Gauss's law
/ GOWSS /
Gauss's law is a shortcut that connects the electric field poking out through any closed surface to the charge trapped inside it. Imagine drawing an imaginary bag around some charges: the law says the total electric flux out of the bag depends only on the charge sealed inside, and not at all on how that charge is arranged or on any charges sitting outside.
Precisely, the net electric flux through any closed surface equals the enclosed charge divided by the constant epsilon_0: Phi_E = Q_enclosed / epsilon_0, where epsilon_0 = 8.85 x 10^-12 C^2/(N m^2) is the permittivity of free space. It is fully equivalent to Coulomb's law, but far more powerful when the situation is symmetric. By choosing a clever imaginary surface (a Gaussian surface) that matches the symmetry, you can pull E outside the sum and solve for the field with almost no calculation.
Gauss's law is one of Maxwell's four equations, the foundation of all electricity and magnetism. It instantly explains several facts: the field outside a uniformly charged sphere looks exactly like that of a point charge at its centre; the field inside a hollow conductor is zero; and excess charge on a conductor lives entirely on its surface. Wherever a problem has spherical, cylindrical, or planar symmetry, Gauss's law is the fastest route to the answer.
Wrap a spherical Gaussian surface of radius r around a point charge Q. By symmetry E is the same everywhere on it and points outward, so E times the area 4 pi r^2 equals Q / epsilon_0, giving E = Q / (4 pi epsilon_0 r^2), exactly Coulomb's law.
For symmetric charges, Gauss's law delivers the field with almost no work.
Gauss's law is always true, but it lets you solve for E only when symmetry makes the field constant over your chosen surface. Without that symmetry the law still holds, but it no longer hands you the field for free.