Categories: Electromagnetism | Multivariate calculus

Gauss's law

In physics, Gauss's law gives the relation between the electric flux flowing out a closed surface and the charge enclosed in the surface. Its integral form is:

$\Phi = \oint_A \mathbf{E} \cdot d\mathbf{A} = \frac{Q_A}{\epsilon_0}$

where $\mathbf{E}$ is the electric field, $d\mathbf{A}$ is the area of a differential square on the surface A with an outward facing surface normal defining its direction, $Q A$ is the charge enclosed by the surface, $ε 0$ is the permittivity of free space and $\oint_A$ is the integral over the surface A.

Its partial differential form is:

$\nabla \cdot \mathbf{D} = \rho$

where $\nabla \cdot$ is the divergence, D is the electric displacement field (in units of C/m²), and ρ is the free electric charge density (in units of C/m³), not including dipole charges bound in a material

In linear materials, the equation becomes:

$\nabla \cdot \epsilon \mathbf{E} = \rho$

where $ε$ is the electrical permittivity

In the special case of a spherical surface with a central charge, the electric field is perpendicular to the surface, with the same magnitude at all points of it, giving the simpler expression:

$E=\frac{Q}{4\pi\epsilon_0r^2}$

where E is the electric field strength at radius r, Q is the enclosed charge, and ε₀ is the permittivity of free space. Thus the familiar inverse-square law dependence of the electric field in Coulomb's law follows from Gauss' law.

Gauss's law can be used to demonstrate that there is no electric field inside a Faraday cage without electric charges. Gauss's law is the electrostatic equivalent of Ampère's law, which deals with magnetism. Both equations were later integrated into Maxwell's equations.

It was formulated by Carl Friedrich Gauss in 1835, but was not published until 1867. Because of the mathematical similarity, Gauss's law has application for other physical quantities governed by an inverse-square law such as gravitation or the intensity of radiation. See also divergence theorem.