Gauss's Law and Choosing a Gaussian Surface

Picture a closed bubble around some charge. Every field line that starts on positive charge inside has to cross the bubble on its way out, and field lines from outside charge pass through once and leave. Count them up with the right weighting and you get the charge trapped inside, no matter how twisted the bubble's shape is. That counting is flux, and Gauss's law states it exactly: the flux through any closed surface equals Q_enc divided by epsilon₀, full stop, for any charge distribution whatsoever.

The law is always true, but it only becomes a calculation trick when the charge distribution has enough symmetry to let you choose a Gaussian surface on which E is either constant in magnitude or perpendicular to the surface everywhere, or both. Then E slides outside the integral, flux becomes E times the surface area, and you solve for E in one line. The three symmetries that work are spherical, cylindrical, and planar, matched to a sphere, cylinder, or pillbox surface respectively. Without matching symmetry, the integral is still true but you cannot pull E out, so it is useless as a tool even though it never breaks as a law.

This is why Gauss's law sits right next to Coulomb's law rather than replacing it. Apply Gauss's law to a single point charge with a spherical surface centered on it: E is constant over the sphere by symmetry, so E times 4 π r² equals q over epsilon₀, giving E = q / (4 π epsilon₀ r²), exactly Coulomb's law. Gauss's law is the deeper statement; Coulomb's law falls out of it as a special case once you add spherical symmetry.