421-424] Potential Energy 369 



Potential Energy of a Shell in a Field of Force. 



423. Consider a shell of which the strength at any point is $, placed 

 in a field of potential O. The element dS of the shell is a magnetic particle 

 of strength $ dS, so that its potential energy in the field of force will, by 

 formula (353), be 



where denotes differentiation along the normal to the shell. Thus the 



dn 



potential energy of the whole shell will be 



d8:i ........................... (356). 



If the shell is of uniform strength, this may be replaced by 



dS ........................... (357). 



Since the normal component of force at a point just outside the shell 

 and on its positive force is -~ , it is clear that 1 1 ^ dS is equal to minus 



the surface integral of normal force taken over the positive face of the shell, 

 and this again is equal to minus the number of unit tubes of force which 

 emerge from the shell on its positive face. Denoting this number of unit 

 tubes by n, equation (357) may be expressed in the form 



TF = -c/m .............................. (358). 



Here it must be noticed that we are concerned only with the original 

 field before the shell is supposed placed in position. Or, in other terms, the 

 number n is the number of tubes which would cross the space occupied by 

 the shell, if the shell were annihilated. Since the tubes are counted on the 

 positive face of the shell, we see that n may be regarded as the number of 

 unit tubes of the external field which cross the shell in the direction of its 

 magnetisation. 



424. Consider a field consisting only of two shells, each of unit strength. 

 Let Wj be the number of tubes from shell 1 which cross the area occupied 

 by 2, and let n 2 be the number of tubes from shell 2 which cross the area 

 occupied by 1. The potential energy of the field may be regarded as being 

 either the energy of shell 1 in the field set up by 2, or as the energy of 

 shell 2 in the field set up by 1. Regarded in the first manner, the energy 

 of the field is found to be - n 2 ; regarded in the second manner, the energy 

 is found to be n^. Hence we see that n l = n 2 . This result, which is of 

 great importance, will be obtained again later ( 446) by a purely geometrical 

 method. 



j. 24 



