﻿Fields due to Motion of Electromagnetic Systems. 1113 



E = E', B = B' (so that also A = A'), Maxwell has shown that 

 the motion produces (to an observer fixed in C) an electric 

 field whose polar part is derivable from the potential 



£ = <f-^'=*(Aw) (6) 



From the assumptions made it is clear that the result is 

 only an approximation ; hut, as pointed oat by Larmor *", who 

 has given another derivation of (6), the error is only of the 

 second order in vjc. 



For a constant electromagnetic system, 



(¥) = (t ) = ' and therefore It = - ( " V > A - • (7 > 



If also the system is unelectrified (in B), as will be assumed 

 henceforth, 



V^' = 0, and V^=^V(Au) (8) 



c 



In this case 



E = -(vV)A- 1 V(Av)=E'= 1 [Bw]. . (9) 

 c c c 



The electromotive intensity of the field produced by the 



motion is thus given completely by - X the vector product 



c 



of B and v. 



§2. The vector potential at a point distant r from an 



element of space (volume, surface, or length) dr in which 



the current density (volume, surface, or line) is i is defined as 



A =H? do) 



We have also the relation 



curlA = B, or ( (Adl) = ( (BdS), . . (11) 



where dl is the element of length of a closec^curve bounding 

 the surface whose area is S. From one or both of these 



equations we can always determine A, and hence <j> = -(Av). 



If o- denotes the electric density (volume, surface, or line; 



* J. Larmor, Phil. Mag-, xvii. p. 1, Jan. 1884. 

 Phil. Mag. S. 6. Vol. 44. No. 264. Dec. 1922. 4 C 



