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Gauss

[gous]
Gauss, Carl Friedrich, born Johann Friederich Carl Gauss, 1777-1855, German mathematician, physicist, and astronomer. Gauss was educated at the Caroline College, Brunswick, and the Univ. of Göttingen, his education and early research being financed by the Duke of Brunswick. Following the death of the duke in 1806, Gauss became director (1807) of the astronomical observatory at Göttingen, a post he held until his death. Considered the greatest mathematician of his time and as the equal of Archimedes and Newton, Gauss showed his genius early and made many of his important discoveries before he was twenty. His greatest work was done in the area of higher arithmetic and number theory; his Disquisitiones Arithmeticae (completed in 1798 but not published until 1801) is one of the masterpieces of mathematical literature.

Gauss was extremely careful and rigorous in all his work, insisting on a complete proof of any result before he would publish it. As a consequence, he made many discoveries that were not credited to him and had to be remade by others later; for example, he anticipated Bolyai and Lobachevsky in non-Euclidean geometry, Jacobi in the double periodicity of elliptic functions, Cauchy in the theory of functions of a complex variable, and Hamilton in quaternions. However, his published works were enough to establish his reputation as one of the greatest mathematicians of all time. Gauss early discovered the law of quadratic reciprocity and, independently of Legendre, the method of least squares. He showed that a regular polygon of n sides can be constructed using only compass and straight edge only if n is of the form 2p(2q+1)(2r+1) … , where 2q + 1, 2r + 1, … are prime numbers.

In 1801, following the discovery of the asteroid Ceres by Piazzi, Gauss calculated its orbit on the basis of very few accurate observations, and it was rediscovered the following year in the precise location he had predicted for it. He tested his method again successfully on the orbits of other asteroids discovered over the next few years and finally presented in his Theoria motus corporum celestium (1809) a complete treatment of the calculation of the orbits of planets and comets from observational data. From 1821, Gauss was engaged by the governments of Hanover and Denmark in connection with geodetic survey work. This led to his extensive investigations in the theory of space curves and surfaces and his important contributions to differential geometry as well as to such practical results as his invention of the heliotrope, a device used to measure distances by means of reflected sunlight.

Gauss was also interested in electric and magnetic phenomena and after about 1830 was involved in research in collaboration with Wilhelm Weber. In 1833 he invented the electric telegraph. He also made studies of terrestrial magnetism and electromagnetic theory. During the last years of his life Gauss was concerned with topics now falling under the general heading of topology, which had not yet been developed at that time, and he correctly predicted that this subject would become of great importance in mathematics.

See biography by T. Hall (tr. 1970).

gauss [for C. F. Gauss], abbr. G, unit of magnetic flux density (see flux, magnetic) equal to 0.0001 (10-4) weber per square meter. Since this unit is derived from the cgs system of units rather than the mks system, it is largely obsolete. See electric and magnetic units.

In physics, Gauss's law, also known as Gauss's 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'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 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'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 and Gauss'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. See the article Divergence 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 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:

$Phi_\left\{E,S\right\} = frac\left\{Q_V\right\}\left\{varepsilon_0\right\}$
where

• $Phi$E,S, called the "electric flux through S", is defined by $Phi_\left\{E,S\right\}=oint_S mathbf\left\{E\right\} cdot mathrm\left\{d\right\}mathbf\left\{A\right\}$, where $mathbf\left\{E\right\}$ is the electric field, and $mathrm\left\{d\right\}mathbf\left\{A\right\}$ is a differential area on the surface $S$ with an outward facing surface normal defining its direction. (See surface integral for more details.) The surface S is the surface bounding the volume V.
• $Q_V$ 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'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:

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

where:

• $mathbf\left\{nabla\right\}cdot$ denotes divergence,
• E is the electric field,
• $mathbfrho$ 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.

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 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. 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:

$Phi_\left\{D,S\right\} = Q_\left\{f,V\right\}$
where

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

Differential form

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

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

where:

• $mathbf\left\{nabla\right\}cdot$ denotes divergence,
• D is the electric displacement field (in units of C/m²), and *$rho_\left\{mathrm\left\{free\right\}\right\},$ 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.

In linear materials

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

$varepsilon mathbf\left\{E\right\} = mathbf\left\{D\right\}$
where $varepsilon$ is the permittivity of the material. Under these circumstances, there is yet another pair of equivalent formulations of Gauss's law:
$Phi_\left\{E,S\right\} = frac\left\{Q_\left\{V,mathrm\left\{free\right\}\right\}\right\}\left\{varepsilon\right\}$
$mathbf\left\{nabla\right\} cdot mathbf\left\{E\right\} = frac\left\{rho_\left\{mathrm\left\{free\right\}\right\}\right\}\left\{varepsilon\right\}$

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:

$mathbf\left\{E\right\}\left(mathbf\left\{r\right\}\right) = frac\left\{q\right\}\left\{4pi epsilon_0\right\} frac\left\{mathbf\left\{e_r\right\}\right\}\left\{r^2\right\}$
where
er is the radial unit vector,
$epsilon_0$ 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

$mathbf\left\{E\right\}\left(mathbf\left\{r\right\}\right) = frac\left\{1\right\}\left\{4piepsilon_0\right\} int frac\left\{rho\left(mathbf\left\{s\right\}\right)\left(mathbf\left\{r\right\}-mathbf\left\{s\right\}\right)\right\}\left\{|mathbf\left\{r\right\}-mathbf\left\{s\right\}|^3\right\} d^3 mathbf\left\{s\right\}$

where $rho$ is the charge density. If we take the divergence of both sides of this equation with respect to r, and use the known theorem

$nabla cdot left\left(frac\left\{mathbf\left\{s\right\}\right\}\left\{|mathbf\left\{s\right\}|^3\right\}right\right) = 4pi delta\left(mathbf\left\{s\right\}\right)$
where δ(s) is the Dirac delta function, the result is

$nablacdotmathbf\left\{E\right\}\left(mathbf\left\{r\right\}\right) = frac\left\{1\right\}\left\{epsilon_0\right\} int rho\left(mathbf\left\{s\right\}\right) delta\left(mathbf\left\{r\right\}-mathbf\left\{s\right\}\right) d^3 mathbf\left\{s\right\}$

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

$nablacdotmathbf\left\{E\right\} = rho/epsilon_0$

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 (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).

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

$oint_\left\{S\right\}mathbf\left\{E\right\}cdot dmathbf\left\{A\right\} = Q/varepsilon_0$
By the assumption of spherical symmetry, the integrand is a constant which can be taken out of the integral. The result is
$4pi r^2hat\left\{mathbf\left\{r\right\}\right\}cdotmathbf\left\{E\right\}\left(mathbf\left\{r\right\}\right) = Q/varepsilon_0$
where $hat\left\{mathbf\left\{r\right\}\right\}$ is a unit vector pointing radially away from the charge. Again by spherical symmetry, E points in the radial direction, and so we get
$mathbf\left\{E\right\}\left(mathbf\left\{r\right\}\right) = frac\left\{Q\right\}\left\{4pi varepsilon_0\right\}frac\left\{hat\left\{mathbf\left\{r\right\}\right\}\right\}\left\{r^2\right\}$
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's law.