20 Dr. J. Larmor on Electromagnetic Induction 



VII. It is interesting to look back over the solutions, and 

 observe what part of the result is due to the mutual action of 

 the induced currents; we shall then be able to form an esti- 

 mate of the cases in which their mutual action may be neglected. 

 Now the effect of neglecting this mutual action is merely to 

 drop P in the formation of the potential functions. In the 

 accurate solutions, for thin shells the part of the induced field 

 which is directly opposed to the original field will disappear. 

 This part is, for a solution of the nth order, the fraction 

 4c7rafc/ (2n + 1)R of the other component ; and the approximate 

 solution would hold to the first order of this small quantity. 

 In the case of thick shells, we would neglect X 2 on the right- 

 hand side of equation (3), and the solution then will be 

 exactly the approximation to which we were conducted on 

 neglecting squares of the above small quantity, as may be 

 easily shown. 



For a bounded plane sheet we have, if the magnetic potential 



dP 



is -y-, and r, 0, z are columnar coordinates, 



R *3> ^(P+P ) djr 



rd.0 dt rdO ■ dr' 



dr " dt dr rdff 



of which the general solution is 



»--*^ + * (D 



where 



d% dyjr 



dr rdd 9 



and therefore 



l( r IO + § =0 ' 00 



which shows that % is a potential function in two dimensions 

 Also 



^=-Tz (3) 



We have thus to solve (1) and (3), with x so determined as 

 to make <E> constant round the boundary of the sheet. 





