capacity. 275, 279
Electrostatic iield, 25.S
Electrostatic induction, 234
Electrostatic lines of force, 238
Electrostatic machines. 2()() ct seq.
Electrostatics, 247 et seq.
Electrotyping, 311
Elevator, hydraulic, 134
Emanation, 424
Energy, 74
conservation of, 74
kinetic, 74
of a charge, 283
of rotatory motion, 76
potential, 74
INDEX
597
Energy, transformation of, 74
units of, 74
Engine, Carnot's ideal lieat, 242
gasoline, 234
Hero's, 230
ideal, 242
steam, 230
Equilibrium, first condition of, 41
second condition of, 41
Equipotential lines, 273
Equipotential surfaces, 273
Equivalent, mechanical, of heat, 232
electrochemical, 339
Erg, 70
Ether, the, 224, 399
Ether waves, 399 et seq
Evaporation, 187, 191
Exchanges, Prevost's theory of, 225
Expansion, coefficient of cubical, 170
coefficient of linear, 165
cuhical, 170
linear, 1G5
of liquids, 174
of water, 175
Expansion tank, 218
Extraordinary ray, 685
Eye, the, 531
farsighted, 532
nearsighted, 531
Eyeglasses, 531, 532
Eyepiece, 527
Factor, proportionality, 38
Fahrenheit thermometer, 163
Falling body, 26
Fall of potential, 292
Faraday's laws, 339
Farsighted eye, 532
Field, electrostatic, 258
intensity of, 260
Field, magnetic, 304
Fire syringe, 230
Fizeau, 496
Flaming arc, 333
Flat field, 539
Flats and sharps, 468
Flexure, 103
Flicker photometer, 563
Floating bodies, 113, 115
Fluids, 108
properties of, 108
Fluid theory, single, 248
two, 249
Fluorescence, of Crookes tube, 410
Fluorescent screen, 414
Flywheel, bursting, 51
Focal length, principal, 519
Foci, conjugate, 520
Focus, principal, 519
universal, 631
Focusing, 530
Fog, 208.
Foot-candle, 565
B'oot-pound, 71
Foot-poundal, 70
Force, 36
between charges, 269
between magnet poles, 303
lines of, electrostatic, 258
lines of, magnetic, 304.
magnetizing, 310
measurement of, 37
units of, 36
Force pump, 130
Forces, balanced, 41
Forge, electric. 334
Foucault's method lor velocity of light, 496
Fountain, luminous, 516
Franklin, Benjamin, 249
Fraunhofer's lines, 547
Freezing point, 189
effect of pressure upon, 190
Frequency of wave motion, 432
Fresnel's biprism, 549
Friction, 81
coefficient of, 82
effects of, 91
head, 139
rolling, 83
Friction machine, 206
Front, wave, 499
Frost, 208
Frost line, 198
Fundamental tone, 453
of vibrating air column, 473
Furnace, electric, 334
Fuses, 335
Fusion, heat of, 183
Fusion point, effect of pressure on, 190
Galvani's experiment, 343
Galvanometer, 353
d'Arsonval, 355
tangent, 353
'Thomson, 354
Gas, 108.
Gas, isothermals of a, 200
pressure of a, 209
Gas atoms, vibratory motion of, 209
Gases, compressibility of, 125
expansibility of, 125
general law of, 171
liquefaction of, 203
specific heat of, 179
Gasoline engine, 234
water-cooled, 218
598
INDEX
Gay Lussac's law, 170
Geissler effect, 408
General law of gases, 171
Generator, alternating current, 381
direct current, 383
Geryk pump, 132
Glaciers, motion of, 191
Glass, object, 527
Gold leaf electroscope, 252
Gradient, temperature, 223
Gram , 4
Gram weight, 37
Graphical method, 14
Grating, diffraction, 553
Gravitational units of work, 71
Gravitational waves on liquids, 429, 432
Gravity, acceleration of, 37
Gravity battery, 346
Gravity, specific, 117
center of, 92
Gridiron pendulum, 167
principle of, 168
Gyration, radius of, 78
Gyroscope, 61
Gyroscopic action, examples of, 65
Hail, 208
Half tone, 467
Hardness, 108
Harmonic motion, simple, 54
examples of, 67
of rotation, 60
Heat, effects of, 160
measurement of, 181
mechanical equivalent of, 232
nature of, 159
specific, 178
transmission of, 216 et seq.
