equation (16) becomes
/- ( 22 )
and equation (18) becomes
That is, the electric flux passing out from charge Q is equal to
47r<2, and the capacity of an air condenser is equal to - in which
ELECTROSTATICS. 3 7
a is the area of the plates (one plate) and x is the distance between
than, provided the various electrical quantities are expressed in
terms of the new units.
These new units are to be chosen so that the factor C may have
the value - L instead of the value 2 ; that is, this factor is to be
471 4/ra 2 '
made v 2 times as large. By inspection of equation (19) this is
seen to require any one of the following changes in the units of
charge and e. m. f. :
(a) The number which expresses a given e. m. f. must be made
v 2 times as small, that is, the unit e. m. f. must be chosen v 2 times
as large as the unit heretofore used, unit charge being un-
changed ; or
(b) The number which expresses a given e. m. f. must be made v
times as small and the number which expresses a given charge
must be made v times as large. That is the unit e. m. f. must be
chosen v times as large and the unit charge v times as small as the
units heretofore used ; or
(c) The number which expresses a given charge must be made
v 2 times as large, that is the unit charge must be made v 2 times
as small as the unit heretofore used, unit e. m. f. being unchanged.
So far as the immediate object, of reducing the value of the
factor C to > is concerned (a), (b) and (c) are equally satisfac-
tory ; but another consideration is important, namely, that the
work, Eq [see . equation (4)], done by an e. m. f. in transferring
a charge may still be equal to the product Eq where e. m. f. and
charge are expressed in terms of the new units. The scheme
(b) satisfies this condition inasmuch as the numerical value of a
charge is increased in the same ratio that the numerical value of
an e. m. f. is decreased. That is the numerical value of the prod-
uct Eq is not affected by this change of units. A simple and
direct definition of the new unit charge will be given later. For
brevity these new electrical units (which are also c g. s. units)
will be called Faraday units to distinguish them from the units
38 ELEMENTS OF PHYSICS.
which have been heretofore, and are to be hereafter, spoken of as
c. g. s. units.
Concerning the Faraday unit of capacity, it is seen to be of the
same dimensions as length from equation (22) ; a given capacity
is therefore expressed as a length, and the unit (Faraday's unit)
capacity is called the centimeter.
Remarks : Many equations in magnetism require the introduc-
tion of the factor C [equation (16)] when the quantities such as
pole strength, magnetic field intensity, etc., are expressed in Fara-
Relative values of Faraday units and units heretofore used :
One Faraday unit charge = c. g. s. units charge
One Faraday unit current c. g. s. units current
One Faraday unit magnetic field = - c. g. s. units magnetic field,
One Faraday unit e. m. f. = v c. g. s. units e. m f.
= o volts,
One Faraday unit electric field = v c. g. s. units electric field
= o volts per centimeter,
One Faraday unit magnetic pole = v c. g. s. units magnetic pole,
One Faraday unit capacity = 2 c. g. s. units capacity
= r farads,
One Faraday unit resistance = z/ 2 c. g. s. units resistance
= 9 ohms,
where v == 3 x i o
If the student wishes to look more minutely into this matter of
electrical units he will find a full discussion of it in Chapter XV.
48. The spherical condenser. Consider two concentric metal
spheres A and B, Fig. 29. Let R be the external radius of A
and R' the internal radius of B.
If the radius of B is very large
the sphere A is said to stand by
itself or to be isolated. Let the
spheres be charged (charge on A
being -f- Q and on inside of
B, Q) by a battery of e. m. f.
E connected to A and B.
It is desired to find the inten-
sity, at each point, of the electric
field in the region between A and
B, and to find the relation be-
tween R, R r , and Q.
Electric field intensity near a charged sphere. The lines of force
of the electric field between A and B are straight lines radiating
from A to B as shown in Fig. 29. Let/ be the electric field in-
tensity at the point / distant r from the center of A. Describe a
spherical surface of radius r y concentric with A. The electric
flux across this spherical surface is qxr 2 ./ according to equation
(14), 47rr 2 being the area of the spherical surface. But this total
flux out from the charge Q which is on A is equal to 471^ from
equation (21) so that 4rrQ = 4;rr 2 /or
/ - i
7 8 '
This equation expresses the electric field intensity at a point
distant r from the. center of the sphere A. The value of the
radius of the sphere A does not affect the validity of equation
(24) so that this radius may be supposed to be very small, in
which case the charge Q is called a concentrated charge. Equa-
tion (24) therefore expresses the electric field intensity at a point
40 ELEMENTS OF PHYSICS.
distant r from a concentrated charge Q. The conception of con-
centrated charge is of the same importance in the study of elec-
trostatics as is the conception of concentrated magnetic pole in
the study of magnetism.
