[NOTES/EM-03020] Proof of Gauss Law from Maxwell's Equations

Gauss Law

The proof of Gauss law, given by Feynman, is simple and intuitive and is based on the properties of Coulomb force. Here we use  divergence theorem of vector calculus to prove Gauss law using one of the Maxwell's equations. We integrate the equation \(\nabla\cdot \vec E= \rho/\epsilon_0\) over a closed surface ,\(S\), enclosing a volume \(V\). This gives
\begin{equation} \int_V \nabla\cdot \vec E, dV  = \int_V \frac{\rho}{\epsilon_0}, dV= \frac{Q_\text{encl}}{\epsilon_0}. \end{equation}
The left hand side of the above equation becomes a surface integral on using divergence theorem of vector calculus.
\begin{equation}\label{eq01} \iint_S \vec E\cdot \hat n dS = \frac{Q_\text{encl}}{\epsilon_0}. \end{equation}
Noting that the left hand side is just the flux of the electric field through the surface \(S\), the above equation is a mathematical statement of the Gauss law.

Reference:
 R. P. Feynman, Robert B. Leighton and Mathew Sands, {\it Lectures on Physics}, Vol-II, B.I. Publications (1964)