94 Systems of Conductors [OH. iv 



There are n coefficients of the type q^, q 22 , ... q nn . These are known as 

 coefficients of capacity. There are ^n (n 1) coefficients of the type q rs , and 

 these are known as coefficients of induction. 



From equations (34), it is clear that q u is the value of E when 

 F,=l, F 2 =F 3 =...=0. This leads to an extended definition of the 

 capacity of a conductor, in which account is taken of the influence of the 

 other conductors in the field. We define the capacity of the conductor 1, 

 when in the presence of conductors 2, 3, 4, ..., to be q n , namely, the charge 

 required to raise conductor 1 to unit potential, all the other conductors being 

 put to earth. 



ENERGY OF A SYSTEM OF CHARGED CONDUCTORS. 



106. Suppose we require to find the energy of a system of conductors, 

 their charges being E lt E^ ... E n , so that their potentials are F 1? F 2 , ... V n 

 given by equations (32). 



Let W denote the energy when the charges are kE l} kE 2 , ... kE n . 

 Corresponding to these charges, the potentials will be kV^ kV 2) ... kV n . If 

 we bring up an additional small charge dk . E from infinity to conductor 1, 

 the work to be done will be dkE 1 .kV 1 ; if we bring up dkE 2 to conductor 2 

 the work will be dkE 2 kV 2 and so on. Let us now bring charges dkE l to 1, 

 dkE z to 2, dkE 3 to 3, ... dkE n to n. The total work done is 



n ) ..................... (36), 



and the final charges are 



(k + dk) E lt (k + dk) E 2 ,...(k + dk) E n . 



The energy in this state is the same function of k + dk as W is of k, and may 

 therefore be expressed as 



Expression (36), the increase in energy, is therefore equal to -^=- dk, whence 



so that on integration 



W = kk*(E l V, + E 2 V 2 + ... + E n V n ). 



No constant of integration is added, since W must vanish when k=0. 

 Taking k = l, we obtain the energy corresponding to the final charges 

 E lf EZ, ... E nt in the form 



(37). 



