122 L. Page — Energy of a Moving Electron. 



how the charge could be in equilibrium otherwise, — the 

 distribution of electricity on the surface of the electron will 

 not be uniform, and consequently the electric forces teudino- 

 to disrupt the electron cannot be counteracted by a hydrostatic 

 pressure. If the pressure is not hydrostatic, but of such a 

 magnitude at each point on the surface as to balance the 

 disruptive forces, it will exert a resultant force in the direction 

 of motion that will exactly balance the retarding force due to 

 the electron's own held, and in order to satisfy the equation of 

 motion for the electron it would be necessary to introduce a 

 mechanical mass. The most obvious way to avoid this difficulty 

 is to deny the existence of mechanical forces per se, and put 

 everything on an electrodynamic basis, at least in so far as the 

 motion of electrons is concerned. Then in dealing with the motion 

 of an accelerated electron we could not eliminate the external 

 electromagnetic field which was responsible for the acceleration. 

 We should have to deal with two overlapping fields, which 

 would render the problem more complicated. So far as 

 mechanical forces are concerned, it would seem that a body 

 which is subject to a mechanical force must have a mechanical 

 mass. 



Third Method. 



We will now calculate the energy of an electron moving with 

 a constant and finite velocity v by a method analogous to that 

 used in finding the electrostatic energy of a stationary charged 

 conductor and without causing the electron to pass through a 

 series of states in which the velocity varies. 



To an observer relative to whom it is at rest, we have 

 assumed the electron to be a uniformly charged spherical shell. 

 Hence the kinematical transformations of relativity 8how that 

 it will be an oblate spheroid to the observer relative to whom 

 it has the constant velocity v. If the origin of moving axes is 

 taken at the center of the electron with the Z axis parallel to 

 the direction of motion, the surface of the electron will be that 

 formed by revolving the ellipse 



i + ^i&W) = l (20) 



about the Zaxis. If 6 is the angle made by any radius vector 

 of the ellipse with the Zaxis, the charge on the annular ring 

 between the cones defined by 6 and 6 + <16 is easily seen 

 to be 



e (l-/n *\»0d6 

 <(e ~ 2 (1-/3* sin* 0jf { ' 



