- Gauss's law
In

physics ,**Gauss's law**, also known as**Gauss's flux theorem**, is a law relating the distribution ofelectric charge to the resultingelectric field . It is one of the fourMaxwell's equations , which form the basis ofclassical electrodynamics , and is also closely related toCoulomb's law . The law was formulated byCarl Friedrich Gauss in1835 , but was not published until1867 . Fact|date=April 2008Gauss's law has two forms, an "integral form" and a "differential form". They are related by the

divergence theorem , also called "Gauss's theorem". Each of these forms can also be expressed two ways: In terms of a relation between theelectric field **E**and the total electric charge, or in terms of theelectric displacement field **D**and the "free" electric charge. (The former are given in sections 1 and 2, the latter in Section 3.)Gauss's law has a close mathematical similarity with a number of laws in other areas of physics. See, for example,

Gauss's law for magnetism andGauss's law for gravity . In fact, any "inverse-square law" can be formulated in a way similar to Gauss's law: For example, Gauss's law itself is essentially equivalent to the inverse-squareCoulomb's law , and Gauss's law for gravity is essentially equivalent to the inverse-squareNewton's law of gravity . See the articleDivergence theorem for more detail.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 something of an electrical analogue ofAmpère's law , which deals with magnetism. Both equations were later integrated intoMaxwell's equations .**Integral form**In its integral form (in SI units), the law states that, for any volume "V" in space, with surface "S", the following equation holds::$Phi\_\{E,S\}\; =\; frac\{Q\_V\}\{varepsilon\_0\}$where

* $Phi$_{"E,S"}, called the "electric flux through "S", is defined by $Phi\_\{E,S\}=oint\_S\; mathbf\{E\}\; cdot\; mathrm\{d\}mathbf\{A\}$, where $mathbf\{E\}$ is theelectric field , and $mathrm\{d\}mathbf\{A\}$ is a differential area on the surface $S$ with an outward facingsurface normal defining its direction. (Seesurface integral for more details.) The surface "S" is the surface bounding the volume "V".

* $Q\_V$ is the totalelectric charge in the volume "V", including bothfree charge andbound charge (bound charge arises in the context ofdielectric materials; see below).

* $varepsilon\_0$ is theelectric constant , a fundamental physical constant.**Applying the integral form**If the electric field is known everywhere, Gauss's law makes it quite easy, in principle, to find the distribution of electric charge: The charge in any given region can be deduced by integrating the electric field to find the flux.

However, much more often, it is the reverse problem that needs to be solved: The electric charge distribution is known, and the electric field needs to be computed. This is much more difficult, since if you know the total flux through a given surface, that gives almost no information about the electric field, which (for all you know) could go in and out of the surface in arbitrarily complicated patterns.

An exception is if there is some "symmetry" in the situation, which mandates that the electric field passes through the surface in a uniform way. Then, if the total flux is known, the field itself can be deduced at every point. Common examples of symmetries which lend themselves to Gauss's law include cylindrical symmetry, planar symmetry, and spherical symmetry. See the article

Gaussian surface for examples where these symmetries are exploited to compute electric fields.**Differential form**In differential form, Gauss's law states:

:$mathbf\{\; abla\}\; cdot\; mathbf\{E\}\; =\; frac\{\; ho\}\{varepsilon\_0\}$

where:

*$mathbf\{\; abla\}cdot$ denotesdivergence ,

***E**is theelectric field ,

*$mathbf\; ho$ is the total electric charge density (in units of C/m³), including both free and bound charge (see below).

*$varepsilon\_0$ is theelectric constant , a fundamental constant of nature.This is mathematically equivalent to the integral form, because of the

divergence theorem .**Gauss's law in terms of free charge****Note on free charge versus bound charge**The electric charge that arises in the simplest textbook situations would be classified as "free charge"—for example, the charge which is transferred in

static electricity , or the charge on acapacitor plate. In contrast, "bound charge" arises only in the context ofdielectric (polarizable) materials. (All materials are polarizable to some extent.) When such materials are placed in an external electric field, the electrons remain bound to their respective atoms, but shift a microscopic distance in response to the field, so that they're more on one side of the atom than the other. All these microscopic displacements add up to give a macroscopic net charge distribution, and this constitutes the "bound charge".Although microscopically, all charge is fundamentally the same, there are often practical reasons for wanting to treat bound charge differently from free charge. The result is that the more "fundamental" Gauss's law, in terms of

**E**, is sometimes put into the equivalent form below, which is in terms of**D**and the free charge only. For a detailed definition of free charge and bound charge, and the proof that the two formulations are equivalent, see the "proof" section below.**Integral form**This formulation of Gauss's law states that, for any volume "V" in space, with surface "S", the following equation holds::$Phi\_\{D,S\}\; =\; Q\_\{f,V\}$where

* $Phi\_\{D,S\}$ is defined by $Phi\_\{D,S\}=oint\_S\; mathbf\{D\}\; cdot\; mathrm\{d\}mathbf\{A\}$, where $mathbf\{D\}$ is theelectric displacement field , and the integration is asurface integral .