Capacity of a spherical condenser. An expression for the e.
m. f., E, between the spheres A and B, Fig. 29, can be obtained
by the use of equation (13), in terms of Q, R and R' '. Substi-
tute the value of/ from equation (24) in equation (13). Choose
a line of force as the path from A to B over which the summa-
tion of equation (13) is to be extended, then the angle is every-
where zero and cos e = i. Equation (13) then becomes :
or <2 = ___. (25)
Comparing this with equation (5) we find that the capacity, J t
of the concentric spheres is
R R 1
If the external sphere is indefinitely large so that the sphere
A stands alone or isolated then -=-,- is negligibly small and equa-
tion (26) becomes
That is, the capacity of an isolated sphere in air is equal to its
49. Electrostatic attraction of concentrated charges. Consider
a charge Q 2 at a distance r from a charge Q v Fig. 30. The
ELECTROSTATICS. 4 1
electric field intensity at QI due to Q l is/= \ (24) and the
charge Q 2 is acted upon by a force which is equal to Q^f accord-
ing to equation (i i), so that the force with which the two charges
repel (or attract) is
The fact expressed by this equation, namely that two charges
attract or repel each other with a force which is inversely propor-
tional to the square of the distance between the charges, was dis-
covered by Coulomb and is called Coulomb's Law. When <2 t
and Q 2 are both positive or both negative their product is posi-
tive and they repel each other. When one is positive and the
other negative their product is negative and they attract. There-
fore when F, equation (28), is positive it is a repulsion and when
it is negative it is an attraction.
The Faraday unit charge. The relation expressed by equation
(26) furnishes the simplest basis for the definition of the unit
charge (Faraday unit). The unit charge is that charge which re-
pels an equal charge at a distance of one centimeter with a force of
50. Electrostatic attraction of parallel plates. Consider two
parallel plates of area a, at distance x apart, with air between,
and charged by an e. m. f., , so that
42 ELEMENTS OF PHYSICS.
in which - is the capacity of the condenser in Faraday units,
equation (22). The energy of this charged condenser is
which, since /= - - , becomes
If the distance x between the plates is increased by the amount
while the plates are insulated so that Q cannot change, then
the energy of the condenser increases by the amount
and this increase of energy comes from the work done in pulling
the plates apart against their electrostatic attraction. Let F be
the force of attraction of the plates, then F.x is the work done
in increasing their distance by A-*", so that
in which F is the force of attraction of parallel air condenser plates
of area a, when charged ; the charge on one plate being + Q and
the charge in the other being Q. It is remarkable that the
force of attraction is independent of the distance between the
plates for given charge, the plates being large compared to the
distance between them.
By using the values J= - - (23) in the above discussion we
UNIVERSITY OF CALIFORI^
DEPARTMENT OF PHYSICS
in which /MS the force of attraction of condenser plates of area a
in a dielectric of which the inductivity is k. Thus condenser
plates with given charge attract with J~ as much force in petro-
leum as in air.
An expression for the force of attraction of parallel plates in
terms of the e. m. f. between them may be found by substituting
for Q its value - E I or - E I in equation (31) (or equa-
tion (32)). This gives for an air condenser
and for a condenser with any other dielectric
Thus the electrostatic attraction of condenser plates with given
e. in. f. is 2.04 times as great in petroleum as it is in air.
It can be shown that the force of attraction of any two charged
bodies is proportional to the square of the e. m. f. between them.
The case of parallel plates is one of the veiy few cases in which
the value of the proportionating factor can be calculated in terms
of the dimensions and distance of the bodies. The equation (33)
is used in connection with the Absolute Electrometer ; and the
fact that any two charged bodies attract with a force which is
proportional to the square of the e. m. f. between them is taken
advantage of in the Quadrant Electrometer.
