Editing Polarization density (section)

==Polarization density in Maxwell's equations==

The behavior of [[electric field]]s ({{math|'''E'''}}, {{math|'''D'''}}), [[magnetic field]]s ({{math|'''B'''}}, {{math|'''H'''}}), [[charge density]] ({{mvar|ρ}}) and [[current density]] ({{math|'''J'''}}) are described by [[Maxwell's equations#.22Microscopic.22 versus .22macroscopic.22|Maxwell's equations in matter]].

===Relations between E, D and P===

In terms of volume charge densities, the '''free''' charge density <math>\rho_f</math> is given by

<math display="block">\rho_f = \rho - \rho_b</math>

where <math>\rho</math> is the total charge density. By considering the relationship of each of the terms of the above equation to the divergence of their corresponding fields (of the [[electric displacement field]] {{math|'''D'''}}, {{math|'''E'''}} and {{math|'''P'''}} in that order), this can be written as:<ref>{{cite book
 | last1 = Saleh | first1 = B.E.A.
 | last2 = Teich+ | first2 = M.C.
 | title = Fundamentals of Photonics | publisher = [[John Wiley & Sons|Wiley]] | year = 2007 | location = Hoboken, NJ | pages = 154 | isbn = 978-0-471-35832-9}}</ref>

<math display="block">\mathbf{D} = \varepsilon_0\mathbf{E} + \mathbf{P}.</math>

This is known as the [[constitutive equation]] for electric fields.  Here {{math|''ε''<sub>0</sub>}} is the [[electric permittivity]] of empty space.  In this equation, '''P''' is the (negative of the) field induced in the material when the "fixed" charges, the dipoles, shift in response to the total underlying field '''E''', whereas '''D''' is the field due to the remaining charges, known as "free" charges.<ref name="def_P_M_Maxwell_eqs"/><ref name=bound_charge_current>{{Cite journal | author = A. Herczynski | title = Bound charges and currents | journal = American Journal of Physics | volume = 81 | issue = 3 | pages = 202–205 | year = 2013 | url = http://www.bc.edu/content/dam/files/schools/cas_sites/physics/pdf/herczynski/AJP-81-202.pdf | bibcode = 2013AmJPh..81..202H | doi = 10.1119/1.4773441}}</ref>

In general, {{math|'''P'''}} varies as a function of {{math|'''E'''}} depending on the medium, as described later in the article. In many problems, it is more convenient to work with {{math|'''D'''}} and the free charges than with {{math|'''E'''}} and the total charge.<ref name="Gri07" />

Therefore, a polarized medium, by way of [[Green's theorem]] can be split into four components.

* The bound volumetric charge density: <math>\rho_b = -\nabla \cdot \mathbf{P}</math>
* The bound surface charge density: <math>\sigma_b = \mathbf{\hat{n}}_\text{out} \cdot \mathbf{P}</math>
* The free volumetric charge density: <math>\rho_f = \nabla \cdot \mathbf{D}</math>
* The free surface charge density: <math>\sigma_f = \mathbf{\hat{n}}_\text{out} \cdot \mathbf{D}</math>

===Time-varying polarization density===
When the polarization density changes with time, the time-dependent bound-charge density creates a ''polarization [[current density]]'' of

<math display="block"> \mathbf{J}_p = \frac{\partial \mathbf{P}}{\partial t} </math>

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

<math display="block"> \mathbf{J} = \mathbf{J}_f + \nabla\times\mathbf{M} + \frac{\partial\mathbf{P}}{\partial t}</math>

where '''J'''<sub>f</sub> is the free-charge current density, and the second term is the [[Magnetization#Magnetization current|magnetization current]] density (also called the ''bound current density''), a contribution from atomic-scale [[Magnet#Two models for magnets: magnetic poles and atomic currents|magnetic dipoles]] (when they are present).