Polarization density

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Electromagnetism

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In classical electromagnetism, the polarization density (or electric polarization, or simply polarization) is the vector field that expresses the density of permanent or induced electric dipole moments in a dielectric material. The polarization vector P is defined as the dipole moment per unit volume. The SI unit of measure is coulombs per square metre.

Polarization density in Maxwell's equations

The behavior of electric fields (E and 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

The polarization density P defines the electric displacement field D as ^[1]

which is convenient for various calculations. Here ε₀ is the electric permittivity.

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:

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

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

where 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 polarization current density of

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

where J_f is the free-charge current density, and the second term is the magnetization current density (also called the bound current density), a contribution from atomic-scale magnetic dipoles (when they are present).

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:

where ε₀ is the electric constant, and χ is the electric susceptibility of the medium.

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:

where ε₀ is the electric constant, and κ_e, 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:

where χ⁽¹⁾ is the linear susceptibility, χ⁽²⁾ is the second-order susceptibility (describing phenomena such as the Pockels effect, optical rectification and second-harmonic generation), and χ⁽³⁾ is the third-order susceptibility (describing third-order effects such as the Kerr effect and electric field-induced optical rectification).

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