Gauss's Law

Theorem: Gauss's Law (Integral Form)

The Electric Flux through any closed surface is given by the Permittivity $\epsilon$ of space and the total Electric Charge $Q_{\text{enc}}$ enclosed within the surface.

$$\oint \mathbf{E}\cdot \mathrm{d}\mathbf{a}=\frac{Q_{\text{enc}}}{\epsilon }$$

PROOF

TODO

Theorem: Gauss's Law (Differential Form)

The divergence of the electric field $\mathbf{E}$ at any point is given by the Permittivity $\epsilon$ of space and the Charge Density $\rho$ at that point.

$$\nabla \cdot \mathbf{E}=\frac{\rho }{\epsilon }$$

PROOF

TODO