Page:TolmanEquations.djvu/5

inverse square law for the force exerted by a stationary charge^[1].

Consider a set of coordinates S(x, y, z, t), and let there be a charge $\epsilon$ in uniform motion along the X axis with the velocity $v$. We desire to know the force acting at the time $t$ on any other charge $\epsilon _{1}$, which has any desired coordinates $x,y,$ and $z$ and any desired velocity ${\dot {x}},{\dot {y}},{\dot {z}}$.

Assume a system of coordinates $S'(x',y',z',t')$ moving with the same velocity as the charge $\epsilon$ which is situated at the origin. To an observer moving with the system S', the charge always appears at rest an to be surrounded by a pure electrostatic field. Hence in system S' the force with which $\epsilon$ acts on $\epsilon _{1}$, will be in accord with Coulomb’s law.

${\mathsf {F}}'={\frac {\epsilon \epsilon _{1}{\mathsf {r}}'}{r^{3}}},$

or

${\mathsf {F}}_{x}={\frac {\epsilon \epsilon _{1}x'}{\left(x'^{2}+y'^{2}+z'^{2}\right)^{3/2}}},$

(18)

${\mathsf {F}}_{y}={\frac {\epsilon \epsilon _{1}y'}{\left(x'^{2}+y'^{2}+z'^{2}\right)^{3/2}}},$

(19)

${\mathsf {F}}_{z}={\frac {\epsilon \epsilon _{1}z'}{\left(x'^{2}+y'^{2}+z'^{2}\right)^{3/2}}},$

(20)

where $x',y',$ and $z'$ are the coordinates of charge $\epsilon _{1}$, at the time $t'$. For simplicity let us consider the force at the time $t'=0$, then from transformation equations (1)-(3) we shall have

$x'=\kappa ^{-1}x,\ y'=y,\ z'=z.$

Substituting into (18), (19), and (20) and also making use of the transformation equations of force (15), (16), and (17), we obtain the following equations for the force acting on $\epsilon _{1}$, as it appears to an observer in system S.

${\mathsf {F}}_{x}={\frac {\epsilon \epsilon _{1}\kappa ^{-1}}{\left(\kappa {}^{-2}x^{2}+y{}^{2}+z{}^{2}\right)^{3/2}}}\left[x+{\frac {v}{c^{2}}}\kappa ^{2}\left(y{\dot {y}}+z{\dot {z}}\right)\right],$

(21)

${\mathsf {F}}_{y}={\frac {\epsilon \epsilon _{1}\kappa \left(1-{\frac {v{\dot {x}}}{c^{2}}}\right)y}{\left(\kappa {}^{-2}x^{2}+y{}^{2}+z{}^{2}\right)^{3/2}}},$

(22)

${\mathsf {F}}_{z}={\frac {\epsilon \epsilon _{1}\kappa \left(1-{\frac {v{\dot {x}}}{c^{2}}}\right)z}{\left(\kappa {}^{-2}x^{2}+y{}^{2}+z{}^{2}\right)^{3/2}}},$

(23)

1. In its simplest form, Coulomb’s law merely states the force acting between two stationary charges. It should be noted that our derivation assumes the same law for the force with which a stationary charge acts on a moving charge.