105-109] Energy 95 



If we substitute for the Fs their values in terms of the charges as given by 

 equations (31), we obtain 



W = \ (p n E^ + 2p l2 EiEz + p^ E + . . . ) (38) 



and similarly from equations (34), 



W = K<?nF 1 2 + 2<7 12 F 1 F 2 + g 22 F 2 2 + ...) (39). 



107. If W is expressed as a function of the E's, we obtain by differ- 

 entiation of (38), 



3Tf F. Pa ff 



Qj=j- = p n ^ l -\- p l2 J^ 2 + ... +p ln & n 



= F! by equation (32). 



This result is clear from other considerations. If we increase the charge 



dW 



on conductor 1 by dE lt the increase of energy is -^~- dE 1} and is also V 1 dE 1 



since this is the work done on bringing up a new charge dE 1 to potential V l . 

 Thus on dividing by dE lt we get 



3TF 



So also ^W = E I 



o YI 



as is at once obvious on differentiation of (39). 



108. In changing the charges from E l , E 2) ... to E^, E 2 , ... let us suppose 

 that the potentials change from F a , F 2 , ... to F/, F 2 ', .... The work done, 

 W W, is given by 



Since, however, by 103, ^EV = 2"F, this expression for the work done 

 can either be written in the form 



i S [E'V - EV-(EV - E'V}}, 

 which leads at once to 



TT- W = &(E'-E)(V'+V) .................. (42); 



or in the form J I {E'V 1 - EV + (EV - E'V)}, 



which leads to W - W = ^(V- V)(E' + E) .................. (43). 



109. If the changes in the charges are only small, we may replace E' by 

 E + dE, and find that equation (42) reduces to 



from which equation (40) is obvious, while equation (43) reduces to 

 leading at once to (41). 