51. The absolute electrometer. If the force of attraction of the
plates of a parallel-plate air condenser is measured the e. m. f.
between the plates may be calculated from equation (33). This
equation gives the e. m. f. in Faraday units ; one Faraday unit
e. m. f. being equal to 300 volts. The absolute electrometer is an
instrument for measuring e. m. f. in this way. The absolute elec-
trometer is described in article 482, text-book.
52. The quadrant electrometer. The absolute electrometer
44 ELEMENTS OF PHYSICS.
can be used for measuring large e. m. f.'s, only, inasmuch as
the force of attraction of two charged plates is too small to
measure by a balance when the e. m. f. is small. For example
the attraction of two plates each of 133 cm. 2 area, at a distance of
one cm. is about one ten-thousandth of a dyne or about one
ten-millionth of the weight of a gram for an e. m. f. of one volt.
For the measurement of small e. m. f.'s by electrostatic attraction
the Quadrant Electrometer is used. The quadrant electrometer
is described in Art. 383, text-book. The two schemes for con-
necting the various parts of a quadrant electrometer are as fol-
First arrangement. In this arrangement one terminal of the
e. m. f, e, to be measured is connected to the needle and to
the quadrants q$ v Fig. 268, and the other terminal of the e. m.
f. is connected with the quadrants q l q l . In this case the force
tending to turn the needle is proportional to e 2 , and the deflection
d of the needle is sensibly proportional to e 2 , or e is proportional
to \/d so that
With this arrangement e. m. f.'s as small as, say, 10 volts may
be measured. This arrangement is always employed when the
electrometer is used to measure the e. m. f. of an alternating dy-
Second arrangement. In this arrangement one terminal of the
e. m. f., e, to be measured is connected to the quadrants q 2 q 2 , Fig.
268, and the other terminal to the quadrants q^q l ; and a very
large auxiliary e. m. f, E, is connected, one terminal to the
needle and the other terminal to the quadrants q v q r Then the
e. m. f. between the needle and quadrants q l q l is E and the needle
is pulled towards these quadrants with a force kE* (proportional
to E 2 ). The e. m. f. between the needle and quadrants q z q z is
E -f e, so that the needle is pulled towards these quadrants with
a force k(E + if. The total force acting to deflect the needle is
k(E -f ef kE 2 2kEe -f ke*-, but e 2 is negligible in comparison
with Ee so that the deflecting force, and, therefore, the deflection
d is sensibly proportional to Ee or to e, since E is supposed con-
stant, so that
e = kd. (36)
With this second arrangement e. m. f.'s as small as, say, o.oi
volt may be measured. The reduction factor k is, of course, en-
tirely different in equations (35) and (36).
53. Energy of the electric field. The whole of the energy of
an electric charge resides in the electric field. Thus the energy
of a parallel -plate air condenser is W =. %JE 2 by equation (8) ;
buty= - by equation (22) so that
W -* -
" 87T * 2 '
But ax is the volume of the region between the plates and - is
the intensity of the electric field between the plates so that
Energy per unit volume I
of an electric field = Sx'S '
Tension along the lines of force in an electric field. The total
a E 2
pull of the plates of an air condenser is F= - - - % from equation
i E 2 E
(33), so that the pull per unit area is - - 2, but is the intensity
of the electric field, so that the pull per unit area is ^ /*. This
pull is necessarily transmitted through something between the
plates, else the plates could not act on each other. Therefore
The pull per unit area I 2
of an electric field = gjf/
The curvature of the lines of force of an electric field as ex-
hibited in Fig. 9 shows, since the lines of force tend to shorten,
46 ELEMENTS OF PHYSICS.
that they also push sidewise on each other. It can be shown
that this side push per unit area is equal to the pull in the direc-
tion of the lines of force, namely, ^ -f 2 .
