Gaussian surface

A Gaussian surface is any closed surface you imagine around a charge distribution in order to apply Gauss's law. The surface is not real — no physical membrane is involved — and you are free to pick whatever shape makes the integral easiest. The art of the method is choosing a surface whose geometry matches the symmetry of the field, so that the flux integral collapses into simple arithmetic.

For a point charge or a spherically symmetric distribution, the right choice is a concentric sphere: on that sphere E is constant in magnitude and perpendicular to the surface, so the flux is just E × (4πr²). For an infinite line of charge, a coaxial cylinder works: E is radial and constant on the side, zero flux through the end caps. For an infinite charged plane, a "pillbox" straddling the plane gives two equal end-cap contributions and no side flux.

The trick is entirely about symmetry. Gauss's law is always true for any closed surface, but the integral is only tractable when you can pull E out of it. If the charge distribution has no useful symmetry, Gauss's law still holds, but you would not use it for calculation — you would use Coulomb's law or a computer instead.