432 The Magnetic Field produced by Electric Currents [CH. xm 



500. There are of course innumerable other possible systems of forces, 

 but none of these seem at all plausible, so that we are almost compelled to 

 give up all attempts at explaining the action between the circuits by theories 

 of action at a distance. We accordingly attempt to construct a theory on 

 the hypothesis that the forces result from the transmission of stresses by the 

 medium. This in turn compels us to assume that the energy of the system 

 of currents resides in the medium. 



ENERGY OF A SYSTEM OF CIRCUITS CARRYING CURRENTS. 

 501. The energy of a magnetic field, as we have seen ( 470), is 



.................. (424). 



If the energy resides in the medium, this expression may be regarded as 

 the energy of the field, no matter how this field is produced. If the field is 

 produced wholly by currents, expression (424) may be regarded as the energy 

 of the system of currents. As we shall now see, it can be transformed in a 

 simple way, so as to express the energy of the field in terms of the currents 

 by which the field is produced. 



The integral through all space, as given by expression (424), may be 

 regarded as the sum of the integrals taken over all the tubes of induction by 

 which space is filled. The lines of induction, as we have seen, will be closed 

 curves, so that the tubes are closed tubular spaces. 



If ds is an element of length, and dS the cross section at any point, of a 

 tube of unit strength, we may replace dxdydz by dSds, and instead of inte- 

 grating with respect to dS we may sum over all tubes. Thus expression (424) 

 becomes 



where the summation is over all unit tubes of induction. If H* = a 2 -f j& -f 7 2 , 

 we have, by the definition of a unit tube, fj,HdS= 1, so that 



and the integral becomes 



Now IHds is the work performed on a unit pole in taking it once round 



the tube of induction, and this we know, is equal to ^ir'Z'i, where 2'i is the 

 sum of all the currents threaded by the tube, taken each with its proper 

 sign. Thus the energy becomes 



