154 Prof. R. C. Tolman on 



inverse square law for the force exerted by a stationary 

 charge*. 



Consider a set of coordinates S(#, y, z, t), and let there be 

 a charge e 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 e 1? which has any desired coordinates 

 a?, y, and z and any desired velocity i?, y, z. 



Assume a system of coordinates S'(a/, y f , z\ t') moving 

 with the same velocity as the charge e which is situated at 

 the origin. To an observer moving with the system S', the 

 charge always appears at rest and to be surrounded by a 

 pure electrostatic field. Hence in system S' the force with 

 which e acts on € 2 will be in accord with Coulomb's law. 



F / = €€ l r ' 



F ^ = (^ + y3 +<2 /2)3/2. ' • ' • ( 18 ) 



F * = (?*+y* + *'3)a/2' • • • • (19) 



/ 

 F * =E ^+^ + ^)M' ' " .' ' (20) 



where x' , y', and z' are the coordinates of charge e 1? at the 

 time t. For simplicity let us consider the force at the time 

 £'=(), then from transformation equations (l)-(3) we shall 

 have 



x' = /c-\v, y'=y, z'=z. 



Substituting into (IS), (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 e l5 

 as it appears to an observer in system S. 



^ f^wf h^^ + ^]' • (21) 



(tc~V+y 2 + z 2 ) 

 _ee 1 «(l-^)/ 



F^=__^ <^1_^, (22) 



(«-V + */ 2 -rs 2 ) 3/2 



€€!* (1—4)2 

 F 2 = V ^L (23) 



( K -\v 2 +y 2 + z 2 f 12 



* 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. 



