ELECTRIC INDUCTION. 



& in flowing in the positive direction, that is, from x towards y. Such a 

 lit is equivalent in ita magnetic effects to a magnetic shell of strength 8<, 



having the circuit for ita edge*. 



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



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



as above, into which it can be divided. The strength of the equivalent complex 



shell at any point will be 



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

 imaginary 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 (, i), C) due to a plane sheet of imaginary magnetic matter whose surface- 

 density is <f>, and which coincides with the plane of xy. The potential due to 



the positive sheet whose surface-density is -, and which is at a distance c on 



C 



the positive side of the plane of xy, is 



dp 



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

 plane .of xy is 



-l 



Hence the magnetic potential of the shell is 



V- -^ 



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 (f, 17, ) on the negative side of the 

 sheet the potential is equal and of opposite sign to that at the point (f, 77, ) 

 on the positive side. 



* W. Thomson, " Mathematical Theory of Magnetism," Phil. Tram. 1850. 



