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tween the two surfaces. If n is the dis-
tance PP', the work done by the force
in conveying a unit quantity from one
surface to the other is F x n. We have
then

Fn= V- V,

F=r^.

n

The electric intensity along a line of force is therefore
the rate at which the potential diminishes per unit length
along that line.







ELECTRICAL POTENTIAL. 191

Reduced to limits, or to infinitesimal values,
F=-



dV



dn

This expression is the strength of field at any point.
The minus sign indicates that the positive direction of the
force is the direction in which the potential diminishes.
In general the intensity of the force in any direction is the
rate of diminution of the potential in that direction.

140. Equilibrium of a Conductor. When a charge
of electricity is imparted to a conductor, it at once distrib-
utes itself over the surface and comes to equilibrium.
The surface of the conductor is therefore an equipoteiitial
surface. Moreover, since there is no force inside a con-
ductor, due to a charge on its surface, there is no difference
of potential throughout its entire volume, since force is the
rate of variation of potential. Hence all points of a charged
conductor have the same potential.

The surface of an insulated conductor under the influence
of a charged one is an equipotential surface, because there
is no electric flow along it. This equality of potential in
the presence of an influencing + charge is brought about
by the negative charge on the near end a (Fig. 57) and the
positive on the remote end b. The potential at a, due to the
-f charge on A, is higher than at the more distant point b ;
but the negative charge near a lowers the potential of the
nearer half of the cylinder, and the positive near b raises
the more distant half to the same level as a. If now the
cylinder be connected with the earth, it will be reduced to
the same potential as the earth, or to zero. The cylinder
will then remain charged negatively, but its potential will
be zero. The positive potential due to A and the negative
due to its own charge then everywhere equal each other,



192 ELECTRICITY AND MAGNETISM.

and the resultant is zero. It is evident that surface density
and potential are not in any sense the equivalents of each
other.

141. Potential equals - Consider the potential
at A., at a distance r from an element q of the charge at



r'

Fig. 73.

(Fig. 73). Let B be at a distance r' from 0. Let the dis-
tance between A and B be divided into n very small ele-
ments, so that the points of division are distant n, n, r a ,
etc., from 0. ^

Then the force at*; is g/r 2 , the force at r } is q /r\, etc.

If r and r l are very nearly equal, we may put without
sensible error q I rr\ as the equivalent force which will do
the same amount of work as the variable force between
the two adjacent points at r and r\ . This force is smaller
than the first expression above and larger than the second
one.

Then to carry unit charge from TI to r. work must be
spent equal to




Similarly the work between r. 2 and r l is q |i _ j . y . i*m

\ l 2 ' ' ,vv

t

From r' to r n _ l the work is . . . q ( ) . If

Vn-i r')



ELECTRICAL POTENTIAL. 193

The whole work done in transferring the unit quantity
from B to A is the sum of all these elements of work ; it
is evident that on adding, all the terms containing the r's
cancel out except the first and the last, or

Work from B to A = Q (



Next suppose the point B moved off to an infinite dis-
tance. Then 1 / r 1 becomes zero, and

Work from infinity to A = -

But by definition this is the potential at J., since it is the
expression for the work spent in bringing unit quantity of
electricity from an infinite distance to the point. There
will be similar expressions for the several elements of the
charge, and the resulting potential at A will be the alge-
braic sum of the potentials due to the several elements, or

'-&*?+: -?

142. Potential of a Sphere. Let the sphere have a
charge Q. Every element q of this charge is at a distance
r from the centre of the sphere ; and the potential at the
centre due to this element is q / r, where ; is the radius.
The potential due to the entire charge is then

q 1 Q

S*-?Zf = F'

But as all points of a conductor in equilibrium have the
same potential, the potential of every point of a sphere
due to a charge Q is Q I r.

Since a charge, uniformly distributed over a sphere, acts
on external points as if it were collected at its centre, the



194



ELECTRICITY AND MAGNETISM.'



potential at any point outside of the sphere and distant d
units from its centre is Q / d.

