A Treatise on Electricity and Magnetism, Vol. 1 (1881) — part 8 of 28

Hence when x^—x^ is diminished without limit this term must ultimately vanish, leaving

where JPX is the value of X on the negative and X2 on the positive side of the surface.

But by Art. 78, ^-^ = -- = 477(7, (18)

so that we may write

Here dydz is the element of the surface, a is the surface-density,

90 ELECTROSTATICS. [80.

and J(X2 + Xl) is the arithmetical mean of the electromotive in- tensity on the two sides of the surface.

Hence an element of a charged surface is acted on by a force, the component of which normal to the surface is equal to the charge of the element into the arithmetical mean of the normal electro- motive intensities on the two sides of the surface.

Since the other two components of the electromagnetic intensity are not discontinuous, there can be no ambiguity in estimating the corresponding components of the force acting on the surface.

We may now suppose the direction of the normal to the surface to be in any direction with respect to the axes, and write the general expres- sions for the components of the force on the element of surface dS,

A = ^(Xi + XJffdS,

5 = l(ri+r2)cr^, (20)

C =

Charged Surface of a Conductor.

80.] We have already shewn (Art. 72) that throughout the sub- stance of a conductor in electric equilibrium X = Y = Z = 0, and therefore V is constant.

dX dY dZ

Hence — + — + — = 4irp = 0,

dx dy dz

and therefore p must be zero throughout the substance of the conductor, or there can be no electricity in the interior of the con- ductor.

Hence a superficial distribution of electricity is the only possible distribution in a conductor in equilibrium.

A distribution throughout the mass of a body can exist only when the body is a non-conductor.

Since the resultant intensity within the conductor is zero, the resultant intensity just outside the conductor must be in the direc- tion of the normal and equal to 4:7(7, acting outwards from th> conductor.

This relation between the surface-density and the resultant in- tensity close to the surface of a conductor is known as Coulomb's Law, Coulomb having ascertained by experiment that the intensity of the electric force near a given point of the surface of a conductor is normal to the surface and proportional to the surface-density at the given point. The numerical relation

R = 477(7

was established by Poisson.

8 1.] CHARGED WIRE. 91

The force acting on an element, dS, of the charged surface of a conductor is, by Art. 79, (since the intensity is zero on the inner side of the surface,)

8 77

This force acts outwards from the conductor, whether the charge of the surface is positive or negative.

Its value in dynes per square centimetre is

\E<T = 27T(72 = — .B2,

8 77

acting as a tension outwards from the surface of the conductor.

81.] If we now suppose an elongated body to be electrified, we may, by diminishing its lateral dimensions, arrive at the conception of an electrified line.

Let ds be the length of a small portion of the elongated body, and let c be its circumference, and a the surface density of the electricity on its surface; then, if A. is the charge per unit of length, A = £cr, and the resultant electrical intensity close to the surface will be \

4 770- = 477-'

c

If, while A remains finite, c be diminished indefinitely, the in- tensity at the surface will be increased indefinitely. Now in every dielectric there is a limit beyond which the intensity cannot be increased without a disruptive discharge. Hence a distribution of electricity in which a finite quantity is placed on a finite portion of a line is inconsistent with the conditions existing in nature.

Even if an insulator could be found such that no discharge could be driven through it by an infinite force, it would be impossible to charge a linear conductor with a finite quantity of electricity, for an infinite electromotive force would be required to bring the electricity to the linear conductor.

In the same way it may be shewn that a point charged with a finite quantity of electricity cannot exist in nature. It is con- venient, however, in certain cases, to speak of electrified lines and points, and we may suppose these represented by electrified wires, and by small bodies of which the dimensions are negligible com- pared with the principal distances concerned.

Since the quantity of electricity on any given portion of a wire at a given potential diminishes indefinitely when the diameter of the wire is indefinitely diminished, the distribution of electricity on bodies of considerable dimensions will not be sensibly affected by

92 ELECTEOSTATICS. [82.

the introduction of very fine metallic wires into the field, such as are used to form electrical connexions between these bodies and the earth, an electrical machine, or an electrometer.

