Gauss's law

Gauss's law is electric-field bookkeeping: it says the total number of field lines bursting out through any closed surface tells you exactly how much charge is trapped inside. Imagine the surface as a fishing net thrown around a region of space; count the field lines poking through the net, and you've counted the charge in the catch — no need to know how the charge is arranged inside. If equal positive and negative charge sit inside, the lines that leave are matched by lines that re-enter and the net flux is zero.

This deceptively simple statement is one of Maxwell's four equations and a workhorse for finding fields quickly when there's symmetry. Want the field around a charged sphere or an infinite charged plate? Draw a Gaussian surface that matches the symmetry, and the law hands you the answer in one line instead of a brutal integral. It also explains why the field inside a hollow conductor is exactly zero — there's no enclosed charge, so no net flux, so no field — which is the physics behind the shielding 'Faraday cage' that keeps your car safe in a lightning storm.