The Mathematical Theory of Electricity and Magnetism (5th ed, 1927) — part 6 of 39

forming a single parallel plate condenser of capacity j— -% , so that the capacity

of the compound condenser is (n — 1) KA/4nrd. By making n large and d small, we can make this capacity large without causing the apparatus to occupy an unduly large amount of space. For this reason standard con- densers are usually made of this pattern.

90. Guard Ring. In both the condensers described the capacity can only be calculated approximately. Lord Kelvin has devised a modification of the parallel plate condenser in which the error caused by the irregularities of the lines of force near the edges is dispensed with, so that it is possible accurately to calculate the capacity from measurements of the plates.

The principle consists in making one plate B of the condenser larger than the second plate A, the remainder of the space opposite B being occupied by a " guard ring " G which fits A so closely as almost to touch, and is in the same plane with it. The guard ring G and the plate A, if at the same potential, may without serious error be regarded as forming a single plate of a parallel plate condenser of which the other plate is B. The irregularities in the tubes of force now occur at the outer edge of the guard ring G, while the lines of force from A to B are perfectly straight and uniform. Thus if A is the area of the plate A its capacity may be supposed, with great accuracy, to be

4ivd'

where d is the distance between the plates A and B.

89-92] Mechanical Force 79

Submarine Gables.

91. Unfortunately for practical electricians, a submarine cable forms a condenser, of which the capacity is frequently very considerable. The effect of this upon the transmission of signals will be discussed later. A cable consists generally of a core of strands of copper wire surrounded by a layer of insulating material, the whole being enclosed in a sheathing of iron wire. This arrangement acts as a condenser of the type of the coaxal cylinders investigated in § 82, the core forming the inner cylinder whilst the iron sheathing and the sea outside form the outer cylinder.

In the capacity formula obtained in § 82, namely

K

»•©■

let us suppose that b = 2a, and that K = 3'2, this being about the value for the insulating material generally used. Using the value loge 2 = "69315, we find a capacity of 231 electrostatic units per unit length. Thus a cable 2000 miles in length has a capacity equal to that of a sphere of radius 2000 x 231 miles, i.e., of a sphere greater than the earth. In practical units, the capacity of such a cable would be about 827 microfarads.

Mechanical Force on a Conducting Surface.

92. Let  Q  be  any  point  on  the  surface  of  a  conductor,  and  let  the 
    

surface-density at the point Q be <x. Let us draw any small area dS

Fig. 36.

enclosing Q. By taking dS sufficiently small, we may regard the area as perfectly plane, and the charge on the area will be adS. The electricity on the remainder of the conductor will exert forces of attraction or repulsion on the charge <rdS, and these forces will shew themselves as a mechanical force acting on the element of area dS of the conductor. We require to find the amount of this mechanical force.

80 Conductors and Condensers [ch. ih

The electric intensity at a point near Q and just outside the conductor is 4-7TO-, by Coulomb's Law, and its direction is normally away from the surface. Of this intensity, part arises from the charge on dS itself, and part from the charges on the remainder of the conductor. As regards the first part, which arises from the charge on dS itself, we may notice that when we are con- sidering a point sufficiently close to the surface, the element dS may be treated as an infinite electrified plane, the electrification being of uniform density <r. The intensity arising from the electrification of dS at such a point is accordingly an intensity 2ircr normally away from the surface. Since the total intensity is 4>tt<t normally away from the surface, it follows that the intensity arising from the electrification of the parts of the conductor other than dS must also be 2ira normally away from the surface. It is the forces composing this intensity which produce the mechanical action on dS. The charge on dS being adS, the total force will be 2ira-dS normally away from the surface. Thus per unit area there is a force 2tto-2 tending to repel the charge normally away from the surface. The charge is prevented from leaving the surface of the conductor by the action between electricity and matter which has already been explained. Action and reaction being equal and opposite, it follows that there is a mechanical force 27rcr2 per unit area acting normally outwards on the material surface of the conductor.

Remembering that R = 4"7rcr, we find that the mechanical force can also

R2

be expressed as ^— per unit area.

