Editing Displacement current (section)

== Explanation ==

The [[electric displacement field]] is defined as:

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

where:
* {{math|''ε''<sub>0</sub>}} is the [[permittivity]] of free space;
* {{math|'''E'''}} is the [[electric field intensity]]; and
* {{math|'''P'''}} is the [[polarization (electrostatics)|polarization]] of the medium.

Differentiating this equation with respect to time defines the ''displacement current density'', which therefore has two components in a [[dielectric]]:<ref name="Jackson">{{cite book |title=Classical Electrodynamics |url=https://archive.org/details/classicalelectro00jack_449 |url-access=limited |author=John D Jackson |edition=3rd |publisher=Wiley |year=1999 |page=[https://archive.org/details/classicalelectro00jack_449/page/n237 238] |isbn=978-0-471-30932-1}}</ref>(see also the "displacement current" section of the article "[[current density]]")

<math display=block>\mathbf{J}_\mathrm{D} = \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t} + \frac{\partial  \mathbf{P}}{\partial t}\,.</math>

The first term on the right hand side is present in material media and in free space. It doesn't necessarily come from any actual movement of charge, but it does have an associated magnetic field, just as a current does due to charge motion. Some authors apply the name ''displacement current'' to the first term by itself.<ref name=Griffiths>For example, see {{cite book |author=David J Griffiths |page=[https://archive.org/details/introductiontoel00grif_0/page/323 323] |title=Introduction to Electrodynamics |edition=3rd |isbn=978-0-13-805326-0 |publisher=Pearson/Addison Wesley |year=1999 |url-access=registration |url=https://archive.org/details/introductiontoel00grif_0/page/323 }} and {{cite book |author=Tai L Chow |title=Introduction to Electromagnetic Theory |page=204 |publisher=Jones & Bartlett |year=2006 |isbn=978-0-7637-3827-3 |url=https://books.google.com/books?id=dpnpMhw1zo8C&pg=PA204}}</ref>

The second term on the right hand side, called polarization current density, comes from the change in [[electric polarization|polarization]] of the individual molecules of the dielectric material. Polarization results when, under the influence of an applied [[electric field]], the charges in molecules have moved from a position of exact cancellation. The positive and negative charges in molecules separate, causing an increase in the state of polarization {{math|'''P'''}}.  A changing state of polarization corresponds to charge movement and so is equivalent to a current, hence the term "polarization current". Thus,

<math display="block">I_\mathrm{D} =\iint_S\mathbf{J}_\mathrm{D}\cdot\operatorname{d}\!\mathbf{S} = \iint_S\frac{\partial \mathbf{D}}{\partial t} \cdot \operatorname{d}\!\mathbf{S}=\frac{\partial}{\partial t}\iint_S \mathbf{D} \cdot \operatorname{d}\!\mathbf{S}=\frac{\partial \Phi_\mathrm{D}}{\partial t}\,.</math>

This polarization is the displacement current as it was originally conceived by Maxwell. Maxwell made no special treatment of the vacuum, treating it as a material medium. For Maxwell, the effect of {{math|'''P'''}} was simply to change the [[relative permittivity]] {{math|''ε''<sub>r</sub>}} in the relation {{math|1= '''D''' = ''ε''<sub>0</sub>''ε''<sub>r</sub> '''E'''}}.

The modern justification of displacement current is explained below.

===Isotropic dielectric case===
In the case of a very simple dielectric material the [[constitutive relation]] holds:

<math display=block> \mathbf{D} = \varepsilon \, \mathbf{E} ~ , </math>

where the [[permittivity]] {{nowrap|<math>\varepsilon = \varepsilon_0 \,  \varepsilon_\mathrm{r}</math>}} is the product of:
* {{math|''ε''<sub>0</sub>}}, the ''[[permittivity of free space]]'', or the ''[[electric constant]]''; and
* {{math|''ε''<sub>r</sub>}}, the [[relative permittivity|''relative'' permittivity]] of the dielectric.

In the equation above, the use of  {{mvar|ε}}  accounts for
the polarization (if any) of the dielectric material.

The [[scalar (physics)|scalar]] value of displacement current may also be expressed in terms of [[electric flux]]:

<math display=block> I_\mathrm{D} = \varepsilon \, \frac{\, \partial \Phi_\mathrm{E}  \, }{\partial t} ~ .</math>

The forms in terms of [[scalar (physics)|scalar]]  {{mvar|ε}}  are correct only for linear [[isotropic]] materials. For linear non-isotropic materials, {{mvar|ε}}  becomes a [[matrix (mathematics)|matrix]]; even more generally,  {{mvar|ε}}  may be replaced by a [[tensor]], which may depend upon the electric field itself, or may exhibit frequency dependence (hence [[Dispersion (optics)|dispersion]]).

For a linear isotropic dielectric, the polarization {{math|'''P'''}} is given by:

<math display=block>\mathbf{P} = \varepsilon_0 \chi_\mathrm{e}  \, \mathbf{E} = \varepsilon_0 (\varepsilon_\mathrm{r} - 1) \, \mathbf{E} ~,</math>

where  {{math|''χ''<sub>e</sub>}} is known as the [[electric susceptibility|''susceptibility'']] of the dielectric to electric fields. Note that

<math display=block>\varepsilon = \varepsilon_\mathrm{r} \, \varepsilon_0 = \left( 1 + \chi_\mathrm{e} \right) \, \varepsilon_0 ~. </math>