Gauss's law

For a point charge q at the centre of a sphere of radius r, E is the same everywhere on the sphere and along the normal, so the flux is E × 4πr² = q/ε₀, whatever the radius.

Gauss's law: the total electric flux through any closed surface equals 1/ε₀ times the net charge enclosed, φ = ∮E·dS = q_enc/ε₀.

It holds for a closed surface of any shape or size; the enclosed charges may lie anywhere inside it.

q_enc counts only charges inside the surface, but the field E on the surface is produced by all charges, inside and outside.

If a closed surface encloses no net charge, the total flux through it is zero, as for a closed cylinder in a uniform field: the flux in through one end equals the flux out through the other, and none crosses the curved side.

The chosen surface is the Gaussian surface. It must not pass through a point charge (the field is undefined there), though it may pass through a continuous distribution.

Gauss's law rests on the inverse-square form of Coulomb's law; it is most useful for finding E when the charge has symmetry.