# Gauss's Law

Gauss's Law states that the electric flux through a closed surface is proporational to the electrical charge inside. This holds true regardless of the volume or shape of the closed surface. This is one of the most fundamental principles of electrodynamics, and is one of Maxwell's Equations. The law is named after Carl Friedrich Gauss.

In integral form, Gauss's Law is this:

$\Phi = \oint_S \vec{E} \cdot \mathrm{d}\vec{A} = {1 \over \varepsilon_0} \int_V \rho\ \mathrm{d}V = \frac{Q_A}{\varepsilon_0}$

where Φ is the electric flux through the surface S, $\vec{E}$ is the electric field, $\mathrm{d}\vec{A}$ is a differential area on the closed surface S with an outward facing surface normal defining its direction, QA is the charge enclosed by the surface, ρ is the charge density at a point in V, $\varepsilon_0$ is a constant for the permittivity of free space and the integral $\oint_S$ is over the surface S enclosing volume V.

It can also be stated as:

$\vec{\nabla} \cdot \vec{E} = \frac{\rho}{\varepsilon_0}$

where ρ is the charge density.