From Freepedia

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.

Contents 1 Integral Form 2 {1 \over \epsilon_o} \int_V \rho \cdot dV 2.1 Differential Form 2.2 Coulomb's Law 2.3 See also 2.4 External links

Integral Form

In its integral form, the law states:

<math>\Phi = \oint_S \mathbf{E} \cdot d\mathbf{A}

{1 \over \epsilon_o} \int_V \rho \cdot dV

\frac{Q_A}{\epsilon_o} </math>

where <math>\mathbf{E}</math> is the electric field, <math>d\mathbf{A}</math> is the area of a differential square on the closed surface S with an outward facing surface normal defining its direction, <math>Q_\mbox{A}</math> is the charge enclosed by the surface, <math>\epsilon_o</math> is the permittivity of free space and <math>\oint_A</math> is the integral over the surface S enclosing volume V.

Differential Form

In differential form, the equation becomes:

<math>\nabla \cdot \mathbf{D} = \rho </math>

where <math> \nabla </math> 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. The differential form derives in part from Gauss's divergence theorem.

And for linear materials, the equation becomes:

<math>\nabla \cdot \epsilon \mathbf{E} = \rho</math>

where <math> \epsilon </math> is the electrical permittivity

Coulomb's Law

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:

<math>E=\frac{Q}{4\pi\epsilon_0r^{2}}</math>

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.

See also

• Maxwell's equations
• Carl Friedrich Gauss
• Divergence theorem
• Flux

External links

• MISN-0-132 Gauss's Law for Spherical Symmetry (PDF file) by Peter Signell for Project PHYSNET.
• MISN-0-133 Gauss's Law Applied to Cylindrical and Planar Charge Distributions (PDF file) by Peter Signell for Project PHYSNET.