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Gauss's law

2007 Schools Wikipedia Selection. Related subjects: Electricity and
Electronics

   Electromagnetism
   Electricity · Magnetism
           Electrostatics
   Electric charge
   Coulomb's law
   Electric field
   Gauss's law
   Electric potential
   Electric dipole moment
           Magnetostatics
   Ampère's law
   Magnetic field
   Magnetic dipole moment
          Electrodynamics
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   Lorentz force law
   Electromotive force
   (EM) Electromagnetic induction
   Faraday-Lenz law
   Displacement current
   Maxwell's equations
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         Electrical Network
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   In physics and mathematical analysis, Gauss's law, developed by Carl
   Friedrich Gauss, closely related to Gauss's theorem, gives the relation
   between the electric or gravitational flux flowing out of a closed
   surface and, respectively, the electric charge or mass enclosed in the
   surface. Gauss's law can be used in any context where the
   inverse-square law holds, where electrostatics and Newtonian
   gravitation are but two examples. It is one of the four equations that
   underpins electromagnetic theory.

Integral form

   In its integral form, the law states:

          \Phi = \oint_S \vec{E} \cdot \mathrm{d}\vec{A} = {1 \over
          \varepsilon_o} \int_V \rho\ \mathrm{d}V =
          \frac{Q_A}{\varepsilon_o}

   where Φ is the electric flux, \vec{E} is the electric field,
   \mathrm{d}\vec{A} is a differential area on the closed surface S with
   an outward facing surface normal defining its direction, Q[A] is the
   charge enclosed by the surface, ρ is the charge density at a point in
   V, \varepsilon_o is the permittivity of free space and \oint_S is the
   integral over the surface S enclosing volume V.

   For information and strategy on the application of Gauss's law, see
   Gaussian surfaces.

Differential form

   In differential form, the equation becomes:

          \vec{\nabla} \cdot \vec{D} = \rho_{\mathrm{free}}

   where \vec{\nabla} is the del operator, representing divergence, D is
   the electric displacement field (in units of C/m²), and ρ[free] 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:

          \vec{\nabla} \cdot \varepsilon \vec{E} = \rho_{\mathrm{free}}

   where \varepsilon is the electric 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:

          E=\frac{Q}{4\pi\varepsilon_0r^{2}}

   where E is the electric field strength at radius r, Q is the enclosed
   charge, and ε[0] 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.

Gravitational analogue

   Since both gravity and electromagnetism have strength that propagates
   relative to the squared distance between two objects, we can relate the
   two using Gauss's Law by examining their respective vector fields
   \vec{g} and \vec{E} , where

          \vec{g} = -G\frac{m}{\vec{r}^2}\hat{r},

   and

          \vec{E} = \frac{1}{4 \pi \varepsilon_{0}}
          \frac{q}{\vec{r}^2}\hat{r},

   where G is the gravitational constant, m is the mass of the point
   source, r is the radius (distance) between the point source and another
   object, \varepsilon_{0} is the permittivity of free space, and q is the
   charge of the electric point source.

   In the same way that we evaluate the surface integral for
   electromagnetism to get the result \frac{q}{\varepsilon_{0}} , we can
   choose a proper Gaussian surface to find an answer for the
   gravitational flux. For a point mass centered at the coordinate system
   origin, the most logical choice for our Gaussian surface is a sphere of
   radius r centered at the origin.

   We start with the integral form of Gauss's law:

          \Phi_{g} = \oint_S \vec{g} \cdot \mathrm{d}\vec{A}.

   An infinitesimal area element is just the area of the infinitesimal
   solid angle, which is defined as

          \mathrm{d}\vec{A} = r^{2} \mathrm{d}\Omega \hat{r}.

   Our Gaussian Surface is wisely chosen since the vector normal to the
   surface is radial from the origin. With

          \Phi_{g} = \oint_S g(r) \hat{r} \cdot \hat{r} r^{2}
          \mathrm{d}\Omega,

   we see the inner product of the two radial vectors is unity and that
   both the magnitude of our field, \vec{g} , and the square of the
   distance between the surface and the point, r^2, remain constant over
   every element of the surface. This gives us the integral

          \Phi_{g} = g(r) r^{2} \oint_S \mathrm{d}\Omega.

   The remaining surface integral is just the surface area of our sphere
   (4πr^2). If we combine this with our gravitational field equation from
   above, we have an expression for the gravitational flux of a point
   mass.

          \Phi_{g} = -\frac{Gm}{r^2} 4 \pi r^{2} = -4\pi Gm

   It is interesting to note that the gravitational flux, like its
   electromagnetic counterpart, does not depend on the radius of the
   sphere.

   Retrieved from " http://en.wikipedia.org/wiki/Gauss%27s_law"
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