Gauss's law

Gauss's Law gives the relation between the electric flux flowing out a closed surface and the charge enclosed in the surface:

$\oint _{A}\mathbf {E} \cdot d\mathbf {A} ={\frac {Q_{\mbox{A}}}{\epsilon _{0}}}$

where $\mathbf {E}$ is the electric field (in units of V/m), $d\mathbf {A}$ is the area of a differential square on the surface A with an outward facing surface normal defining its direction, $Q_{\mbox{A}}$ is the charge enclosed by the surface, $\epsilon _{0}$ is the permittivity of free space and $\oint _{A}$ is the integral over the surface A.

In the case of a spherical surface with a central charge, the electric field is perpendicular to the surface, with the same magnitude at all points of it, giving the simpler expresion:

$E={Q \over \epsilon _{0}A}$

where E is the electric field strength, Q is the enclosed charge, A is the area of the sphere, and &epsilon₀ is the permittivity of free space.

Gauss's law can be used to demonstrate that there is no electric field inside a Faraday cage. Gauss's law is the electrical equivalent of Ampere's law, which deals with magnetism. Both equations were later integrated into Maxwell's equations.