Gauss' law

In physics, Gauss' law, also known as Gauss' flux theorem, is a law relating the distribution of electric charge to the resulting electric field. 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. The law was formulated by Carl Friedrich Gauss in 1835, but was not published until 1867.

Gauss' law has two forms, an integral form and a differential form. They are related by the divergence theorem, also called "Gauss' 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. (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 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. See the article Divergence theorem for more detail.

Gauss' law can be used to demonstrate that there is no electric field inside a Faraday cage with no electric charges. 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.

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 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. (See surface integral for more details.)
 * $$Q_V$$ 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).
 * $$\varepsilon_0$$ is the electric constant, a fundamental physical constant.

Applying the integral form
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.

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.

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,
 * $$\rho$$ is the total electric charge density (in units of C/m³), including both free and bound charge (see below).
 * $$\varepsilon_0$$ is the electric constant, a fundamental constant of nature.

This is mathematically equivalent to the integral form, because of the divergence theorem.

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 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 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:
 * $$\Phi_{D,S} = Q_{f,V}$$

where
 * $$\Phi_{D,S}$$ is defined by $$\Phi_{D,S}=\oint_S \mathbf{D} \cdot \mathrm{d}\mathbf{A}$$, where $$\mathbf{D}$$ is the electric displacement field, and the integration is a surface integral.
 * $$Q_{f,V}$$ is the free electric charge in the volume V, not including bound charge (see below).

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.

The differential form and integral form are mathematically equivalent. The proof primarily involves the divergence theorem.

Proof of equivalence

 * {| class="toccolours collapsible collapsed" width="80%" style="text-align:left"

!Proof that the formulations of Gauss' law in terms of free charge are equivalent to the formulations involving total charge.
 * In this proof, we will show that the equation
 * $$\nabla\cdot \mathbf{E} = \rho/\epsilon_0$$
 * $$\nabla\cdot \mathbf{E} = \rho/\epsilon_0$$

is equivalent to the equation
 * $$\nabla\cdot\mathbf{D} = \rho_{\mathrm{free}}$$

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:
 * $$\mathbf{D}=\epsilon_0 \mathbf{E} + \mathbf{P}$$

and the following relation to the bound charge:
 * $$\rho_{\mathrm{bound}} = -\nabla\cdot \mathbf{P}$$

Now, consider the three equations:
 * $$\rho_{\mathrm{bound}} = \nabla\cdot (-\mathbf{P})$$
 * $$\rho_{\mathrm{free}} = \nabla\cdot \mathbf{D}$$
 * $$\rho = \nabla \cdot(\epsilon_0\mathbf{E})$$

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

where $$\varepsilon$$ is the permittivity of the material. Under these circumstances, there is yet another pair of equivalent formulations of Gauss' 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). 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).

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. 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.