Page:TolmanEquations.djvu/5

 inverse square law for the force exerted by a stationary charge.

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.

or

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

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.