CHAP, iv.] ELECTRICITY AND MAGNETISM. 195 



by a unit of + electricity when moved up to the given 

 point P from an infinite distance, according as the 

 potential at P is positive or negative. 



Proof. First determine the difference of potential between 

 point P and point Q due to a charge of electricity q on a small 

 sphere at A. 



Fig. 96. 



Call distance AP = r, and AQ = r 1 . Then PQ = 

 r 1 r. The difference of potential between Q and P is the 

 work done in moving a + unit from Q to P against the force ; 

 and since 



work = (average) force x distance through which it 

 is overcome 



The force at P exerted by q on a + unit = - 2 , 

 and tHe force at Q exerted by q on a + unit = '4%. 



Suppose now that the distance PQ be divided into any 

 number (n) of equal parts rr lt r^r^ r z r B , ;- n _ a r. 



The force at r -^. 



> r i = "2 etc - 



Now since r^ may be made as close to r as we choose, if we 

 only take n a large enough number, we shall commit no serious 

 error in supposing that r x r^ is a fair mean between r* and 

 r-f ; hence we may assume the average force over the short 



length from r to r^ to be -- 

 > 



