AND THE LINES OF FORCE OF A CIRCULAR CURRENT. 



and if <r''=^^ . CIL) 



then since 



"b X "à X <:> Z "b Z 



çA= const. 

 must represent lines of force. 



The potential of a circular current is equivalent to that of a 

 magnetic shell, which is derivable from the Newtonian potential of a 

 uniform disc, by differentiation with respect to the normal. 



§ 3. Potential of a uniform circular disc. — Let the surface 

 density = 1 ; then taking x, y axes in the plane of the disc, and z axis 

 perpendicular to it through the centre, the potential at point x, o, z 

 is evidently given by 



u=f I ^' ^ '"^ ^ (1) 



./ ./ aV X'^ — 'IX p cos d + (J^ + z- 



[> and d being polar coordinates, and a the radius of the disc. 



Writing •B^= x'—^xpcosd + fr, and making use of Lipschitz's 

 integral concerning the J vessel's function, we obtain 



1 



^ =fe-'-'J, {B X) d À. (2) 



y/ x'—2xpcosd + p' + z' ^B' + z- 



u 



The addition theorem of I^essel's functions gives 



JIX B)=JJik x) JoW + 'i 2 Jn(^- x) JJ,kp) cos n d. (3) 



Substituting in (2), we get 



V =1' ^ f' fp'e-'' [Jllx) JlA p) + 22 Jri^ X) J„(Àp) cos nd]dÀ dp dd. 



=27üf fe- '■' p Jo (Xx) JlXp) dl dp. 



