223, 224] Images 197 



We shall shew that the field in air is the same as that due to a charge 

 e at P and a certain charge e at P', the image of P, while the field in the 

 dielectric is the same as that due to a certain charge e" at P, if the whole 

 field were occupied by air. 



Let PP' be taken for axis of x, the origin being in the boundary 

 of the dielectric, and let OP = a. Then we have to shew that the potential 

 V A in air is 



V A ~- -* = +- J= 

 *J(x 4- of + y 2 + z 2 V ( - a) 2 4- 1 



while that in the dielectric is 



e" 



V(# 4- a) 2 + y 2 + z 2 



These potentials, we notice, satisfy Laplace's equation in each medium, 

 everywhere except at the point P, and they arise from a distribution of 

 charges which consists of a single point charge e at P. The potential in air 

 at the point 0, y, z on the boundary is 



F = e + e 



<Ja? + y 2 +z z> 



while that in the dielectric at the same point is 



Thus the condition that the potential shall be continuous at each point 

 of the boundary can be satisfied by taking 



e" = e + e' (129). 



The remaining condition to be satisfied is that at every point of the 



3F 3F 



boundary, ^ in air shall be equal to K ^ in the dielectric ; i.e. that 



K^^WI 



Now, when x = 0, 



* dVn Ke"a 



ea 



(a 2 + f + * 2 ) (a 2 -f i/ 2 + * 2 ) 

 so that this last condition is satisfied by taking 



Ke" = e-e' (130). 



