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direction and magnitude of the resultant electric force.

The direction of the force is best expressed by the device

of lines of force. A line of force must be conceived so

drawn in the electric field that a tangent to it at any point

represents the direction of the electric intensity at the

point. For brevity the expression "force at a point" is

used to signify the intensity of the force sustained by unit

quantity of the active agent at the point, or the electric

intensity at the point.

Lines of electric force always spring from a

positively electrified surface and end in a nega-

tively electrified one. The stress along these

lines is a tension, tending to shorten them. It

is accompanied by a pressure at right angles to

the lines and tending to separate them.

When one electrified body attracts another,

the two are drawn together by these taut lines

of force stretching between them. When two

plates oppositely electrified face each other (Fig.

47), lines of electric force stretch across from

the positive to the negative, and the tension in

the medium tends to draw the plates together.

111. Equal Charges of Opposite Sign. When a body

is electrically excited by friction, the body rubbed and the

rubber are equally electrified, but with charges of opposite

sign. The equality consists in the ability of the one

ELECTUK CHARGES. 157

charge to exactly neutralize the other. If a stick of seal-

ing-wax, provided with a flannel cap with a silk cord

attached (Fig. 48), be excited by turning it around a few

times inside the cap, it will not attract a positively electri-

fied pith-ball if the cap be left on ; but if the cap be with-

drawn by the cord, the sealing-wax will attract the pith

and the cap will repel it.

The electrification of a body consists

in the separation of two equal charges

of opposite sign against their mutual at-

traction. Hence the medium between

them is strained by the operation, and

work is done. A positively charged _

conductor, insulated by supports of glass,

shellac, silk, or other non-conductors, is connected to other

bodies by invisible lines of electric force, springing from the

positive charge and extending to the equal negative one on

surrounding bodies. The slightest charge of positive elec-

tricity at one point always means an equal charge of the

opposite sign as near to it as the conductivity of the

dielectrics permits.

Whatever operations of electrically exciting, discharg-

ing, and the like, may be carried on within an insulated

conducting chamber, no signs of excitation will be ex-

hibited without. The positive and negative excitations

exactly neutralize each other outside the chamber.

112. Electroscopes. An electroscope is an instru-

ment for detecting electric charges. The simplest one,

which was employed by Gilbert, consists of a long straw,

turning freely on a sharp point, which must be insulated

from the earth. A pith-ball suspended by a silk thread is

also a convenient sensitive electroscope.

158

ELECTRICITY AND MAGNETISM.

-The Gold-leaf Electroscope is still more sensitive.

Through the top of a glass jar passes a brass rod, terminat-

ing in a ball above, and bent at right angles below to

receive two strips of gold leaf (Fig. 49). The top of the

jar should be coated with shellac both within and without.

Two strips of tin foil are pasted inside the jar from the

bottom up to the lower level

of the gold leaves to prevent

the latter from sticking to

the glass when they are vio-

lently repelled.

If the knob be touched

with a positively electrified

glass tube the leaves will be

mutually repelled with -f

charges. The approach of

any other charged body will

cause them to diverge more

widely if the charge pre-

Fi g . 49. sented is +, and to approach

each other if it is .

113. Charge External. - When a conductor is electri-

fied by friction or by electricity conveyed to it from some

external source, the charge always resides on the outside.

Biot devised a direct demonstration by fitting to an insu-

lated copper ball two hemispherical copper shells. When

the whole was charged and the shells were then deftly

removed by glass handles, the charge was found to be

entirely removed with them. A simple demonstration

of the law is afforded by a hollow metal sphere with a

hole at the top and insulated on a glass stem (Fig. 50).

It may be tested by means of a proof-plane, which is

ELECTRIC CHARGES.

159

composed of a small metal disk with a shellac or ebonite

handle. If the proof -plane be applied to the outside of the

charged sphere, a small charge

may be removed and tested by an

electroscope. If the proof-plane

l>e passed through the hole in the

sphere and applied to the inner

surface, it will be found on with-

drawal to exhibit no trace of elec-

trification. The proof-plane may

be charged from the outside of

the sphere, and then be made to

touch the interior. It will lose

all its charge and will show none

on withdrawal.

Faraday constructed a cube 12

feet on each side and covered it

with tin foil. He went inside of

it with his electroscopes ; but

while it was charged so that long

flashes were given off from the

outside, he could detect no signs of electrification within.

114. Distribution of Charge. The quantity of elec-

tricity (119) on unit surface of a conductor, or the ratio

of the quantity on any small area to the area itself, is

called the surface density. The distribution of an electric

charge is not such as to give uniform surface density over

an insulated conductor, except in the case of a sphere

remote from other conductors and electrified bodies. The

distribution on conductors of various shapes was investi-

gated by Coulomb by means of the proof-plane and torsion

balance (116). The following is a summary of results:

Fig. 50.

160 ELECTRICITY AND MAGNETISM.

(1) On a cylinder with rounded ends the surface den-

sity is greatest at the ends.

(2) On a flat disk the density is much greater at the

edges than on the flat surfaces, but over the latter the dis_

tribution is fairly uniform except near the edges.

