Gauss's law
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In physics, Gauss' law, also known as Gauss' flux theorem, is a law relating the distribution of electric charge to the resulting electric field. Gauss' law states that:
The law was formulated by Carl Friedrich Gauss in 1835, but was not published until 1867. It is one of four of Maxwell's 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.
Gauss's law may be expressed in its integral form:
where the lefthand side of the equation is a surface integral denoting the electric flux through a closed surface S, and the righthand side of the equation is the total charge enclosed by S divided by the electric constant.
Gauss's law also has a differential form:
where ∇ · E is the divergence of the electric field, and ρ is the charge density.
The integral and differential forms 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 the electric field E and the total electric charge, or in terms of the electric displacement field D and the free electric charge.
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 and Gauss's law for gravity. In fact, any " inversesquare law" can be formulated in a way similar to Gauss's law: For example, Gauss's law itself is essentially equivalent to the inversesquare Coulomb's law, and Gauss's law for gravity is essentially equivalent to the inversesquare Newton's law of gravity.
Gauss's law can be used to demonstrate that all electric fields inside a Faraday cage have an electric charge. Gauss's law is something of an electrical analogue of Ampère's law, which deals with magnetism.
In terms of total charge
Integral form
For a volume V with surface S, Gauss's law states that
where Φ_{E,S} is the electric flux through S, Q is total charge inside V, and ε_{0} is the electric constant. The electric flux is given by a surface integral over S:
where E is the electric field, dA is a vector representing an infinitesimal element of area, and · represents the dot product..
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:
where ∇ · denotes divergence, E is the electric field, and ρ is the total electric charge density (including both free and bound charge), and ε_{0} is the electric constant. This is mathematically equivalent to the integral form, because of the divergence theorem.
Equivalence of integral and differential forms
The integral and differential forms are mathematically equivalent, by the divergence theorem. Here is the argument more specifically:
The integral form of Gauss's law is:
for any closed surface S containing charge Q. By the divergence theorem, this equation is equivalent to:
for any volume V containing charge Q. By the relation between charge and charge density, this equation is equivalent to:
for any volume V. In order for this equation to be simultaneously true for every possible volume V, it is necessary (and sufficient) for the integrands to be equal everywhere. Therefore, this equation is equivalent to:
Thus the integral and differential forms are equivalent.
In terms of free charge
Free 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 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, 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 that, for any volume V in space, with surface S, the following equation holds:
where Φ_{D,S} is the flux of the electric displacement field D through S, and Q_{free} is the free charge contained in V. The flux Φ_{D,S} is defined analogously to the flux Φ_{E,S} of the electric field E through S. Specifically, it is given by the surface integral
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 ρ_{free} is the free electric charge density.
The differential form and integral form are mathematically equivalent. The proof primarily involves the divergence theorem.
Equivalence of total and free charge statements

Proof that the formulations of Gauss's law in terms of free charge are equivalent to the formulations involving total charge. In this proof, we will show that the equation is equivalent to the equation
Note that we're only dealing with the differential forms, not the integral forms, but that is sufficient since the differential and integral forms are equivalent in each case, by the divergence theorem.
We introduce the polarization density P, which has the following relation to E and D:
and the following relation to the bound charge:
Now, consider the three equations:
The key insight is that the sum of the first two equations is the third equation. This completes the proof: The first equation is true by definition, and therefore the second equation is true if and only if the third equation is true. So the second and third equations are equivalent, which is what we wanted to prove.
In linear materials
In homogeneous, isotropic, nondispersive, linear materials, there is a nice, simple relationship between E and D:
where ε is the permittivity of the material. Under these circumstances, there is yet another pair of equivalent formulations of Gauss's law:
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 stationary point charge is:
where
 e_{r} is the radial unit vector,
 r is the radius, r,
 is the electric 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
where is the charge density. If we take the divergence of both sides of this equation with respect to r, and use the known theorem
where δ(s) is the Dirac delta function, the result is
Using the " sifting property" of the Dirac delta function, we arrive at
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 (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 sphericallysymmetric (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
By the assumption of spherical symmetry, the integrand is a constant which can be taken out of the integral. The result is
where is a unit vector pointing radially away from the charge. Again by spherical symmetry, E points in the radial direction, and so we get
which is essentially equivalent to Coulomb's law. Thus the inversesquare law dependence of the electric field in Coulomb's law follows from Gauss's law.