486-488] 



Magnetic Potential of Field 



419 



Let us take the line itself for axis of z. Any semi-infinite plane termi- 

 nated by this line may be regarded as an equivalent magnetic shell. Let us 

 fix on any plane and take it as the plane of xz. 



Consider any point P such that OP, the shortest distance from-P to 

 the axis of z t makes an angle with Ox. The cone 

 through P which is subtended by the semi-infinite 

 plane Ox, is bounded by two planes one a plane 



through P and the axis of z ; the other a plane through ^ /$\ 

 P parallel to the plane zOx. These contain an angle 

 TT 0, so that the solid angle subtended by the plane 

 zOx at P is 2 (TT 6). Giving this value to co in 

 formula (411), we obtain as the magnetic potential at P p IG 



r)Q 



Since -^ = it is clear that there is no radial magnetic force, and the 

 force at any point in the direction of increasing 



This result is otherwise obvious. If the work done in taking a unit pole 

 round a circle of circumference 2?rr is to be 4-Tn, the tangential force at 



4. V 2 * 



every point must be . 



488. This result admits of a simple experimental confirmation. 



Let PQR be a disc suspended in such a way that the only motion of 

 which it is capable is one of pure rotation about a 

 long straight wire in which a current is flowing. 

 On this disc let us suppose that an imaginary unit 

 pole is placed at a distance r from the wire. There 

 will be a couple tending to turn the disc, the 



2i 

 moment of this couple being x r or 2i. Similarly 



if we place a unit negative pole on the disc there is 

 a couple 2i. 



On placing a magnetised body on the disc, there 

 will be a system of couples consisting of one of 

 moment 2z for every positive pole and one of moment FlG 12 5 



2i for every negative pole. Since the total charge 



in any magnet is nil, it appears that the resultant couple must vanish, so 

 that the disc will shew no tendency to rotate. This can easily be verified. 



272 