07T

93. Let us try to form some estimate of the magnitude of this mechanical force as compared with other mechanical forces with which we are more familiar. We have already mentioned Maxwell's estimate that a gramme of gold, beaten into a gold-leaf one square metre in area, can hold a charge of 60,000 electrostatic units. This gives 3 units per square centimetre as the charge on each face, giving for the intensity at the surface,

R = 4nra = 38 C.G.S. units,

and for the mechanical force

i?2

2-77- cr2 = ^— = 56 dynes per sq. cm.

Lord Kelvin, however, found that air was capable of sustaining a tension of 9600 grains wt. per sq. foot, or about 700 dynes per sq. cm. This gives R = 130, a = 10.

7?3 Taking R = 100 as a large value of R, we find ^— = 400 dynes per

sq. cm. The pressure of a normal atmosphere is

1,013,570 dynes per sq. cm.,

92-94] Electrified Soap-Bubble 81

so that the force on the conducting surface would be only about ^^ of an atmosphere : say *3 mm. of mercury.

If a gold-leaf is beaten so thin that 1 gm. occupies 1 sq. metre of area, the weight of this is '0981 dyne per sq. cm. In order that 2-rra2 may be equal to '0981, we must have <r = -1249. Thus a small piece of gold-leaf would be lifted up from a charged surface on which it rested as soon as the surface acquired a charge of about | of a unit per sq. cm.

Electrified Soap-Bubble.

94. As has already been said, this mechanical force shews itself well on electrifying a soap-bubble.

Let us first suppose a closed soap-bubble blown, of radius a. If the atmospheric pressure is IT, the pressure inside will be somewhat greater than II, the resulting outward force being just balanced by the tension of the surface of the bubble. If, however, the bubble is electrified there will be an additional force acting normally outwards on the surface of the bubble, namely the force of amount lira3, per unit area just investigated, and the bubble will expand until equilibrium is reached between this and the other forces acting on the surface.

As the electrification and consequently the radius change, the pressure inside will vary inversely as the volume, and therefore inversely as a3. Let

Fig. 37.

us, then, suppose the pressure to be «/a3. Consider the equilibrium of the small element of surface cut off by a circular cone through the centre, of small semi-vertical angle 6. This element is a circle of radius a6, and therefore of area ira}Q%. The forces acting are :

(i) The atmospheric pressure IT7ra2#2 normally inwards.

(ii) The internal pressure — ird262 normally outwards. j. 6

82 Conductors and Condensers [ch. m

(iii) The mechanical force due to electrification, 2ira2 x 7ra202 normally outwards.

(iv) The system of tensions acting in the surface of the bubble across the boundary of the element.

If T is the tension per unit length, the tension across any element of length ds of the small circle will be Tds acting at an angle 6 with the tangent plane at P, the centre of the circle. This may be resolved into Tds cos 6 in the tangent plane, and Tds sin 6 along PO. Combining the forces all round the small circle of circumference 2ira6, we find that the components in the tangent plane destroy one another, while those along PO combine into a resultant 2irad x Tsin 6. To a sufficient approximation this may be written as 2ira6*T.

The equation of equilibrium of the element of area is accordingly

n-Tra2^2 - -, Tra2^ - 27r<727ra202 + 2-rradiT = 0, a3

k 2T or, simplifying, II - - -2ira2 + — -0 (28).

Let a0 be the radius when the bubble is uncharged, and let the radius be a, when the bubble has a charge e, so that

Then n-— ,+ — = 0,

a03

tt * e2 2T .

n - — - g — t + — = o.

We can without serious error assume T to be the same in the two cases. If we eliminate T from these two equations, we obtain

II (ax - a0) - k ( —. : - — 2 } = ■

^Ox2 a02/ 87TO!3 ' giving the charge in terms of the radii in the charged and uncharged states.

95. We have seen (§ 93) that the maximum pressure on the surface which electrification can produce is only about ^^ atmosphere : thus it is not possible for electrification to change the pressure inside by more than about ^^y atmosphere, so that the increase in the size of the bubble is necessarily very slight.

If, however, the bubble is blown on a tube which is open to the air, equation (28) becomes

7TCT2 =

T

a '

94-97] Energy 83

As a rough approximation, we may still regard the bubble as a uniformly charged sphere, so that if V is its potential,

o- = V/4nra, and the relation is V2 = 1§ttcl T,

giving V in terms of the radius of the bubble, if the tension T is known. In this case the electrification can be made to produce a large change in the radius, by using films for which T is very small.

