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XXI. On Electromagnetic Induction in Plane, Cylindrical, and 

 Spherical Cur rent- Sheets, and its Representation by Moving 

 Trails of Images. By G. H. Bryan, M.A.* 



Part I. — General Equations — Plane Sheets. 

 Introduction. 



1. T T is well known that if a very thin, indefinitely extended 



•A- plane sheet of metal of finite conductivity is placed 

 in a varying magnetic field due to the presence of moving 

 magnetic poles in its neighbourhood, induction-currents 

 are set up in the sheet and the field of force due to these 

 currents may be represented by a moving trail of images. 



In the present paper the surface-conditions which hold at 

 the surface of a plane, cylindrical, spherical, or other con- 

 ducting sheet of uniform small thickness are deduced directly 

 from the fundamental laws of electromagnetic induction. 

 By working directly with the scalar magnetic potential, and 

 avoiding the introduction of the vector potential and the 

 quantity which Maxwell denotes by P, the investigations are 

 much simplified. Moreover, in at least one comparatively 

 recently published paper the boundary conditions satisfied by 

 the vector potential at the surface of separation of two different 

 media have been erroneously stated, and for this reason it 

 is advantageous to employ a method which obviates the 

 difficulty. 



The results will be employed to show how the field due to 

 the presence of a magnetic pole of varying intensity in the 

 neighbourhood of a plane, cylindrical, or spherical current- 

 sheet may be represented by means of a moving trail of 

 images. By the principle of superposition the effect of any 

 number of poles of varying intensity can be deduced, and the 

 corresponding expressions for the field can thus be obtained 

 when the variable inducing system of magnets is of the most 

 general possible character, as, for example, one (or more) 

 magnetic poles moving about in any manner whatever. 



In this way a synthetic solution of the problem of induction 

 in current sheets is obtained. The phenomena of induction 

 in spherical and other current sheets have been treated at 

 considerable length from an analytical point of view by 

 Larmor, Lamb, and Niven, and the last-named writer has 

 made some attempt, in the case of a sphere, to interpret the 

 results by means of images ; but the present investigation 



* Communicated by the Physical Society: read May II, 1894. 



