187-190] Comparison of different fields 163 



an equipotential. Calling the corresponding energies W and W, we 

 have 



f , 



= Q~ H/i(-5 --- -o }+... \dxdyd 



STT JJJ V 9# 8a? 



If we put 4>= F, ^= F'- 7, in equation (100), we find that the last 

 integral becomes 



or, since V is by hypothesis constant over each conductor, 



and this vanishes since each. total charge llo-'dS is the same as the corre- 

 sponding total charge lla-dS. Thus 



This integral is essentially positive, so that W is greater than W, which 

 proves the theorem. 



If any distribution is suddenly set free and allowed to flow so that the 

 surface of each conductor becomes an equipotential, the loss of energy 

 W W is seen to be equal to the energy of a field of potential V V at 

 any point. 



190. THEOREM. The introduction of a new conductor lessens the energy 

 of the field. 



Let accented symbols refer to the field after a new conductor 8 has been 

 introduced, insulated and uncharged. Then 



W-W'= ~ fff&dxdydg through the field before S is introduced 



- - \\\R*dxdydz through the field after 8 is introduced 

 == - I \\R z dxdydz through the space ultimately occupied by S 



+ ^- IJI(R 2 -R'") through the field after 8 is introduced. 



112 



