Gauss's Law

Gauss's law states that the net flux through any closed surface equals the enclosed net charge over $\varepsilon_0$, where the field being integrated is the total field, outside charges included. It yields a value of $E$ only when symmetry makes $E$ uniform and normal on the flux-carrying part of the surface: a concentric sphere for spherical charge, a coaxial cylinder for a line, a pillbox for a sheet. Inside a uniform shell $E = 0$; inside a uniform solid sphere $E = kQr/R^3$; an infinite line gives $2k\lambda/r$; an infinite sheet gives $\sigma/2\varepsilon_0$ at any distance, and a conductor's surface gives $\sigma/\varepsilon_0$.

Five errors recur. Erasing an outside charge from the field on the surface because it does not count in the flux. Factoring $E$ out of the integral over a cube face or around a dipole, where $E$ is neither uniform nor normal. Swapping the shell's zero interior for the solid sphere's linear rise, or using total $Q$ inside the solid sphere. Treating the imagined surface as a physical object, so that a bigger sphere seems to change the field. And making the infinite sheet's field fade with distance, or handing a free-standing sheet the conductor's $\sigma/\varepsilon_0$.