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# Gauss's law

In physics, 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, $d\mathbf{A}$ 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 ε0 is the permittivity of free space.

Gauss's law can be used to demonstrate that there is no electric field inside a Faraday cage without electric charges. Gauss's law is the electrical equivalent of Ampere's law, which deals with magnetism. Both equations were later integrated into Maxwell's equations.

It was formulated by Carl Friedrich Gauss in 1835, but was not published until 1867.

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