Polarization density in Maxwell's equations

The behavior of electric fields (E, D),magnetic fields (B, H), charge density (ρ) and current density (J) are described by Maxwell's equations. The role of the polarization density P is described below.

Relations between E, D and P

$mathbf\{D\}\; =\; epsilon\_0mathbf\{E\}\; +\; mathbf\{P\}$

which is convenient for various calculations.

A relation between P and E exists in many materials, as described later in the article.

Bound charge

Electric polarization corresponds to a rearrangement of the bound electrons in the material, which creates an additional charge density, known as the bound charge density ρ_b:

$rho\_b\; =\; -nablacdotmathbf\{P\}$

so that the total charge density that enters Maxwell's equations is given by

$rho\; =\; rho\_f\; +\; rho\_b\; ,$

where ρ_f is the free charge density (describing charges brought from outside).

At the surface of the polarized material, the bound charge appears as a surface charge density

$sigma\_b\; =\; mathbf\{P\}cdotmathbf\{hat\; n\}\_mathrm\{out\}\; ,$

where $mathbf\{hat\; n\}\_mathrm\{out\},$ is the normal vector. If P is uniform inside the material, this surface charge is the only bound charge.

When the polarization density changes with time, the time-dependent bound-charge density creates a current density of

$mathbf\{J\}\_b\; =\; frac\{partial\; mathbf\{P\}\}\{partial\; t\}$

so that the total current density that enters Maxwell's equations is given by

$mathbf\{J\}\; =\; mathbf\{J\_f\}\; +\; nablatimesmathbf\{M\}\; +\; frac\{partialmathbf\{P\}\}\{partial\; t\}$

where J_f is the free-charge current density, and the second term is a contribution from the magnetization (when it exists).

Relation between P and E in various materials

In a homogeneous linear and isotropic dielectric medium, the polarization is aligned with and proportional to the electric field E. In an anisotropic material, the polarization and the field are not necessarily in the same direction. Then, the i^th component of the polarization is related to the j^th component of the electric field according to:

$P\_i\; =\; sum\_j\; epsilon\_0\; chi\_\{ij\}\; E\_j\; ,\; ,!$

where ε₀ is the permittivity of free space, and χ is the electric susceptibility tensor of the medium. The case of an anisotropic dielectric medium is described by the field of crystal optics.

As in most electromagnetism, this relation deals with macroscopic averages of the fields and dipole density, so that one has a continuum approximation of the dielectric materials that neglects atomic-scale behaviors. The polarizability of individual particles in the medium can be related to the average susceptibility and polarization density by the Clausius-Mossotti relation.

In general, the susceptibility is a function of the frequency ω of the applied field. When the field is an arbitrary function of time t, the polarization is a convolution of the Fourier transform of χ(ω) with the E(t). This reflects the fact that the dipoles in the material cannot respond instantaneously to the applied field, and causality considerations lead to the Kramers–Kronig relations.

If the polarization P is not linearly proportional to the electric field E, the medium is termed nonlinear and is described by the field of nonlinear optics. To a good approximation (for sufficiently weak fields, assuming no permanent dipole moments are present), P is usually given by a Taylor series in E whose coefficients are the nonlinear susceptibilities:

$P\_i\; /\; epsilon\_0\; =\; sum\_j\; chi^\{(1)\}\_\{ij\}\; E\_j\; +\; sum\_\{jk\}\; chi\_\{ijk\}^\{(2)\}\; E\_j\; E\_k\; +\; sum\_\{jkell\}\; chi\_\{ijkell\}^\{(3)\}\; E\_j\; E\_k\; E\_ell\; +\; cdots\; !$

where $chi^\{(1)\}$ is the linear susceptibility, $chi^\{(2)\}$ gives the Pockels effect, and $chi^\{(3)\}$ gives the Kerr effect.

In ferroelectric materials, there is no one-to-one correspondence between P and E at all because of hysteresis.

References and notes

See also

• Electric field
• Electric susceptibility
• Electric displacement field
• Electret