[NOTES/EM-03009]-Coulomb's Law from Maxwell's Equations --- An Outline

We will outline the derivation of Coulomb's law from Maxwell's equations for electrostatics. \begin{eqnarray}\label{EQ01} \nabla\cdot\bar{E}&=&\rho/\epsilon_0 \\ \nabla\times\bar{E}&=&0 \label{EQ02} \end{eqnarray} The equation \eqRef{EQ02} implies the existence of electric potential \(\phi\), such that \(\vec{E}=-\nabla \phi\). Substituting this in \EqRef{EQ01} we get \begin{equation}\label{EQ03} \nabla ^2 \phi = -\frac{\rho}{\epsilon_0}. \end{equation} This equation can be solve by first finding the Green function Details?. The Green function satisfies the equation \begin{equation}\label{EQ04} \nabla^2 G(\vec{r}, \vec{r}{'} ) = \delta^{(3)}(\vec{r}-\vec{r}{'}), \end{equation} where \(\delta^{(3)}(\vec{r}-\vec{r}{'})\) is the Dirac delta function in three dimensions. The Green function turns out to be Details \begin{equation}\label{EQ05} G(\vec{r}, \vec{r}{'} ) =- \frac{1}{4\pi}\, \frac{1}{|\vec{r}-\vec{r}{'}|}. \end{equation} and is, apart from overall constant, the electric potential of a point charge. In terms of the Green function the solution of Poisson equation \eqRef{EQ03} is given by Details \begin{equation} \phi(|\vec{r}-\vec{r}{'}|)= -\frac{1}{\epsilon_0}\iiint_V G(\vec{r}-\vec{r}{'}) \rho(\vec{r}{'}) \, d^3r{'}. = \frac{1}{4\pi\epsilon_0}\iiint_V \frac{\rho(\vec{r}{'})}{|\vec{r}-\vec{r}{'}|}\, d^3r{'}. \end{equation} Therefore \begin{equation} \vec{E} = -\nabla \phi = \frac{1}{4\pi\epsilon_0}\iiint_V \rho(\vec{r}{'}) \frac{(\vec{r}-\vec{r}{'})}{|\vec{r}-\vec{r}{'}|^3}\, d^3r{'}. \end{equation} This is just the Coulomb's law.