﻿200 Mr. G. H. Bryan on Electromagnetic Induction in 



point to the origin. Since the thickness of the sheet is small, 

 the terms contributed to the line-integral of the magnetic 

 force by the passage of the circuit from one side of the sheet 

 to the other may be neglected and we obtain * 



n 2 — n 1 = 47r<£ + constant (1) 



Now apply Law II. to any circuit s drawn in the plane of 

 the sheet. Let S,, S 2 be two surfaces drawn in the dielectric 

 infinitely near to the positive and negative faces of the sheet 

 respectively and both bounded by the curve s, and let P, Q 

 be the components of electromotive force at any point. Then, 

 assuming the magnetic permeability of the dielectric to be 

 unity, we have 



]>* + <*) round .- |jp£ ,S 1= |jjttj ><*S S , 



and since corresponding elements of the near surfaces S,, S 2 

 are equal, it is evident that 



dn 1 _dn 1 m 



dz dz K } 



at the surface, so that the normal component of magnetic 

 induction is continuous, as it should be f. 

 Again, the equations of conduction give 



dy dx 



Hence 



Ccjxdy dn dx dn; Cc J dn 



CoJJV 



d*<j> d*4> 



+ ^W 



> 



t dx* dy* 



where dn is the element of the outward-drawn normal to s 

 and the surface-integral taken over the area S of the sheet 

 bounded by the curve 5. 



* Maxwell, ' Electricity and Magnetism,' ii. § 653. 



t The magnetic permeability of the sheet itself will not affect the 

 conditions of the problem unless it is required to proceed to a higher 

 order of approximation by taking into account first powers of the thick- 

 ness (or unless the sheet is formed of soft iron whose magnetic perme- 

 ability may be large). 



