Gauss's law

In vacuum, the integral form of Gauss's law is ∮ₛ E · dA = Qₑₙc/ε₀. Here E is the total electric field on the closed surface S, dA is an outward-directed area element, Qₑₙc is the net charge within the surface, and ε₀ is the vacuum permittivity. Electric flux is the surface integral of the field through the surface. The law says that the net outward flux depends on the enclosed net charge, not on the shape or size of the chosen closed surface.

A Gaussian surface is an imaginary closed surface chosen to evaluate electric flux; it need not coincide with a physical object. Charges inside contribute to the enclosed-charge term. Charges outside do not change the net flux through the closed surface, although their fields can contribute to the total electric field at points on it. If the net enclosed charge is zero, the net flux is zero, but the electric field need not be zero everywhere on the surface.

Gauss's law is valid regardless of symmetry, but solving for the electric field becomes simpler when the charge distribution has spherical, cylindrical, or planar symmetry. A suitable Gaussian surface can then make the field magnitude constant on part or all of the surface, or make the field perpendicular to selected surface elements, simplifying the flux integral. For an arbitrary irregular charge distribution, the law alone generally does not determine the field at every point without additional information or methods.

Using the divergence theorem, the integral law can be written in differential form as ∇ · E = ρ/ε₀, where ρ is the local charge density. This states locally that electric charge density is a source of electric field. The relation is one of Maxwell's equations, a set of foundational equations for classical electromagnetism. In matter, equivalent formulations can use the electric displacement field and free charge, with the chosen form depending on how material polarization is represented.

For electrostatic charge distributions, Gauss's law and Coulomb's inverse-square law are closely related: under the usual assumptions, either can be derived from the other. The symmetry method provides a particularly direct route from Gauss's law to electric fields around idealized charge distributions such as spherical shells or uniformly charged infinite lines and planes.