SOME MAGNETIC FIELDS 



due to the element is <pdw. Now sum up for all the elements of 

 the shell and V = 2<c?to = 0Sd<o = 0o>, where w is the solid angle 

 subtended at P by the rim of the shell. 



At an infinite distance the angle subtended is zero and the 

 potential is therefore zero. 



If we bring a unit NSP from an infinite distance to the point 

 on the positive side of the shell where the angle subtended is w, 

 then we do more work against the near side than is done for us by 

 the more remote side and the potential is -f <w. 



If we approach from an infinite distance to a point on the 

 negative side where w is subtended the potential is evidently <w. 

 The potential depends only on the strength </> and the angle 

 subtended by the rim. Hence all shells of the same strength and 

 the same contour have the same external magnetic field. 



Difference of potential of two neighbouring points, 

 one on each side of the shell. Let PQ, Fig. 170, be two 



. 170. 



Fio. 1706. 



neighbouring points : P on the + side, Q on the side. Let us sup- 

 pose them so near that they subtend the same solid angle w at the 

 rim. In reckoning the potential, if P subtends o>, Q must be con- 

 sidered to subtend 4?r w. For take a point o, Fig. 1706, and draw a 

 sphere of unit radius with o as centre. Let a line from P 5 Fig. ITOrt, 

 trace out the rim A BCD of the shell and let a parallel line from o, 

 Fig. 1706, trace out the curve abed on the sphere. The area abode 

 represents w. And as the pole moves from an infinite distance on 

 the 4- side to P the angle increases from zero to w and the potential 

 at P is rf)tt). Now let the pole move from an infinite distance 

 on the side to P. The angle subtended increases from zero to 

 abcdf = whole area of sphere abcde : 



= 4>7T W 



and the potential at Q is ^ (4-Tr a>), 

 Hence potential at P potential at Q 



= 4-7T0 



