L. Page — Energy of a Moving Electron. 123 



In order to find the energy of the moving electron we shall 

 charge it by shrinking down from infinite size to its surface a 

 series of moving shells. Each shell will carry an infinitesimal 

 charge de distributed over its surface in such a way as to give 

 rise to no field inside the shell, and will maintain throughout 

 the process of contraction the same velocity as the electron. 

 The work done in shrinking these shells against the forces 

 exerted by the charge already on the electron will be equal to 

 the energy of the electron in its final state. 



First it is necessary to show that the energy of n shells of 



radius B, and charge de, where R is infinitely great and 



nde = e is finite, is negligible compared to that of an electron 



of charge e and finite radius a. A consideration of dimensions 



alone shows that the energy of one of these shells must be 



(deY 

 proportional to — &~ while that of the electron itself must be 



proportional to — . Hence the energy of the n shells will be 



proportional to -77 which is an infinitesimal of the second 



order compared to — . This must be true quite irrespective 



of the velocities of the shells and of the electron, provided 

 these velocities are less than the velocity of light. Hence the 

 energy of the n shells of infinite radius moving with velocity v 

 is negligible compared to the work done in contracting them. 

 Moreover in contracting a shell, the work done against the 

 electromagnetic forces due to the shell itself will be of the 



ordel . <^ ana he „ce ne gligM e spared to tLe work dooe 



against the forces due to that part of the charge which is 

 already on the surface of the electron. So in calculating the 

 energy we need consider only the work done against the forces 

 exerted by the charge already on the electron's surface. 



Let (fig. 1) be the center of the electron and P a point on 

 the contracting shell. The electric and magnetic forces at P 

 are given by the familiar expressions 



E=^ (^ s (22 ) 



e 





(1-/8 2 ) 





47rr* 



(1 



-/3'sin 2 



6$ 



e 



£ 



sin 0(1 



-F) 



H 4^ (l-^sin^jf (23) 



In order that the contracting shell shall give rise to no field 

 in its interior, it is obviously sufficient that relative to an 