On Lines of Force.

82.] If a line be drawn whose direction at every point of its course coincides with that of the resultant intensity at that point, the line is called a Line of Force.

In every part of the course of a line of force, it is proceeding from a place of higher potential to a place of lower potential.

Hence a line of force cannot return into itself, but must have a beginning and an end. The beginning of a line of force must be in a positively charged surface, and the end of a line of force must be in a negatively charged surface.

The beginning and the end of the line are called corresponding points on the positive and negative surface respectively.

If the line of force moves so that its beginning traces a closed curve on the positive surface, its end will trace a corresponding closed curve on the negative surface, and the line of force itself will generate a tubular surface called a tube of induction. Such a tube is called a Solenoid *.

Since the force at any point of the tubular surface is in the tangent plane, there is no induction across the surface. Hence if the tube does not contain any electrified matter, by Art. 77 the total induction through the closed surface formed by the tubular surface and the two ends is zero, and the values of

/ /

R cos € dS for the two ends must be equal in magnitude

but opposite in sign.

If these surfaces aro the surfaces of conductors e = 0 and R = — lira,

and RcosedS becomes — 4?r / / a dS, or the charge of the sur-

face multiplied by 47r.

Hence the positive charge of the surface enclosed within the closed curve at the beginning of the tube is numerically equal to the negative charge enclosed within the corresponding closed curve at the end of the tube.

• From acaXrjv, a tube. Faraday uses (3271) the terra ' Sphondyloid ' in the same sense. rfa

82.] LINES OF FORCE. 93

Several important results may be deduced from the properties of lines of force.

The interior surface of a closed conducting- vessel is entirely free from charge, and the potential at every point within it is the same as that of the conductor, provided there is no insulated and charged body within the vessel.

For since a line of force must begin at a positively charged surface and end at a negatively charged surface, and since no charged body is within the vessel, a line of force, if it exists within the vessel, must begin and end on the interior surface of the vessel itself.

But the potential must be higher at the beginning of a line of force than at the end of the line, whereas we have proved that the potential at all points of a conductor is the same.

Hence no line of force can exist in the space within a hollow vessel, provided no charged body be placed inside it.

If a conductor within a closed hollow vessel is placed in com- munication with the vessel, its potential becomes the same as that of the vessel, and its surface becomes continuous with the inner surface of the vessel. The conductor is therefore free from charge.

If we suppose any charged surface divided into elementary por- tions such that the charge of each element is unity, and if solenoids having these elements for their bases are drawn through the field of force, then the surface-integral for any other surface will be re- presented by the number of solenoids which it cuts. It is in this sense that Faraday uses his conception of lines of force to indicate not only the direction but the amount of the force at any place in the field.

We have used the phrase Lines of Force because it has been used by Faraday and others. In strictness, however, these lines should be called Lines of Electric Induction.

In the ordinary cases the lines of induction indicate the direction and magnitude of the resultant electromotive intensity at every point, because the intensity and the induction are in the same direction and in a constant ratio. There are other cases, how- ever, in which it is important to remember that these lines indi- cate primarily the induction, and that the intensity is directly indicated by the equipotential surfaces, being normal to these surfaces and inversely proportional to the distances of consecutive surfaces.

94: ELECTROSTATICS. [83 a.

On Specific Inductive Capacity.

83# .] In the preceding investigation of surface-integrals we have adopted the ordinary conception of direct action f a£ a distance, and have not taken into consideration any effects Depending on the nature of the dielectric medium in which the forces are observed.

But Faraday has observed that the quantity of electricity in- duced by a given electromotive force on the surface of a conductor which bounds a dielectric is not the same for all dielectrics. The induced electricity is greater for most solid and liquid dielectrics than for air and gases. Hence these bodies are said to have a greater specific inductive capacity than air, which he adopted as the standard medium.

