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Encyclopedia > Gauss's law
Electrostatics Electromagnetism Electricity · Magnetism Electric charge Coulomb's law Electric field Gauss's law Electric potential Electric dipole moment Ampère's law Magnetic field Magnetic flux Biot-Savart law Magnetic dipole moment Electric current Lorentz force law Electromotive force (EM) Electromagnetic induction Faraday-Lenz law Displacement current Maxwell's equations (EMF) Electromagnetic field (EM) Electromagnetic radiation Electrical conduction Electrical resistance Capacitance Inductance Impedance Resonant cavities Waveguides This box: view • talk • edit

## Contents

In its integral form, the law states:

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

where Φ is the electric flux, $mathbf{E}$ is the electric field, $mathrm{d}mathbf{A}$ is a differential area on the closed surface S with an outward facing surface normal defining its direction, QA 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. It has been suggested that optical field be merged into this article or section. ... A surface normal, or just normal to a flat surface is a three-dimensional vector which is perpendicular to that surface. ... Permittivity is a physical quantity that describes how an electric field affects and is affected by a dielectric medium and is determined by the ability of a material to polarize in response to an applied electric field, and thereby to cancel, partially, the field inside the material. ...

For information and strategy on the application of Gauss's law, see Gaussian surfaces. A cylindrical Gaussian surface is commonly used to calculate the electric charge of an infinitely long, straight, ideal wire. ...

## Differential form

In differential form, the equation becomes: In mathematics, a partial differential equation (PDE) is a relation involving an unknown function of several independent variables and its partial derivatives with respect to those variables. ...

$mathbf{nabla} cdot mathbf{D} = rho_{mathrm{free}}$

where $mathbf{nabla}$ is the del operator, representing divergence, D is the electric displacement field (in units of C/m²), and ρfree 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. In vector calculus, del is a vector differential operator represented by the symbol âˆ‡. This symbol is sometimes called the nabla operator, after the Greek word for a kind of harp with a similar shape (with related words in Aramaic and Hebrew). ... In vector calculus, the divergence is an operator that measures a vector fields tendency to originate from or converge upon a given point. ... In physics, the electric displacement field or electric flux density is a vector-valued field that appears in Maxwells equations. ... The Earths magnetic field, which is approximately a dipole. ... In classical electromagnetism, the polarization density (or electric polarization, or simply polarization) is the vector field that expresses the density of permanent or induced electric dipole moments in a dielectric material. ... In vector calculus, the divergence theorem, also known as Gauss theorem, Ostrogradskys theorem, or Ostrogradskyâ€“Gauss theorem is a result that links the divergence of a vector field to the value of surface integrals of the flow defined by the field. ...

And for linear materials, the equation becomes:

$mathbf{nabla} cdot varepsilon mathbf{E} = rho_{mathrm{free}}$

where $varepsilon$ is the electric permittivity. Permittivity is a physical quantity that describes how an electric field affects and is affected by a dielectric medium and is determined by the ability of a material to polarize in response to an applied electric field, and thereby to cancel, partially, the field inside the material. ...

## 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: It has been suggested that optical field be merged into this article or section. ...

where E is the electric field strength at radius r, Q is the enclosed charge, and ε0 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. It has been suggested that optical field be merged into this article or section. ... This diagram shows how the law works. ... Coulombs torsion balance In physics, Coulombs law is an inverse-square law indicating the magnitude and direction of electrostatic force that one stationary, electrically charged object of small dimensions (ideally, a point source) exerts on another. ...

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. Entrance to a Faraday room A Faraday cage or Faraday shield is an enclosure formed by conducting material, or by a mesh of such material. ... An electric current produces a magnetic field. ... In electromagnetism, Maxwells equations are a set of equations first presented as a distinct group in the later half of the nineteenth century by James Clerk Maxwell. ...

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. Johann Carl Friedrich Gauss or GauÃŸ ( ; Latin: ) (30 April 1777 â€“ 23 February 1855) was a German mathematician and scientist of profound genius who contributed significantly to many fields, including number theory, analysis, differential geometry, geodesy, magnetism, astronomy, and optics. ... | Come and take it, slogan of the Texas Revolution 1835 was a common year starting on Thursday (see link for calendar). ... Cunt BAg Twat Fuk suck my penis ring 0778851865!!!!!!Year 1867 (MDCCCLXVII) was a common year starting on Tuesday (link will display the full calendar) of the Gregorian calendar (or a common year starting on Thursday of the of the 12-day slower Julian calendar). ... This diagram shows how the law works. ... â€œGravityâ€ redirects here. ... In physics, intensity is a measure of the time-averaged energy flux. ... Radiation as used in physics, is energy in the form of waves or moving subatomic particles. ... In vector calculus, the divergence theorem, also known as Gauss theorem, Ostrogradskys theorem, or Ostrogradskyâ€“Gauss theorem is a result that links the divergence of a vector field to the value of surface integrals of the flow defined by the field. ...

## Application to Gravity

The gravitational form of Gauss's Law is largely a theoretical curiosity, but can be used by analogy to the electrostatic form of Gauss's Law to prove that the gravitational force of any body on any other body can be treated as though both masses were concentrated at their centers.

In applying the above form of Gauss's Law to prove, for example, that the force of the Earth acting on the Moon does not depend on a detailed treatment of the Earth's composition, one encloses the Earth in a spherical Gaussian surface, whose area is r2.

Since the field lines of the Earth extend out equally in all directions and fall off as $frac{1}{r^{2}}$ (which can be proven independently from Newtonian mechanics and the force law so derived), the gravitational field must be constant at a given radius.

Trivially, multiplying through by m yields the familiar force equation. If the only assumption being made is that gravitational field lines look like electrostatic ones then no prior knowledge of Newton's work is needed. While no reference can be formally found for this, it is often remarked casually in introductory physics classes that Isaac Newton took several pages of calculus to prove that mass distributions act as though their mass were concentrated at a point in their center as far as their interactions with other bodies are concerned, and that had he had Gauss's Law, much of the cumbersome work he undertook would have been shortened dramatically.

In electromagnetism, Maxwells equations are a set of equations first presented as a distinct group in the later half of the nineteenth century by James Clerk Maxwell. ... A cylindrical Gaussian surface is commonly used to calculate the electric charge of an infinitely long, straight, ideal wire. ... Johann Carl Friedrich Gauss or GauÃŸ ( ; Latin: ) (30 April 1777 â€“ 23 February 1855) was a German mathematician and scientist of profound genius who contributed significantly to many fields, including number theory, analysis, differential geometry, geodesy, magnetism, astronomy, and optics. ... In vector calculus, the divergence theorem, also known as Gauss theorem, Ostrogradskys theorem, or Ostrogradskyâ€“Gauss theorem is a result that links the divergence of a vector field to the value of surface integrals of the flow defined by the field. ... flux in science and mathematics. ... The method of image charges (also known as the method of images and method of mirror charges) is a basic problem solving tool in electrostatics. ...

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