192 Methods for the Solution of Special Problems [OH. vni 



At a small distance a from the line charge which represents the telegraph- 

 wire, we may put r 2h, so that the potential is 



2h 

 2, log-, 



from which it appears that a cylinder of small radius a surrounding the 

 wire is an equipotential. We may now suppose the wire to have a finite 

 radius a, and to coincide with this equipotential. Thus the capacity of the 

 wire per unit length is 



1 



Infinite series of Images. 



221. Suppose we have two spheres, centres A, B and radii a, b, of which 

 the centres are at distance c apart, and that we require to find the field when 



FIG. 65. 



both are charged. We can obtain this field by superposing an infinite series 

 of separate fields (cf. 116). 



Suppose first that A is at potential V while B is at potential zero. As a 

 first field we can take that of a charge Va at A. This gives a uniform 

 potential V over A, but does not give zero potential over B. We can reduce 

 the potential over B to zero by superposing a second field arising from 



the image of the original charge in sphere B, namely a charge - at B f 



c 



where BB' '= . This new field has, however, disturbed the potential over 



c 



A. To reduce this to its original value we superpose a new field arising 

 from the image of the charge at B' in A, namely a charge - - . - -p at A' y 



c -- 

 c 



a~ 

 where A A' = - r^. This field in turn disturbs the potential over B, and so 



c -- 

 c 



we superpose another field, and so on indefinitely. The strengths of the 