We may express the theory of Faraday in mathematical language by saying that in a dielectric medium the induction across any surface is the product of the normal electric force into the coefficient of specific inductive capacity of that medium. If we denote this coefficient by K, then in every part of the investigation of sur- face-integrals we must multiply JT, Y, and Z by K, so that the equation of Poisson will become

d ^dV d -dV d _dV

-j-.K-j- + -r-K-r + T-'K-T~ +^P = 0. (l)

dx dx dy dy dz dz

At the surface of separation of two media whose inductive capa- cities are K± and K2, and in which the potentials are T[ and ^2, the characteristic equation may be written

where v19 v2 are the normals drawn in the two media, and a- is the true surface-density on the surface of separation ; that is to say, the quantity of electricity which is actually on the surface in the form of a charge, and which can be altered only by con- veying electricity to or from the spot.

Apparent distribution of Electricity.

835.] If we begin with the actual distribution of the potential and deduce from it the volume density p' and the surface density <r' on the hypothesis that K is everywhere equal to unity, we may call p' the apparent volume density and </ the apparent surface density, because a distribution of electricity thus defined would account for the actual distribution of potential, on the hypothesis that the law

83 &.] SPECIFIC INDUCTIVE CAPACITY. 95

of electric force as given in Art. 66 requires no modification on account of the different properties of dielectrics.

The apparent charge of electricity within a given region may increase or diminish without any passage of electricity through the bounding surface of the region. We must therefore distinguish it from the true charge, which satisfies the equation of continuity.

In a heterogeneous dielectric in which K varies continuously, if p' be the apparent volume-density,

d2F

• -j-T + TT + 47T/ = 0- (3)

dy1 dzz v '

Comparing this with the equation above, we find

d-Kd7 dKdV dKdY

4w(p— JT/) + — — +—— + — — =0. (4)

^ ' dx dx dy dy dz dz

The true electrification, indicated by p, in the dielectric whose variable inductive capacity is denoted by K, will produce the same potential at every point as the apparent electrification, denoted by //, would produce in a dielectric whose- inductive capacity is every- where equal to unity.

The apparent surface charge, a-', is that deduced from the electrical forces in the neighbourhood of the surface, using the ordinary characteristic equation

Ji+*S+4,y-0. (5)

rf&i dv2

If a solid dielectric of any form is a perfect insulator, and if its surface receives no charge, then the true electrification remains zero, whatever be the electrical forces acting on it.

T d Hence K -=-

The surface-density v is that of the apparent electrification produced at the surface of the solid dielectric by induction. It disappears entirely when the inducing force is removed, but if during the action of the inducing force the apparent electrification of the surface is discharged by passing a flame over the surface, then, when the inducing force is taken away, there will appear a true electrification opposite to c/ *.

• See Faraday's • Remarks on Static Induction/ Proceedings of the Royal In- stitution, Feb. 12, 1858.

CHAPTEE III.

ON ELECTRICAL WORK AND ENERGY IN A SYSTEM OF CONDUCTORS.

84.] On the J7orJc whicJi must be done by an external agent in order to charge an electrified system in a given manner.

The work spent in bringing a quantity of electricity be from an infinite distance (or from any place where the potential is zero) to a given part of the system where the potential is 7, is, by the defi- nition of potential (Art. 70), 7 be.

The effect of this operation is to increase the charge of the given part of the system by be, so that if it was e before, it will become e + be after the operation.

"We may therefore express the work done in producing a given alteration in the charges of the system by the integral

; (1)

where the summation, (2), is to be extended to all parts of the electrified system.

It appears from the expression for the potential in Art. 73, that the potential at a given point may be considered as the sum of a number of parts, each of these parts being the potential due to a corresponding part of the charge of the system.

Hence if 7 is the potential at a given point due to a system of charges which we may call 2 (e\ and V the potential at the same point due to another system of charges which we may call 2 (/), the potential at the same point due to both systems of charges existing together would be 7+ 7'.

