(Redirected from Polarization (electrostatics))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 (
,
),
magnetic fields (
,
),
charge density (
) and
current density (
) are described by
Maxwell's equations. The role of the polarization density
is described below.
Relations between 'E', 'D' and 'P'
The polarization density
defines the
electric displacement field as
:
which is convenient for various calculations.
A relation between
and
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'
:
:
so that the total charge density that enters Maxwell's equations is given by
:
where
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
current density of
:
so that the total current density that enters Maxwell's equations is given by
:
where
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:
:
where ε
0 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:
:
where
is the linear susceptibility,
gives the
Pockels effect, and
gives the
Kerr effect.
In
ferroelectric materials, there is no one-to-one correspondence between 'P' and 'E' at all because of
hysteresis.
See also
★
Electric field
★
Electric susceptibility
★
Electric displacement field
★
Electret