Derivation of eq of motion of q in static E?

[jason12345]

The equation of motion for q in a static E is given by:

$$d/dt \gamma mv = qE$$

Some textbooks use the above equation in deriving Maxwell's equtions, but is there a way of deriving this equation from elementary assumptions such as Newton's law in q's frame and Coulomb's law?

Thanks.

 

[bcrowell]

If you know about tensors, then the following may help. If we're going to define something like an electromagnetic field in terms of what it does to a test charge, then that definition has to be expressed in terms of a tensor equation, or else it wouldn't have the same form in all frames of reference. The tensor we want to predict is $dp^j/d\tau$, and the only thing it can depend on besides the field is the particle's motion, described by its four-velocity $v^k$. Given these facts, the most general tensor equation we can have is of the form $dp^j/d\tau = F^{jk}v_k$. If all of this is going to match up with Newton's laws in the nonrelativistic limit, then the time-space components of F have to be the electric field.