L. Page — Energy of a Moving Electron. 125 



From (3) we find for the rate at which work is done on the 

 charge contained on an element of the surface of the contract- 

 ing shell 



dW . r_ _ *T?r~i /oeX 



_ = _ A |_ E .V + E . W J (25) 



Putting 



W=U 3 +2\ 





where 









dU 3 , _ dr 

 — z- 3 - = — c?e E • -r- 



eft f//; 



(26) 



and 









-^t- 3 = — de E • V 



CM 



(27) 



we find, 



on substituting from (21) and (22) 



er?e(l-/3 2 ) 2 /» /* sin 6d6dr 







8tt 7 ,/ r 2 (L-/3 2 sin* 0) 3 



o 6 







ede 3— j8 2 

 12ira Vl-/i 2 ' 



(28) 





where o == — ==r--- 



Vl-i8"sin»0 ' 





Integrating with respect to <? to find the work done in shrink- 

 ing all the shells clown to the surface of the electron, we find 

 for the total potential energy 



e 2 3- /3 2 m c 2 3-/3' 



3 ~ 24tt« Vi^/S 2 ~~ * V^W 



which is the same as the electric energy of the electron's 

 field, i.e. 



lT=~pS'dr (30) 



Subtracting from Z7 3 the electrostatic energy of a charged con- 

 ducting sphere, we find for the increase in potential energy due 

 to the electron's motion 



AZ7 S =-^— i , P -3 (31) 



Equation (27) indicates that there is a resultant electric 

 force on the contracting shell in such a direction as to oppose 



