"INVERSE SQUARE" SYSTEMS 45 



\ ll Mippo-e that all the charges are in position except q 2 , 

 which i> at an infinite distance. The work done in bringing </ 2 

 into po>ition is 



23 



when- V 2 istlu- potential at H dm- to the rot of the system. 



ng thus with each element in turn, the "total work is 

 '/i^i + f/i^t + 7^3 + *>'<-' = 2>/V. Hut in the work thus clone each 



product such as *-^lz occurs twice, and twice only, once when q m 



' | :. 



i> brought up, and once when tj n is brought up, and therefore the 

 total work done ^ twice the potential energy of the .system. We 

 have tin 



Potential energy of the system =-^-2/yV, 



where e.-n-h i-Iement of charge is multiplied by it.s potential. 

 \Yc may illiistiat. tln-r i. Milts bv considering 

 The potential due to a uniformly charged sphere at 



points without and within its surface. Ix;t a charge Q be 



distributed imiformU uHlM O.\ , l'ig. J5J). \Ye 



ha\e shown 'p. ;> ( .)) that the intensity due to the charge at any 

 I ; i if the charge wen concentrated at O. 



II done iii bringing up unit charge from an infinite 



H i> the sune M if <> were concentrated at O and 



\ l \ (l OH. Within the sphere, as we have shown, 



90 that the potential is constant, 



