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This article is about Gauss' law concerning the electric field. Electromagnetism is the Physics of the Electromagnetic field: a field which exerts a Force on particles that possess the property of In Physics, magnetism is one of the Phenomena by which Materials exert attractive or repulsive Forces on other Materials. Electrostatics is the branch of Science that deals with the Phenomena arising from what seems to be stationary Electric charges Since Classical Electric charge is a fundamental conserved property of some Subatomic particles which determines their Electromagnetic interaction. ---- Bold text Coulomb's law', developed in the 1780s by French physicist Charles Augustin de Coulomb, may be stated in scalar form In Physics, the space surrounding an Electric charge or in the presence of a time-varying Magnetic field has a property called an electric field (that can In Electromagnetism, electric flux is Flux of the Electric field. At a point in space the electric potential is the Potential energy per unit of charge that is associated with a static (time-invariant Electric field Electrostatic induction is a redistribution of Electrical charge in an object caused by the influence of nearby charges In Physics, the electric dipole moment (or electric dipole for short is a measure of the polarity of a system of Electric charges. In Physics, the space surrounding an Electric charge or in the presence of a time-varying Magnetic field has a property called an electric field (that can For an analogous law concerning the magnetic field, see Gauss' law for magnetism. In Physics, a magnetic field is a Vector field that permeates space and which can exert a magnetic force on moving Electric charges For an analogous law concerning the gravitational field, see Gauss' law for gravity. A gravitational field is a model used within Physics to explain how gravity exists in the universe In Physics, Gauss' law for gravity, also known as Gauss' flux theorem for gravity, is a law of physics which is essentially equivalent to Newton's law of universal For Gauss' theorem, a general theorem relevant to all of these laws, see Divergence theorem. In Vector calculus, the divergence theorem, also known as Gauss&rsquos theorem ( Carl Friedrich Gauss) Ostrogradsky&rsquos theorem ( Mikhail

In physics, Gauss' law, also known as Gauss' flux theorem, is a law relating the distribution of electric charge to the resulting electric field. Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. Electric charge is a fundamental conserved property of some Subatomic particles which determines their Electromagnetic interaction. In Physics, the space surrounding an Electric charge or in the presence of a time-varying Magnetic field has a property called an electric field (that can It is one of the four Maxwell's equations, which form the basis of classical electrodynamics, and is also closely related to Coulomb's law. In Classical electromagnetism, Maxwell's equations are a set of four Partial differential equations that describe the properties of the electric Classical electromagnetism (or classical electrodynamics) is a theory of Electromagnetism that was developed over the course of the 19th century most prominently ---- Bold text Coulomb's law', developed in the 1780s by French physicist Charles Augustin de Coulomb, may be stated in scalar form The law was formulated by Carl Friedrich Gauss in 1835, but was not published until 1867. Johann Carl Friedrich Gauss (ˈɡaʊs, Gauß Carolus Fridericus Gauss ( 30 April 1777 – 23 February 1855) was a German Year 1835 ( MDCCCXXXV) was a Common year starting on Thursday (link will display the full calendar of the Gregorian Calendar (or a Common 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

Gauss' law has two forms, an integral form and a differential form. They are related by the divergence theorem, also called "Gauss' theorem". In Vector calculus, the divergence theorem, also known as Gauss&rsquos theorem ( Carl Friedrich Gauss) Ostrogradsky&rsquos theorem ( Mikhail Each of these forms can also be expressed two ways: In terms of a relation between the electric field E and the total electric charge, or in terms of the electric displacement field D and the free electric charge. In Physics, the space surrounding an Electric charge or in the presence of a time-varying Magnetic field has a property called an electric field (that can In Physics, the electric displacement field (also called electrical field/flux density is a Vector field \mathbf{D} that appears in Maxwell's equations In Classical electromagnetism, the polarization density (or electric polarization, or simply polarization) is the Vector field that expresses (The former are given in sections 1 and 2, the latter in Section 3. )

Gauss' law has a close mathematical similarity with a number of laws in other areas of physics. See, for example, Gauss' law for magnetism and Gauss' law for gravity. In Physics, Gauss' law for gravity, also known as Gauss' flux theorem for gravity, is a law of physics which is essentially equivalent to Newton's law of universal In fact, any "inverse-square law" can be formulated in a way similar to Gauss' law: For example, Gauss' law itself follows from the inverse-square Coulomb's law, and Gauss' law for gravity follows from the inverse-square Newton's law of gravity. ---- Bold text Coulomb's law', developed in the 1780s by French physicist Charles Augustin de Coulomb, may be stated in scalar form Newton 's law of universal Gravitation is a physical law describing the gravitational attraction between bodies with mass See the article Divergence theorem for more detail. In Vector calculus, the divergence theorem, also known as Gauss&rsquos theorem ( Carl Friedrich Gauss) Ostrogradsky&rsquos theorem ( Mikhail

