Exploring Coulombs Law Extended: Derivations, Uses, and Limitations

[bicabone]

My physics professor gave us an extended version of coulombs law which includes terms to account for moving charges. He then used (or rather compared) this law to the Lorentz force equation to give us a feel for the electrical forces generated by moving charges as he introduced us to magnetism. Does anyone know where I can find more information regarding this following equation (i.e. its derivations, uses, limitations, etc.)

$$F_{on 2 by 1} = \frac{1}{4\pi\epsilon_{0}} \frac{q_{1}q_{2}}{r^{2}}[ \widehat{r} + \frac{1}{c^{2}} \cdot \vec{v_{2}} \times ( \vec{v_{1}} \times \widehat{r})]$$

 

[vanhees71]

This should come from the nonrelativistic limit of the field of a moving point charge, which you most easily obtain by a Lorentz boost of the four-vector potential of a charge at rest:

$${A'}^{\mu}=(q/(4 \pi |\vec{x}'|),0,0,0)$$.

 

[Meir Achuz]

That equation is never right.
Even in the non-relativistic limit, the second term is of order v^2, so order v^2 corrections to the first term are required.

 

[Bill_K]

That equation is never right.

Perhaps you can tell us what it should be.

 

[Meir Achuz]

$$\frac{d{\bf p}}{dt}= \frac{-qq'[{\bf r}+{\bf v\times(v'\times r)}]} {\gamma_{v'}^2[{\bf r}^2-({\bf v'\times r)^2}]^{\frac{3}{2}}},$$