If, therefore, every one of the charges of the system is altered in the ratio of n to 1 , the potential at any given point in the system will also be altered in the ratio of n to 1 .

85 a.] WORK DONE IN CHARGING A SYSTEM. 97

Let us, therefore, suppose that the operation of charging the system is conducted in the following manner. Let the system be originally free from charge and at potential zero, and let the different portions of the system be charged simultaneously, each at a rate proportional to its final charge.

Thus if e is the final charge, and V the final potential of any part of the system, then, if at any stage of the operation the charge is ne, the potential will be nF, and we may represent the process of charging by supposing n to increase continuously from 0 to 1.

While n increases from n to n + bn, any portion of the system whose final charge is e, and whose final potential is F, receives an increment of charge ebn, its potential being nFt so that the work done on it during this operation is eFnbn.

Hence the whole work done in charging the system is

^.eV \ ndn-^(eV\ (2)

M

or half the sum of the products of the charges of the different portions of the system into their respective potentials.

This is the work which must be done by an external agent in order to charge the system in the manner described, but since the system is a conservative system, the work required to bring the system into the same state by any other process must be the same.

"We may therefore call

r=-|2(*r) (3)

the electric energy of the system, expressed in terms of the charges of the different parts of the system and their potentials.

85 a.~\ Let us next suppose that the system passes from the state (e, F) to the state (/, V) by a process in which the different charges increase simultaneously at rates proportional for each to its total increment / — e.

If at any instant the charge of a given portion of the system is e + n(J—e\ its potential will be V+n(V'—V\ and the work done in altering the charge of this portion will be

e'-e) [7+n(7'- F)]dn = *(/-<?) (F+ F);

o

so that if we denote by W the energy of the system in the state

(*'> n

W'-W=^(e'-e)(Y'+r). (4)

VOL. I. H

98 SYSTEM OF CONDUCTOKS.

But r=*S(*F)»

and r'=|S(/r).

Substituting these values in equation (4) we find

S(«F) = S(/F). (5)

Hence if, in the same fixed system of electrified conductors, we consider two different states of electrification, the sum of the products of the charges in the first state into the potentials of the corresponding portions of the conductors in the second state, is equal to the sum of the products of the charges in the second state into the potentials of the corresponding conductors in the first state.

This result corresponds, in the elementary theory of electricity, to Green's Theorem in the analytical theory. By properly choosing the initial and final state of the system, we may deduce a number of useful results.

85 $.] From (4) and (5) we find another expression for the in- crement of the energy, in which it is expressed in terms of the increments of potential,

w-w-=^(e'+e)(V'-v). (6)

If the increments are infinitesimal, we may write (4) and (6)

dr=2(F6tf) = S(<?dF), (7)

and if we denote by We and Wv the expressions for W in terms of the charges and the potentials of the system respectively, and by Ar, er, and Vr a particular conductor of the system, its charge, and its potential, then

<•)

86.] If in any fixed system of conductors, any one of them, which we may denote by Ati is without charge, both in the initial and final state, then for that conductor et — 0, and e{ = 0, so that the terms depending on At vanish from both members of equation (5).

If another conductor, say Au, is at potential zero in both states of the system, then ?u = 0 and 7U' = 0, so that the terms depending on Au vanish from both members of equation (5).

If, therefore, all the conductors except two, Ar and Js, are either

86] RECIPROCAL RELATIONS. 99

insulated and without charge, or else connected to the earth, equation (5) is reduced to the form

•rr?+«.V = 'T'rr+et'rf (10)

If in the initial state

er = 1 and et = 0, and in the final state

<=0 and */=!,

equation (10) becomes p?= ?9; (11)

or if a unit charge communicated to Ar raises A8 to a potential V, then a unit charge communicated to As will raise Ar to the same potential F, provided that every one of the other conductors of the system is either insulated and without charge, or else connected to earth so that its potential is zero.

This is the first instance we have met with in electricity of a reciprocal relation. Such reciprocal relations occur in every branch of science, and often enable us to deduce the solution of new problems from those of simpler problems already solved.

