Nikiya Anton Bettey 2021: 09-25-2021-1752 - polarization density (or electric polarization, or simply polarization)

Saturday, September 25, 2021

09-25-2021-1752 - polarization density (or electric polarization, or simply polarization)

In classical electromagnetism, 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. When a dielectric is placed in an external electric field, its molecules gain electric dipole moment and the dielectric is said to be polarized. The electric dipole moment induced per unit volume of the dielectric material is called the electric polarization of the dielectric.^[1]^[2]

Polarization density also describes how a material responds to an applied electric field as well as the way the material changes the electric field, and can be used to calculate the forces that result from those interactions. It can be compared to magnetization, which is the measure of the corresponding response of a material to a magnetic field in magnetism. The SI unit of measure is coulombs per square meter, and polarization density is represented by a vector P.^[2]

Relationship between the fields of P and E[edit]

Homogeneous, isotropic dielectrics[edit]

Field lines of the D-field in a dielectric sphere with greater susceptibility than its surroundings, placed in a previously-uniform field.^[6] The field lines of the E-field are not shown: These point in the same directions, but many field lines start and end on the surface of the sphere, where there is bound charge. As a result, the density of E-field lines is lower inside the sphere than outside, which corresponds to the fact that the E-field is weaker inside the sphere than outside.

In a homogeneous, linear, non-dispersive and isotropic dielectric medium, the polarization is aligned with and proportional to the electric field E:^[7]

\mathbf {P} =\chi \varepsilon _{0}\mathbf {E} ,

where ε₀ is the electric constant, and χ is the electric susceptibility of the medium. Note that in this case χ simplifies to a scalar, although more generally it is a tensor. This is a particular case due to the isotropy of the dielectric.

Taking into account this relation between P and E, equation (3) becomes:^[3]

-Q_{b}=\chi \varepsilon _{0}\

$\oiint$

\scriptstyle {S}

\mathbf{E} \cdot \mathrm{d}\mathbf{A}

The expression in the integral is Gauss's law for the field E which yields the total charge, both free $(Q_{f})$ and bound $(Q_{b})$ , in the volume V enclosed by S.^[3] Therefore,

{\begin{aligned}-Q_{b}&=\chi Q_{\text{total}}\\&=\chi \left(Q_{f}+Q_{b}\right)\\[3pt]\Rightarrow Q_{b}&=-{\frac {\chi }{1+\chi }}Q_{f},\end{aligned}}

which can be written in terms of free charge and bound charge densities (by considering the relationship between the charges, their volume charge densities and the given volume):

\rho _{b}=-{\frac {\chi }{1+\chi }}\rho _{f}

Since within a homogeneous dielectric there can be no free charges $(\rho _{f}=0)$ , by the last equation it follows that there is no bulk bound charge in the material $(\rho _{b}=0)$ . And since free charges can get as close to the dielectric as to its topmost surface, it follows that polarization only gives rise to surface bound charge density (denoted $\sigma_b$ to avoid ambiguity with the volume bound charge density $\rho_b$ ).^[3]

$\sigma_b$ may be related to P by the following equation:^[8]

\sigma _{b}=\mathbf {\hat {n}} _{\text{out}}\cdot \mathbf {P}

where $\mathbf {\hat {n}} _{\text{out}}$ is the normal vector to the surface S pointing outwards. (see charge density for the rigorous proof)

Anisotropic dielectrics[edit]

The class of dielectrics where the polarization density and the electric field are not in the same direction are known as anisotropic materials.

In such materials, the ith component of the polarization is related to the jth component of the electric field according to:^[7]

P_{i}=\sum _{j}\epsilon _{0}\chi _{ij}E_{j},

This relation shows, for example, that a material can polarize in the x direction by applying a field in the z direction, and so on. 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:

{\frac {P_{i}}{\epsilon _{0}}}=\sum _{j}\chi _{ij}^{(1)}E_{j}+\sum _{jk}\chi _{ijk}^{(2)}E_{j}E_{k}+\sum _{jk\ell }\chi _{ijk\ell }^{(3)}E_{j}E_{k}E_{\ell }+\cdots

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