Gauss' law can be used to demonstrate that there is no electric field inside a Faraday cage with no electric charges. A Faraday cage or Faraday shield is an enclosure formed by conducting material, or by a mesh of such material Gauss' law is something of an electrical analogue of Ampère's law, which deals with magnetism. Both equations were later integrated into Maxwell's equations. In Classical electromagnetism, Maxwell's equations are a set of four Partial differential equations that describe the properties of the electric

1 Integral form
- 1.1 Applying the integral form
2 Differential form
3 Gauss' law in terms of free charge
4 Deriving Coulomb's law from Gauss' law
5 See also
6 References
7 External links

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

$Φ$ _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 the electric field, and $\mathrm{d}\mathbf{A}$ is a differential area on the surface $S$ with an outward facing surface normal defining its direction. Phi (uppercase Φ, lowercase φ or ϕ) pronounced in modern Greek and as in English is the 21st letter of the Greek alphabet In Electromagnetism, electric flux is Flux of the Electric field. In Physics, the space surrounding an Electric charge or in the presence of a time-varying Magnetic field has a property called an electric field (that can (See surface integral for more details. In Mathematics, a surface integral is a Definite integral taken over a Surface (which may be a curved set in Space) it can be thought )
$Q V$ is the total electric charge in the volume V, including both free charge and bound charge (bound charge arises in the context of dielectric materials; see below). Electric charge is a fundamental conserved property of some Subatomic particles which determines their Electromagnetic interaction. In Classical electromagnetism, the polarization density (or electric polarization, or simply polarization) is the Vector field that expresses In Classical electromagnetism, the polarization density (or electric polarization, or simply polarization) is the Vector field that expresses A dielectric is a nonconducting substance ie an insulator. The term was coined by William Whewell in response to a request from Michael Faraday.
$\varepsilon_0$ is the electric constant, a fundamental physical constant. Vacuum permittivity, referred to by international standards organizations as the electric constant, and denoted by the symbol ε0 is a fundamental Physical

Applying the integral form

Main article: Gaussian surface

If the electric field is known everywhere, Gauss' 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. A Gaussian surface is a closed two-dimensional Surface through which a Flux or Electric field is to be calculated

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' 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. A Gaussian surface is a closed two-dimensional Surface through which a Flux or Electric field is to be calculated

Differential form

In differential form, Gauss' law states:

$\mathbf{\nabla} \cdot \mathbf{E} = \frac{\rho}{\varepsilon_0}$

where:

$\mathbf{\nabla}\cdot$ denotes divergence,
E is the electric field,
$ρ$ is the total electric charge density (in units of C/m³), including both free and bound charge (see below). In Mathematics, partial differential equations ( PDE) are a type of Differential equation, i In Vector calculus, the divergence is an Operator that measures the magnitude of a Vector field &rsquos source or sink at a given point the In Physics, the space surrounding an Electric charge or in the presence of a time-varying Magnetic field has a property called an electric field (that can
$\varepsilon_0$ is the electric constant, a fundamental constant of nature. Vacuum permittivity, referred to by international standards organizations as the electric constant, and denoted by the symbol ε0 is a fundamental Physical

This is mathematically equivalent to the integral form, because of the divergence theorem. In Vector calculus, the divergence theorem, also known as Gauss&rsquos theorem ( Carl Friedrich Gauss) Ostrogradsky&rsquos theorem ( Mikhail

Gauss' law in terms of free charge

Note on free charge versus bound charge

Main article: Electric polarization

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 a capacitor plate. In Classical electromagnetism, the polarization density (or electric polarization, or simply polarization) is the Vector field that expresses For the science of static charges see Electrostatics Static electricity refers to the accumulation of excess Electric charge in a A capacitor is a passive electrical component that can store Energy in the Electric field between a pair of conductors In contrast, "bound charge" arises only in the context of dielectric (polarizable) materials. A dielectric is a nonconducting substance ie an insulator. The term was coined by William Whewell in response to a request from Michael Faraday. (All materials are polarizable to some extent. ) When such a 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' 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' law states that, for any volume V in space, with surface S, the following equation holds:

Φ D, S = Q f, V

where

$Φ D, S$ is defined by $\Phi_{D,S}=\oint_S \mathbf{E} \cdot \mathrm{d}\mathbf{A}$ , where $\mathbf{D}$ is the electric displacement field, and the integration is a surface integral. In Physics, the electric displacement field (also called electrical field/flux density is a Vector field \mathbf{D} that appears in Maxwell's equations In Mathematics, a surface integral is a Definite integral taken over a Surface (which may be a curved set in Space) it can be thought
$Q f, V$ is the free electric charge in the volume V, not including bound charge (see below). In Classical electromagnetism, the polarization density (or electric polarization, or simply polarization) is the Vector field that expresses In Classical electromagnetism, the polarization density (or electric polarization, or simply polarization) is the Vector field that expresses

Differential form

The differential form of Gauss' law, involving free charge only, states:

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

where:

