These equations give the force acting on at the time . From equation (4) we have since . At this time, the charge which is moving with the uniform velocity along the X axis will evidently have the position
For convenience we may now refer our results to a system of coordinates whose origin coincides with the position of the charge at the instant under consideration. If X, Y, and Z are the coordinates of with respect to this new system, we evidently have the relations
Substituting into (21), (22), and (23) we may obtain:—
(24)
(25)
(26)
where for simplicity we have placed , and
These same equations could also be obtained by substituting the well-known formula for the strength of the electric and magnetic field around a moving point charge into the fifth fundamental equation of the Maxwell-Lorentz theory . It is interesting to see that they can be obtained so directly, merely from Coulomb's law.
If we consider the particular case that the charge is stationary (i. e.) and equal to unity, equations (24), (25) and (26) should give us the strength of the electric field produced by the moving point charge , and in fact they do reduce as expected to the known expression