Page:TolmanEquations.djvu/6

 These equations give the force acting on $$\epsilon_1$$ at the time $$t$$. From equation (4) we have $$t=\frac{v}{c^{2}}x$$ since $$t'=0$$. At this time, the charge $$\epsilon$$ which is moving with the uniform velocity $$v$$ along the X axis will evidently have the position

$x_{\epsilon}=\frac{v^{2}}{c^{2}}x,\ y_{\epsilon}=0,\ z_{\epsilon}=0.$|undefined

For convenience we may now refer our results to a system of coordinates whose origin coincides with the position of the charge $$\epsilon$$ at the instant under consideration. If X, Y, and Z are the coordinates of $$\epsilon_1$$ with respect to this new system, we evidently have the relations

$\mathsf{X}=x-\frac{v^{2}}{c^{2}}x=\kappa^{-2}x,\ \mathsf{Y}=y,\ \mathsf{Z}=z,\ \mathsf{\dot{X}}=\dot{x},\ \mathsf{\dot{Y}}=\dot{y},\ \mathsf{\dot{Z}}=\dot{z}.$|undefined

Substituting into (21), (22), and (23) we may obtain:—

where for simplicity we have placed $$\beta=\tfrac{v}{c}$$, and

$s=\sqrt{\mathsf{X}^{2}+\left(1-\beta^{2}\right)\left(\mathsf{Y}^{2}+\mathsf{Z}^{2}\right)}.$

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 $$\mathsf{F}=\mathsf{E}+1/c\ \mathsf{v}\times\mathsf{H}$$. 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 $$\epsilon_1$$ is stationary (i. e. $$\mathsf{\dot{X}}=\mathsf{\dot{Y}}=\mathsf{Z}=0$$) and equal to unity, equations (24), (25) and (26) should give us the strength of the electric field produced by the moving point charge $$\epsilon$$, and in fact they do reduce as expected to the known expression

$\mathsf{F}=\mathsf{E}=\frac{\epsilon}{s^{3}}\left(1-\beta^{2}\right)\mathsf{r}$|undefined

where

$\mathsf{r}=\mathsf{Xi}+\mathsf{Yj}+\mathsf{Zk},$