SOME MAGNETIC FIELDS 227 



but with its centre at O', a small distance from O, O'O being in 

 the direction of magnetisation. Then the two spheres will neutralise 

 each other everywhere where they overlap. But on one hemisphere 

 there is left a positive layer unneutralised, and on the other a 

 negative layer unneutralised. The total quantity thus left acting at 

 P on area dS, Fig. 174, will be pPQdS = pRQ cos 6dS = pOO' cos 0dS. 

 If we make pOO' = I, this quantity is o-dS, where a- = I cos 0. 



That is, the two superposed spheres give the actual distribution 

 which exists on a uniformly magnetised sphere. Now the potential of 

 a sphere at an external point is the same as if it were collected at its 

 centre. Hence the potential of the two at an external point, distant r 



in direction with OO', is the same as that of a pole = ira?p at O, 



4 

 and a pole ^ 7ra 3 p at O', where a is the radius of the sphere, or is 



the same as that of a short magnet length 

 OO' with moment 



We can calculate the field within the 

 sphere on the supposition that the two 

 surface layers act as if the space within 

 were air. The force due to a uniform sphere at a point within it 

 is due to the part of the sphere included within a concentric sphere 

 drawn through the point. Hence at a point P, Fig. 175, it will be 



4 OP 3 4 



along OP, due to the positive sphere, and 



4 PO /3 4 



- wp pT-7^ = - TrpPO' along PO', due to the negative sphere. 



The resultant of these two is by the triangle of forces equal to 

 q TrpOO' = ^ Ti-I parallel to OO', or the field within the sphere is 



everywhere uniform and in the direction parallel to OO'. 



Force on a small magnet placed at a point in a non- 

 uniform field. If a small magnet, length /, pole strength m, and 

 therefore with moment M = ml, is placed in a field which is not 

 uniform, the magnet will set in the direction of the line of force 

 through its centre, but the forces on the two poles will not be quite 

 equal and opposite, and the magnet will move in the direction of 

 the resultant of the two forces. 



If, for instance, it is placed in the field radiating from the NS 

 pole of a long magnet, it will ultimately set in a line of force and 

 its SSP will be nearer the pole producing the field than its NPS. 

 It will therefore tend to move inwards, that is, to the stronger- 

 part of the field. 



Or if it is placed in the circular field round a straight current 