(3) With two spheres in contact the charge is nothing at

the point of contact, increases rapidly between 30 and 60

from that point, and becomes greatest at 180. When the

spheres are of unequal size, the density at corresponding

points is greater on the small sphere than on the large one.

The density is greatest on those parts of a conductor

which project most and have the greatest convexity.

Hence at sharp points, such as that of a needle, the density

is very great, and as a consequence the charge escapes

rapidly from them. It is therefore necessary to round

off all edges of insulated conductors and to make them

smooth.

115. Redistribution of Charge. Coulomb demon-

strated that when a charged conducting sphere is brought

into contact with an identical one in the neutral state,

each will then possess a quantity equal to half of the

original charge. If the second sphere, instead of being

neutral, is itself charged, the final charges are again equal.

Each of them is half the algebraic sum of the initial

charges, so that both spheres will be neutral if those

charges were equal and of opposite sign.

The result will be the same with two like conductors of

any form whatever when one touches the other, provided

they are symmetrical with respect to the point of contact.

If this condition of symmetry is not fulfilled, the charges

will divide unequally, but so that their algebraic sum

always equals that of the initial charge*.

ELECTRIC CHARGE*.

161

Since the charge resides on the outside, if a small charged

sphere be introduced into a larger hollow one, it will give

up its charge entirely to the larger sphere. By this. means

a conductor may be charged by successive additions of

small quantities, or one can increase or decrease the electric

charge on the outside of a closed surface by introducing

within small positive or negative charges.

116. Coulomb's Torsion Balance. The torsion balance

was invented by Coulomb for the pur-

pose of investigating the law of at-

traction and repulsion between two

charges of electricity. The instru-

ment is now obsolete, but it illus-

trates the meaning of the law of

inverse squares which was established

by Coulomb's elaborate experiments.

From a torsion head 7i (Fig. 51)

is suspended a very fine wire, carry-

ing at its lower end a light shellac

rod with a gilt pith-ball b. The shel-

lac rod swings inside a protecting glass

case, around which is a graduated

scale s at the level of the gilt ball. A

shellac rod, carrying another gilt ball

'-. can be introduced through a hole in

the top of the case. The torsion head

is divided into degrees, and is pro-

vided with an index. The rod carrying the torsion wire

can be turned independently of the rest of the head, so that

the index can be held at zero, while the rod and wire are

turned till the movable ball just touches the fixed one

without anv torsion of the wire. Calcium chloride, or

Fig. 51.

162 ELECTRICITY AND MAGNETISM.

some other drying agent, is placed in the case to keep

the air dry.

117. Law of Inverse Squares. When the instrument

has been set as described, the vertical rod is removed, the

attached ball is charged, and is then replaced in the instru-

ment. It touches the ball -b and divides its charge with it.

Repulsion follows, and the ball b moves away till the tor-

sion couple of the suspending wire equals the moment of

the force due to the mutual repulsion. The distance

between the balls is not sensibly different from the arc

of the circle separating them, if the balls are not many

degrees apart. The balls are now made to approach each

other by turning the torsion head and twisting the wire.

The two divided circles then give the whole angle of tor-

sion of the wire. The principle employed in comparing

the forces is that when a wire is twisted, the couple of

torsion is proportional to the angle through which the wire

is twisted. For example, if the moments of the couples

required to twist a wire through 10 and through 20 are

measured, the latter will be found to be twice as great as

the former.

The following data belong to one of Coulomb's experi-

ments : The first deflection of the movable ball was 36.

To reduce it to 18 it was found necessary to turn the head

through 126; and for a further reduction to 8. 5 an addi-

tional rotation of 441 was required. The several relative

distances of the balls were then about as 1 to J to J , and

the torsion of the wire was 36 for the first distance,

126 + 18 = 144 for the second, and 441 + 126 + 8.5 = 575.5

for the third. But 144 is 4 x 36, and 575.5 is nearly 16 x 36 ;

so that as the distance is reduced successively to $ and J ,

the force is increased to 4 and 16 times respectively.

ELECTRIC CHARGES. 163

The law of attraction was also investigated by a similar

method, and was found to hold within the same limits.

Also the dependence of the force on the charge was

examined by touching one of the balls with an insulated

one of the same size. Half of the charge was thus

removed, and the force was found to be reduced to one-

half. If the charge of either ball was reduced, the mutual

force was reduced in the same ratio.

118. Second Law of Electrostatics. The second law

of electrostatic action, established by the experiments of

Coulomb, may be enunciated as follows: The force between

twv ///'/ ///<''-? bodies is directly proportional to the product of

the tiro charges, and inversely proportional to the square of

the distance between them.

The law of distance does not hold unless the dimensions

of the charged conductors are very small in comparison

with the distance between them. The charge on a sphere

acts as if it were collected at its centre (121) only when

the distribution of this charge is not affected by the

influence of neighboring charges. In Coulomb's experi-

ment the actual mean distance of the two charges when

the balls were brought as near together as 8 C .5 was greater

than the distance between the centres of the spheres. The

force between two flat disks near each other does not vary

appreciably with a moderate change in the distance.

If the t\\o quantities q and '/are on infinitesimal spheres,

the distance oi' whose centres is r, then the force between

them may he expressed by the formula

The positive sign corresponds to similar charges, and there-

fore to repulsion, and the negative sign to attraction.

