298-302] Two-dimensional Problems 253 



PKOBLEMS IN TWO DIMENSIONS. 



300. Often when a solution of a three-dimensional problem cannot be 

 obtained, it is found possible to solve a similar but simpler two-dimensional 

 problem, and to infer the main physical features of the three-dimensional 

 problem from those of the two-dimensional problem. We are accordingly 

 led to examine methods for the solution of electrostatic problems in two 

 dimensions. 



At the outset we notice that the unit is no longer the point-charge, but 

 the uniform line-charge, a line-charge of line-density a having a potential 

 (cf.75) 



C -Zv log r. 



Method of Images. 



301. The method of images is available in two dimensions, but presents 

 no special features. An example of its use has already been given in 220. 



Method of Inversion. 



302. In two dimensions the inversion is of course about a line. Let this 

 be represented by the point in fig. 81. 



Let PP f , QQ' be two pairs of inverse points. Let a line-charge e at Q 

 produce potential V P at P, and let a p 



line-charge e at Q produce potential Vp> 

 at P', so that 



V P =C-2elogPQ- 



If we take e e', we obtain FIG. 81. 



(224). 



Let P be a point on an equipotential when there are charges e l at Q lt 

 e z at Q 2 > etc., and let V denote the potential of this equipotential. Let V 

 denote the potential at P' under the influence of charges e l} e 2 , ... at the 

 inverse points of Q lf Q 2 , Then, by summation of equations such as (224), 



V- V = - 2 (2e log OP') + 2 (2e log OQ) + constants, 

 or F=constants -2(2e)logOP' .................. (225). 



