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An electromagnetic wave has circular polarization when its transverse electric field has constant magnitude and rotates once during each period. In complex-amplitude notation, two orthogonal components have equal magnitude and a relative phase of one quarter-cycle.

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.

The Debye–Hückel approximation linearizes mobile-ion Boltzmann densities about a neutral bulk electrolyte. It is valid for ionic potential energies small compared with $k_{B}T$. The inverse Debye–Hückel screening length satisfies ${\kappa }_D^2={\sum }_i{n}_i{z}_i^2{e}^2/(\epsilon k_{B}T)$. The quadratic screened charging free energy includes both electric field energy and the linearized ionic-response term $\epsilon {\kappa }_D^2{\varphi }^2/2$.

Electrical electrical conductivity relates current density to the local electric field in the rest frame of an isotropic ohmic medium: $\mathbf{J}=\sigma \mathbf{E}$. It is a bulk material coefficient, distinct from a sheet surface electrical conductivity. A moving conductor instead uses the moving-conductor Ohm law.

The leading scalar coupling of an electric field to the nematic order parameter is quadratic in the field and linear in the tensor. For uniaxial nematic order, it is $-3\chi \lambda E^2({\cos }^2\vartheta -1/3)/2$. When $\chi \lambda >0$, minimization aligns the nematic director with the field axis and adds $-\chi E^2\lambda$ to the scalar Landau free energy.

The combined electric field and magnetic field that determine the Lorentz force on a charged particle.

Electromagnetic radiation consists of propagating oscillations of the electric field and magnetic field that carry energy and momentum.

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.

Across a charged dielectric interface the tangential electric field is continuous and the normal displacement jump is the fixed surface charge density. Without a dipole sheet the electrostatic potential is continuous. For a charge sheet with equal permittivity on both sides, its normal potential derivative has jump $-{\sigma }_{\mathrm{fixed}}/\epsilon$. Setting the field to zero on one side is an additional physical boundary assumption, as in a conductor or a negligible-exterior-admittance idealization.

A wave is linearly polarized when its oscillating electric field always lies along one fixed line. For a transverse plane wave, that line is perpendicular to the wavevector. Unlike circular polarization, the field oscillates along the line rather than rotating through directions.

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.

Confine the counterions of a uniformly charged infinite cylinder within a coaxial cylinder of radius $R$, with total mobile line charge $\lambda$. The electric field vanishes at $R$, while its surface value at $a$ is fixed by the bare line charge. This gives $a{\varphi }^{\prime }(a)=2\lambda /ϵ$ and $R{\varphi }^{\prime }(R)=0$ in Gaussian units. Sending $R/a\to \infty$ defines the infinite-dilution limit of cylindrical counterions.

A material whose current density is proportional to its electric field, with proportionality given by its electrical conductivity.