Energy in the Electromagnetic Field. 287 



vector potentials and arising from the incompleteness of the 

 definitions of these potentials, it must not be allowed to stand 

 without a strict scrutiny of the reasons for its introduction. 

 A careful examination of Mr. Biedermann's argument will, 

 however, soon show that his evidence of justification is con- 

 vincing only for the special case when divA = 0, wlien the 

 modified expression is equivalent to the older one. 



Mr. Biedermann commences by an examination of the 

 field of a number of very small spherical surface charges, the 

 typical one of which, with charge e s and radius a s , may be 

 taken with its centre at the point (x s i/sZ s ) and to be moving 

 with the comparatively small velocity v s . The electric force 

 at the point (<#,?/, -) in the field is then derived as the gradient 

 of the scalar potential 



? s 



where 9 . ... , X9 . N9 



r* = C* - *0 2 + (y -ys) 2 + (~ - z*) 2 ; 



■whilst the magnetic force is 



H = -2|> S EJ, E.= -grad^„ 



c 



the notation denoting the vector product, and c is the 



velocity of radiation : this last vector may also be derived 



as the curl of the vector potential 



A = I 2 — = %-^ s . 

 c r s c 



The potential energy in the field, when the motions are 

 slow enough, is then 



J- iwdv = i j^{2E s 2 + 22(E s E,,)} 



a s r ss < ) 



where 



rj,= (*,—x*y + (y.— y*) s + (zs—z s >) 2 > 



The kinetic energy is, on the other hand, equal to 



-H 



07TC 1 



-g^2 \ *{W+2(*)M.)(. 