143. Electrometers. An electrometer is an instru-
ment designed to measure differences of electrostatic
potential. Its indications depend on the attraction be-
tween an electrified and an unelec-
trified plate, or on the action between
two conductors electrified to different
potentials. Sir Snow Harris was the
first to construct such an instrument.
It was made like a balance, with a
small pan P (Fig. 74) on one end
of the beam, and a small round disk
d on the other, just above a fixed
insulated plate a. When a was
electrified it attracted t7, and the
attraction was counterbalanced by
weights in the pan P. But the
plate d was not protected from in-
ductive influence, and no precise ab-
solute measurements involving the
dimensions of the disks could be made, because the surface
density was not uniform over the whole disk (see Fig. 47),
but was greatest at the edges, where the lines of force were
not parallel to one another, but curved outward. This
difficulty was overcome by Lord Kelvin, to whom we are
indebted for modern electrometers.

The essential addition of Lord Kelvin is the guard ring
shown in Fig. 75. The suspended disk fits, without
contact, an aperture in the guard ring A, to which it is
electrically connected. The disk C is the only part of the
area utilized ; the surface density over it is uniform and
the lines of force between it and B are parallel.




Fig. 74,



ELECTRICAL POTENTIAL.



195



i



144. Attracted-disk Electrometer. In the attracted-
disk electrometer the attraction between two parallel disks
at different potentials is counterbalanced by a weight D
(Fig. 75). The disk G, when in position, is adjusted so
that its lower face is as nearly as possible in the same plane
\vith the lower surface of the guard ring A. The lever L
is pivoted on a torsion wire stretched between two insulated
pillars EE. A lens G- is mounted so as to observe an
index hair at the end of the lever L relative to two dote
on the plate F. The
plate is in posi-
tion when the hair
is between the two
dote. The disk B is
insulated and can
be raised or low-
ered by means of a
micrometer screw T
not shown.

The counterpoise
I) is such that when
E and C are at the same potential, the index hair rises above
the sighted position. The force required to bring the hair
down to the sighted position is determined by placing a
small weight on C and a "rider " on the arm L. But when
B and C are at different potentials, the attraction between
them draws down ; the plate B is then adjusted in height
till the index hair comes to the sighted position. The
attraction between the plates is then equal to the force of
gravity on the weights previously determined.




Fig. 75.



145. Theory and Use of the Instrument. - Let 7"

bt- the potential of the movable disk G, which is charged



196 ELECTUICITY AND MAGNETISM.

positively to a surface density a ; and let FT, be the poten-
tial of the plate B. Since the lines of force between the
two plates are parallel, the surface densities of the plates
are of opposite sign and numerically equal. Then the
electric intensity, or the force on a positive unit, between
the plates is 4-Trcr, an attraction of 2?ro- by the fixed plate,
and a repulsion of 2?ro- by the movable plate. The two
plates are equi potential surfaces and T 7 ! - V 2 is the work
which must be done on a positive unit to convey it from
to B. Therefore, since work equals the product of force
and distance,



where D is the distance between the fixed and movable
plates.

The electric intensity at due to the charge on B is
27TO-. If 8 is the area of the movable plate (7, the charge on
it is Sa. Therefore the normal mechanical force pulling
the plate downward is .

F = 2-TTcr XiSa- = 27T(T 2 iS.



Whence "



w



By substitution in the equation above we have



Now F is known from the weights previously applied, and

/o rr

8 can be measured ; ^ 77- is therefore the constant of

the instrument. If F is measured in dynes, S in square
centimetres, and D in centimetres, the measurement of .Z>
determines the difference of potential in absolute measure.



ELECTRICAL POTENTIAL



197



Practically there is great difficulty in measuring D with
sufficient accuracy. Hence a different method of measure-
ment is adopted. The plate B is kept charged to a definite
potential, and the disk C is first connected to the earth,
whose potential is zero, and B is adjusted in height till C is
in the sighted position; a reading of the micrometer is
then taken. The conductor to be tested is then connected
with C and another adjustment of B is made and a reading
is taken. Let the distances between B and C for the two
adjustments be D and D'. Then we have for the potential
of C






It is then necessary to determine the difference DD 1
only, and this can always be done with great accuracy.

In the most elaborate modern instruments the disk C is
suspended by small springs, and both are protected from
inductive influence by a cylindrical metal cover.

146. The Quadrant Electrometer (J. J. T., 98).-
The force F measured by the instrument just described





Fig. 76.



varies as the square of the potential difference. When
this potential difference diminishes, the force falls off very
rapidly. For this reason the instrument is not suitable for



198



ELECTRICITY AND MAGNETISM.



the measurement of very small potential differences ; for
these Lord Kelvin devised the quadrant electrometer.