Heat of fusion, 183
Heat developed by electric current, 330
Heat of vaporization, 183
Heat units, 178
Heat waves, 227, 228
Heating effect of electric current, 329 et seq.
Heating, electric, 332
Heating system, hot water, 218
Hefner lamp, 556
Hemispheres, Magdeburg, 124
Hero's engine, 230
Hertz's experiments, 399
High altitudes, boiling point at, 188
Hollow conductor, screening effect of, 261
Holmgren test for color blindness, 578
Hooke'3 law, 102
Horizontal intensity of earth's magnetic
field, 316
Horse power, 95
Hotbed, 228
Hot box, 230
Hot-water heating system, 218
Hot wire instrument, 358
Hue, 572
Humidity, absolute, 205
relative, 205
Huyghen's construction for reflected
wave, 500, 502, 505, 508
Huyghen's principle, 500
Hydraulic elevator, 134
Hydraulic press, 133
Hydraulic ram, 142
Hydraulic transmission of power, 135
Hydrometer, 119
Beaume's, 119
Hydrostatic paradox. 111
Hydrostatic pressure, 101, 110, 111, 112
Hygrometer, chemical, 205
dew point, 206
wet and dry bulb, 207
Hygrometry, 205 et seq.
Ice, lowering of melting point by pressure,
189
Ice calorimeter, 184
Ice clouds, 208
Ice line, 198
Ice pail experiment, 255
Ideal engine, 242
Illumination, 565
Image formed by concave lens, 522
Image formed by convex lens, 521
Image formed by pinhole, 499
Image, real, 505
virtual, 505
Impact, 104
elastic and inelastic, 105
Incandescent lamp, 332
Incidence, angle of, 437, 513
Inclined plane, 89
Independence of forces, principle of, 33
Index of refraction, 498, 513, 541
Indicator card, 243
Induced current, 361
Induced electromotive force, 361
direction of, 366
law of, 365
in a revolving coil, 379
Induction coil, 374
Induction motor, 385
Induction, electrostatic, 254
electromagnetic, 361 et seq.
magnetic, 311
Inelastic bodies, 100
Inertia, moment of, 39, 77
Instruments, electrical measuring, 353 et
seq.
optical, 525 et seq.
INDEX
599
Insulators, 251
Intensity flicker, 565
Intensity ol field, electrostatic, 200
magnetic, 310
Intensity, horizontal, of earth's field,
31G
Interference, 440, 549 et seq.
International candle, 556
International pitch, 466
Intervals, musical, 463
addition and subtraction of, 4()4
Intervals of the major scale, 467
Inverse squares, law of, 558
Ionization, 406, 414
Ions, 337
Isothermal, 201
at critical temperature, 202
of a gas, 200
of a vapor, 201
Isothermal process, 239
Jar, Leyden, 277
unit, ^82
Jet, ball and, 146
Joule (unit of energy), 70
Joule's law, 32!)
Jupiter, occultation of satellites, 492
Kathode {see Cathode)
Kilogram, 4
Kilogram-meter, 71
Kilowatt, 95
Kinetic energy, 74
Kinetic theory of gases, 209 et seq.
Kite problem, 18
Kundt's experiment, 478
Laminations, 371
Lamp, arc, 333
Carcel. 556
Hefner, 556
incandescent, 332
Lantern, projection, 529
Law of inverse squares, 558
Law of the simple pendulum, 58
Laws of motion, 43
Left-hand rule, 333
Length, units of, 3
measurement of, 5
Lens, achromatic, 533
anastigmatic, 537
apian atic, 536
concave, 519
convex, 517
projecting, 529
rectilinear, 538
Lenses, defects of, 533 et seq.
