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

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

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

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

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

${\mathsf {F}}_{x}={\frac {\epsilon \epsilon _{1}}{x^{3}}}\left(1-\beta ^{2}\right)\left\{{\mathsf {X}}+{\frac {v}{c^{2}}}\left({\mathsf {Y{\dot {Y}}}}+{\mathsf {Z{\dot {Z}}}}\right)\right\},$

(24)

${\mathsf {F}}_{y}={\frac {\epsilon \epsilon _{1}}{x^{3}}}\left(1-\beta ^{2}\right)\left(1-{\frac {v{\mathsf {\dot {X}}}}{c^{2}}}\right){\mathsf {Y}},$

(25)

${\mathsf {F}}_{z}={\frac {\epsilon \epsilon _{1}}{x^{3}}}\left(1-\beta ^{2}\right)\left(1-{\frac {v{\mathsf {\dot {X}}}}{c^{2}}}\right){\mathsf {Z}},$

(26)

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

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

where

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