The most essential parts are the cage, or quadrants, and
the needle (Fig. 76). The needle, a thin oblong piece of
aluminium with broad rounded ends, shown in dotted out-
line in the figure, is suspended by a very fine wire or fibre

so as to turn in a horizontal plane
around a vertical axis. It swings
centrally within four quadrants, a,
5, <?, d, which together form a short
hollow cylinder with parallel ends.
Opposite quadrants, as a and <?, and
b and 6?, are connected electrically.
The needle is supported on a stiff
wire carrying a mirror M (Fig. 77)
at the top, and connecting at the
bottom with the jar B by a fine
platinum wire dipping into sul-
phuric acid

Consider the needle charged posi-
tively. If all the quadrants are at
the same potential, the needle will
take a position depending only on
the torsion of the suspending fibre ; but if a and , for ex-
ample, be at a higher potential than b and df, the forces
acting on both ends of the needle form a couple which will
turn it opposite to watch-hands. If the potential of a and
c is lower than that of the other pair of quadrants, the
needle will turn the other way ; it will come to rest when
there is equilibrium between the. two couples, the one due
to the electrical forces, and the other to the torsion of the
suspending fibre.

Let Fo denote the potential of the needle, V\ and V>> the




Fig. 77.



ELECTRICAL POTENTIAL. 199

potentials of the two pairs of quadrants, and 6 the angular
deflection of the needle; then the equation of equilib-
rium is

= C'(F 1 -F 2 ){F;,-4(F"i+ r s )}, . . (a)

where is a constant. 1

If K, be very large in comparison with the other poten-
tials, the term J ( Fj + F^) may be neglected in comparison
with it, and

0=<7(F,-r,) K, (6)

or the deflection is proportional to the difference of poten-
tial to be measured. The sensibility is proportional to Fi,
the potential of the needle.

When the needle is thus charged from a source inde-
pendent of the quadrants, the instrument is said to be used
heterostatically.

147. Quadrant Electrometer used Idiostatically. -
For the measurements of larger potential differences the
needle is connected with one pair of quadrants, so that
there is only one source of electrification, and this use of
the electrometer is called idiostatic. We may then put V Q
equal to V\ , and equation (# ) becomes

i0'(r l -F3?,

or the deflection 01 the needle is proportional to the square
of the potential difference of the quadrants. The physical
explanation is that doubling the potential doubles the
charges on the quadrants and the needle; and since the
force is proportional to the product of these charges,
the force is quadrupled.

For measuring large potential differences the quadrant

1 J. J. Thomson's Elements of Electricity and Magnetism, p. 103.



200 ELECTRICITY AND MAGNETISM.

electrometer, or electrostatic voltmeter, may be used idio-
statically in a different way. 1 If the suspension is provided
with a torsion head and a horizontal scale, graduated in
equal divisions, the charged needle may be brought back
to its initial or zero position by turning the torsion head
and twisting the suspending fibre. This adjustment is
made either by a telescope, or by means of a beam of light
reflected from the mirror M. The forces are. then propor-
tional to the angular twist of the suspending fibre, and the
potential difference to the square root of this twist. In
this way potentials from 10 volts upwards may be readily
measured.

PROBLEMS.

1. What would be the potential difference between A and B
(Fig. 73) if O were charged with 100 units of -f- electricity, the dis-
tance r being 10 cms. and r' 15 cms. ?

2. Positive charges of 150, 424, and 300 units are placed at the
three corners A, B, C, of a square 30 cms. on a side. Calculate the
potential at the fourth corner D.

3. Positive charges of 50 units are placed at the three corners of
an equilateral triangle whose sides are 50 cms. Find the potential
at the centre of the circumscribing circle.

4. What would be the potential at the same point in the last prob-
lem if the charges were placed at the middle points of the three
sides?

5. Find the potential at the centre of the square in problem 2,
and the work to be done to bring a -f- unit from D to the centre.

6. A sphere 10 cms. in diameter is charged with 50 units of posi-
tive electricity. Find the potential at the surface of the sphere, and
at a point 20 cms. distant from its surface.