54. Contact e. m. f. When any two substances are left in
contact they settle to a state of equilibrium * with a definite e. m.
f. between them. This e. m. f. is called the contact e. m. /. of the
55. Charging by contact and separation. Consider plates of
zinc and copper at a distance x apart. When the plates have
settled to equilibrium there will be an electric field between them
of which the intensity is where E is the contact e. m. f. of zinc
and copper. The lines of force of this
field are shown in Fig. 31. If the plates
COPPER are connected by a wire and moved apart,
the e. m. f. between them will remain
equal to E and the field intensity be-
tween them will, of course, decrease as x increases. If, however,
the plates are insulated the electric field intensity between them
does not decrease as they are moved apart inasmuch as the
charges cannot escape and given charges signify a fixed number
of lines of force ending on the plates. Therefore, the e. m. f. be-
tween the plates increases in proportion to the increase of x.
This production of large e. m. f.'s by separating bodies from very
near juxtaposition is called charging by contact and separation.
Example. The contact e. m. f. of zinc and copper is about
0.9 volt. When zinc and copper plates settle into equilibrium
at a distance of, say, o.oi cm. apart, the field intensity between
them is 90 volts per cm. If now the. plates are insulated and
separated to a distance of 10 cm. the e. m. f. between them will
be 900 volts.
The phenomenon of charging by contact and separation seems
* That is to say a state in which there is no tendency to further change of any
to be exhibited by all substances. Thus glass and silk when
rubbed together and separated are charged, the glass positively
and the silk negatively; rosin and fur when rubbed together and
separated are charged, the rosin negatively and the fur positively.
When metal plates are charged by contact and separation
great care must be exercised to prevent the plates from touching
after they are slightly separated, for this will allow the e. m. f.
between the plates to drop immediately to the value of the con-
tact e. m. f. No such difficulty exists in the charging of non-
conductors by contact and separation.
56. The Magnetic Field as a Seat of Kinetic Energy. The mag-
netic field is a kind of obscure motion of the all-pervading me-
dium, the ether ; and this motion represents energy. The amount
of energy* in a given portion of a magnetic field is proportional
to the square of the intensity of the field. This is analogous to
the fact that the kinetic energy of a portion of a moving liquid
is proportional to the square of the velocity of the liquid.
57. Kinetic Energy of the Electric Current in a Coil. Defini-
tion of Inductance. The kinetic energy of an electric current is
the energy which resides in the magnetic field produced by the
current. The field intensity is at each point proportional to the
current and the energy at each point is proportional to the square
of the field intensity, that is, to the square of the current.
Therefore the total kinetic energy of the field is proportional to
the square of the current. That is
in which W is the total energy of a current i in a given coil, and
* A mechanical conception of the magnetic field is given in Chapter XVI. The
kinetic energy per unit volume (in air) of a magnetic field is ^ f* where^ is the in-
tensity of the field. Compare article 53, this pamphlet, concerning energy (potential)
per unit volume of an electric field.
48 ELEMENTS OF PHYSICS.
is the proportionality factor. The quantity L is called the
inductance of the given coil.
Units of inductance. When, in equation (39), H^is expressed
in joules and i in amperes, then L is expressed in terms of a unit
called the henry. When W is expressed in ergs and i in c. g. s.
units of current, then L is expressed in c. g. s. units of induc-
tance. This c. g. s. unit of inductance is called the centimeter,
for the reason that the square of a current must be multiplied by
a length to give energy or work ; that is, inductance is expressed
as a length and the unit of inductance is of course the unit of
length. The henry is equal to io 9 centimeters of inductance.
Example. A given coil with a current of 0.8 c. g. s. units
produces a magnetic field of which the total energy is 6,400,000
ergs, so that the value of L for this coil is 20,000,000 centi-
meters. If current is expressed in amperes and energy in joules
then the total energy corresponding to 8 amperes would be 0.64
joules and the value of L would be 0.02 henry.
58. Noninductive circuits. A circuit of which the inductance is
negligibly small is called a noninductive circuit. Since the in-
ductance of a circuit depends upon the energy of magnetic field,
therefore a noninductive circuit is one which produces only a
weak magnetic field, or a magnetic field which is confined to a
very small region. Thus the two wires, Fig. 32, constitute a
noninductive circuit, especially if they are near together ; for,
these two wires with opposite currents produce only very feeble
magnetic field in the surrounding region. The wires used in
resistance boxes are usually arranged noninductively. This may
be done by doubling the wire back on itself, as in Fig. 32, and
winding this double wire on a spool. In this case the e. m. f. be-
tween adjacent wires at db t Fig. 32, is great and the wires have
considerable electrostatic capacity ; or the wire may be wound in
one layer on a thin paper cylinder, and this cylinder flattened so
as to reduce the region (inside) in which the magnetic field is in-
tense. This gives a noninductive coil of which the electrostatic
capacity is inconsiderable.