164 ELECTRICITY AND MAGNETISM.

119. The Unit of Quantity. - - The definition of the

electrostatic unit of quantity is derived from the law of

attraction and repulsion. If the force in the foregoing

proportion is to become unity when the distance and the

charges are unity, unit quantity must be defined as fol-

lows: The electrostatic unit of electricity is that quantity

which repels an equal and similar quantity, at a distance of

one centimetre in air, with a force of one dyne.

Since the intensity of an electric force is the force

exerted on unit quantity, it follows that the electric inten-

sity at a point distant r centimetres in air from a charge q

is q /r z . The reason for inserting the expression "in air"

will appear later (165).

120. Indirect Proof of the Law of Inverse Squares.

It has already been pointed out

that no electric force can be detected

inside a hollow conductor. This ex-

perimental fact furnishes the basis

of the most conclusive proof that

the force varies inversely as the

square of the distance.

The following may be considered

Fig 52 as an illustration of the principle

rather than a rigid mathematical dem-

onstration : Let P (Fig. 52) be any point within a charged

conducting sphere, and let a narrow cone of two sheets be

described with P as the apex, and cutting the sphere in

two areas and s / at ab and a'l' respectively. Then, since

the surface density is supposed to be uniform, the quantities

on the two areas are proportional to those areas; but the

areas are proportional to the squares of their respective

distances from P. To prove this latter relation, it must

ELECTKTC CHARGES. 165

be noted, first, that the two areas are sections of the cone

equally inclined to its axis. Let ab and a'b' (Fig. 53) be

oblique sections of a cone making the same angle with the

axis. Their linear dimensions are directly proportional to

the distances PA and PB ; and since the areas of similar

figures are proportional to the squares of their homologous

dimensions, the areas of the two sections are proportional

to the squares of PA and PB.

It follows that the two quantities on s and s' P

are proportional to the squares of Pa and Pa'-

Hence the two forces acting on P are directly

proportional to the squares of Pa and Pa', and

inversely proportional to some function of these

distances. But since there is no force inside a

charged sphere, and since the whole surface may

be divided into a series of such pairs of sections Fj 53

made by a cone, and what is true of the whole

is true of each pair, it follows that the forces due to the

charges on s and s' are equal to each other. But the only

function of the distances which will satisfy this condition

is the inverse square. The forces are proportional to the

acting quantities, which are directly proportional to the

squares of the distances ; the forces are also inversely pro-

portional to the squares of the same distances ; and, being

opposite in direction, the resultant is zero.

121. Force Outside a Charged Sphere. The force or

electric intensity at any point outside a charged sphere,

over which the distribution is uniform, is the same as if

the entire charge were collected at its centre. This propo-

sition admits of simple demonstration.

Let P be the point at a distance D from the centre of

the sphere (Fig. 54). Let a be the surface density, and

160

ELECTRICITY AND MAGNETISM.

let s be the area of a very small element of the surface at

the point B. The quantity on it is so-, and if p is the

distance PB, the force at P due to this element of the

charge is scr/p'. Since the entire surface of the sphere is

Fig. 54.

symmetrical with respect to the line PO, the resultant of

all the forces due to the several elements of the charge

must be along P 0. The component of the. force scr/p' along

this line is g(r

f cos a,

P*

where a is the angle OPB.

Draw BA, making the angle AB equal

to a. Also let o) be the solid angle which

the area s subtends at A. The projection s'

of the area s at right angles to AB subtends

the same angle co at A. Since the angle be-

tween s and s / is a (Fig. 55), we may write

or

s'= (or- = s cos a.

cor

cos a

Substituting this value of s in the expression for/, and

ELECTRIC CHARGE*. 167

The triangles OB A and OBP are similar, and therefore

rR

where D is the distiince J0. Hence by substituting above,

, R 2

f = &-

This is the force due to a single element of the surface.

For the entire surface the force is the sum of the small

forces due to all such elements, or

*=&,*.

The expression 2&> is the entire solid angle subtended by

the surface of the sphere at any point within it, and this is

4?r. Hence

But 7rR 2 (T is the product of surface of the sphere and the

surface density, or the whole charge on the sphere, and D

is the distance between the point P and the centre of the

sphere. Therefore the expression for F is precisely the

same as would be obtained for the force at P if the whole

charge were at the centre of the sphere. It is worth noting

that this demonstration applies equally well to the force of

gravity due to a thin shell of matter, when the shell is of

uniform thickness and density.

122. Force very near a Charged Sphere. If the

point P in Fig. 54 is made to approach the sphere, the point

A also moves toward the surface to meet P ; and when P

is at the surface D equals R and

F= 47TO-,

or the electric intensity is independent of the size of the

168 ELECTRICITY AND MAGNETISM.

sphere, and is numerically equal to 4?r times the surface

density. This result, which is known as Coulomb's Law,

requires modification when the sphere is surrounded by

some other dielectric than air. It applies to any charged

conductor. Since there is no force inside the sphere, the

change of force in passing from a point just outside to the

interior is 4?rcr.

