Application of Gauss' Law

Gauss' law, $\phi_E=\oint\vec{E}\cdot d\vec{S}=q_{enc}/\epsilon_0$, is exact for any closed surface, but the flux integral on the left is only easy to evaluate by hand when the chosen Gaussian surface matches the SYMMETRY of the charge distribution being studied -- ideally a surface over which $\vec{E}$ is either constant in magnitude and everywhere parallel to the surface's own outward normal (so the dot product reduces to a simple product $E\,dS$), or else exactly perpendicular to the normal (so that portion contributes nothing at all).

The common recipe used in every application below has five steps: (1) identify whether the given charge distribution is effectively linear (a thin wire), cylindrical, planar (a sheet), or spherical in its symmetry; (2) visualise an imaginary Gaussian surface of matching symmetry passing through the point where the field is wanted; (3) write down the LEFT side of Gauss' law, $q_{enc}/\epsilon_0$, using the actual enclosed charge; (4) separately work out the flux GEOMETRICALLY as $E\times(\text{effective area})$ using the chosen surface's shape; and (5) equate the two expressions from steps 3 and 4 and solve algebraically for $E$. …