420 The Magnetic Field produced by Electric Currents [CH. xm 



Circular Current. 



489. Let us find the potential due to a current of strength i flowing in a 

 circle of radius a. The equivalent magnetic shell may be supposed to be a 

 hemisphere of radius a bounded by this circle. 



The potential at any point on the axis of the circle can readily be found. 

 For at a point on the axis distant r from the centre 

 of the circle, the solid angle o> subtended by the 

 circle is given by 



ft> = 27r(l-cosa)=27r(l ..-= V 



so that the potential at this point is 



/ r \ 



n - 2wt ( i - -== = } . 



v a 2 + r 8 ' 



This expression can be expanded in powers of r 

 by the binomial theorem. We obtain the following 

 expansions : 



if r < a, 



FIG. 126. 



n 



if r > a, 



...(412), 



la 2 

 2r 2 



2.4... 2n 



.(413). 



From this it is possible to deduce the potential at any point in space. 

 Let us take spherical polar coordinates, taking the centre of the circle as 

 origin, and the axis of the circle as the initial line = 0. Inside the sphere 

 r = a, the potential is a solution of V 2 H = which is symmetrical about the 

 axis 6=0, and remains finite at the origin. It is therefore capable of 

 expansion in the form 



Along the axis we have = 0, so that this assumed value of H becomes 



and the coefficients may be determined by comparison with equation (412). 

 Thus we obtain for the potentials, 



} a 2 a 3 



