Gauss's law

In physics and mathematical analysis, Gauss' law is the electrostatic application of the generalized Gauss's theorem giving the equivalence relation between any flux, e.g. of liquids, electric or gravitational, flowing out of any closed surface and the result of inner sources and sinks, such as electric charges or masses enclosed within the closed surface. The law was developed by Carl Friedrich Gauss. By the divergence theorem, a generalized Gauss's law can be used in any context where the inverse-square law holds. Electrostatics and Newtonian gravitation are two examples. The differential form of Maxwell's equations underpins electromagnetic theory.

Integral form

In its integral form (in SI units), the law states:

$\Phi =\oint _{S}\mathbf {E} \cdot \mathrm {d} \mathbf {A} ={1 \over \varepsilon _{0}}\int _{V}\rho \ \mathrm {d} V={\frac {Q_{A}}{\varepsilon _{0}}}$

where $\Phi$ is the electric flux, $\mathbf {E}$ is the electric field, $\mathrm {d} \mathbf {A}$ is a differential area on the closed surface $S$ with an outward facing surface normal defining its direction, $Q_{\mathrm {A} }$ is the charge enclosed by the surface, $\rho$ is the charge density at a point in $V$, $\varepsilon _{0}$ is the permittivity of free space and $\oint _{S}$ is the integral over the surface $S$ enclosing volume $V$.

For information and strategy on the application of Gauss's law, see Gaussian surfaces.

Differential form

In differential form, the equation becomes:

$\mathbf {\nabla } \cdot \mathbf {D} =\rho _{\mathrm {free} }$

where $\mathbf {\nabla }$ is the del operator, representing 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 dipole charges bound in a material. The differential form derives in part from Gauss's divergence theorem.

And for linear materials, the equation becomes:

$\mathbf {\nabla } \cdot \varepsilon \mathbf {E} =\rho _{\mathrm {free} }$

where $\varepsilon$ is the electric permittivity.

Coulomb's law

In the special case of a spherical surface with a central charge, the electric field is perpendicular to the surface, with the same magnitude at all points of it, giving the simpler expression:

$E={\frac {1}{\varepsilon _{0}}}{\frac {Q}{4\pi r^{2}}}$

where $E$ is the electric field strength at radius $r$, $Q$ is the enclosed charge, and $\varepsilon _{0}$ is the permitivity of free space. Thus the familiar inverse-square law dependence of the electric field in Coulomb's law follows from Gauss's law.

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 the electrostatic equivalent of Ampère's law, which deals with magnetism. Both equations were later integrated into Maxwell's equations.

It was formulated by Carl Friedrich Gauss in 1835, but was not published until 1867. Because of the mathematical similarity, Gauss's law has application for other physical quantities governed by an inverse-square law such as gravitation or the intensity of radiation. See also divergence theorem.

Application to magnetism

In the static case of a bar magnet or other situation where the generator of a magnetic field is at rest with respect to the observer, the integral form of Gauss's Law can be proven using a heuristic argument regarding the net flux proportionality to the number of field lines that enter and leave a Gaussian surface.

With such an argument it can be shown that in all static cases, the net magnetic flux is zero. As many field lines enter any Gaussian surface as leave a Gaussian surface, and so there is no "source" of the magnetic field to enclose.

$\Phi _{B}=\oint _{S}\mathbf {B} \cdot \mathrm {d} \mathbf {A} =0.$

The above expression is one of Maxwell's Equations in integral form. Both in integral and differential form, this expression asserts that unlike electric monopoles, magnetic monopoles do not exist, and indeed, no experiment has yet provided conclusive evidence of their existence.

Application to gravity

The gravitational form of Gauss's Law is largely a theoretical curiosity, but can be used by analogy to the electrostatic form of Gauss's Law to prove that the gravitational force of any spherically symmetric body on any other spherically symmetric body can be treated as though both masses were concentrated at their centers.

$\Phi _{g}=\oint _{S}\mathbf {g} \cdot \mathrm {d} \mathbf {A} =-4\pi G\int _{V}\rho _{m}\ \mathrm {d} V=-4\pi GM$

In applying the above form of Gauss's Law to prove, for example, that the force of the Earth acting on the Moon does not depend on a detailed treatment of the Earth's composition, one encloses the Earth in a spherical Gaussian surface, whose area is $4\pi r^{2}$. Note that the gravitational "flux" out of a closed surface is never positive, because gravitational fields always point inward towards mass.

Since the field lines of the Earth extend out equally in all directions and fall off as ${\frac {1}{r^{2}}}$ (which can be proven independently from Newtonian mechanics and the force law so derived), the gravitational field must be constant at a given radius.

$\oint _{S}\mathbf {g} \cdot \mathrm {d} \mathbf {A} =-4\pi GM$

$-g\oint _{S}\mathrm {d} A=-4\pi GM$

$g\cdot 4\pi r^{2}=4\pi GM$

$g={\frac {GM}{r^{2}}}$

Trivially, multiplying through by m yields the familiar force equation. If the only assumption being made is that gravitational field lines look like electrostatic ones then no prior knowledge of Newton's work is needed. While no reference can be formally found for this, it is often remarked casually in introductory physics classes that Isaac Newton took several pages of calculus to prove that mass distributions act as though their mass were concentrated at a point in their center as far as their interactions with other bodies are concerned, and that had he had Gauss's Law, much of the cumbersome work he undertook would have been shortened dramatically.

See also

• Maxwell's equations
• Gaussian surface
• Carl Friedrich Gauss
• Divergence theorem
• Flux
• Method of image charges

References

Jackson, John David (1999). 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.
• section on Gauss's law in an online textbook
• MISN-0-132 Gauss's Law for Spherical Symmetry (PDF file) by Peter Signell for Project PHYSNET.
• MISN-0-133 Gauss's Law Applied to Cylindrical and Planar Charge Distributions (PDF file) by Peter Signell for Project PHYSNET.