Gauss's law

Gauss's law makes a statement that sounds almost too simple to be useful: the total electric flux passing out through any closed surface — a sphere, a cube, any shape at all — depends on nothing but the total charge enclosed inside it. Charge outside the surface contributes flux going in on one side and exactly the same flux going out the other, so it cancels completely and never counts. Where the charge sits inside the surface does not matter either; only the total does.

That total independence from shape is precisely what makes the law powerful rather than merely elegant. Direct calculation of the electric field from Coulomb's law means adding up the contribution of every piece of charge individually, which is often intractable. Gauss's law sidesteps this entirely when a charge distribution has enough symmetry to choose a surface — called a Gaussian surface — on which the field is constant by symmetry and can be pulled outside the flux integral. A point charge suggests a sphere centred on it; an infinite charged line suggests a cylinder around it; an infinite charged sheet suggests a flat pillbox through it. In each of these cases what would be a difficult integral collapses to simple algebra.

None of this restricts the law to symmetric situations — Gauss's law holds for every charge distribution and every closed surface, always. It is only useful for finding the field directly when symmetry provides a convenient surface; for anything else it remains true but is put to work differently, as one of Maxwell's four equations governing the field everywhere at once.