1 Carhart and Patterson's Electrical Measurements, p. 200.



OU/MC7/T AND CONDENSERS. 201



CHAPTER XIV.

CAPACITY AND CONDENSERS.

148. Definition of Capacity. The electrical capacity
of a conductor is denned as the numerical value of its
charge when its potential is unity, all other conductors
within its field being at zero potential. Since the potential
of such a conductor is directly proportional to its charge,
the charge corresponding to unit potential, or its capacity,
may be found by dividing its total charge by the number
of units of potential to which it is raised ; or, in symbols,

c-Q

F*

wl it-re C denotes the capacity. Also

Q= GTand r=-^-.

C

149. Capacity of an Insulated Sphere. The capacity
of a sphere at a great distance from all other conductors is
numerically equal to its radius in centimetres. For the
potential of such a sphere is Q / r.

Hence C= = Q = r.

V ' )

The radius must be expressed in centimetres because the
centimetre is the unit of length employed in defining the
unit of quantity.

Two spheres of unequal radii when charged to the same



202 ELECTRICITY AND MAGNETISM.

potential have surface densities inversely as their radii.
For

Q QV__ r
47T/-' 47rr" ~~ 4?rr



Therefore a varies inversely as r, or,, for the same
potential,

<r\ r.,
o-> r {

If a small sphere is connected to a large one by a fine wire,
and if it is then supposed to diminish in size while its po-
tential remains unchanged, the surface density on it will
vary inversely as its radius. If it becomes indefinitely
small, its surface density becomes indefinitely great. The
electric intensity just at its surface increases therefore as its
diameter decreases. This relation explains the discharging
power of points.

15O. Condensers. Two conductors placed near to-
gether, but insulated from each other, form with the
dielectric a condenser. The effect of the additional con-
ductor is to increase the charge without any increase of
potential. In other words, the capacity of the one conductor
is greatly increased by the presence of the other. If the
charges are equal and opposite in sign, the charge on either
conductor when the potential difference between the two is
unity is called the capacity of the condenser.

Let a horizontal brass plate with rounded edges be
mounted on an insulating glass standard, and let a plate
of glass CD (Fig. 78), larger than the brass plate, be placed
on the latter. On this place another brass plate of the
same dimensions as the lower one. Connect one plate with
one electrode of an influence machine, and the other plate



.
UNIVukS!

( '. I rACITY\dftI) CQjfAEfpdERS.



203




\



A



with the other electrode, and charge them. If now they
are disconnected from the machine and the upper one be
touched with the finger, the attached pith-balls, which must
be hung with linen threads, will
collapse. But if the upper plate
l)e lifted by its insulating stem,
the pith-balls will again diverge
and a small spark may be drawn
from the plate. The two metal
disks and the glass plate consti-
tute a condenser.

If the upper plate be charged
positively, its positive charge at-
tracts a nearly equal negative
el large on the lower plate, and the
two are bound so long as the
plates remain in position close to-
gether. The induction takes place
through the glass, better in fact
than through air.

Let the plates be again charged
as before. If then one end of a
bent wire be placed in contact
with one plate and the other end
be made to touch the other plate, a bright electric spark
will pass just before the second contact is made. The
charge of either plate is evidently greatly augmented by
tin- presence 1 of the other. If one plate be connected to the
source of electrification and the other to the earth, then the
former is called the collecting plate and the latter the con-
<L .<ln <j plate ; the insulator between them is the dielectric,
or the medium through which the mutual electric action
between the plates takes place.




Fig. 78.



204



ELECTRICITY AND MAGNETISM.



151. Capacity of Two Concentric Spheres. Let the
radius of the inner sphere be r and that of the inner surface
of the outer one r 1 (Fig. 79), and let the outer sphere be
connected to the earth. Then its potential
and that of all other neighboring bodies is
zero. Hence, since lines of force connect
only bodies of different potentials, all the
lines of force from the insulated charged
sphere A run to the outer sphere B. Their
charges are then equal and of opposite sign,
+ Q and - Q.

The potential at 0, the common centre
of the two spheres, is




Fig. 79.