Energy of Discharge.

96. On discharging a conductor or condenser, a certain amount of energy is set free. This may shew itself in various ways, e.g. as a spark or sound (as in lightning and thunder), the heating of a wire, or the piercing of a hole through a solid dielectric. The energy thus liberated has been previously stored up in charging the conductor or condenser.

To calculate the amount of this energy, let us suppose that one plate of a condenser is to earth, and that the other plate has a charge e and is at potential V, so that if C is the capacity of the condenser,

e = CV (29).

If we bring up an additional charge de from infinity, the work to be done is, in accordance with the definition of potential, Vde. This is equal to dW, where W denotes the total work done in charging the condenser up to this stage, so that

dW= Vde

= -jr by equation (29). On integration we obtain

W=k% (30),

no constant of integration being added since W must vanish when e = 0. This expression gives the work done in charging a condenser, and therefore gives also the energy of discharge, which may be used in creating a spark, in heating a wire, etc.

Clearly an exactly similar investigation will apply to a single conductor, so that expression (30) gives the energy either of a condenser or of a single conductor. Using the relation e = CV, the energy may be expressed in any one of the forms

e-

W=^=\eV=\GV* (31).

97. As  an  example  of  the  use  of  this  formula,  let  us  suppose  that  we 
    

have a parallel plate condenser, the area of each plate being A, and the

6—2

84 Conductors and Condensers [oh. in

distance of the plates being d, so that G = A/4nrd, by § 83. Let a be the surface density of the high potential plate, so that e = a A. Let the low potential plate be at zero potential, then the potential of the high potential plate is

V = ^ = 4nrda, \j

and the electrical energy is

W=±eV=27rda2A.

Now let us pull the plates apart, so that d is increased to d'. The electrical energy is now 2ird'a2A, so that there has been an increase of electrical energy of amount

2tt<t*A (d' - d).

It is easy to see that this exactly represents the work done in separating the two plates. The mechanical force on either plate is 2ira2 per unit area, so that the total mechanical force on a plate is 2tt<t-A. Obviously, then, the above is the work done in separating the plates through a distance d'-d.

It appears from this that a parallel plate condenser affords a ready means of obtaining electrical energy at the expense of mechanical. A more valuable property of such a condenser is that it enables us to increase an initial difference of potential. The initial difference of potential

Girder

is increased, by the separation, to

4f7rd'cr.

By taking d small and d' large, an initial small difference of potential may be multiplied almost indefinitely, and a potential difference which is too small to observe may be increased until it is sufficiently great to affect an instrument. By making use of this principle, Volta first succeeded in detecting the difference of electrostatic potential between the two terminals of an electric battery.

There are practical difficulties which restrict the application of the principle. For if the initial distance d is made too small the condenser may discharge itself by a spark passing directly between the plates, while if d! is made large com- pared with the size of the plates the formulae we have used are no longer true.

EXAMPLES.

1. The two plates of a parallel plate condenser are each of area A, and the distance between them is d, this distance being small compared with the size of the plates. Find the attraction between them when charged to potential difference V, neglecting the irregularities caused by the edges of the plates. Find also the energy set free when the plates are connected by a wire.

97] Examples 85

2. A sheet of metal of thickness t is introduced between the two plates of a parallel plate condenser which are at a distance d apart, and is placed so as to be parallel to the plates. Shew that the capacity of the condenser is increased by an amount

t 4nd(d-t)

per unit area. Examine the case in which t is very nearly equal to d.

3. A high-pressure main consists first of a central conductor, which is a copper tube of inner and outer diameters of ^ and % inches. The outer conductor is a second copper tube coaxal with the first, from which it is separated by insulating material, and of diameters 1§£ and \% inches. Outside this is more insulating material, and enclosing the whole is an iron tube of internal diameter 2^ inches. The capacity of the conductor is found to be "367 microfarad per mile : calculate the inductive capacity of the insulating material.