Lenz's law, 366
Lever, 85, 91
Leyden jar, 277
oscillatory discharge of, 284
Lift pump, 129
Light, corpuscular, and wave theories of,
491
monochromatic, 550
nature of, 491 et seq.
polarized, 582
rectilinear propagation of, 498
standards of, 566
velocity of, 492, 498
Light waves. 228, 491
energy of, 430
interference of, 549
Lighting, electric, 332
Lightning, 284
Lightning rod, 286
essentials of, 288
protection afforded by, 286
Limit, elastic, 102
Limits of audition, 456
Linde's liquid air machine, 214
Linear and angular motion compared,
67
Linear exiDansion, 165
Lines, Fraunhofer's, 547
Lines, equipotential, 273
Lines of force, electrostatic, 258, 263
magnetic, 304
Liquefaction of gases, 203
Liquid, 108
Liquid air machine, 214
Liquids, flow of, 137
exi^ansion of, 174
Local action, 345
Lodge's experiment, 402
Long distance telephone, 395
Longitudinal vibration of rods and
strings, 482
Loops and nodes, 434
Loop, circular, magnetic field of, 325
Loudness, 452
Luminosity, 573
Luminous fountain, 516
Lummer-Brodhun photometer, 562
Machine, definition of, 85
Machine, simple, efficiency of, 85
mechanical advantage of, 85
Machines, simple, 85 et seq.
Magdeburg hemispheres, 124
Magic lantern, see projection lantern
Magnet pole, unit, 304
Magnet poles, force between, 304
Magnetic circuit, 325
Magnetic detector, Marconi's, 405
Magnetic dip, 315
600
INDEX
Magnetic effect of the electric current,
319
Magnetic field, 304
about a wire carrying current, 319,
320
intensity of, 310
of a circular loop, 325
of the earth, 315, 316
of a solenoid, 324
uniforna, 313
Magnetic lines of force, 304
Magnetic moment, 314
Magnetic substances, 305
Magnetism, 303 et seq.
of the earth, 315
retention of, 307
theory of, 309
Magnetite, 303
Magnetite arc, 333
Magnetization, 306
Magnetization curve, 312
Magnetizing force, 310
Magneto, 396
Magnets, artificial and natural, 303
permanent, 307
Magnifying power, of compound micro-
scope, 527
of simple microscope, 526
of telescope, 528
Major chord, 464
Major scale, 465
Manometer, 127
Mass, measurement of, 8
unit of, 4
Mass and velocity of a and /3 particles, 419
Mass and weight compared, 9, 37
Matter, 8
three forms of, 108
general properties of, 108
Maximum density of water, 175
Slaxwell's color diagram, 574
Maxwell's theory, 399
Measuring Instruments, electrical, 353 et
seq.
Mechanical advantage, 85
Mechanical equivalent of heat, 232
Melting point, see Fusion point
Mercury air pump, 132
Meter, 4
Meter candle, 565
Method of mixtures, 181
Metric system, 3
Microscope, compound, 527
simple, 525
Mirror, concave, 502, 504, 605, 536
convex, 507
parabolic, 536
plane, 501, 507
Mirrors, defects of, 536
Mixing colored lights, 570
Jlixiug pigments, 569
Jlixtures, method of, 181
Modulus, bulk, 102
Young's, 102
Modulus of elasticity, 102
how used, 102
Molecular force action, 149
sphere of, 150
Molecules, 108
Moment of inertia, 39, 76, 77, 80
and kinetic energy, 80
and mass compared, 77
Momentum, 67, 104
Monochromatic light, 550
Jlotion, 25 et seq.
uniform and accelerated, 25
uniformly accelerated, 26
Motor, electric, 384
Jlousson, 190
JIultiple echo, 450
Musical intervals, 463
addition and subtraction of, 464
Jlusical scale, 463 et seq.
Musical sound, 439
characteristics of, 452
Musical tone, 439
Nature of light, 491 et seq.
Nature of sound, 439 et seq.
Nearsighted eye, 531
Negative electricity, 247
Nernst lamp, 333
Neutral layer in flexure, 104
Newton's laws of motion, 43
Nickel, magnetic properties of, 306, 309
Nicol's prism, 587
Noises, 439
Non-conductors, see Insulators
Nozzle, ball, 145
Object glass, 527
Octave, 463
Oersted's experiment, 319
Ohm, definition of, 292
Ohm's law, 292
Oil, Colza, 556
Oil on water, behavior of, 155
Opaque bodies, color of, 568
Open pipes, 473
Optic axis of crystals, 586
Optical instruments, 525 et seq.