If a plane perpendicular to P be drawn through J., it

will divide the spherical surface into two parts, each of

which subtends at A the same angle 2?r. Hence half the

force is due to the charge to the right of this dividing plane,

and the other half to the charge to the left of it. At the

surface of the sphere one of these charges is contained on

an infinitesimal area, and the other is the charge on all the

rest of the sphere. The force is then the same as that due

to a plane of indefinite extent, tangent at C and charged

on both sides.

123. Force near a Charged Plane Conductor (Th.,

262). Imagine a plane of indefinite extent charged

positively on one side to

a density a. Let P be

the point at which the

force is to be determined

(Fig, 56), and PO the

normal to the plane. Let

8 be any small surface on

the plane, and co the solid

angle which it subtends

Fig . 55 at P. It is the solid

angle at the apex of the

cone made by drawing lines from the boundary of s to

the point P. The force at P due to the charge on this

ELEi'TlUC f'lfMWKS. 169

element is sa/r. and the component of this force along the

normal PO is

/. scr

/=__ cos a,

where a is the angle between the normal and the axis of

the cone.

As in Art. 121, the orthogonal section of the cone

x' = cor and

$' = cor = s cos a.

Therefore s

cos a

Substituting in the equation for/, we have

f= aco.

Since the resultant of all the forces due to the elementary

charges is along the normal, the total intensity of the force

at Pis

But 2o> is the solid angle subtended at P by a plane of

indefinite extent, and this is the angle subtended by a

whole hemisphere, or 2?r. Therefore

In the C.<T.S. system the force is in dynes.

If the plane is limited and the point P indefinitely near

it, the force is again ZTTCT.

Since the force on a + unit above the plane is directed

upward and below the plane downward, in passing through

the plane the force changes by the quantity 4?ro-.

PROBLEMS.

1. Two equal small balls are charged with -|-30 and 6 units of

electricity respectively. Find the mutual force between them when

their centres are C cms. apart, before and after contact with each

other.

170 ELECTRICITY AND MAGNETISM.

2. A charge of 100 units is applied to a sphere of 10 cms. radius.

What is the surface density ?

3. In the last problem, Avhat is the value of the electric intensity

at the surface ?

4. Two small balls, each one gm. in mass, are suspended from

the same point by silk fibres 490 cms. long. If g is 980 dynes, show

that the balls will diverge to a distance of one cm. if each is charged

with one unit of electricity.

5. Two small spheres 10 cms. apart are charged with -}- 5 units

and 5 units respectively. Find the direction and magnitude of the

force acting on a -j- unit at a distance of 10 cms. from both charges.

ELECTRIFICATION BY ISFLUESCE. 171

CHAPTER XII.

ELECTRIFICATION BY INFLUENCE.

124. Fundamental Phenomena. - A charged con-

ductor exerts influence, or acts inductively, on all neigh-

boring bodies. If it be positively charged, lines of electric

force start from it and proceed to an equal negative quan-

tity on adjacent bodies. The influence is exerted along

these lines of force, or lines of tension.

Let an insulated

sphere A (Fig. 57), ^ ^"

charged positively,

be placed near an in-

sulated cylindrical

conductor 5. Light

pith-balls suspended by linen threads at either end of B

will diverge, and the nearer A approaches B the wider the

divergence, unless the charges on A and B unite by a

spark across the air-gap. If A and both ends of B be now

examined by means of a proof-plane and an electroscope,

it will be found that the charge on A has been redistributed,

so that the surface density on the side toward B is greater

than on the remote side ; also the end a of the cylinder will

be found to be negatively charged, the central portion will

be neutral, and the end b will be positively charged. The

density at b will be less than at #, and the neutral line will

be somewhat nearer a than b.

172 ELECTRICITY AND MAGNETISM.

When A is removed or discharged by connecting with

the earth, all signs of electrification on B disappear. The

separation of the positive and negative charges on B

through the influence of the charge on A is called electro-

static induction, or electrification by influence.

125. Charging by Influence. If the conductor B be

connected with the earth while under the inductive influ-

ence of A, the repelled charge will pass off, leaving only

the attracted electricity. This latter charge is said to be

" bound " in distinction from the " free " charge which goes

to the earth. If now A be removed while B remains insu-

lated, the charge on the latter will be distributed over the

whole conductor, and B is said to have been charged by

influence or induction.

The electrification of B represents energy. Work has

been done in removing A against the attraction of the

charge on B. If B uninsulated were to be brought up to

A from a distance, and then removed after insulating it, the

work done by mutual attraction during the approach would

be less than that done against the attraction during the

withdrawal, because the acting charge on B in the latter

movement remains constant, while during the approach of

B to A the charge on B increases from nothing to the

maximum. The working force is then less during the

approach than during the recession.

If when the charge has been insulated on B the posi-

tive on A is discharged to earth, the electrification of B

still represents energy. The energy of the discharge of A

under these conditions is less than that required to charge

it when removed from inductive action on other bodies.

This will be better understood after studying the relation

between energy and potential.