4. An infinite plane is charged to surface density o-, and P is a point distant half an inch from the plane. Shew that of the total intensity 271-0- at P, half is due to the charges at points which are within one inch of P, and half to the charges beyond.

5. A disc of vulcanite (non-conducting) of radius 5 inches, is charged to a uniform surface density <r by friction. Find the electric intensities at points on the axis of the disc distant respectively 1, 3, 5, 7 inches from the surface.

6. A condenser consists of a sphere of radius a surrounded by a concentric spherical

shell of radius b. The inner sphere is put to earth, and the outer shell is insulated.

b2 Shew that the capacity of the condenser so formed is y — -.

7. Four equal large conducting plates A, B, C, D are fixed parallel to one another. A and D are connected to earth, B has a charge E per unit area, and C a charge E' per unit area. The distance between A and B is a, between B and C is b, and between C and D is c. Find the potentials of B and C.

8. A circular gold-leaf of radius b is laid on the surface of a charged conducting sphere of radius a, a being large compared to b. Prove that the loss of electrical energy in removing the leaf from the conductor — assuming that it carries away its whole charge — is approximately \b2E2\o?, where E is the charge of the conductor, and the capacity of the leaf is comparable to b.

9. Two condensers of capacities Cl and C2, and possessing initially charges #1 and Qit are connected in parallel. Shew that there is a loss of energy of amount

2C1Ca(C1 + C2)'

10. Two Leyden Jars A, B have capacities Clf C-2 respectively. A is charged and a spark taken : it is then charged as before and a spark passed between the knobs of A and B. A and B are then separated and are each discharged by a spark. Shew that the energies of the four sparks are in the ratio

(C1 + C2)2 : (Ci + C2)tf2 = Ci~ ■ Wi-

11. Assuming an adequate number of condensers of equal capacity C, shew how a compound condenser can be formed of equivalent capacity 8C, where 6 is any rational number.

8G Conductors and Condensers [ch. in

12. Three insulated concentric spherical conductors, whose radii in ascending order of magnitude are a, b, c, have charges et, e2, e3 respectively, find their potentials and shew that if the innermost sphere be connected to earth the potential of the outermost is diminished by

« /fi + £2 + £3

13. A conducting sphere of radius a is surrounded by two thin concentric spherical conducting shells of radii b and c, the intervening spaces being filled with dielectrics of inductive capacities E and L respectively. If the shell b receives a charge E, the other two being uncharged, determine the loss of energy and the potential at any point when the spheres A and C are connected by a wire.

14. Three thin conducting sheets are in the form of concentric spheres of radii a + d, a, a — c respectively. The dielectric between the outer and middle sheet is of inductive capacity E, that between the middle and inner sheet is air. At first the outer sheet is uninsulated, the inner sheet is uncharged and insulated, the middle sheet is charged to potential V and insulated. The inner sheet is now uninsulated without connection with the middle sheet. Prove that the potential of the middle sheet falls to

E Vc (a + d) Ec(a + d) + d(a-c)'

15. Two insulated conductors A and B are geometrically similar, the ratio of their linear dimensions being as L to L'. The conductors are placed so as to be out of each other's field of induction. The potential of A is V and its charge is E, the potential of B is V and its charge is E'. The conductors are then connected by a thin wire. Prove that, after electrostatic equilibrium has been restored, the loss of electrostatic

energy is

, (EL'-E'L)(Y- V)

• Z + L'

16. If two surfaces be taken in any family of equipotentials in free space, and two

metal conductors formed so as to occupy their positions, then the capacity of the

C C- condenser thus formed is n 1 ^-, where d, C2 are the capacities of the external and

internal conductors when existing alone in an infinite field.

17. A conductor (B) with one internal cavity of radius b is kept at potential U. A conducting sphere (A), of radius a, at great height above B contains in a cavity water which leaks down a very thin wire passing without contact into the cavity of B through a hole in the top of B. At the end of the wire spherical drops are formed, concentric with the cavity ; and, when of radius d, they fall passing without contact through a small hole in the bottom of B, and are received in a cavity of a third conductor (C) of capacity c at a great distance below B. Initially, before leaking commences, the conductors A and C are uncharged. Prove that after the rth drop has fallen the potential of C is

f ar(b-dy ,\an.

