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In physics and mathematical analysis, Gauss's law is the electrostatic application of the generalized Gauss's theorem giving the equivalence relation between any flux, e.g. of liquids, electric or gravitational, flowing out of any closed surface and the result of inner sources and sinks, such as electric charges or masses enclosed within the closed surface. The law was developed by Carl Friedrich Gauss. By Divergence theorem generalized Gauss's law can be used in any context where the inverse-square law holds. Electrostatics and Newtonian gravitation are two examples. The differential form of four equations underpins electromagnetic theory.
In its integral form, the law states:
where is the electric flux, is the electric field, is a differential area on the closed surface S with an outward facing surface normal defining its direction, is the charge enclosed by the surface, is the charge density at a point in , is the permittivity of free space and is the integral over the surface S enclosing volume V.
For information and strategy on the application of Gauss's law, see Gaussian surfaces.
In differential form, the equation becomes:
where is the del operator, representing divergence, is the electric displacement field (in units of C/m²), and is the free electric charge density (in units of C/m³), not including the dipole charges bound in a material. The differential form derives in part from Gauss's divergence theorem.
And for linear materials, the equation becomes:
where is the electric 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:
where E is the electric field strength at radius r, Q is the enclosed charge, and ε<sub>0</sub> is the permitivity of free space. Thus the familiar inverse-square law dependence of the electric field in Coulomb's law follows from Gauss's law.
Gauss's law can be used to demonstrate that there is no electric field inside a Faraday cage with no 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.