726 Prof. Harkins and Mr. Wilson on Energy Relations 



where the analysis is given in vector notation, and Heaviside 

 units are used. 



The electromagnetic momentum, G, due to a system of 

 charges is 



[EH] [(2E.X2H,)] STEvHJ 2[E«H,-] 

 q. _ j j _ j |_ (y) 



c c c c 



where the summation % is the vector product of each i with 

 each j. ^ 



The first summation gives the electromagnetic momentum 

 which would he due to the particles if their fields did not 

 overlap, and the second term gives the effect of the over- 

 lapping of the fields. This may he called the "mutual 

 electromagnetic momentum/' and is designated by G. 



For point charges, 



_ (1-^ 



1_ 47rr 2 (l-^ 2 sin 2 ^ 1 )^ 2 * 



Let (l-u 2 )=k 2 , and (1-u 2 sin 2 l )=/3 1 \ 



The transverse component of E due to the two particles 

 1 and 2 is 



V - — f sin ^1 r sin 0* \ 



where the sign is positive if the charges have the same sign, 

 and negative if they are of opposite sign. As only the longi- 

 tudinal component of the vector G is desired, only the 

 transverse component of E is needed. Then H = wE sin 0/c, 

 where <j> = the angle between E and the direction of u. 

 If E, is used, cf> =90°, and H = (Ej sin ^ + E 2 sin 3 )/c. 

 Hence 



G L = L^S. = - 2 (E x sin 1 ± E 2 sin ff 2 ) (E x sin d l ± E 2 sin S ) , 

 c c 



and 



— '2u r_ _ . Q . a 7 . 2u fc^e 2 Tsin 6 X sin 2 i 



Now r 2 /3 2 = r 2 -it 2 (r 2 sin 2 <9) and r 2 sin 2 <9 = ?/ 2 . 



Let a =1/2 the distance apart of the electrons. Neglecting 

 all terms in u 2 , and placing dr=-2ydydx, we have 



— _ uK^e 2 9(9 , rp y z dyda 



- ± 83V- l ""'J, J v/{L( a -a) 2 +^JL(*' + «) 2 +/]} 3 



which is obtained by making use of the symmetry of the 