Optically active substances, 592
Ordinary and extraordinary rays, 585
Organ pipes, 476
Origin of color, 567
Oscillator, Hertz's, 399
INDEX
601
Oscillatory discharge, 284, 400, 401
Overtones, 453
of air columns, 474
Parabolic mirror, 536
Paradox, hydrostatic, 111
Parallel currents, force between, 323
Parallelogram law, 13
Pascal's law, 133, 135
Pendulum, ballistic, 105
gridiron, 167
simple, 58
law of the simple, 60
principle of gridiron, 168
Permanent magnets, 307
Permeability, magnetic, 311
Period of wave motion, 432
Persistence of vision, 571
Phase, 431
Photographic camera, 530
Photographic lens, 530
Photometer, Bunsen, 661
flicker, 663
Lummer-Brodhun, 562
Eumford, 660
Photometry, 656 et seq.
Pigments, mixing of, 569
Pinhole images, 499
Pipes, open and closed, 473
organ, 476
Pitch, 452
international, 466
measurement of, 454
Pitch of a screw, 7
Pith ball electroscope, 252
Plane of polarization, 583
Plane polarized light, 583
Plane wave, reflected by plane mirror,
501
reflected by concave mirror, 502
Plates, colors of thin, 551
vibration of metal, 485
Plunger instrument, 366
Point, discharging action of, 264
Point discharge, 406
theory of, 407
Points, conjugate, 606
Polariscope, 689
Polarization, electric, 345
Polarization of light, 580 et seq.
by reflection, 583
Polarization, plane of, 583
Polarized light, circular, 592
plane, 583
Polarizer, 583
Polarizing angle, 583
Pole, magnet, 303
unit, 304
Porous plug experiment, 212
Positive electricity, 247
Potential, 271
fall of, 292
high, of thunder storms, 285
of a point distant r from a charge Q,
271
Potential energy, 74
Pound, 4
Pound weight', 37
Poundal, 36
Power, 96
candle, 556
expended in heating a conductor, 331
hydraulic transmission of, 135
measurement of, 96
P = EI, 331
units of, 96
Precession, 62
direction of, 62
explanation of, 63
velocity of, 63
Precipitation, 207
Press, hydraulic, 133
Pressure, atmospheric, 121
hydrostatic, 101, 110, 111, 112
Pressure, diminution of, 144
Pressure, effect of,
on boiling point, 187
on discharge, 408
on freezing point, 190
on melting point, 190
Prevention of eddy currents, 370
Prevost's theory of exchanges, 225
Primary battery, see Voltaic cell
Primary colors, 671
Principal focal length, 619
Principal focus, 619
Principle, Avogadro's, 211
Principle of Archimedes, 112, 113, 115, 120
Prism, 640
double image, 587
Nicol's, 587
Processes, adiabatic and isothermal, 239
Production of sound, 439
Products, radioactive, 424
Projectile, 29
Projecting lens, 529
Projection lantern, 529
Prony brake, 98
Propagation of sound, 440
Proportionality factor, 38
Pulley, 87
Pump, air, 131
force, 130
Geryk, 132
lift, 129
Sprengle, 132
602
INDEX
Quality of sound, 452
Quantity of heat, 178, 181
Radian, 40
Radiation emitted by radium, 417 et seq.
Radiation of iieat, 210, 224, 226.
Radioactive bodies, 416
products, 424
substances, 416
transformations, 424
Radioactivity, 415 et seq.
Radium, 416
Radius of curvature, 503
Rain, 208
Ram, hydraulic, 142
Range of a projectile, 31
Rarefaction in sound wave, 444
Ray, extraordinary, 585
of light, 499
ordinary, 585
Rays, canal, 413
Rays, cathode, 410
properties of, 411, 412
a, /3, and y, 417 et seq.
Reaction, action and, 43
Real image, 505
Rectilinear lens, 538
Rectilinear propagation of light, 498
Reflection of heat, 228
Reflection of light, 500, 502, 504, 507
Reflection of sound, 449.
Reflection of water waves, 436
Reflection, polarization by, 583
Refracting angle, 543.
Refraction, angle of, 513
double, 585
index of, 498, 513, 541
law of, 513
Refraction of light, 511 et seq.
