Gauss's Theorem

Gauss's theorem (sometimes called Gauss's law) is one of the four fundamental equations of electromagnetism, and it states an exact, general relationship between the electric flux through any closed surface and the charge enclosed by that surface:

The total electric flux through any closed surface (a "Gaussian surface", which may be any shape whatsoever, real or purely imaginary) is equal to $1/\epsilon_0$ times the total electric charge enclosed within that surface.

Here $q_{\text{enc}}$ is the algebraic sum of all the charge lying strictly inside the closed surface (charges outside the surface, however close, are excluded); $\epsilon_0$ is the same permittivity of free space that appears in Coulomb's law.

Why the theorem is true for any surface. The theorem is not an independent new law of nature; it is a direct mathematical consequence of Coulomb's inverse-square dependence on distance. Consider a single point charge $q$ enclosed by a Gaussian sphere of radius $r$ centred on it: the field at every point of the sphere's surface has the same magnitude $E=kq/r^2$ (by symmetry) and points radially outward, exactly parallel to the outward normal everywhere, so the flux is simply $\Phi=E\times(\text{area of sphere})=\frac{kq}{r^2}\times4\pi r^2=4\pi kq=q/\epsilon_0$ (using $k=1/4\pi\epsilon_0$) -- and, remarkably, the radius $r$ has cancelled out completely, so the flux is exactly $q/\epsilon_0$ regardless of how large or small the sphere is. A more general argument (using the idea of solid angle) shows that the same result, $q/\epsilon_0$, holds even for a Gaussian surface of any other shape, not just a sphere, and even if the charge is not exactly at the surface's centre: every field line leaving the charge crosses the surface exactly once (assuming the surface is not weirdly self-intersecting), so the total flux depends only on how much charge is enclosed, never on the surface's shape or size.

Charges outside the surface contribute nothing. A charge lying outside a closed Gaussian surface contributes exactly zero to the net flux through that surface: every field line from an external charge that enters the surface on one side must also leave it again on another side (since the surface does not enclose that charge), so the flux it contributes going in is exactly cancelled by the flux it contributes coming back out. …