28 ELECTRICITY [ELECTKOSTATICAL THEORT. may construct then a series of states of equilibrium repre sented thus : Potential, J Vj | | | | Charge, | q^ 1 V 1 g, ^V 1 | ft 3 v i I ... | q l n^ Potential, [ | V 2 | | ... | 0__ Charge | 21 V 8 j q 23 V a g s3 V, | | g- 2n V a and so on. Superposing all these, we get a system in equilibrium, in which the potentials are V lf V 2 , . . . V n , and the charges EI =?n"V 1 + j lll V a + . . . +q ni Vn ) E a =2 ii.V 1 + 2 2i! V 2 + . . . + ? nS V n [ . . .(18). It appears therefore that the 2 n quantities E ; , <tc., V 1? &c., are connected by n linear equations ; so that when n of them are given, the rest can be determined in terms of these in a definite manner. Returning then to our general problem, we see that, when either the charge or the potential is given for each conductor, the electrical problem is determinate, and a solution is given by the linear equations of (18). The potential at any point of the field can be written down very easily. Suppose in fact v l to be the value at the point P of the function V which we determined in solving the case where the potentials 1, 0, 0, .... are given for 1, 2, ... n, v. 2 the corresponding function for the case 0, 1, 0, ... 0, and so on. Then the potential at P in the general case is obviously V - V^ + > 2 + + Y n r rt . . (1 9), whero v v v. 2 , ... v n are all known functions, and V t , V.,, . . . V n are all either given, or determined in terms of given quantities by the equations (18). It is very easy to show that there is no other solution of the problem than the one we have found. Suppose in fact that V is a function different from V, which satisfies all the conditions of the problem. Consider the function U" = V- V, since V and V both satisfy the equation v 2 V=0 , we have V 2 U 0. Also at surfaces where V is given U = 0. At surfaces where V is not given, we have U = constant - constant = constant ; and, since in this case the charge wil" be given, we shall have tS = // --jdS; and therefore // -7 rfS = 0. JJ ( v J J dv rrr (<w dv^,. , , , II ) + &c. dxd.ydz IJJ (dx dx J fff Now we have dU dz dU dx The first term vanishes for all the surfaces, for some because U = 0, for others because U is constant and / / r~dS = Q ; and the second term vanishes because v"U = 0. Hence the integral on the left hand must vanish, and that too element by element, since every element is positive. Hence we must have dV__dV dV _dV dV dV dx ~dx dy ~ dy dz = ~dz Hence V and V can only differ by a constant. But such differ ence is precluded by the boundary conditions. Hence the func tions are identical ; in other words, there is but one solution to the problem we have proposed. It is very easy to show, by methods of which we have already had an example, that the value of V thus found makes the integral ix dV dy dz
dxdydt / gral represents the potential energy of the system. It follows, therefore, that the distribution which we have found is in stable equilibrium. If we solve the equations (18), we shall get (20). A set of equations which we might obviously have Coeffi- arrived at by first principles. The physical meaning of the cients of coefficients p l v p^ v and p l is very obvious ; they are pot the potentials, corresponding to a state of equilibrium, in which the charges on 1, 2, 3, .... n are 1, 0, 0, ... 0, and so on. p l 1? p l 2 , . . &c., are called coefficients of potential; and, mutatis mutandis, all the remarks already made about ?n> ?i2 & c -> apply to them. Many interesting and im portant theorems have been proved about these coefficients, for which we refer the reader to Maxwell (Electricity, vol. i. chap. 2), whose treatment of the subject we have in the main been, folio wing. One of these, of great importance. we shall prove here, because it leads us to state a very important general theorem, which we shall have occasion to use again. The mutual potential energy of two electrical systems, Theorem A and B, is the work done in removing the two systems of mutual to an infinite distance from each other, the internal arrange- P ote " tia 6ncr fr y ment of each system being supposed unaltered during the process. It is clear that we may suppose either that A is fixed and B moves off to infinity, or that B is fixed and A moves ; the work done in both cases is, by Newton s third law of motion, the same. This is sometimes expressed by saying that the potential of A on B is the same as that of B on A. In fact, the expression for the mutual potential energy is 1) (21), where q is any element of electricity belonging to A, and a[ any element belonging to B, and D is the distance between them, the summation being extended so as to include every pair of elements. We may arrange (21 ) as follows : vi -i JMJ raa* yy each group belonging to a point in B, or, as we may write it, We may also arrange (21) in the form each group belonging to a point in A. ing equalities : Hence we have the follow- a minimum. Now, we shall show directly that this inte- (22). The first and last of these expressions are called respectively tho potential of A on B, and the potential of B on A, and this equality explains the statement made above. The two systems A and B may be different states of equi librium of the same system, if we choose. In this case we may still farther modify the expression in (22), and write So that we may state the proposition thus : If E n E 2 , . . . E,,, V lf V 2 , . . . V n , and E/, E, , . . . E n , V T , V 2 , . . . V n be the respective charges and potentials of the conductors in two different states of equilibrium, then we have If we take for the two states of the system and equation (23) becomes
(24),