Thus from the fact that at a point outside a conducting sphere whose charge is 1 the potential is r"1, where r is the distance from the centre, we conclude that if a small body whose charge is 1 is placed at a distance r from the centre of a conducting sphere without charge, it will raise the potential of the sphere to r"1.

Let us next suppose that in the initial state

Pr = 1 and F8 = 0, and in the final state

J7= 0 and 77= 1,

equation (10) becomes ea = £/; (12)

or if, when Ar is raised to unit potential, a charge e is induced on At9 then if A8 is raised to unit potential, an equal charge e will be induced on Ar.

Let us suppose in the third place, that in the initial state

Tr — 1 and e8 = 0, and that in the final state

J?= 0 and */= 1, equation (10) becomes in this case

<V'+7.= 0. (13)

Hence if when A8 is without charge, the operation of charging Ar to potential unity raises A8 to potential F, then if Ar is kept

H %

100 SYSTEM OF CONDUCTOKS. [87.

at potential zero, a unit charge communicated to As will induce on Ar a negative charge, the numerical value of which is V.

In all these cases we may suppose some of the other conductors to be insulated and without charge, and the rest to be connected to earth.

The third case is an elementary form of one of Green's theorems. As an example of its use let us suppose that we have ascertained the distribution of electric charge on the different elements of a conducting system at potential zero, induced by a charge unity communicated to a given body As of the system.

Let rjr be the charge of Ar under these circumstances. Then if we suppose As without charge, and the other bodies raised each to a different potential, the potential of As will be

F. = -2(r,r7r). (U)

Thus if we have ascertained the surface density at any given point of a hollow conducting vessel due to a unit charge placed at a given point within it, then, if we know the value of the potential at every point of a surface of the same size and form as the interior surface of the vessel, we can deduce the potential at a point within it the position of which corresponds to that of the unit charge.

Hence if the potential is known for all points of a closed surface it may be determined for any point within the surface, if there be no electrified body within it, and for any point outside, if there be no electrified body outside.

Theory of a system of conductors.

87.] Let Alt A2t ... An be n conductors of any form; let elf e^ ,., en be their charges; and 7J, 7J, ... Vn their potentials.

Let us suppose that the dielectric medium which separates the conductors remains the same, and does not become charged with electricity during the operations to be considered.

"We have shown in Art. 84 that the potential of each conductor is a homogeneous linear function of the n charges.

Hence since the electric energy of the system is half the sum of the products of the potential of each conductor into its charge, the electric energy must be a homogeneous quadratic function of the n charges, of the form

(l 5)

The suffix e indicates that W is to be expressed as a function

87.] COEFFICIENTS OF POTENTIAL AND OF INDUCTION. 101

of the charges. When W is written without a suffix it denotes the expression (3), in which both charges and potentials occur.

From this expression we can deduce the potential of any one of the conductors. For since the potential is denned as the work which must be done to bring a unit of electricity from potential zero to the given potential, and since this work is spent in increasing W, we have only to differentiate We with respect to the charge of the given conductor to obtain its potential. We thus obtain

T - «--- +jPri«r.-. +Al«i» "I

(16)

Vn = plnel ... + prner ... +pnnen, J

a system of n linear equations which express the n potentials in terms of the n charges.

The coefficients prs &c., are called coefficients of potential. Each has two suffixes, the first corresponding with that of the charge, and the second with that of the potential.

The coefficient prr, in which the two suffixes are the same, denotes the potential of Ar when its charge is unity, that of all the other conductors being zero. There are n coefficients of this kind, one for each conductor.

The coefficient jors, in which the two suffixes are different, denotes the potential of A8 when Ar receives a charge unity, the charge of each of the other conductors, except Ar , being zero.

We have already proved in Art. 86 thatj?rg = psr, but we may prove it more briefly by considering that

~~ der ~ der des ~~ de8 der ~~ de9 ~^s>"

The number of different coefficients with double suffix is there- fore J n (n— 1), being one for each pair of conductors.

