432 THE ELECTROMAGNETIC FIELD. [PT. III. CH. XI. 



while mutually perpendicular elements exert no action on each 

 other if either is perpendicular to the line joining them. 



222. Currents distributed in three Dimensions. We 



have seen in 210 that a current I is equivalent to a magnetic 

 shell of strength 



* = ^ 



47T' 



where J is the line integral of the magnetic force around a circuit 

 positively linked once with the circuit, and in the electromagnetic 

 system / = <!>. Accordingly 



J=47T/. 



If now we consider steady currents distributed in any manner in a 

 conducting body with current density q, the integral of magnetic 

 force around any closed curve depends only on the tubes of flow 

 with which it is linked, being equal to 4?r times the total current 

 through the curve. Consequently 



(i) ^Ldx + Mdy + Ndz 



= 4?r M {u cos (nx) + v cos (ny) + w cos (nz)} dS, 



the surface integral being taken over any surface bounded by the 

 curve. But by Stokes's theorem 



lldx + Mdy + Ndz 



dN\ 



- cos (nx} + - asv cos (ny} 



The surface integrals can be equal for all surfaces bounded by any 

 curve whatsoever only if we have everywhere 



8^ dM 

 -= -- -5- , 

 dy dz 



dL dN 



(2) 4>7TV =-= --- -5 , 



dz dx 



dM dL 



-5 -- -5- . 



ex cy 



