- Polarization density
In

classical electromagnetism , the**polarization density**(or**electric polarization**, or simply**polarization**) is thevector field that expresses the density of permanent or inducedelectric dipole moment s in adielectric material. The polarization vector**P**is defined as the dipole moment per unit volume. TheSI unit of measure iscoulomb s persquare metre .**Polarization density in Maxwell's equations**The behavior of

electric fields (**E**,**D**),magnetic field s (**B**,**H**),charge density (ρ) andcurrent density (**J**) are described byMaxwell's equations . The role of the polarization density**P**is described below.**Relations between****E**,**D**and**P**The polarization density

**P**defines theelectric displacement field **D**as [*cite book*]

last = Saleh | first = B.E.A. | last2 = Teich | first2 = M.C.

title = Fundamentals of Photonics | publisher =Wiley | date = 2007 | location = Hoboken, NJ

pages = pp. 154 | isbn = 9780471358329:$mathbf\{D\}\; =\; epsilon\_0mathbf\{E\}\; +\; mathbf\{P\}$

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 additionalcharge density , known as the**bound charge density**ρ_{b}::$ho\_b\; =\; -\; ablacdotmathbf\{P\}$

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

:$ho\; =\; ho\_f\; +\; ho\_b\; ,$

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

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

where

**J**_{f}is the free-charge current density, and the second term is a contribution from themagnetization (when it exists).**Relation between****P**and**E**in various materialsIn 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 ε

_{0}is thepermittivity of free space, and χ is theelectric susceptibility tensor of the medium. The case of an anisotropic dielectric medium is described by the field ofcrystal 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 theClausius-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 aconvolution 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, andcausality considerations lead to theKramers–Kronig relation s.If the polarization

**P**is not linearly proportional to the electric field**E**, the medium is termed "nonlinear" and is described by the field ofnonlinear optics . To a good approximation (for sufficiently weak fields, assuming no permanent dipole moments are present),**P**is usually given by aTaylor 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 theKerr effect .In

ferroelectric materials, there is no one-to-one correspondence between**P**and**E**at all because ofhysteresis .**References and notes****ee also***

Electric field

*Electric susceptibility

*Electric displacement field

*Electret

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