Mb maxwell, ON FARADAY'S LINES OF FORCE. 



73 



IV. Two Spheres in uniform Jield. 



Let two spheres of radius a be connected together so that their centres are kept at a dis- 

 tance ft, and let them be suspended in a uniform magnetic field, then, although each sphere by 

 itself would have been in equilibrium at any part of the field, the disturbance of the field will 

 produce forces tending to make the balls set in a particular direction. 



Let the centre of one of the spheres be taken as origin, then the undisturbed potential is 



p — I r cos 9, 

 and the potential due to the sphere is 



, k-k'a^ 



p = I —. ;^ — cos U. 



2k + k' r* 



The whole potential is therefore equal to 



^ f k- k a\ 



V 2k 



1 dp 

 r dO 



2k + K 



dp , / k — k' a?\ 



dr V 2k + k T^j 



k — k' a^ 



cos 9, 



, /■ k- k a^\ . . dp 



»-=^ 



dr 



+ 



1 dp 



t^ dd 



r sm 



dp 

 '9 d^ 



= r4i + 



k — k' a? 

 2k + k' r^ 



d(p 



(l-3cos=0) + 



k-k' 

 2k +k' 



:(1 +3cos' 



'■e)]- 



This is the value of the square of the intensity at any point. The moment of the couple 

 tending to turn the combination of balls in the direction of the original force 



L = l 



d 



k-k' 



T — 3 ri 



^2 2k + k' 



7/1 - 1 7 7' ^'1 ^l^en r = ft, 

 d9 \2k+k J 



k-k' ^a^ ( k-k' a?\ . ^ 



ft-^l^-sTTl'ft-j^^"^^- 



This expression, which must be positive, since ft is greater than a, gives the moment of 

 a force tending to turn the line joining the centres of the spheres towards the original lines of 

 force. 



Whether the spheres are magnetic or diamagnetic they tend to set in the axial direction, 

 and that without distinction of north and south. If, however, one sphere be magnetic and 

 the other diamagnetic, the line of centres will set equatoreally. The magnitude of the force 

 depends on the square of {k — k'), and is therefore quite insensible except in iron *. 



V. Two Spheres between the poles of a Magnet. 



Let us next take the case of the same balls placed not in a uniform field but between a 

 north and a south pole, ± M, distant 2c from each other in the direction of x. 



Vol. X. Part I. 



• See Prof. Thomson in Phil. Mag. March, 18S1. 



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