Refraction of sound waves, 450
Refraction of water waves, 437
Refrigerating machine, 233, 235
Refrigeration, mechanical, 235
Regelation, 189
Regnault's method for cubical expansion,
175
Relative humidity, 205
Relay, 390
differential, 391
polarized, 392
Residual charge, 278
Resistance, 292
specific, 294
Resistance thermometer, 301
Resistance of conductors in parallel, 296
temperature coefficient of, 296
Resistances compared by fall of potential,
291
Resolution of a vector, 20
Resonance, 402, 460
Resonator, electric, 400
Resultant, 14
Retentivity, 308
Reversible cycle, 243
Reversibility of voltaic cell, 349
Revolving coil, induced e. m. f. in, 379
Rifle ball, flight of, 31
Right-hand rule, 367
Rigidity, coefficient of simple, 102
Ripples, 432
Rods, longitudinal vibration of, 482
transverse vibration of, 481
Roemer's method for velocity of light,
492
Roentgen rays, 411, 412
Rotation of plane of polarization, 591
Rotation, simple harmonic motion of, 60
Rumford photometer, 560
Rutherford, 415
Sagitta, 503
Sample, confusion, 578
Saturated vapor, 192, 193
pressure-temperature curve of, 196
Saturation of a color, 573
Scalars, 12
Scale, major, 465
musical, 463 et seq.
of equal temperament, 469
Screen, fluorescent, 414
Screening effect of hollow conductor, 261
Screw, 90
pitch of a, 7
Seat of charge, 278
Second, 4
Secondary battery, see Storage cell
Selective absorption, 567
Self-induced e. m. 1., 371
Self-induction, 371
coefficient of, 373
Separator, cream, 52
Shadow picture. X-ray, 414
Shallowing effect, 513
Sharps and flats, 468
Shearing stress, 101
Ships, problem of the two, 23
Shunt, the, 297
Simple harmonic motion, 54
examples of, 57
Simple harmonic motion of rotation, 60
Simple machine, efficiency of, 85
mechanical advantage of, 85
Simple machines, 85 et seq.
Simple microscope, 525
magnifying power of, 526
Single fluid tlieory, 248
INDEX
603
Siphon, 136
Siphon barometer, 122
Siren, 455
Skiagraph, 414
Snow, 208
Soap bubble, color,s of, 551
pressure inside of, 154
Solar day, mean, 4
Solar spectrum, 540, 547
Solenoid, magnetic field of, 324
Solid, 108
Solidification, 189
Sonorous bodies, 471 et seq.
Sound, how produced, 439
musical, 439
nature of, 439 et seq.
velocity of, 443
Sound waves, combinations of, 445
energy of, 430
general character of, 443
graphical representation of, 444
medium of propagation of, 440
reflection of, 449
refraction of, 450
Sounder, 389
Specific gravity, 117
Specific heat, 178
Specific inductive capacity, 277
Specific resistance, 294
Spectra, different kinds of, 545
Spectrometer, 545
Spectroscope, 544
Spectrum , 542
bright line, 545
colors of, 540
continuous, 545
dark line, 546
pure, 644
Sphere, electrostatic capacity of, 279
Sphere of molecular action, 150
Spherical aberration, 535
Spherical waves, 520, 558
Spin, axis of, 62
Sprengle air pump, 132
Spring balance, 38
Standard candle, British, 556
Standard cell, Clark, 348
Standards of light, 556
Stationary -n'aves, 433
in air columns, 472
Steady strain, 287
Steam calorimeter, 185
Steam engine. 233
Steam line, 197
Stops, 536
Storage battery, 351
Storage cell, lead, 351
Strain, 101
Stress, 100
shearing, 101
tensile, 101
Stretch, 101
Strings, longitudinal vibration of, 482
transverse vibration of, 479
Subjective colors, 576
Substances, magnetic, 305
therinometric, 162
Subtraction of musical intervals, 464
Successive changes, theory of, 424
Sudden strain, 287
Surface density of charge, 264
Surface tension, 151
effect of temperature on, 156
measurement of, 155
Synchronous motor, 385
Syringe, fire, 230
Tangent galvanometer, 353
Tantalum lamp, 333
Telegraph, 389
Telegraphy, 389 et seq.
diplex, .393
duplex, 391
quadruplex, 394
wireless, 403
Telephone, central energy, 397
long distance, 395
simple, 394
Telephony, 394 et seq.
Telescope, 527
magnifying power of, 528
Temperature, 159
critical, 196, 308
effect of, on surface tension, 156
gradient, 223
sense, 159
Temperature coefiicient of resistance, 296
Tempered scale, 469
Tension, surface, 151
Terrestrial magnetism, 315
Tesla coil, 376
Theory, kinetic, of gases, 209 et seq.
