Overview and Gauss's Law

A historical unification

In 1865, James Clerk Maxwell brought together the experimental laws of electricity and magnetism (Coulomb, Gauss, Faraday, Ampère) into four equations, adding a missing term: the displacement current. This complete system predicts the existence of electromagnetic waves propagating at the speed of light, thereby unifying electricity, magnetism, and optics.

Gauss's law for electricity

In integral form, it relates the flux of the electric field E through a closed surface S to the electric charge Qint it encloses:

closed_integral(E . dS) = Qint / epsilon0

Example: for a point charge q surrounded by a sphere of radius r, the flux equals q/epsilon0, regardless of r. The field E therefore decreases as 1/r^2, since the sphere's surface area grows as r^2.

Gauss's law for magnetism

closed_integral(B . dS) = 0

The flux of the magnetic field B through any closed surface is zero: no isolated magnetic charge exists (no magnetic monopole). B's field lines always close on themselves, unlike those of E, which originate from positive charges.

Law	Source quantity	Closed flux
Gauss (electric)	charge Qint	Qint/epsilon0
Gauss (magnetic)	none (no monopole)	0

Common pitfall

Do not confuse Qint (the charge actually enclosed by the surface) with the total charge of the system: only the charges inside the Gaussian surface contribute to the flux.