Field of a Circular Current. 361 



of potential only, so in the electromagnetic field we are 

 concerned with Vector differences only. 



Let us, then, calculate for the infinite straight current AB 

 the vector difference betw een P and a point on the per- 

 pendicular, Pp, at a constant distance Op = a from the line. 



Let Yp=r, and let an element, ds, of the line AB be taken 

 at any point, Q ; let /_pPQ l = 0. Then the vector difference, 



due to this element, at P is jyp — j^. , or 



\ X V^ + (a 2 -^ 2 )cos 2 0J 



dO 

 cos 6' 



IT 



Double the integral of this from 0=0 to 0= -^ is the vector 

 difference at P due to unit current in AB. Expanding the 

 radical in ascending powers of X ( = — % — )? we nave the 

 vector equal to 



andthis=log e (l+X)=:21og- . Thus, then, the vector dif- 

 ference at any point, P, is measured by 



C-21ogr, 

 where C is a constant ; and this gives the known value of 

 the magnetic force at P, viz., — ^- (where G is the vector 



k 

 potential), perpendicular to the plane PAB, i.e. -, where k 



r 

 is a constant. In this way, then, the inconvenience of deal- 

 ing with an infinite vector potential in presence of an infinitely 

 long (or very long) straight current is avoided. 



The lines of constant magnetic potential, or the loci of 

 points, P, at which the given circular current subtends a 

 constant conical (" solid ") angle, are the orthogonal trajec- 

 tories of the lines of force, and can be drawn when these 

 lines are drawn. 



It is not easy to draw these equipotential curves independ- 

 ently, or even to deduce their typical equation from that of 



