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where is the electric field, is the area of a differential square on the surface A with an outward facing surface normal defining its direction, is the charge enclosed by the surface, is the permittivity of free space and is the integral over the surface A.

Its partial differential form is:

where 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:

where 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:

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 GaussJohann Carl Friedrich Gauss (Gauss ( April 30, 1777 February 23, 1855) was a legendary German mathematician, astronomer and physicist with a very wide range of contributions; he is considered to be one of the greatest mathematicians of all time. His name in 1835Events January 1 Ole Pedersen Hoiland breaks into the Bank of Norway and steals 64. 000 dalers January 7 HMS Beagle anchors off the Chonos Archipelago. January 30 Unsuccessful assassination attempt against President Andrew Jackson in the United States Cap, but was not published until 1867Events January 8 African-American men granted the right to vote in the District of Columbia January 11 Benito Juarez becomes Mexican president again January 30 Emperor Komei of Japan dies. Crown Prince Mutsuhito is expected to become the next Emperor of J. 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 intensityIn physics, intensity is a measure of the time-averaged energy flux. To find the intensity, take the energy density (that is, the energy per unit volume) and multiply it by the velocity at which the energy is moving. The resulting vector has the units of of radiationRadiation generally means the transmission of objects or information from a source into a surrounding medium or destination. Within physics, related concepts are: Ionizing radiation is a stream of particles (photons or other particles) with sufficient ene. See also divergence theorem.