How does Gauss’s law connect electric flux and enclosed charge?

Gauss’s law describes the relationship between electric flux and the net electric charge enclosed by a closed surface. It states that the total electric flux $\Phi$ passing through any closed surface is directly proportional to the net charge $q$ contained within that surface.12

Mathematically, this is expressed as:
$$\Phi =\frac{q}{{\epsilon }_0}$$
In this equation, ${\epsilon }_0$ represents the permittivity of free space.13

The law implies that electric field lines originate from positive charges and terminate on negative charges. If no net charge is enclosed, the total flux is zero because any field lines entering the surface must also exit it. Conversely, if a net charge is present, the flux is positive for a net positive charge and negative for a net negative charge. This principle is a fundamental component of Maxwell’s equations, which form the basis of electromagnetic theory.231