ELECTRIFICATION BY INFLUENCE. 173

126. Electrification with like Charges by Influence.

- Wl leu a body is charged by influence as explained in

the last article, the repelled charge always becomes free,

and the conductor is charged so that the inducing and the

induced charges are of opposite sign. In this case pro-

The direction of the force is best expressed by the device

of lines of force. A line of force must be conceived so

drawn in the electric field that a tangent to it at any point

represents the direction of the electric intensity at the

point. For brevity the expression "force at a point" is

used to signify the intensity of the force sustained by unit

quantity of the active agent at the point, or the electric

intensity at the point.

Lines of electric force always spring from a

positively electrified surface and end in a nega-

tively electrified one. The stress along these

lines is a tension, tending to shorten them. It

is accompanied by a pressure at right angles to

the lines and tending to separate them.

When one electrified body attracts another,

the two are drawn together by these taut lines

of force stretching between them. When two

plates oppositely electrified face each other (Fig.

47), lines of electric force stretch across from

the positive to the negative, and the tension in

the medium tends to draw the plates together.

111. Equal Charges of Opposite Sign. When a body

is electrically excited by friction, the body rubbed and the

rubber are equally electrified, but with charges of opposite

sign. The equality consists in the ability of the one

ELECTUK CHARGES. 157

charge to exactly neutralize the other. If a stick of seal-

ing-wax, provided with a flannel cap with a silk cord

attached (Fig. 48), be excited by turning it around a few

times inside the cap, it will not attract a positively electri-

fied pith-ball if the cap be left on ; but if the cap be with-

drawn by the cord, the sealing-wax will attract the pith

and the cap will repel it.

The electrification of a body consists

in the separation of two equal charges

of opposite sign against their mutual at-

traction. Hence the medium between

them is strained by the operation, and

work is done. A positively charged _

conductor, insulated by supports of glass,

shellac, silk, or other non-conductors, is connected to other

bodies by invisible lines of electric force, springing from the

positive charge and extending to the equal negative one on

surrounding bodies. The slightest charge of positive elec-

tricity at one point always means an equal charge of the

opposite sign as near to it as the conductivity of the

dielectrics permits.

Whatever operations of electrically exciting, discharg-

ing, and the like, may be carried on within an insulated

conducting chamber, no signs of excitation will be ex-

hibited without. The positive and negative excitations

exactly neutralize each other outside the chamber.

112. Electroscopes. An electroscope is an instru-

ment for detecting electric charges. The simplest one,

which was employed by Gilbert, consists of a long straw,

turning freely on a sharp point, which must be insulated

from the earth. A pith-ball suspended by a silk thread is

also a convenient sensitive electroscope.

158

ELECTRICITY AND MAGNETISM.

-The Gold-leaf Electroscope is still more sensitive.

Through the top of a glass jar passes a brass rod, terminat-

ing in a ball above, and bent at right angles below to

receive two strips of gold leaf (Fig. 49). The top of the

jar should be coated with shellac both within and without.

Two strips of tin foil are pasted inside the jar from the

bottom up to the lower level

of the gold leaves to prevent

the latter from sticking to

the glass when they are vio-

lently repelled.

If the knob be touched

with a positively electrified

glass tube the leaves will be

mutually repelled with -f

charges. The approach of

any other charged body will

cause them to diverge more

widely if the charge pre-

Fi g . 49. sented is +, and to approach

each other if it is .

113. Charge External. - When a conductor is electri-

fied by friction or by electricity conveyed to it from some

external source, the charge always resides on the outside.

Biot devised a direct demonstration by fitting to an insu-

lated copper ball two hemispherical copper shells. When

the whole was charged and the shells were then deftly

removed by glass handles, the charge was found to be

entirely removed with them. A simple demonstration

of the law is afforded by a hollow metal sphere with a

hole at the top and insulated on a glass stem (Fig. 50).

It may be tested by means of a proof-plane, which is

ELECTRIC CHARGES.

159

composed of a small metal disk with a shellac or ebonite

handle. If the proof -plane be applied to the outside of the

charged sphere, a small charge

may be removed and tested by an

electroscope. If the proof-plane

l>e passed through the hole in the

sphere and applied to the inner

surface, it will be found on with-

drawal to exhibit no trace of elec-

trification. The proof-plane may

be charged from the outside of

the sphere, and then be made to

touch the interior. It will lose

all its charge and will show none

on withdrawal.

Faraday constructed a cube 12

feet on each side and covered it

with tin foil. He went inside of

it with his electroscopes ; but

while it was charged so that long

flashes were given off from the

outside, he could detect no signs of electrification within.

114. Distribution of Charge. The quantity of elec-

tricity (119) on unit surface of a conductor, or the ratio

of the quantity on any small area to the area itself, is

called the surface density. The distribution of an electric

charge is not such as to give uniform surface density over

an insulated conductor, except in the case of a sphere

remote from other conductors and electrified bodies. The

distribution on conductors of various shapes was investi-

gated by Coulomb by means of the proof-plane and torsion

balance (116). The following is a summary of results:

Fig. 50.

160 ELECTRICITY AND MAGNETISM.

(1) On a cylinder with rounded ends the surface den-

sity is greatest at the ends.

(2) On a flat disk the density is much greater at the

edges than on the flat surfaces, but over the latter the dis_

tribution is fairly uniform except near the edges.

