Gauss's law

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

Related Topics:
Physics - Mathematical analysis - Flux - Electric charge

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: Phi = oint_A mathbf{E} cdot dmathbf{A} = rac{Q_A}{epsilon_0}

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where mathbf{E} is the electric field, dmathbf{A} is the area of a differential square on the surface A with an outward facing surface normal defining its direction, Q_mbox{A} is the charge enclosed by the surface, epsilon_0 is the permittivity of free space and oint_A is the integral over the surface A.

Related Topics:
Surface normal - Permittivity

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Its partial differential form is:

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: abla cdot mathbf{D} = ho

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where abla is the del operator, representing 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.

Related Topics:
Del operator - Divergence - Electric displacement field - Dipole

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In linear materials, the equation becomes:

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: abla cdot epsilon mathbf{E} = ho

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where epsilon is the electrical permittivity

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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:

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:E=rac{Q}{4piepsilon_0r^{2}}

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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.

Related Topics:
Electric field - Inverse-square law - Coulomb's law

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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.

Related Topics:
Faraday cage - Ampère's law - Maxwell's equations

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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.

Related Topics:
Carl Friedrich Gauss - 1835 - 1867 - Inverse-square law - Gravitation - Intensity - Radiation - Divergence theorem

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