* $Q\_\{f,V\}$ is the free electric charge in the volume "V", not includingbound charge (see below).**Differential form**The differential form of Gauss's law, involving free charge only, states::$mathbf\{\; abla\}\; cdot\; mathbf\{D\}\; =\; ho\_\{mathrm\{free$

where:

*$mathbf\{\; abla\}cdot$ denotesdivergence ,

***D**is theelectric displacement field (in units of C/m²), and *$ho\_\{mathrm\{free,$ is the "free" electric charge density (in units of C/m³), not including thebound charge s in a material.The differential form and integral form are mathematically equivalent. The proof primarily involves the

divergence theorem .**Proof of equivalence**:

**In linear materials**In homogeneous, isotropic, nondispersive,

linear material s, there is a nice, simple relationship between**E**and**D**::$varepsilon\; mathbf\{E\}\; =\; mathbf\{D\}$where $varepsilon$ is the "permittivity " of the material. Under these circumstances, there is yet another pair of equivalent formulations of Gauss's law::$Phi\_\{E,S\}\; =\; frac\{Q\_\{V,mathrm\{free\}\{varepsilon\}$:$mathbf\{\; abla\}\; cdot\; mathbf\{E\}\; =\; frac\{\; ho\_\{mathrm\{free\}\{varepsilon\}$**Relation to Coulomb's law****Deriving Gauss's law from Coulomb's law**Gauss's law can be derived from

Coulomb's law , which states that the electric field due to a stationarypoint charge is::$mathbf\{E\}(mathbf\{r\})\; =\; frac\{q\}\{4pi\; epsilon\_0\}\; frac\{mathbf\{e\_r\{r^2\}$where:

**e**is the radial_{r}unit vector ,:"r" is the radius, |**r**|,:$epsilon\_0$ is theelectric constant ,:"q" is the charge of the particle, which is assumed to be located at the origin.Using the expression from Coulomb's law, we get the total field at

**r**by using an integral to sum the field at**r**due to the infinitesimal charge at each other point**s**in space, to give:$mathbf\{E\}(mathbf\{r\})\; =\; frac\{1\}\{4piepsilon\_0\}\; int\; frac\{\; ho(mathbf\{s\})(mathbf\{r\}-mathbf\{s\})\}\{|mathbf\{r\}-mathbf\{s\}|^3\}\; d^3\; mathbf\{s\}$

where $ho$ is the charge density. If we take the divergence of both sides of this equation with respect to

**r**, and use the known theorem [*See, for example, cite book | author=Griffiths, David J. | title=Introduction to Electrodynamics (3rd ed.) | publisher=Prentice Hall | year=1998 | id=ISBN 0-13-805326-X | page=50*]:$abla\; cdot\; left(frac\{mathbf\{s\{|mathbf\{s\}|^3\}\; ight)\; =\; 4pi\; delta(mathbf\{s\})$where δ(

**s**) is theDirac delta function , the result is:$ablacdotmathbf\{E\}(mathbf\{r\})\; =\; frac\{1\}\{epsilon\_0\}\; int\; ho(mathbf\{s\})\; delta(mathbf\{r\}-mathbf\{s\})\; d^3\; mathbf\{s\}$

Using the "sifting property" of the Dirac delta function, we arrive at

:$ablacdotmathbf\{E\}\; =\; ho/epsilon\_0$

which is the differential form of Gauss's law, as desired.

Note that since Coulomb's law only applies to "stationary" charges, there is no reason to expect Gauss's law to hold for moving charges "based on this derivation alone". In fact, Gauss's law "does" hold for moving charges, and in this respect Gauss's law is more general than Coulomb's law.

**Deriving Coulomb's law from Gauss's law**Strictly speaking,

Coulomb's law cannot be derived from Gauss's law alone, since Gauss's law does not give any information regarding the curl of**E**(seeHelmholtz decomposition and Faraday's law). However, Coulomb's law "can" be proven from Gauss's law if it is assumed, in addition, that the electric field from apoint charge is spherically-symmetric (this assumption, like Coulomb's law itself, is exactly true if the charge is stationary, and approximately true if the charge is in motion).Taking "S" in the integral form of Gauss's law to be a spherical surface of radius "r", centered at the point charge "Q", we have: $oint\_\{S\}mathbf\{E\}cdot\; dmathbf\{A\}\; =\; Q/varepsilon\_0$By the assumption of spherical symmetry, the integrand is a constant which can be taken out of the integral. The result is: $4pi\; r^2hat\{mathbf\{rcdotmathbf\{E\}(mathbf\{r\})\; =\; Q/varepsilon\_0$where $hat\{mathbf\{r$ is a

unit vector pointing radially away from the charge. Again by spherical symmetry,**E**points in the radial direction, and so we get: $mathbf\{E\}(mathbf\{r\})\; =\; frac\{Q\}\{4pi\; varepsilon\_0\}frac\{hat\{mathbf\{r\}\{r^2\}$which is essentially equivalent to Coulomb's law.Thus the

inverse-square law dependence of the electric field inCoulomb's law follows from Gauss's law.**ee also***

Maxwell's equations

*Gaussian surface

*Carl Friedrich Gauss

*Divergence theorem

*Flux

*Method of image charges **References**Jackson, John David (1999). Classical Electrodynamics, 3rd ed., New York: Wiley. ISBN 0-471-30932-X.

**External links*** [

*http://ocw.mit.edu/OcwWeb/Physics/8-02Electricity-and-MagnetismSpring2002/VideoAndCaptions/ MIT Video Lecture Series (30 x 50 minute lectures)- Electricity and Magnetism*] Taught by ProfessorWalter Lewin .

* [*http://www.lightandmatter.com/html_books/0sn/ch10/ch10.html#Section10.6 section on Gauss's law in an online textbook*]

* [*http://35.9.69.219/home/modules/pdf_modules/m132.pdf MISN-0-132 "Gauss's Law for Spherical Symmetry"*] (PDF file) by Peter Signell for [*http://www.physnet.org Project PHYSNET*] .

* [*http://35.9.69.219/home/modules/pdf_modules/m133.pdf MISN-0-133 "Gauss's Law Applied to Cylindrical and Planar Charge Distributions*] (PDF file) by Peter Signell for Project PHYSNET.

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