(3) With two spheres in contact the charge is nothing at

the point of contact, increases rapidly between 30 and 60

from that point, and becomes greatest at 180. When the

spheres are of unequal size, the density at corresponding

points is greater on the small sphere than on the large one.

The density is greatest on those parts of a conductor

which project most and have the greatest convexity.

Hence at sharp points, such as that of a needle, the density

is very great, and as a consequence the charge escapes

rapidly from them. It is therefore necessary to round

off all edges of insulated conductors and to make them

smooth.

115. Redistribution of Charge. Coulomb demon-

strated that when a charged conducting sphere is brought

into contact with an identical one in the neutral state,

each will then possess a quantity equal to half of the

original charge. If the second sphere, instead of being

neutral, is itself charged, the final charges are again equal.

Each of them is half the algebraic sum of the initial

charges, so that both spheres will be neutral if those

charges were equal and of opposite sign.

The result will be the same with two like conductors of

any form whatever when one touches the other, provided

they are symmetrical with respect to the point of contact.

If this condition of symmetry is not fulfilled, the charges

will divide unequally, but so that their algebraic sum

always equals that of the initial charge*.

ELECTRIC CHARGE*.

161

Since the charge resides on the outside, if a small charged

sphere be introduced into a larger hollow one, it will give

up its charge entirely to the larger sphere. By this. means

a conductor may be charged by successive additions of

small quantities, or one can increase or decrease the electric

charge on the outside of a closed surface by introducing

within small positive or negative charges.

116. Coulomb's Torsion Balance. The torsion balance

was invented by Coulomb for the pur-

pose of investigating the law of at-

traction and repulsion between two

charges of electricity. The instru-

ment is now obsolete, but it illus-

trates the meaning of the law of

inverse squares which was established

by Coulomb's elaborate experiments.

From a torsion head 7i (Fig. 51)

is suspended a very fine wire, carry-

ing at its lower end a light shellac

rod with a gilt pith-ball b. The shel-

lac rod swings inside a protecting glass

case, around which is a graduated

scale s at the level of the gilt ball. A

shellac rod, carrying another gilt ball

'-. can be introduced through a hole in

the top of the case. The torsion head

is divided into degrees, and is pro-

vided with an index. The rod carrying the torsion wire

can be turned independently of the rest of the head, so that

the index can be held at zero, while the rod and wire are

turned till the movable ball just touches the fixed one

without anv torsion of the wire. Calcium chloride, or

Fig. 51.

162 ELECTRICITY AND MAGNETISM.

some other drying agent, is placed in the case to keep

the air dry.

117. Law of Inverse Squares. When the instrument

has been set as described, the vertical rod is removed, the

attached ball is charged, and is then replaced in the instru-

ment. It touches the ball -b and divides its charge with it.

Repulsion follows, and the ball b moves away till the tor-

sion couple of the suspending wire equals the moment of

the force due to the mutual repulsion. The distance

between the balls is not sensibly different from the arc

of the circle separating them, if the balls are not many

degrees apart. The balls are now made to approach each

other by turning the torsion head and twisting the wire.

The two divided circles then give the whole angle of tor-

sion of the wire. The principle employed in comparing

the forces is that when a wire is twisted, the couple of

torsion is proportional to the angle through which the wire

is twisted. For example, if the moments of the couples

required to twist a wire through 10 and through 20 are

measured, the latter will be found to be twice as great as

the former.

The following data belong to one of Coulomb's experi-

ments : The first deflection of the movable ball was 36.

To reduce it to 18 it was found necessary to turn the head

through 126; and for a further reduction to 8. 5 an addi-

tional rotation of 441 was required. The several relative

distances of the balls were then about as 1 to J to J , and

the torsion of the wire was 36 for the first distance,

126 + 18 = 144 for the second, and 441 + 126 + 8.5 = 575.5

for the third. But 144 is 4 x 36, and 575.5 is nearly 16 x 36 ;

so that as the distance is reduced successively to $ and J ,

the force is increased to 4 and 16 times respectively.

ELECTRIC CHARGES. 163

The law of attraction was also investigated by a similar

method, and was found to hold within the same limits.

Also the dependence of the force on the charge was

examined by touching one of the balls with an insulated

one of the same size. Half of the charge was thus

removed, and the force was found to be reduced to one-

half. If the charge of either ball was reduced, the mutual

force was reduced in the same ratio.

118. Second Law of Electrostatics. The second law

of electrostatic action, established by the experiments of

Coulomb, may be enunciated as follows: The force between

twv ///'/ ///<''-? bodies is directly proportional to the product of

the tiro charges, and inversely proportional to the square of

the distance between them.

The law of distance does not hold unless the dimensions

of the charged conductors are very small in comparison

with the distance between them. The charge on a sphere

acts as if it were collected at its centre (121) only when

the distribution of this charge is not affected by the

influence of neighboring charges. In Coulomb's experi-

ment the actual mean distance of the two charges when

the balls were brought as near together as 8 C .5 was greater

than the distance between the centres of the spheres. The

force between two flat disks near each other does not vary

appreciably with a moderate change in the distance.

If the t\\o quantities q and '/are on infinitesimal spheres,

the distance oi' whose centres is r, then the force between

them may he expressed by the formula

The positive sign corresponds to similar charges, and there-

fore to repulsion, and the negative sign to attraction.

