# Displacement current

In electromagnetism, displacement current density is the quantity D/∂t appearing in Maxwell's equations that is defined in terms of the rate of change of D, the electric displacement field. Displacement current density has the same units as electric current density, and it is a source of the magnetic field just as actual current is. However it is not an electric current of moving charges, but a time-varying electric field. In physical materials (as opposed to vacuum), there is also a contribution from the slight motion of charges bound in atoms, called dielectric polarization.

The idea was conceived by On Physical Lines of Force, Part III in connection with the displacement of electric particles in a dielectric medium. Maxwell added displacement current to the electric current term in Ampère's Circuital Law. In his 1865 paper A Dynamical Theory of the Electromagnetic Field Maxwell used this amended version of Ampère's Circuital Law to derive the electromagnetic wave equation. This derivation is now generally accepted as a historical landmark in physics by virtue of uniting electricity, magnetism and optics into one single unified theory. The displacement current term is now seen as a crucial addition that completed Maxwell's equations and is necessary to explain many phenomena, most particularly the existence of electromagnetic waves.

## Explanation

The electric displacement field is defined as:

${\displaystyle {\boldsymbol {D}}=\varepsilon _{0}{\boldsymbol {E}}+{\boldsymbol {P}}\ .}$

where:

ε0 is the permittivity of free space
E is the electric field intensity
P is the polarization of the medium

Differentiating this equation with respect to time defines the displacement current density, which therefore has two components in a dielectric:[1](see also the "displacement current" section of the article "current density")

${\displaystyle {\boldsymbol {J}}_{\boldsymbol {D}}=\varepsilon _{0}{\frac {\partial {\boldsymbol {E}}}{\partial t}}+{\frac {\partial {\boldsymbol {P}}}{\partial t}}\ .}$

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.[2]

The second term on the right hand side, called polarization current density, comes from the change in 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 P. A changing state of polarization corresponds to charge movement and so is equivalent to a current, hence the term "polarization current".

Thus, ${\displaystyle {\boldsymbol {I}}_{\boldsymbol {D}}=\iint _{\mathcal {S}}{\boldsymbol {J}}_{\boldsymbol {D}}\cdot \operatorname {d} \!{\boldsymbol {S}}=\iint _{\mathcal {S}}{\frac {\partial {\boldsymbol {D}}}{\partial t}}\cdot \operatorname {d} \!{\boldsymbol {S}}={\frac {\partial }{\partial t}}\iint _{\mathcal {S}}{\boldsymbol {D}}\cdot \operatorname {d} \!{\boldsymbol {S}}={\frac {\partial \Phi _{D}}{\partial t}}\!}$

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 P was simply to change the relative permittivity εr in the relation D = εrε0 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:

${\displaystyle {\boldsymbol {D}}=\varepsilon {\boldsymbol {E}}\ ,}$

where the permittivity ε = ε0 εr,

In this equation the use of ε accounts for the polarization of the dielectric.

The scalar value of displacement current may also be expressed in terms of electric flux:

${\displaystyle I_{\mathrm {D} }=\varepsilon {\frac {\partial \Phi _{E}}{\partial t}}.}$

The forms in terms of ε are correct only for linear isotropic materials. More generally ε may be replaced by a tensor, may depend upon the electric field itself, and may exhibit frequency dependence (dispersion).

For a linear isotropic dielectric, the polarization P is given by:

${\displaystyle {\boldsymbol {P}}=\varepsilon _{0}\chi _{e}{\boldsymbol {E}}=\varepsilon _{0}(\varepsilon _{r}-1){\boldsymbol {E}}}$

where χe is known as the electric susceptibility of the dielectric. Note that:

${\displaystyle \varepsilon =\varepsilon _{r}\varepsilon _{0}=(1+\chi _{e})\varepsilon _{0}.}$
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