Energy in the Electromagnetic Field, 149 



in ( a ? + a ?\ 



Neglecting terms containing the factor I 1— ), as we 



may do owing to the extreme smallness of the charges con- 

 cerned, the mutual energy of the two circuits is seen to be 



T m = j£$ e ^{u i u 2 + v 1 v 2 + 2w 1 w 2 ), . . (6 a) 



the suffix 1 referring to charges in one circuit, the suffix 2 

 to those in the other. 



The above expression is equivalent to 



™ ... (V C0S 6 j j , . . (V COS CC, COS « 2 7 j /n \ 



T m =ihHU — <M*« + i*i»sir — - — - d Sl ds 2 , . (7) 



where e is the angle between the elements ds x , ds 2 of the two 

 circuits, and « 1? a 2 are the angles these elements make with 

 the line joining them. 



But it is well known as the result of experiment, or as an 

 immediate deduction from experiment, that the mutual 

 energy of two circuits is given by 



T m=^2 }\ °~ds l ds 2 . 



. . (8) 



Expressions (7) and (8) are, however, not in general 

 equivalent. This discrepancy leads one to question whether 

 H : 787r is a true representation of the kinetic energy density. 

 This value is apparently an immediate deduction from the 

 two fundamental circuital laws of electromagnetism. From 

 the law of electromagnetic induction Ave deduce that the 

 energy in the field due to a system of circuits is %%i<j>, 

 where <f> is the magnetic flux linked with the circuit carry- 

 ing the current i. The circuital law connecting magnetic 

 force and current then shows that this expression is equal 



JH 2 

 ^— dv taken throughout all space. But what does 



H here really represent ? It represents not the actual 

 magnetic force at a point at a given instant, but the mean 

 value of this force at the point, for it is now known that 

 the charge whose motion constitutes a current is not dis- 

 tributed uniformly throughout the conducting substance, 

 but is concentrated in a number of discrete particles, so 

 that the magnetic force at a point is not constant. In 



