102 Systems of Conductors [OH. TV 



giving the ratio in which the charge E will distribute itself between the 

 two conductors 1 and 2. If the conductors 3, 4, ... are either absent or 

 uncharged, 



which is independent of E and always positive. It is to be noticed that E l 

 vanishes only if ^22 = ^12, i.e., if 2 entirely surrounds 1. 



MECHANICAL FORCES ON CONDUCTORS. 



118. We have already seen that the mechanical force on a conductor is 

 the resultant of a system of tensions over its surface of amount 27rcr 2 per unit 

 area. The results of the present Chapter enable us to find the resultant 

 force on any conductor in terms of the electrical coefficients of the system. 



Suppose that the positions of the conductors are specified by any co- 

 ordinates fj, f 2 , ..., so that p u , p wt ..., q u , q lz , ..., and consequently also W, 

 are functions of the f's. If f l is increased to fi + dfi, without the charges on 



dW 

 the conductors being altered, the increase in electrical energy is ^- rffj, and 



tffi 



this increase must represent mechanical work done in moving the conductors. 

 The force tending to increase fj is accordingly 



_d_w 

 % 



Since the charges on the conductors are to be kept constant, it will of 

 course be most convenient to use the form of W given by equation (38), and 

 the force is obtained in the form 



It is however possible, by joining the conductors to the terminals of 

 electric batteries, to keep their potentials constant. In this case, however, 

 we must not use the expression (39) for W, and so obtain for the force 



(49), 



for the batteries are now capable of supplying energy, and an increase of 

 electrical energy does not necessarily mean an equal expenditure of mechanical 

 energy, for we must not neglect the work done by the batteries. Since the 

 resultant mechanical force on any conductor may be regarded as the resultant 

 of tensions 2-Tra 2 per unit area acting over its surface, it is clear that this 

 resultant force in any position depends solely on the charges in this position. 

 It is therefore the same whether the charges or potentials are kept constant, 

 and expression (48) will give this force whether the conductors are connected 

 to batteries or not. 



