118 L. Page — Energy of a Moving Electron. 



the diameter of the electron is all that is necessary in order 

 that the expressions given above shall hold. While the ex- 

 pression for the longitudinal mass has not been verified experi- 

 mental]) 7 , that for the transverse mass has been found to hold 

 very exactly for high speed /3 particles, first by Kaufmann* 

 and Bucherer,f and more recently to a high degree of preci- 

 sion by G. Neumann.:}; 



First Ifethod. 



To calculate the energy of a moving electron by the custom- 

 ary method use is made of the familiar energy equation which 

 follows at once from the equations of the electro-magnetic field 

 and the transformation equation for the force. If vectors be 

 denoted by Gothic letters, the force equation, in Gibbs' notation, 

 has the form 



F=p[e+4vxh] (3) 



where F denotes the force per unit volume, due to the electric 

 strain E and the magnetic strain H, on a charge of density p 

 moving with velocity v. The energy equation is 



d_ 



Jt 



^~J{E' + ir)dr~j +cJ(ExH) ■ ds +/F • Wdr=0 (4) 



The first term represents the rate of increase in the energy of 

 the electron's field, the second term the rate of radiation, and 

 the third the rate at which work is done on the electron by its 

 own field. The first term gives for the potential energy of the 

 moving electron's field 



l\=- E 2 dr = — =g, = -^- . — (5) 



2«-/ 24tt« Vl- /3 2 4 Vl /3 2 



and for the kinetic energy 



t=± fwd, = .*- -JL = ^ -JL=. (6) 



2i/ \2ira*/i — p 2 Vl — ^ 



Hence the total energy of the moving electron is 



7T7 e 2 3 + ff' ,r 1 1 , "I .„, 



W= - _ = m c* , — — ^1 — B 1 ( 7 ) 



* Kaufmann, Ann. d. Physik., xix, p. 487, 1906. 

 fBucherer, Phys. Zeitschr., ix. p. 755, 1908. 

 ^Neumann, Ann. d.. Physik., xlv, p. 529, 1914. 



