227-229] Inversion 201 



Take ' 



e 

 V' OP K 



Now let Q be a point of a conducting surface, and replace e by crdS, 

 the charge on the element of surface dS at Q. Let V P denote the potential 

 of the whole surface at P, and let V P ' denote the potential at P' due to a 

 charge e on each element dS' of the inverse surface, such that 



_^_ _ OQ f 

 crdS K ' 



TT 



Then, since V P ' = V P ~, for each element of charge, we have by addition 



v 



IT ' IT 



OP' 



v ' V 



'P ' y^S^FT/ 



Thus charges e' on dS', etc. produce a potential 



V P K p , 



OF * 



Now suppose that P is a point on the conducting surface Q, so that 

 V P becomes simply the potential of this surface, say V. The charges e on 

 dS', etc. now produce a potential 



- 



so that if with these charges we combine a charge VK at 0, the potential 

 produced at P' is zero. Thus the given system of charges spread over the 

 surface P'Q' ..., together with a charge - VK at the origin, make the 

 surface P'Q ... an equipotential of potential zero. In other words, from a 

 knowledge of the distribution which raises PQ... to potential V, we can 

 find the distribution on the inverse surface P'Q' . . . when it is put to earth 

 under the influence of a charge VK at the centre of inversion. 



If e, e are the charges on corresponding elements dS, dS' at Q, Q', we 

 have seen that 



__K__OQ' _ / 

 ~O~ K ~V 







giving the ratio of the surface densities on the two conductors. 



