366 Permanent Magnetism [OH. xi 



The potential of the element dS of the shell at a point Q distant r from dS 

 is then 



so that the potential of the whole shell at Q is given by 



where 6 is the angle between the normal at dS and the line joining dS to P. 

 Clearly dS cos 6 is the projection of the element dS on a plane perpen- 

 dicular to the line joining dS to P, so that - - is the solid angle sub- 

 tended by dS at Q. Denoting this by day, we have the potential in the form 



lot (351). 



419. Uniform shell. If the shell is of uniform strength, <f> may be taken 

 outside the sign of integration in equation (351), so that we obtain 



.^n (352), 



where H is the total solid angle subtended by the shell at Q. 



POTENTIAL ENERGY OF A MAGNET IN A FIELD OF FORCE. 



420. The potential energy of a magnet in an external field of force is 

 equal to the work done in bringing up the magnet from infinity, the field of 

 force being supposed to remain unaltered during the process. 



Consider first the potential energy of a single particle, consisting of a pole 

 of strength m^ at arid a pole of strength + m l at P. Let 

 the potential of the field of force at be fl o and at P be flp. 

 Then the amounts of work done on the two poles in bringing 

 up this particle from infinity are respectively m^o and 

 miflp, so that the potential energy of the particle when in FlG * 

 the position OP 



= m l . OP -^- , in the notation already used, 



an = /^an an air 



ds \ dx dy dz, 

 The potential energy of any magnetised body can be found by integration 

 of expression (353), the body being regarded as an aggregation of magnetic 

 particles. 



