This article is about Gauss's law concerning the electric field. For analogous law concerning different fields, see Gauss's law for magnetismand Gauss's law for gravity. For Gauss's theorem, a mathematical theorem relevant to all of these laws, see Divergence theorem.
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In physics, Gauss's law, also known as Gauss's flux theorem, is a law relating the distribution ofelectric charge to the resulting electric field.
The law was formulated by Carl Friedrich Gauss in 1835, but was not published until 1867.[1] It is one of Maxwell's four equations, which form the basis of classical electrodynamics, the other three being Gauss's law for magnetism, Faraday's law of induction, and Ampère's law with Maxwell's correction. Gauss's law can be used to derive Coulomb's law, and vice versa.
Qualitative description
In words, Gauss's law states that:
- The net electric flux through any closed surface is equal to 1⁄ε times the net electric charge within that closed surface.[3]
Gauss's law has a close mathematical similarity with a number of laws in other areas of physics, such as 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-square Coulomb's law, and Gauss's law for gravity is essentially equivalent to the inverse-square Newton's law of gravity.
Gauss's law is something of an electrical analogue of Ampère's law, which deals with magnetism.
The law can be expressed mathematically using vector calculus in integral form and differential form, both are equivalent since they are related by the divergence theorem, also called Gauss's theorem. Each of these forms in turn 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.[4]
Equation involving E field
Gauss's law can be stated using either the electric field E or the electric displacement field D. This section shows some of the forms with E; the form with D is below, as are other forms with E.
Integral form
Gauss's law may be expressed as:[5]
where ΦE is the electric flux through a closed surface S enclosing any volume V, Q is the total charge enclosed within S, and ε0 is the electric constant. The electric flux ΦE is defined as a surface integral of the electric field:
where E is the electric field, dA is a vector representing an infinitesimal element of area of the surface,[note 1] and · represents the dot product of two vectors.
Since the flux is defined as an integral of the electric field, this expression of Gauss's law is called the integral form.
Applying the integral form
Main article: Gaussian surface
See also Capacitance (Gauss's law)
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
By the divergence theorem, Gauss's law can alternatively be written in the differential form:
where ∇ · E is the divergence of the electric field, ε0 is the electric constant, and ρ is the total electric charge density (charge per unit volume).
Equivalence of integral and differential forms
Main article: Divergence theorem
The integral and differential forms are mathematically equivalent, by the divergence theorem. Here is the argument more specifically.
[show]Outline of proof
Equation involving D field
See also: Maxwell's equations
Free, bound, and total 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 contrast, "bound charge" arises only in the context of dielectric (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 (above), is sometimes put into the equivalent form below, which is in terms of D and the free charge only.
Integral form
This formulation of Gauss's law states the total charge form:
where ΦD is the D-field flux through a surface S which encloses a volume V, and Qfree is the free charge contained in V. The flux ΦD is defined analogously to the flux ΦE of the electric field E through S:
Differential form
The differential form of Gauss's law, involving free charge only, states:
where ∇ · D is the divergence of the electric displacement field, and ρfree is the free electric charge density.
Equivalence of total and free charge statements
[show]Proof that the formulations of Gauss's law in terms of free charge are equivalent to the formulations involving total charge.
Equation for linear materials[edit]
In homogeneous, isotropic, nondispersive, linear materials, there is a simple relationship between E and D:
where ε is the permittivity of the material. For the case of vacuum (aka free space), ε = ε0. Under these circumstances, Gauss's law modifies to
for the integral form, and
for the differential form.
Relation to Coulomb's law
Deriving Gauss's law from Coulomb's law
Gauss's law can be derived from Coulomb's law.
[show]Outline of proof
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 curlof E (see Helmholtz 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 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).
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