1872.] 



Prof. J. C. Maxwell on Electiic Induction. 



165 



The whole system of electric currents in the sheet will therefore be 

 equivalent to a complex magnetic shell, consisting of all the simple shells, 

 denned as above, into which it can be divided. The strength of the equi- 

 valent complex shell at any point will be (p. 



We may suppose this shell to consist of two parallel plane sheets of ima- 

 ginary magnetic matter at a very small distance c, the surface- density 



being on the positive sheet, and — - on the negative sheet. 



21. To find the magnetic potential due to this complex plane shell at 

 any point not in its substance, let us begin by finding P, the potential at 

 the point (£, rj, £) due to a plane sheet of imaginary magnetic matter 

 whose surface -density is <2>, and which coincides with the plane of ccy. 



The potential due to the positive sheet whose surface-density is and 



which is at a distance \c on the positive side of the plane of xy, is 



That due to the negative sheet, at a distance |c on the negative side of the 

 plane of ccy, is 



Hence the magnetic potential of the shell is 



V=-^. .......... (2) 



d£ w 



This, therefore, is the value of the magnetic potential of the current-sheet 

 at any given point on the positive side of it. "Within the sheet there is no 

 magnetic potential, and at any point (£, 77, — £) on the negative side of the 

 sheet the potential is equal and of opposite sign to that at the point (£, rj, £) 

 on the positive side. 



22. At the positive surface the magnetic potential is 



V=-f = 2. f (3) 



At the negative surface 



I = 2 »* w 



The normal component of magnetic force at the positive surface is 



dV d 2 P 



^-irr^c : • v m 



In the case of the magnetic shell, the magnetic force is discontinuous at 

 the surface ; but in the case of the current-sheet this expression gives the 

 value of y within the sheet itself, as well as in the space outside. 



23. Let F, G, H be the components of the electromagnetic momentum at 

 any point in the sheet, due to external electromagnetic action as well as to 



