500-502] Energy of Field 433 



This indicates that for every time that a unit tube threads a current i, 

 a contribution \i is added to the energy. Thus the whole energy is 



where the summation is over all the currents in the field, and N is the 

 number of unit tubes which thread the current i. 



502. We have seen that a shell of strength (f> is equivalent, as regards 

 the field produced at all external points, to a current i, if $ = i. The energy 

 of a system of currents has however been found to be 



^iiV ................................. (425), 



whereas the energy of a system of shells was found ( 450) to be 



.............................. (426). 



The difference of sign can readily be accounted for. Let us consider a 

 single shell of strength <, and let dS be an element of area, and dn an element 

 of length inside the shell measured normally to the shell. At any point just 

 outside the shell, let the three components of magnetic force be a, ft, 7, the 

 first being a component normal to the shell, and the others being components 

 in directions which lie in the shell. On passing to the inside of the shell, the 

 normal induction is discontinuous owing to the permanent magnetism which 

 must be supposed to reside on the surface of the shell. Thus inside the shell, 



we may suppose the components of force to be $+-, ft, 7, where ^ is the 



f* 



permeability of the matter of which the shell is composed, and S is the 

 force originating from the permanent magnetism of the shell. 



The contribution to the energy of the field which is made by the space 

 inside the shell is 



- U US+-) + p + <f\dxdydz, 



87rJJJ r (V f*J } 



where the integral is taken throughout the interior of the shell; or 



This can be regarded as the sum of three integrals, 



(427). 



| 



(iii) -fjfSadndS 



28 



