Editing Polarization density (section)

==Relationship between the fields of ''P'' and ''E''==

===Homogeneous, isotropic dielectrics===

[[File:Dielectric sphere.svg|thumb|250px|[[Field line]]s of the [[electric displacement field|'''D'''-field]] in a dielectric sphere with greater susceptibility than its surroundings, placed in a previously uniform field.<ref name=Gray>Based upon equations from {{cite book |title=The theory and practice of absolute measurements in electricity and magnetism |author=Gray, Andrew |year=1888 |publisher=Macmillan & Co. |pages= [https://archive.org/details/theoryandpracti07graygoog/page/n158 126]–127 |url=https://archive.org/details/theoryandpracti07graygoog}}, which refers to papers by Sir W. Thomson.</ref> The [[field line]]s of the [[electric field|'''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 [[homogeneity (physics)|homogeneous]], linear, non-dispersive and [[isotropic]] [[dielectric]] medium, the '''polarization''' is aligned with and [[Proportionality (mathematics)|proportional]] to the electric field '''E''':<ref name="Fay64">{{ cite book | last1 = Feynman | first1 = R.P. | last2 = Leighton | first2 = R.B. | last3 = Sands | first3 = M | year = 1964 | title = Feynman Lectures on Physics: Volume 2 | publisher = Addison-Wesley | isbn = 0-201-02117-X }}</ref>
<math display="block">\mathbf{P} = \chi\varepsilon_0 \mathbf E,</math>

where {{math|''ε''<sub>0</sub>}} is the [[electric constant]], and {{mvar|χ}} is the [[electric susceptibility]] of the medium. Note that in this case {{mvar|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 ({{EquationNote|3}}) becomes:<ref name="Irodov" />

:{{oiint
 | preintegral = <math>-Q_b = \chi\varepsilon_0\ </math>
 | intsubscpt = <math>\scriptstyle{S}</math>
 | integrand = <math>\mathbf{E} \cdot \mathrm{d}\mathbf{A}</math>
}}

The expression in the integral is [[Gauss's law]] for the field {{math|'''E'''}} which yields the total charge, both free <math>(Q_f)</math> and bound <math>(Q_b)</math>, in the volume {{mvar|V}} enclosed by {{mvar|S}}.<ref name="Irodov" /> Therefore,

<math display="block">\begin{align}
             -Q_b &=  \chi Q_\text{total} \\
                  &=  \chi \left(Q_f + Q_b\right) \\[3pt]
  \Rightarrow Q_b &= -\frac{\chi}{1 + \chi} Q_f,
\end{align}</math>

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):
<math display="block">\rho_b = -\frac{\chi}{1 + \chi} \rho_f</math>

Since within a homogeneous dielectric there can be no free charges <math>(\rho_f = 0)</math>, by the last equation it follows that there is no bulk bound charge in the material <math>(\rho_b = 0)</math>. 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 <math>\sigma_b</math> to avoid ambiguity with the volume bound charge density <math>\rho_b</math>).<ref name="Irodov" />

<math>\sigma_b</math> may be related to {{math|'''P'''}} by the following equation:<ref name="grant08">{{ cite book | title = Electromagnetism | edition = 2nd | first1 = I.S. | last1 = Grant | first2 = W.R. | last2 = Phillips | publisher = Manchester Physics, John Wiley & Sons | year = 2008 | isbn = 978-0-471-92712-9 }}</ref>
<math display="block">\sigma_b = \mathbf{\hat{n}}_\text{out} \cdot \mathbf{P}</math>
where <math>\mathbf{\hat{n}}_\text{out}</math> is the [[normal vector]] to the surface {{math|''S''}} pointing outwards. (see [[charge density]] for the rigorous proof)

===Anisotropic dielectrics===

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 {{mvar|i}}-th component of the polarization is related to the {{mvar|j}}-th component of the electric field according to:<ref name="Fay64"/>

<math display="block">P_i = \sum_j \varepsilon_0 \chi_{ij} E_j ,</math>

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]] {{mvar|ω}} of the applied field.  When the field is an arbitrary function of time {{mvar|t}}, the polarization is a [[convolution]] of the [[continuous Fourier transform|Fourier transform]] of {{math|''χ''(''ω'')}} with the {{math|'''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 relation]]s.

If the polarization '''P''' is not linearly proportional to the electric field {{math|'''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 {{math|'''E'''}} whose coefficients are the nonlinear susceptibilities:

<math display="block">\frac{P_i}{\varepsilon_0} = \sum_j \chi^{(1)}_{ij} 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 </math>

where <math>\chi^{(1)}</math> is the linear susceptibility, <math>\chi^{(2)}</math> is the second-order susceptibility (describing phenomena such as the [[Pockels effect]], [[optical rectification]] and [[second-harmonic generation]]), and <math>\chi^{(3)}</math> 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]].