(ab + bd-ad)r \c ' where the disturbing effect of the wire and hole on the capacities is neglected.

18. An insulated spherical conductor, formed of two hemispherical shells in contact, whose inner and outer radii are b and b', has within it a concentric spherical conductor of radius a, and without it another spherical conductor of which the internal radius is c. These two conductors are earth-connected and the middle one receives a charge. Shew that the two shells will not separate if

2ac>bc + b'a.

Examples 8*7

19. Outside a spherical charged conductor there is a concentric insulated but un- charged conducting spherical shell, which consists of two segments. Prove that the two segments will not separate if the distance of the separating plane from the centre is less than

ab

(a2+J2)J'

where a, b are the internal and external radii of the shell.

20. A soap-bubble of radius a is formed by a film of tension T, the external atmospheric pressure being II. The bubble is touched by a wire from a large conductor at potential V, and the film is an electrical conductor. Prove that its radius increases to r, given by

n (r» - a") + 2 T (r2 - a2) = -^ .

21. If the radius and tension of a spherical soap-bubble be a and T respectively, shew that the charge of electricity required to expand the bubble to twice its linear dimensions would be

n being the atmospheric pressure.

22. A thin spherical conducting envelope, of tension T for all magnitudes of its radius, and with no air inside or outside, is insulated and charged with a quantity Q of electricity. Prove that the total gain in mechanical energy involved in bringing a charge q from an infinite distance and placing it on the envelope, which both initially and finally is in mechanical equilibrium, is

23. A spherical soap-bubble is blown inside another concentric with it, and the former has a charge E of electricity, the latter being originally uncharged. The latter now has a small charge given to it. Shew that if a and 2a were the original radii, the new radii will be approximately a +x, 2a +y, where

/ 101 7E2\

\2y{Ua+T)=x{2AUa + ^r T+±—^%

where n is the atmospheric pressure, and T is the surface-tension of each bubble.

24. Shew that the electric capacity of a conductor is less than that of any other conductor which can completely surround it.

25. If the inner sphere of a concentric spherical condenser is moved slightly out of position, so that the two spheres are no longer concentric, shew that the capacity is increased.

CHAPTEE IV

SYSTEMS OF CONDUCTORS

98. In the present Chapter we discuss the general theory of an electro- static field in which there are any number of conductors. The charge on each conductor will of course influence the distribution of charges on the other conductors by induction, and the problem is to investigate the distributions of electricity which are to be expected after allowing for this mutual induction.

We have seen that in an electrostatic field the potential cannot be a maximum or a minimum except at points where electric charges occur. It follows that the highest potential in the field must occur on a conductor, or else at infinity, the latter case occurring only when the potential of every conductor is negative. Excluding this case for the moment, there must be one conductor of which the potential is higher than that anywhere else in the field. Since lines of force run only from higher to lower potential (§ 36), it follows that no lines of force can enter this conductor, there being no higher potential from which they can come, so that lines of force must leave it at every point of its surface. In other words, its electrification must be positive at every point.

So also, except when the potential of every conductor is positive, there must be one conductor of which the potential is lower than that anywhere else in the field, and the electrification at every point of this conductor must be negative.

If the total charge on a conductor is nil, the total strength of the tubes of force which enter it must be exactly equal to the total strength of the tubes which leave it. There must therefore be both tubes which enter and tubes which leave its surface, so that its potential must be intermediate between the highest and lowest potentials in the field. For if its potential were the highest in the field, no tubes could enter it, and vice versa. On any such conductor the regions of positive electrification are separated from regions of negative electrification by " lines of no electrification," these lines being loci along which a = 0. In general the resultant intensity at any

98, 99] Systems of Conductors 89

point of a conductor is 4nrcr. At any point of a line of no electrification, this intensity vanishes, so that every point of a " line of no electrification " is also a point of equilibrium.

At a point of equilibrium we have already seen that the equipotential through the point cuts itself. A line of no electrification, however, lies entirely on a single equipotential, so that this equipotential must cut itself along the line of no electrification. Moreover, by § 69, it must cut itself at right angles, except when it consists of more than two sheets.

99. We  can  prove  the  two  following  propositions : 
    

I. If the potential of every conductor in the field is given, there is only one distribution of electric charges which will produce this distribution of potential.