Thermal capacity, 179
Thermal conductivity, 220
Thermal units, 178
Thermo couple, 176
Thermodynamics, 230 et seq.
first law of, 232
second law of, 233
Thermoelectric effect, 161, 176
Thermometer, 161
air, theory of, 173
resistance, 301
scales, 163
simple air, 164
Thermometric substances, 162
604
INDEX
Thin plates, colors of, 551
double reiracting, in polarized light,
589
Thomson galvanometer, 354
Thorinm-X, 423
Thunder storms, high potentials of, 285
Time, measurement ol, 10
unit of, 4
Time of vibration of magnet in uniform
field, 314
Toepler-Holtz machine, 267
reversibility of, 270
Tone, fundamental, 453
musical, 439
over-, 453
Tonic, 4«5
Torque, 39
Torque action, of a force, 39
of balanced forces, 11
Torque on magnet in uniform field,
313
Torques, balanced, 41
Torricelh's theorem, 138
Total reflection, 515
Tourmalines, experiment with crossed,
581
Trade winds, 217
Transformations, electrolytic, 338
radioactive, 424
Transformer, 382
Transmission of heat, 216, 227 ,
Transmission of power, hydraulic, 135
Transparent bodies, color of, 567
Transposition, 467
Transverse vibration, of strings, 479
of rods, 481
Triple point, 199
Tube, Crookes, 409
Geissler, 408
Tungsten lamp, 333
Tuning fork, 485
Two fluid theory, 249
Uniform circular motion, 47
central force required in, 50
radical acceleration in, 49
Uniform magnetic field, 313
Unison, 463
Unit jar, 282
Units, c. g. s. electrostatic, of charge, 260
u. g. d. electrostatic, of current, 291
c. g. s. electromagnetic, of current, 322
Units, gravitational, of work, 71
Universal locus camera, 531
Uranium-X, 423
Vacuum, measurement of, 133
Van der Wall's equation, 214
Vapor, formation of, 191
isothermal of a, 201
saturated, 1!I2, 193
Vaporization, 187
cooling effect of, 194
heat of, 183
Variation, magnetic, 317
Vector difference, 22
A'ector polygon, 15
closed, 16
Vector quantity, 12
addition of, 13
represented by a line, 12
resolution of, 20
Vector sum, 13
Vectors, 12 et seq.
Velocity, 25
average, 25
Velocity of light, 492, 498
Bradley's method, 494
Foucault's method, 496
Eoemer's method, 492
Velocity of sound, 443
Vernier, 5
\'ernier caliper, 5
Vertical intensity of the earth's magnetic
field, 317
Vibrating air columns, fundamental tone
of, 473
laws of, 474
overtones of, 474
Vibrating strings, law of, 480
Vibration of air columns, 471
Vibration of places, 485
Vibration of rods, longitudinal, 482
transverse, 481
Vibration of strings, longitudinal, 482
transverse, 479
Virtual image, 505
Vision, persistence of, 571
Volt, definition of, 291
Volta, 343
Voltaic cell, 343 et seq.
reversibility of, 349
Voltmeter, 358
Volume, unit of, 4
Watch, 10
Water-cooled gasoline engine, 218
Water, expansion of, 175, 176
maximum density of, 176, 216
Water jacket, 218
Water waves, 429
form of, 431
reflection of, 436
refraction of, 437
velocity of, 433
Watt, 95
INDEX
605
Wattmeter, 359
Watt's diagram, 237
Wave front, 499
Wave theory of light, 491
Wavelets, secondary, 500, 353
Wave length, 432
of light waves, 547
of light waves, measurement of, 555
Waves, 429 et seq.
electromagnetic, 389 et seq.
energy of light and sound, 430
light and heat, 228
stationary, 433
water, 429
Wedge, 90
Weighing, 9
Weighing machines, 93
Weight, 9
Weight and mass, 9
Welding, 149
electric, 334
Wet- and dry-bulh thermometers, 207
Wheatstone's bridge, 299
Wheel and axle, 86
White light, decomposition of, S40
Wind and sail, problem of, 22
Wireless telegraphy, 403
Wollaston, 547
Work, 70
units of, 70, 71
X-rays, 411, 412, 413
Yard, 3
Young-Helmholtz theory, 575
Young's modulus, 102
Zero, absolute, 172
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