﻿Electromagnetic Induction in Plane Current-Sheets. 109 



seemed desirable for many reasons, and it is hoped that it 

 will overcome some of the difficulties, and elucidate some of 

 the obscurities which present themselves in most treatments 

 of this interesting application of the principles of electro- 

 magnetism. 



Fundamental Assumptions. 



2. The laws of electromagnetic induction assert that in 

 bodies at rest 



I. The total current across any enclosed portion of a sur- 

 face which always contains the same particles is equal to l/Aw 

 of the line-integral of magnetic force round the curve bounding 

 the surface. 



II. The rate of decrease of the surface-integral of magnetic 

 induction across any enclosed surface which ahcays contains 

 the same particles is equal to the line-integral of electromotive 

 force round the curve bounding the surface. 



In applying these laws to an infinite dielectric separated 

 into two portions by a thin conducting sheet, it is usually 

 assumed that the disturbance produced by the inducing system 

 is not a very rapidly alternating one, so that displacement 

 currents in the dielectric have no appreciable magnetic 

 effect*. With this assumption, the magnetic force in the 

 dielectric w T ill always be derivable from a potential which will 

 only depend on the inducing system and the currents in the 

 sheet. In other words, the state of the dielectric will be 

 given by an " equilibrium theory." 



It is also assumed that the induction -currents at any point 

 distribute themselves uniformly throughout the thickness of 

 the sheet. This requires that the disturbance shall not be a 

 very rapidly alternating one, and also that the thickness of 

 the sheet shall be very small compared with the other linear 

 dimensions of the system (such as the distances of the moving 

 poles, the radius of the sheet if spherical, &c.) . 



Surface Conditions at a Plane- Current Sheet, 



3. Let the plane of the sheet be taken as the plane of .r, yj 

 let the thickness of the sheet be c, and specific conductivity C« 



Let flj, fl 2 be the magnetic potentials on the negative and 

 positive side of the sheet respectively, <f> the current function 

 in the sheet at any point. Apply Law I. to the circuit formed 

 by going along the positive side of the sheet from the origin 

 to any point and returning along the negative side from that 



* Watson and Burbury, ■ Mathematical Theory of Electricitv and 

 M.ignetism,' ii. § 405. 



