78 WORK OF ELECTRICAL FORCES. 



The increase dW of the energy of the conductor is therefore 



When the mass of electricity changes from M to M I} the in 

 crease of energy is 



As the energy vanishes with the mass, we see that the energy 

 which corresponds to the mass M is 



W = = -CV2 = -MV 



2C 2 2 



Thus the electrical energy of a single conductor is proportional to the 

 square of the charge, or to the square of the potential. 



90. ENERGY OF A SYSTEM OF CONDUCTORS. Let there be any 

 number of conductors A v A 2 , A 3 , . . . having charges M 1? M 2 , M 3 , .... 

 with the potentials V p V 2 , V 8 , . . . 



If the density of each point is multiplied by x, a new state of 

 equilibrium is obtained, in which the potentials are multiplied by the 

 same factor x. There is the charge xM 1 on A x at the potential 

 xV v xM% on A 2 at the potential #V 2 , etc. 



If we increase x by dx, the masses and the potentials are 

 multiplied by x + dx, and the increase of charge in the conductor 

 A 1 is M^x. The corresponding work lies between M 1 dx.xV 1 and 

 'M. l dx(x + dx)V-^ it is therefore, within an infinitely small expression 

 of the second order, equal to M-^^dx. This is also the case with 

 the other conductors, so that the variation of energy of the system 

 is 



dW = (MjVj + M 2 V 2 + ..... )xdx = 

 Between the two values X Q and x 1 the increase of energy is 



If we make ^ = and ^=1, which amounts to supposing that 



