# Derivation of eq of motion of q in static E?

• jason12345

#### 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.

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.