By solving the equations (16) for elt e2 &c., we obtain n equations giving the charges in terms of the potentials

(18)

102 SYSTEM OF CONDUCTORS. [87.

We have in this case also qr8 = qsr, for

der d dWy d dWv des

, .

By substituting the values of the charges in the equation for the electric energy

r=iOi*i+-+«r5. ..+«.£], (20)

we obtain an expression for the energy in terms of the potentials

(21)

A coefficient in which the two suffixes are the same is called the Electric Capacity of the conductor to which it belongs.

Definition. The Capacity of a conductor is its charge when its own potential is unity, and that of all the other conductors is zero.

This is the proper definition of the capacity of a conductor when no further specification is made. But it is sometimes convenient to specify the condition of some or all of the other conductors in a different manner, as for instance to suppose that the charge of certain of them is zero, and we may then define the capacity of the conductor under these conditions as its charge when its potential is unity.

The other coefficients are called coefficients of induction. Any one of them, as qra denotes the charge of Ar when Ag is raised to potential unity, the potential of all the conductors except A8 being zero.

The mathematical calculation of the coefficients of potential and of capacity is in general difficult. We shall afterwards prove that they have always determinate values, and in certain special cases we shall calculate these values. We shall also shew how they may be determined by experiment.

When the capacity of a conductor is spoken of without specifying the form and position of any other conductor in the same system, it is to be interpreted as the capacity of the conductor when no other conductor or electrified body is within a finite distance of the conductor referred to.

It is sometimes convenient, when we are dealing with capacities and coefficients of induction only, to write them in the form [ A . P], this symbol being understood to denote the charge on A when P is raised to unit potential.

In like manner [(A + H) . (P + Q)] would denote the charge on

89 #•] PROPERTIES OF THE COEFFICIENTS. 103

A + B when P and Q are both raised to potential 1, and it is manifest that since

[(A+S)(P+Q)-\ =

the compound symbols may be combined by addition and multipli- cation as if they were symbols of quantity.

The symbol [A . A] denotes the charge on A when the potential of A is 1, that is to say, the capacity of A.

In like manner [( A + B)(A + Q)] denotes the sum of the charges on A and B when A and Q are raised to potential 1, the potential of all the conductors except A and Q being zero.

It may be decomposed into

[A.A-] + [A.£] + [A.Q] + [B.Q].

The coefficients of potential cannot be dealt with in this way. The coefficients of induction represent charges, and these charges can be combined by addition, but the coefficients of potential represent potentials, and if the potential of A is \ and that of B is T'g, the sum ^-t-7J has no physical meaning bearing on the phenomena, though V^— P2 represents the electromotive force from A to B.

The coefficients of induction between two conductors may be expressed in terms of the capacities of the conductors and that of the two conductors together, thus :

[A.S] =

Dimensions of the coefficients.

& 88.] Since the potential of a charge e at a distance r is -,

the dimensions of a charge of electricity are equal to those of the product of a potential into a line.

The coefficients of capacity and induction have therefore the same dimensions as a line, and each of them may be represented by a straight line, the length of which is independent of the system of units which we employ.

For the same reason, any coefficient of potential may be repre- sented as the reciprocal of a line.

On certain conditions which the coefficients must satisfy.

890.] In the first place, since the electric energy of a system is an essentially positive quantity, its expression as a quadratic

104

SYSTEM OF CONDUCTORS.

function of the charges or of the potentials must be positive, whatever values, positive or negative, are given to the charges or the potentials.

Now the conditions that a homogeneous quadratic function of n variables shall be always positive are n in number, and may be written

0,

Pll '"Pin ,

• • • • >  0. 
        

(22)

Pnl ' ' ' Pnn ,

These n conditions are necessary and sufficient to ensure that W shall be essentially positive *.

But since in equation (16) we may arrange the conductors in any order, every determinant must be positive which is formed sym- metrically from the coefficients belonging to any combination of the n conductors, and the number of these combinations is 2M— 1.