II. If the total charge of every conductor in the field is given, there is only one way in which these charges can distribute themselves so as to be in equilibrium.

If proposition I. is not true, let us suppose that there are two different distributions of electricity which will produce the required potentials. Let <r denote the surface density at any point in the first distribution, and a in the second. Consider an imaginary distribution of electricity such that the surface density at any point is <r — a. The potential of this distribution at any point P is

where the integration extends over the surfaces of all the conductors, and r is the distance from P to the element dS. If P is a point on the surface of any conductor,

ff°dS and \~dS

are by hypothesis equal, each being equal to the given potential of the conductor on which P lies. Thus

v-lfe** -!!'>-.

so that the supposed distribution of density a — a' is such that the potential vanishes over all the surfaces of the conductors. There can therefore be no lines of force, so that there can be no charges, i.e., cr — a' = 0 everywhere, so that the two distributions are the same.

And again, if proposition II. is not true, let us suppose that there are two different distributions a- and a such that the total charge on each conductor has the assigned value. A distribution cr — cr' now gives zero as the total charge on each conductor. It follows, as in § 98, that the

90 Systems of Conductors [ch. iv

potential of every conductor must be intermediate between the highest and lowest potentials in the field, a conclusion which is obviously absurd, as it prevents every conductor from having either the highest or the lowest potential. It follows that the potentials of all the conductors must be equal, so that again there can be no lines of force and no charges at any point, i.e., g = cr' everywhere.

It is clear from this that the distribution of electricity in the field is fully specified when we know either

(i) the total charge on each conductor,

or (ii) the potential of each conductor.

SUPEEPOSITION OF EFFECTS.

100. Suppose  we  have  two  equilibrium  distributions: 
     

(i) A distribution of which the surface density is o- at any point, giving total charges Elf E2, ... on the different conductors, and potentials

'1 > '2 > • • • •

(ii) A distribution of surface density a, giving total charges Ex', E2, ... and potentials V/, V2, ....

Consider a distribution of surface density a + <r'. Clearly the total charges on the conductors will be Ej + E^, E2 + E2, ..., and if VP is the potential at any point P,

W/'-T^

where the notation is the same as before. If P is on the first conductor, however, we know that

// //

-dS = Vlt r

-dS^V/, r

so that Vp = Tf + VI' ; and similarly when P is on any other conductor. Thus the imaginary distribution of surface density is an equilibrium distribution, since it makes the surface of each conductor an equipotential, and the potentials are

K+K, v+v/, ....

The total charges, as we have seen, are Ex + E/, E2 + E2, ..., and from the proposition previously proved, it follows that the distribution of surface- density <r + a' is the only distribution corresponding to these charges.

We have accordingly arrived at the following proposition :

If charges Elt E2> ... give rise to potentials K> V2> ..., and if charges

99-101] Superposition of Effects 91

Ex, E2, . . . give rise to potentials Vx, V2', ..., then charges Ex + Ex, E2 + E2, . . . will give rise to potentials VX + Vx, V2 + V2, ....

In words: if we superpose two systems of charges, the potentials produced can be obtained by adding together the potentials corresponding to the two component systems.

Clearly the proposition can be extended so as to apply to the superposition of any number of systems.

We can obviously deduce the following :

If charges Ex, E2, ... give rise to potentials Vx, V, ..., then charges KEX, KE2, ... give rise to potentials KVX, KV2, . ...

101. Suppose now that we have n conductors fixed in position and uncharged. Let us refer to these conductors as conductor (1), conductor (2), etc. Suppose that the result of placing unit charge on conductor (1) and leaving the others uncharged is to produce potentials

^11 j Pm • •- Pin,

on the n conductors respectively, then the result of placing EX on (1) and having the others uncharged is to produce potentials

P\EX, p12E1} ...pmEx.

Similarly, if placing unit charge on (2) and leaving the others uncharged gives potentials

P2U P?2> •'• Pint

then placing E2 on (2) and leaving the others uncharged gives potentials

p2XE2, p2iE2, ...pmE2.

In the same way we can calculate the result of placing E3 on (3), Et on (4), and so on.