59. Moment of inertia, analogue of inductance. The kinetic
energy of a rotating wheel resides in the various moving particles
of the wheel. The velocity (linear) of each particle of the
wheel is proportional to the speed (angular velocity) of the
wheel, and the energy of each particle is proportional to the
square of the velocity, that is, to the square of the speed.
Therefore, the total kinetic energy of the wheel is proportional
to the square of the speed. That is,
W^ Kto 2 (40)
in which W is the total energy of a wheel rotating at angular
velocity to and (iAT) is the proportionality factor. The quantity
K is called the moment of inertia of the wheel.
60. Electromotive force required to make a current in a coil
change. A current once established in a coil of zero resistance
would continue to flow without the help of an e. m. f. to maintain
it, just as a wheel continues to turn when once started, provided
there is no resistance to the motion of the wheel. To increase
the speed of the wheel a torque must act upon it in the direction
of its rotation, and to increase the current in the coil an e. m. f.
must act on the coil in the direction of the current.
When an e. m. f., e (over and above the e. m. f. required to
overcome the resistance of the coil), acts upon a coil the current
is made to increase at a definite rate, -. such that
Proof of equation (41). Multiplying both members of this equation by the cur-
rent i we have ei=Li . Now ei is the rate, 3-, at which work is done on the
coil in addition to work used to overcome resistance, and this must be equal to the
50 ELEMENTS OF PHYSICS.
rate at which the kinetic energy of the current in the coil increases. Differentiatin
equation (39) we have ,- = Li -. Therefore, equation (41) is proven.
Torque required to make the speed of a wheel increase. When
a torque, T (over and above the torque required to overcome the
frictional resistance), acts upon a wheel, then the angular velocity,
co, of the wheel is made to increase at a definite rate, -j-, such that
Proof of equation (42). Multiplying both members of this equation by the an-
gular velocity, u, of the wheel we have 71>= A" - . Now 71> is the rate at
which work is done on the wheel and this must be equal to the rate at which the
kinetic energy of the wheel increases. Differentiating equation (40) we have
- - - = Ku . Therefore, equation (42) is proven.
61. Magnetic flux through a coil due to a current in the coil.
In dealing with coils it is usual to speak of the magnetic flux through
the coil as the product of the flux through the opening of the coil,
or flux through a mean turn, multiplied by the number of turns.
in which N' is the flux through the opening of a coil (through a
mean turn), Z is the number of turns of wire in the coil and N is
what is called the flux through the coti.
When a current i flows in a coil of wire the magnetic flux, N t
through the coil is proportional to i so that
N- Li (44)*
in which L is the proportionality factor. It is shown in the next
article that this L is the same as the inductance of the coil as
defined in article 57.
Remark: When the current i in a coil changes at a rate -j-
* In this equation L and i must be expressed in c. g. s. units because the unit of
flux corresponding to the ampere-henry is not much used.
INDUCTANCE. 5 1
the magnetic flux through the coil due to the current changes
also so that
according to equation (44), and this changing flux induces an
e. m. f. in the coil such that c = - = or
This induced e. m. f. is equal and opposite to the outside
e. m. f. which, according to equation (41), must act upon the coil
to make the current change. The relation between these two
e. m. f s. viz., -f L-r and L -= is discussed in the next article.
62. Self-induced e. m. f. Reaction of a changing current. When
one pushes on a wheel causing its speed to increase, the wheel
reacts and pushes back against the hand. This reacting torque
is equal and opposite to the acting torque Kr [equation (42)]
which is causing the increase of speed. Thus when the speed of
the wheel is increasing the reacting torque is in a direction oppo-
site to the speed and when the speed is decreasing the reacting
torque is in the same direction as the speed.
Similarly when an e. m. f. acts upon a circuit * causing the
current to increase, the increasing current reacts. This reacting
e. m. f. is equal and opposite to the acting e. m. f. L -r [equation
(41)] which is causing the increase of current. This reacting