But this is the potential of the inner sphere because the
potential inside a charged conductor is the same as at any
point on its surface. From the last equation



Q=V



rr
1 r



When V becomes unity the charge by definition is the
capacity, or rr/



r > _ r



When r' r is very small, that is, when the two spherical
surfaces are very near together, the capacity becomes very
large. The expression for the capacity is then
rr' _ r (r + t)

~r~~ ~~r

where t is the thickness of the dielectric. When t is very
small compared with r, this expression becomes
r 47TT 2 S



CAPACITY AND CONDENSE!;*. 205

where S is the surface of the sphere. The capacity per
unit area in air is therefore 1 / 4?r times the reciprocal of
the distance between the conductors.

If the outer sphere be supposed to expand indefinitely,
or to be removed, while the inner one is insulated, the
potential of the inner sphere will increase ; for



Now if r and r 1 are very nearly the same, the potential for
a given charge Q may be small ; but as r 1 increases, 1 / / '
becomes smaller, and at an infinite distance V= Q I r.
The condensing plate decreases the potential, therefore, in
the ratio of r' r to r 7 , the charge on the collecting plate
remaining the same. Or conversely, for the same potential,
the condensing plate increases the charge in the ratio of
r 7 to r 7 _ r.



152. Capacity of Two Parallel Plates. When the
plates are so close together that the curved lines at the
edges are negligible in comparison with the others, all
the lines may be conceived as straight and at right angles
to the plates. The capacity is then easily calculated. If
t is the distance separating the plates, or the thickness of
the air film as the dielectric, then the electric intensity
between the plates is uniform, and the work done in con-
veying a unit charge from the plate of higher potential to
the other is

V=Ft,

where V is the potential difference between the plates and
F is the electric intensity.

The surface densities will be equal and of opposite sign,
-f cr on the one of higher potential and a on the other.
Then the electric intensity between the plates is 4?rcr, half



206



ELECTRICITY AND MAGNETISM.



of this expression being due to the charge on one plate and
the other half to the other, as before explained. Therefore

F= 47TO-,

and from the last equation V=
If A is the area of each plate,



O



When Fis unity the charge on one of the plates of ami
A is A I 47r, and this by definition is the capacity. This
expression is the same as that for the capacity of two con-
centric spheres.

153. The Leyden Jar. The Leyden jar was the earli-
est form of condenser. It was discovered accidentally by
Cuneus at Leyden in 1746 while attempting to collect
"electric fluid" in a bottle half filled with
water and held in the hand. The water was
connected with an electric machine. While
holding the bottle in one hand and attempting
to remove the connecting chain with the other
Cuneus received an unexpected shock, from
which it took two days to recover his mental
equilibrium. It is evident that the water in
the bottle served as the collecting plate and
the hand as the condensing plate, the glass
being the dielectric.

As now made a Leyden jar consists of a
wide-mouthed jar of thin flint glass, coated within and with-
out with tin foil for about three-fourths of its height (Fig.
80). The metal knob is connected to the inner coating by
a rod terminating in a short piece of chain. The jar may
be charged by holding it in the hand, touching the knob to



Fig. 80.



CAPACITY AND CONDENSERS. 207

one electrode of an influence machine, and bringing the

outer coating so near the other electrode that a series of

sparks will pass across. If charged too

highly it will discharge along the glass

<>ver the top. A hissing, crackling sound

indicates a partial brush discharge over

the surface of the glass above the tin foil.

It may be safely discharged by a dis-

charger (Fig. 81) held by the glass

handles, one ball being brought into con-

tact with the outer coating and the other

with the knob.

If A is the area of the tin foil and t the thickness of the
glass, then if the space between the tin-foil coatings were
tilled with air, the capacity would be approximately




since this case is practically the same as that of two parallel
plates, if t is small in comparison with the radius of the jar.
It will be explained shortly that the effect of interposing
the glass instead of air between the two coatings is to
increase the capacity by a factor K, so that



K is a constant depending on the kind of glass, and varies
from about 4 to 10 for different specimens.

154. Residual Charge. - - If a Leyden jar be left
standing for a few minutes after it has been discharged, the
two coatings will gradually acquire a small potential differ-
ence and a small discharge can be again obtained from it.
This is called the /v#/7Mt// charge. Several of them, of



208 ELECTRICITY AND MAGNETISM.

decreasing potentials, may sometimes be observed. The
magnitude of the residual charge depends upon the original
potential difference to which the jar was charged, the length
of time it is left charged, and the kind of glass of which
it is made.


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