If we now superpose the solutions we have obtained, we find that the effect of simultaneous charges Ex, E2, ... En is to give potentials VX,V2, ... Vn, where

Vx =pnEx +p2XE2 +p31E3 + ... '

V2 = px2Ex +p22E2+p3i E3 + ... I (32).

etc.

These equations give the potentials in terms of the charges. The coefficients pxx, p2X, ... do not depend on either the potentials or charges, being purely geometrical quantities, which depend on the size, shape and position of the different conductors.

92 Systems of Conductors [ch. iv

Green's Reciprocation Theorem.

102. Let us suppose that charges eP, eQ, ... on elements of conducting surfaces at P, Q, ... produce potentials VP, VQ, ... at P, Q, ..., and that similarly charges eP', eQ', ... produce potentials VP, VQ', .... Then Green's Theorem states that

Jll&pVp ~— ^*&p Vpy

the summation extending in each case over all the charges in the field. To prove the theorem, we need only notice that

Vp~ZPQ' the summation extending over all charges except eP, so that in ^ePVP the

coefficient of -p-^ is eP'eQ from the term ePVP, and ePe<i' from the term eQ'VQ. Thus

ePeQ' + eQeP PQ

^e 'V - S2 F q

= 2ePTp', from symmetry.

103. The following theorem follows at once :

If total charges Elt E2 on the separate conductors of a system produce potentials Tf, V2> ..., and if charges E-[, E2', ... produce potentials Vx ', V/, ..., then

2EV' = ZE'V (33),

the summation extending in each case over all the conductors.

To see the truth of this, we need only divide up the charges Ely E2, ... into small charges eP, eQ, ... on the different small elements of the surfaces of the conductors, and the proposition becomes identical with that just proved.

104. Let us now consider the special case in which

E, = l, Ea = Es = E4 = ... = Q,

so that ^ = Pn, K=Pvn etc-5

and #/ = 0, Et' = l, E3' = EJ= ... =0.

so that K' =Pn, V/=P-22> etc.

Then 'SEV =p21 and HE'V=p12, so that the theorem just proved becomes

Pi2= Pa- in words: the potential to which (1) is raised by putting unit charge on (2), all the other conductors being uncharged, is equal to the potential to which (2) is raised by putting unit charge on (1), all the other conductors being uncharged.

102-105]

Coefficients of Potential

93

As a special case, let us reduce conductor (2) to a point P, and suppose that the system contains in addition only one other conductor (1). Then

The potential to which the conductor is raised by placing a unit charge at P, the conductor itself being uncharged, is equal to the potential at P when unit charge is placed on the conductor.

For instance, let the conductor be a sphere, and let the point P be at a distance r from its centre. Unit charge on the sphere produces potential

• at P, so that unit charge at P raises the sphere to potential -.

Coefficients of Potential, Capacity and Induction.

105. The relations p12 = p2l , etc. reduce the number of the coefficients Pn, Pn> ••• Pnn> which occur in equations (32), to %n(n + l). These coeffi- cients are called the coefficients of potential of the n conductors. Knowing the values of these coefficients, equations (31) give the potentials in terms of the charges.

If we know the potentials V1} V2, ..., we can obtain the values of the charges by solving equations (32). We obtain a system of equations of the form

^ = guK+gnK+ — 1

E* = quK + q22V2 + etc.

.(34).

The values of the q's obtained by actual solution of the equations (32), are

" A

7^22 7-'32 P'23 P-33

Pm

Pn3

where

Pin Pzn • ■ • Pnn

#21 Pzi

P-2Z P33

Pm

Pn3

pin Pan • • • Pnn Pn P21 • • • Pni P12 P-22 • • • Pni

•(35),

Pin Pin • • ' Pnn

Thus qrs is the co-factor of prs in A, divided by A.

The relation qrs = q^

follows as an algebraical consequence of the relation prs = psr, or is at once obvious from the relation

2EV' = 2E'V,

and equations (34), on taking the same sets of values as in § 104.

94 Systems of Conductors [ch. iv

There are n coefficients of the type qn, q22) ... qnn- These are known as coefficients of capacity. There are ^n(n — 1) coefficients of the type qrs, and these are known as coefficients of induction.