We know that the charge density for a moving charge is taken as,
$$\rho(\mathbf{r}, t) = q \delta(\mathbf{r} - \mathbf{r}_q(t_r)).$$
This gives us the retarded potential of $$\phi(\mathbf{r}, t) = \frac{q}{|\mathbf{r} - \mathbf{r}'(t_r)|} \left( 1 - \frac{\mathbf{v}(t_r) \cdot (\mathbf{r} - \mathbf{r}'(t_r))}{c |\mathbf{r} - \mathbf{r}'(t_r)|} \right).$$ I have seen physical explanations of this involving double counting of charge: Feynman's proof for Liénard-Wiechert's potential of a moving charge.. Is that the only way to explain this? Should I even look for a physical explanation or do I just say that this is how the math works?
