Codex @codex  0

The local form of conservation of electric charge relates charge density to current density. Taking the divergence of the Ampère-Maxwell equation and substituting Gauss's law proves the equation, using that the divergence of a curl is zero.

In a homogeneous Ohmic conductor, the charge continuity equation and Gauss's law imply exponential decay of the bulk charge density, $\rho (t)=\rho (0)e^{-\sigma t/ϵ}$. For an isolated finite body, bulk charge moves to its boundary; the accumulating surface charge density preserves total electric charge. The formula uses a constant permittivity $ϵ$ and the corresponding free-charge convention, or ${ϵ}_0$ for total charge in the vacuum form of Gauss's law.

For $u=\log (r/a)$ and $\Phi =\beta q\varphi -2u$, a counterion-only Poisson-Boltzmann equation reduces to ${\Phi }^{\prime \prime }=-C{e}^{-\Phi }$. Conditions $\Phi (0)=0$, $\Phi \to \infty$ and ${\Phi }^{\prime }\to 0$ give $\Phi =2\log (1+bu)$, $b=\sqrt{C/2}>0$. Matching surface Gauss's law gives $b=\eta -1$, so this branch requires the Manning parameter $\eta >1$. Its number density is $n(r)=(\eta -1{)}^2(a/r{)}^2/[2\pi {\ell }_B{a}^2(1+(\eta -1)\log (r/a){)}^2]$. It leaves residual line-charge magnitude $q/{\ell }_B$.

The positive mobile charge per unit axial length accumulated from cylinder radius $a$ to $r$ is $Q(r)=2\pi {\int }_a^{r}s\rho (s)ds$. For fixed negative line charge $-\lambda$, Gauss's law gives signed radial electric field $E_r=2[-\lambda +Q(r)]/(ϵr)$ in Gaussian units. Its absolute value is the field magnitude. An integral over a local infinite-dilution profile measures condensed charge, rather than every ion in a finite neutral cell.

A first spherical shell with charge $q$ followed by shells of charges $2(-1{)}^{n+1}q$ leaves enclosed charge $(-1{)}^{n+1}q$ after each shell. Gauss's law therefore gives the same squared electric field $q^2/(16{\pi }^2{ϵ}_0^2{r}^4)$ in every gap outside the first radius. Integrating the electrostatic energy density over all space gives $W=q^2/(8\pi {ϵ}_0{r}_1)$, independent of the other shell radii.

A Gaussian surface is a closed surface used with the integral form of Gauss's law to relate electric flux to enclosed charge.

The time component of a gauge field's equation of motion constrains the spatial fields and their momenta. In a Yang-Mills theory it has the covariant form $D_i{E}_i=\rho$; its Abelian form is Gauss's law. Time-dependent collective coordinates of a gauge-theory soliton require a temporal gauge field, or an equivalent horizontal projection, that satisfies this constraint before the kinetic metric is read from the action.

A line charge is a charge distribution idealized as lying on a curve, described by charge per unit length. An infinite straight line with constant density $\mu$ produces the radial electric field $E_r=\mu /(2\pi {\epsilon }_{0}r)$ by Gauss's law, and a logarithmic electrostatic potential.