Maxwell Equations in the Presence of Dielectrics and/or Magnetics

Consider an arbitrary moving point with place \(\textbf{r}(t)\)  and velocity \(\frac{d\textbf{r}}{dt}(t)\) and let \(\textbf{u}(\textbf{x},t)\) be a speed-like vector field. Next, consider a totally neutral system of n point charges \(\sigma _1,\ldots ,\sigma _n\) , with places \(\textbf{r}_1(t),\ldots ,\textbf{r}_n(t)\) and velocities \(\frac{d\textbf{r}_1}{dt}(t),\ldots , \frac{d\textbf{r}_n}{dt}(t)\) satisfying \(\sum _{k=1}^{n}\sigma _k=0.\) Then, denoting, as usual, the charge and the current densities as: \(\rho (\textbf{x},t):=\sum _{k=1}^{n}\sigma _k\delta \left( \textbf{x}-\textbf{r}_k(t)\right) ,\quad \text {and}\quad \textbf{j}(\textbf{x},t) :=\sum _{k=1}^{n}\sigma _k\frac{d\textbf{r}_k}{dt}(t)\,\delta \left( \textbf{x}-\textbf{r}_k(t)\right) ,\)