CURRENTS IN CONDUCTING SHELLS OF SMALL THICKNESS. 319 



Cnse of an Infinite Plane Sheet. 



36. The case of an infinite plane sheet differs somewhat in its practical treatment 

 from that of a sphere, and requires independent investigation. It has been fully 

 treated by MAXWELL, MASOART and JOUBERT, and other writers. We here regard it 

 from a somewhat different point of view. 



Let the plane be that of xy. 



If then for any system of currents in it the condition F/V = G/V be satisfied, we 

 can always make the plane sheet self-inductive by suitably choosing cr/h. 



For instance, let the system of currents in the plane be induced by the variation of 

 an infinitely small circular current i, parallel to the plane, and of radius a, and distant 

 z from the plane. In that case we see at once by symmetry that the induced currents 

 flow in circles round 0, the foot of the perpendicular from the centre of the circular 

 current on the plane. The same is the case with the vector potential of all the 

 currents ; and therefore the resultant of F and G coincides with the resultant current. 

 Also t/ = in this case. We have then F/U = G/V at every point. 



But 



'7 =-2,17, f=-2,V. 

 dz dz 



Therefore, in order that the sheet may be self-inductive, we must make 



<r 2-irF 2-irG 



h~~ ' * rfF : " * dG ' 

 dz dz 



But, if r be the distance of a point on the sheet from the circular current 



*- 1 , 

 dy r' 



G = 2ai 

 and therefore 



. d 1 



G = 2ai - - ; 

 dx r 



d .yz 



= 6ai *T , 

 dz r* 



dG 

 ^= 



and 



ff_ 2ir/c r* 



A = T~ z ' 

 or h CK. 1/r*. 



That is the condition that the currents excited in a plane sheet under the influence 

 of an external field of the kind in question may be self-inductive. 



