If Fig. 40 represents a magnet of pole area , whose two poles are cm. apart, and have a total magnetization , a density of magnetization , and a field strength between them of , then the work necessary to carry a unit pole from to is , and the work necessary to create the poles and , i.e., to carry units of magnetism across against a mean held strength is . Hence the total energy of the magnetic held is given by but since or since is the volume of the held the energy per unit volume of the magnetic held is given by Now the strength of the magnetic held at a distance from a moving charge in the plane of the charge is , if is the charge and its speed. Also the magnetic field strength at a point distant from the charge, being the angle between and the direction of motion, is given by Hence the total energy of the magnetic field created by the moving charge is in which is an element of volume and the integration is extended over all space. But in terms of , , and . Since kinetic energy = , the mass-equivalent of the moving charge is given by setting The radius of the spherical charge which would have a mass equal to the observed mass of the negative electron is found by inserting in the last equation and . This gives .

The expression just obtained for obviously holds only so long as the magnetic field is symmetrically distributed about the moving charge, as assumed in the integration, that is, so long as is small compared with the velocity of light. When exceeds .1 the speed of light , the mass of the charge begins to increase measurably and becomes infinite at the speed of light. According to the theory developed by Lorentz, if the mass for slow speeds is called and the mass at any speed is called , then This was the formula which Bucherer found to hold accurately for the masses of negative electrons whose speeds ranged from .3 to .8 that of light.