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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.

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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