Only n, however, of the conditions so found can be independent.

The coefficients of capacity and induction are subject to con- ditions of the same form.

89 £.] The coefficients of potential are all positive , lut none of the coefficients prs is greater than prr or pss.

JPor let a charge unity be communicated to Ar, the other con- ductors being uncharged. A system of equipotential surfaces will be formed. Of these one will be the surface of Ar, and its potential will be prr. If As is placed in a hollow excavated in Ar so as to be completely enclosed by it, then the potential of As will also be prr.

If, however, A8 is outside of Ar its potential prs will lie between prr and zero.

For consider the lines of force issuing from the charged con- ductor Ar. The charge is measured by the excess of the number of lines which issue from it over those which terminate in it. i Hence, if the conductor has no charge, the number of lines which \ enter the conductor must be equal to the number which issue from it. The lines which enter the conductor come from places of greater potential, and those which issue from it go to places of less poten-

• See Williamson's Differential Calculus, 3rd edition, p. 407.

89 dJ] PROPERTIES OF THE COEFFICIENTS. 105

tial. Hence the potential of an uncharged conductor must be intermediate between the highest and lowest potentials in the field, and therefore the highest and lowest potentials cannot belong to any of the uncharged bodies.

The highest potential must therefore be prr, that of the charged body Ar, the lowest must be that of space at an infinite distance, which is zero, and all the other potentials such as pr8 must lie between prr and zero.

If As completely surrounds At, thenj9r8 =prt.

89<?.] None of the coefficients of induction are positive ', and the sum of all those belonging to a single conductor is not numerically greater than the coefficient of capacity of that conductor ; which is always positive.

For let Ar be maintained at potential unity while all the other conductors are kept at potential zero, then the charge on Ar is qrr) and that on any other conductor As is qr8 .

The number of lines of force which issue from Ar is qrr . Of these some terminate in the other conductors, and some may proceed to infinity, but no lines of force can pass between any of the other conductors or from them to infinity, because they are all at poten- tial zero.

No line of force can issue from any of the other conductors such as A8, because no part of the field has a lower potential than As. If Ag is completely cut off from Ar by the closed surface of one of the conductors, then qrs is zero. If A8 is not thus cut off, qrs is a negative quantity.

If one of the conductors At completely surrounds ArJ then all the lines of force from Ar fall on At and the conductors within it, and the sum of the coefficients of induction of these conductors with respect to Ar will be equal to qrr with its sign changed. But if Ar is not completely surrounded by a conductor the arithmetical sum of the coefficients of induction qrs, &c. will be less than qrr.

We have deduced these two theorems independently by means of electrical considerations. We may leave it to the mathematical student to determine whether one is a mathematical consequence of the other.

89 d.~\ When there is only one conductor in the field its coefficient of potential on itself is the reciprocal of its capacity.

The centre of mass of the electricity when there are no external forces is called the electric centre of the conductor. If the conductor

106 SYSTEM OF CONDUCTOKS. [896.

is symmetrical about a centre of figure, this point is the electric centre. If the dimensions of the conductor are small compared with the distances considered, the position of the electric centre may be estimated sufficiently nearly by conjecture.

The potential at a distance c from the electric centre must be

between e az e # 2

_(l + _.) and -(l-i—);

c \ C2 / c \ C* '

where e is the charge, and a is the greatest distance of any part of the surface of the body from the electric centre.

For if the charge be concentrated in two points at distances a on opposite sides of the electric centre, the first of these expressions .is the potential at a point in the line joining the charges, and the second at a point in a line perpendicular to the line joining the charges. For all other distributions within the sphere whose radius is a the potential is intermediate between those values.

If there are two conductors in the field, their mutual coefficient

of potential is - , where c' cannot differ from c, the distance between

a? --l2 the electric centres, by more than - — ; a and b being the greatest

distances of any part of the surfaces of the bodies from their re- spective electric centres.