FORCES ON MAGNETISED BODIES 255 



Draw a narrow tube bounded by lines of induction through the 

 body from AB to CD, Fig.198. We may describe the body as mag- 

 netised throughout with intensity of magnetisation given at any point 

 by I = /cH, where H is the actual intensity of field at the point. 

 If a is the cross-section at any point 

 of the tube, la is constant and is equal 

 to the total surface polarity at AB or 

 CD. Let us then cut the tube into 

 short lengths and on each section sup- 

 pose poles la. These poles may be 

 supposed to be acted on with in- 

 tensity H, the actual intensity at the 

 cross-section. Thus at each section 

 we have a pair of forces in equilibrium, 

 and the whole system of pairs may be 

 superposed on the system consisting 

 of the two end forces on AB and CD. 



But now we have a series of short Fio. 198. 



magnets each placed in a magnetic 



field of intensity H 2 , the actually existing field where each is 

 situated. If / is the length of one of these and a its cross-section, 

 its moment is la/ = *Ha/ = *H X volume. 



But the force on a small magnet in any direction x placed in a 

 field H is (p. 228, Chapter XVIII) 



i d\\ ., dH 



moment X -7 = /cH - - x volume, 

 dx dx 



or is - K y per unit volume. 



% da: 



We may therefore replace the surface system Ho- by a volume 



, 1 dW 

 system ^ AC -= per unit volume in any direction x. 



We may further represent the volume system, if we choose, as 

 due to a pressure within the body : 



where C is some unknown constant. For this pressure will give 

 the force on any element in direction x. 



The forces, then, on a magnetised body in air may be calculated 

 by supposing that we have 



1. A tension 2?r/c 2 H 2 2 cos 2 2 outwards along the normal. 



2. A volume force ~ K j per unit volume in any direction x 



at each point in the interior. 



For all but the ferromagnetic bodies K is very small, and the 