164 ELECTRICITY AND MAGNETISM.

119. The Unit of Quantity. - - The definition of the

electrostatic unit of quantity is derived from the law of

attraction and repulsion. If the force in the foregoing

proportion is to become unity when the distance and the

charges are unity, unit quantity must be defined as fol-

lows: The electrostatic unit of electricity is that quantity

which repels an equal and similar quantity, at a distance of

one centimetre in air, with a force of one dyne.

Since the intensity of an electric force is the force

exerted on unit quantity, it follows that the electric inten-

sity at a point distant r centimetres in air from a charge q

is q /r z . The reason for inserting the expression "in air"

will appear later (165).

120. Indirect Proof of the Law of Inverse Squares.

It has already been pointed out

that no electric force can be detected

inside a hollow conductor. This ex-

perimental fact furnishes the basis

of the most conclusive proof that

the force varies inversely as the

square of the distance.

The following may be considered

Fig 52 as an illustration of the principle

rather than a rigid mathematical dem-

onstration : Let P (Fig. 52) be any point within a charged

conducting sphere, and let a narrow cone of two sheets be

described with P as the apex, and cutting the sphere in

two areas and s / at ab and a'l' respectively. Then, since

the surface density is supposed to be uniform, the quantities

on the two areas are proportional to those areas; but the

areas are proportional to the squares of their respective

distances from P. To prove this latter relation, it must

ELECTKTC CHARGES. 165

be noted, first, that the two areas are sections of the cone

equally inclined to its axis. Let ab and a'b' (Fig. 53) be

oblique sections of a cone making the same angle with the

axis. Their linear dimensions are directly proportional to

the distances PA and PB ; and since the areas of similar

figures are proportional to the squares of their homologous

dimensions, the areas of the two sections are proportional

to the squares of PA and PB.

It follows that the two quantities on s and s' P

are proportional to the squares of Pa and Pa'-

Hence the two forces acting on P are directly

proportional to the squares of Pa and Pa', and

inversely proportional to some function of these

distances. But since there is no force inside a

charged sphere, and since the whole surface may

be divided into a series of such pairs of sections Fj 53

made by a cone, and what is true of the whole

is true of each pair, it follows that the forces due to the

charges on s and s' are equal to each other. But the only

function of the distances which will satisfy this condition

is the inverse square. The forces are proportional to the

acting quantities, which are directly proportional to the

squares of the distances ; the forces are also inversely pro-

portional to the squares of the same distances ; and, being

opposite in direction, the resultant is zero.

121. Force Outside a Charged Sphere. The force or

electric intensity at any point outside a charged sphere,

over which the distribution is uniform, is the same as if

the entire charge were collected at its centre. This propo-

sition admits of simple demonstration.

Let P be the point at a distance D from the centre of

the sphere (Fig. 54). Let a be the surface density, and

160

ELECTRICITY AND MAGNETISM.

let s be the area of a very small element of the surface at

the point B. The quantity on it is so-, and if p is the

distance PB, the force at P due to this element of the

charge is scr/p'. Since the entire surface of the sphere is

Fig. 54.

symmetrical with respect to the line PO, the resultant of

all the forces due to the several elements of the charge

must be along P 0. The component of the. force scr/p' along

this line is g(r

f cos a,

P*

where a is the angle OPB.

Draw BA, making the angle AB equal

to a. Also let o) be the solid angle which

the area s subtends at A. The projection s'

of the area s at right angles to AB subtends

the same angle co at A. Since the angle be-

tween s and s / is a (Fig. 55), we may write

or

s'= (or- = s cos a.

cor

cos a

Substituting this value of s in the expression for/, and

ELECTRIC CHARGE*. 167

The triangles OB A and OBP are similar, and therefore

rR

where D is the distiince J0. Hence by substituting above,

, R 2

f = &-

This is the force due to a single element of the surface.

For the entire surface the force is the sum of the small

forces due to all such elements, or

*=&,*.

The expression 2&> is the entire solid angle subtended by

the surface of the sphere at any point within it, and this is

4?r. Hence

But 7rR 2 (T is the product of surface of the sphere and the

surface density, or the whole charge on the sphere, and D

is the distance between the point P and the centre of the

sphere. Therefore the expression for F is precisely the

same as would be obtained for the force at P if the whole

charge were at the centre of the sphere. It is worth noting

that this demonstration applies equally well to the force of

gravity due to a thin shell of matter, when the shell is of

uniform thickness and density.

122. Force very near a Charged Sphere. If the

point P in Fig. 54 is made to approach the sphere, the point

A also moves toward the surface to meet P ; and when P

is at the surface D equals R and

F= 47TO-,

or the electric intensity is independent of the size of the

168 ELECTRICITY AND MAGNETISM.

sphere, and is numerically equal to 4?r times the surface

density. This result, which is known as Coulomb's Law,

requires modification when the sphere is surrounded by

some other dielectric than air. It applies to any charged

conductor. Since there is no force inside the sphere, the

change of force in passing from a point just outside to the

interior is 4?rcr.