$\mathbf{\nabla}\cdot$ denotes divergence,
D is the electric displacement field (in units of C/m²), and * $\rho_{\mathrm{free}}\,$ is the free electric charge density (in units of C/m³), not including the bound charges in a material. In Vector calculus, the divergence is an Operator that measures the magnitude of a Vector field &rsquos source or sink at a given point the In Physics, the electric displacement field (also called electrical field/flux density is a Vector field \mathbf{D} that appears in Maxwell's equations In Classical electromagnetism, the polarization density (or electric polarization, or simply polarization) is the Vector field that expresses

The differential form and integral form are mathematically equivalent. The proof primarily involves the divergence theorem. In Vector calculus, the divergence theorem, also known as Gauss&rsquos theorem ( Carl Friedrich Gauss) Ostrogradsky&rsquos theorem ( Mikhail

Proof of equivalence

In linear materials

In homogeneous, isotropic, nondispersive, linear materials, there is a nice, simple relationship between E and D:

$\varepsilon \mathbf{E} = \mathbf{D}$

where $\varepsilon$ is the permittivity of the material. Linear elasticity is the mathematical study of how solid objects deform and become internally stressed due to prescribed loading conditions 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 Under these circumstances, there is yet another pair of equivalent formulations of Gauss' law:

$\Phi_{E,S} = \frac{Q_{V,\mathrm{free}}}{\varepsilon}$

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

Deriving Coulomb's law from Gauss' law

Strictly speaking, Coulomb's law cannot be derived from Gauss' law alone, since Gauss' law does not give any information regarding the curl of E (see Helmholtz decomposition and Faraday's law). ---- Bold text Coulomb's law', developed in the 1780s by French physicist Charles Augustin de Coulomb, may be stated in scalar form cURL is a Command line tool for transferring files with URL syntax. In Mathematics, in the area of Vector calculus, Helmholtz's theorem, also known as the fundamental theorem of vector calculus, states that any sufficiently Faraday's law of induction describes an important basic law of electromagnetism which is involved in the working of Transformers Inductors and many forms of However, Coulomb's law can be proven from Gauss' law if it is assumed, in addition, that the electric field from a point 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). A point charge is an idealized model of a particle which has an Electric charge.

Taking S in the integral form of Gauss' law to be a spherical surface of radius r, centered at the point charge Q, we have

$\oint_{S}\mathbf{E}\cdot d\mathbf{A} = Q/\varepsilon_0$

By the assumption of spherical symmetry, the integrand is a constant which can be taken out of the integral. The result is

$4\pi r^2\hat{\mathbf{r}}\cdot\mathbf{E}(\mathbf{r}) = Q/\varepsilon_0$

where $\hat{\mathbf{r}}$ is a unit vector pointing radially away from the charge. In Mathematics, a unit vector in a Normed vector space is a vector (often a spatial vector) whose length is 1 (the unit length Again by spherical symmetry, E points in the radial direction, and so we get

$\mathbf{E}(\mathbf{r}) = \frac{Q}{4\pi \varepsilon_0}\frac{\hat{\mathbf{r}}}{r^2}$

which is essentially equivalent to Coulomb's law.

Thus the inverse-square law dependence of the electric field in Coulomb's law follows from Gauss' law. In Physics, an inverse-square law is any Physical law stating that some physical Quantity or strength is inversely proportional ---- Bold text Coulomb's law', developed in the 1780s by French physicist Charles Augustin de Coulomb, may be stated in scalar form

References

Jackson, John David (1999). In Classical electromagnetism, Maxwell's equations are a set of four Partial differential equations that describe the properties of the electric A Gaussian surface is a closed two-dimensional Surface through which a Flux or Electric field is to be calculated Johann Carl Friedrich Gauss (ˈɡaʊs, Gauß Carolus Fridericus Gauss ( 30 April 1777 – 23 February 1855) was a German In Vector calculus, the divergence theorem, also known as Gauss&rsquos theorem ( Carl Friedrich Gauss) Ostrogradsky&rsquos theorem ( Mikhail In the various subfields of Physics, there exist two common usages of the term flux, both with rigorous mathematical frameworks 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. Classical Electrodynamics, 3rd ed. , New York: Wiley. ISBN 0-471-30932-X.

External links

MIT Video Lecture Series (30 x 50 minute lectures)- Electricity and Magnetism Taught by Professor Walter Lewin. Walter H G Lewin is currently a professor of Physics at the Massachusetts Institute of Technology (MIT
section on Gauss' law in an online textbook
MISN-0-132 Gauss' Law for Spherical Symmetry (PDF file) by Peter Signell for Project PHYSNET.
MISN-0-133 Gauss' Law Applied to Cylindrical and Planar Charge Distributions (PDF file) by Peter Signell for Project PHYSNET.

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Contents

Integral form

Applying the integral form

Differential form

Gauss' law in terms of free charge

Note on free charge versus bound charge

Integral form

Differential form

Proof of equivalence

In linear materials

Deriving Coulomb's law from Gauss' law

See also

References

External links