Energy in the Electromagnetic Field. 

 surface charge 



T .-=if(V+# + 7, s > 



?«=*«=.. V* 



147 



(2) 



where e r is expressed in E.M.U. 



To calculate the mutual energy of any two charges e x , e 2i 

 of radii a 1} a 2 , the distance between them being r, refer 

 them to axes with origin at e l9 the z-axis coinciding with 

 the line joining the charges. Let (u lf i? l9 10,), (w 2 , v 2 , w 2 ) De 

 the component velocities of the charges, r l3 r 2 their respec- 

 tive distances from any point (#, y, z), and (« lf A, 7^, 

 ( a 2> A? 72) the components of magnetic force at that point 

 due to the charges respectively. We then have 



£1 = -^ {xw x — zi fl ), /3 2 ={xw 2 -(z—r)u 2 \ y . (2) 

 7i= 3 (y M i — *»]), 72= {^2—^2} 



/ 1 



^Denoting by T m the mutual energy of the charges, we 

 obtain 



T m= ^ (( a i*2 + AA-r-7r72>^ 



= 4-^ Ul (^2 + w 1 w s )a 2 + (u ] u 2 + w 1 iv 2 )y 2 



— {i'iif 2 + "ai' 2 ).V£— (»t'iWj + t*ii0j)#;s 



dx dy dz 



+ w x u 2 rx -f WiV 2 rg 



/r/v 



the integration extending throughout all space external to 

 the charges. Since the axis of z coincides with the line 

 joining the charges, most of the integrals evidently vanish 

 by symmetry and the expression reduces to 



e\e 2 



T m = 4" {(^ u 2 + v 1 v 2 + 2w l w 2 )I 1 +(u 1 u 2 + v 1 v 2 )I 2 }, . (4) 

 L2 