If a plane perpendicular to P be drawn through J., it

will divide the spherical surface into two parts, each of

which subtends at A the same angle 2?r. Hence half the

force is due to the charge to the right of this dividing plane,

and the other half to the charge to the left of it. At the

surface of the sphere one of these charges is contained on

an infinitesimal area, and the other is the charge on all the

rest of the sphere. The force is then the same as that due

to a plane of indefinite extent, tangent at C and charged

on both sides.

123. Force near a Charged Plane Conductor (Th.,

262). Imagine a plane of indefinite extent charged

positively on one side to

a density a. Let P be

the point at which the

force is to be determined

(Fig, 56), and PO the

normal to the plane. Let

8 be any small surface on

the plane, and co the solid

angle which it subtends

Fig . 55 at P. It is the solid

angle at the apex of the

cone made by drawing lines from the boundary of s to

the point P. The force at P due to the charge on this

ELEi'TlUC f'lfMWKS. 169

element is sa/r. and the component of this force along the

normal PO is

/. scr

/=__ cos a,

where a is the angle between the normal and the axis of

the cone.

As in Art. 121, the orthogonal section of the cone

x' = cor and

$' = cor = s cos a.

Therefore s

cos a

Substituting in the equation for/, we have

f= aco.

Since the resultant of all the forces due to the elementary

charges is along the normal, the total intensity of the force

at Pis

But 2o> is the solid angle subtended at P by a plane of

indefinite extent, and this is the angle subtended by a

whole hemisphere, or 2?r. Therefore

In the C.<T.S. system the force is in dynes.

If the plane is limited and the point P indefinitely near

it, the force is again ZTTCT.

Since the force on a + unit above the plane is directed

upward and below the plane downward, in passing through

the plane the force changes by the quantity 4?ro-.

PROBLEMS.

1. Two equal small balls are charged with -|-30 and 6 units of

electricity respectively. Find the mutual force between them when

their centres are C cms. apart, before and after contact with each

other.

170 ELECTRICITY AND MAGNETISM.

2. A charge of 100 units is applied to a sphere of 10 cms. radius.

What is the surface density ?

3. In the last problem, Avhat is the value of the electric intensity

at the surface ?

4. Two small balls, each one gm. in mass, are suspended from

the same point by silk fibres 490 cms. long. If g is 980 dynes, show

that the balls will diverge to a distance of one cm. if each is charged

with one unit of electricity.

5. Two small spheres 10 cms. apart are charged with -}- 5 units

and 5 units respectively. Find the direction and magnitude of the

force acting on a -j- unit at a distance of 10 cms. from both charges.

ELECTRIFICATION BY ISFLUESCE. 171

CHAPTER XII.

ELECTRIFICATION BY INFLUENCE.

124. Fundamental Phenomena. - A charged con-

ductor exerts influence, or acts inductively, on all neigh-

boring bodies. If it be positively charged, lines of electric

force start from it and proceed to an equal negative quan-

tity on adjacent bodies. The influence is exerted along

these lines of force, or lines of tension.

Let an insulated

sphere A (Fig. 57), ^ ^"

charged positively,

be placed near an in-

sulated cylindrical

conductor 5. Light

pith-balls suspended by linen threads at either end of B

will diverge, and the nearer A approaches B the wider the

divergence, unless the charges on A and B unite by a

spark across the air-gap. If A and both ends of B be now

examined by means of a proof-plane and an electroscope,

it will be found that the charge on A has been redistributed,

so that the surface density on the side toward B is greater

than on the remote side ; also the end a of the cylinder will

be found to be negatively charged, the central portion will

be neutral, and the end b will be positively charged. The

density at b will be less than at #, and the neutral line will

be somewhat nearer a than b.

172 ELECTRICITY AND MAGNETISM.

When A is removed or discharged by connecting with

the earth, all signs of electrification on B disappear. The

separation of the positive and negative charges on B

through the influence of the charge on A is called electro-

static induction, or electrification by influence.

125. Charging by Influence. If the conductor B be

connected with the earth while under the inductive influ-

ence of A, the repelled charge will pass off, leaving only

the attracted electricity. This latter charge is said to be

" bound " in distinction from the " free " charge which goes

to the earth. If now A be removed while B remains insu-

lated, the charge on the latter will be distributed over the

whole conductor, and B is said to have been charged by

influence or induction.

The electrification of B represents energy. Work has

been done in removing A against the attraction of the

charge on B. If B uninsulated were to be brought up to

A from a distance, and then removed after insulating it, the

work done by mutual attraction during the approach would

be less than that done against the attraction during the

withdrawal, because the acting charge on B in the latter

movement remains constant, while during the approach of

B to A the charge on B increases from nothing to the

maximum. The working force is then less during the

approach than during the recession.

If when the charge has been insulated on B the posi-

tive on A is discharged to earth, the electrification of B

still represents energy. The energy of the discharge of A

under these conditions is less than that required to charge

it when removed from inductive action on other bodies.

This will be better understood after studying the relation

between energy and potential.

ELECTRIFICATION BY INFLUENCE. 173

126. Electrification with like Charges by Influence.

- Wl leu a body is charged by influence as explained in

the last article, the repelled charge always becomes free,

and the conductor is charged so that the inducing and the

induced charges are of opposite sign. In this case pro-

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