Displacement current

From Wikipedia, the free encyclopedia

Electromagnetism
Electricity · Magnetism
Electrodynamics
 · Free space · Lorentz force law · EMF · Electromagnetic induction · Faraday’s law · Displacement current · Maxwell’s equations · EM field · Electromagnetic radiation · Liénard-Wiechert Potentials · Maxwell tensor · Eddy current ·
This box: view • talk • edit

Displacement current is a quantity related to a changing electric field. It occurs in the vacuum or other dielectric and is not a physical current in the sense that no physical charge is transported. It nevertheless has the units of electric current and it has an associated magnetic field. It appears in James Clerk Maxwell's 1861 paper entitled On Physical Lines of Force, equation (112), where he added it as an additional term to the electric current term in Ampère's Circuital Law.

Contents

[edit] Explanation

The displacement current was introduced by Maxwell as the rate of change of the electric displacement, D:

\mathbf{J}_\mathrm{D} = \frac{\partial \mathbf{D}} {\partial t}\ ,

where D is the electric displacement field that enters Maxwell's equations. The electric displacement field is defined as:

\mathbf{D} = \varepsilon_0 \mathbf{E} + \mathbf{P}

Taking the time derivative of this, we find that displacement current has two components in a dielectric:

\mathbf{J}_\mathrm{D} = \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t} + \frac{\partial \mathbf{P}}{\partial t}

The first part is present everywhere, even in a vacuum. It is believed not to involve any actual movement of charge, but to nevertheless have an associated magnetic field, as if it were an actual current. The second part is caused by the linear polarization of the individual molecules of the dielectric material. Even though charges cannot flow freely in a dielectric, their limited but elastically self restoring movements produce a polarization current.

[edit] Simplifications

In the case of a very simple dielectric material the constitutive relation holds:

\mathbf{D} = \epsilon \mathbf{E} \ ,

where the permittivity ε = ε0 εr,

  • εr is the relative permittivity of the dielectric and
  • ε0 is the electric constant.

In this equation the use of ε, accounts for the polarisation of the dielectric.

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

I_\mathrm{D} =\varepsilon \frac{d\Phi_E}{dt}

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

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

\mathbf{P} = \varepsilon_0 \chi_e \mathbf{E} = \varepsilon_0 (\varepsilon_r - 1) \mathbf{E}

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

\varepsilon = \varepsilon_r \varepsilon_0 = (1+\chi_e)\varepsilon_0

[edit] History and interpretation

Prior to Maxwell's work, it was thought that the magnetic field was generated solely by electric charge in motion. This idea is expressed mathematically with Ampère's Circuital Law.

As in the case of Kirchhoff's Current law, Ampère's Circuital Law applies only to situations in which there is no variation in charge density. This fact can be seen by considering the divergence of the differential form of Ampère's Circuital Law in conjunction with the equation of continuity of charge. The divergence of a curl is always zero and hence the rate of change of charge density must necessarily be zero for Ampère's Circuital Law to hold true.

If we substitute Gauss's law into the equation of continuity of charge in the above scenario we can see the mathematical justification for Maxwell's displacement current.

Kirchhoff used the above interrelationships when he derived his 'Telegraphy Equation' in 1857, without any explicit mention of displacement current. Maxwell on the other hand, explicitly used displacement current in his 1864 paper A Dynamical Theory of the Electromagnetic Field, in order to derive the Electromagnetic wave equation. The Electromagnetic wave equation is very closely related to the 'Telegraphy Equation'.

Maxwell's displacement current was postulated in part III of his 1861 paper 'On Physical Lines of Force'.

It appears in the preamble and then again formally at equation (111). It is the time differential of the elasticity equation. Maxwell interpreted the displacement current as a real motion of electrical particles in a sea of aethereal vortices. This interpretation has been abandoned in modern physics, although Maxwell's correction to Ampère's circuital law remains valid (a changing electric field produces a magnetic field).

Using the concept of electrical displacement, Maxwell concluded, using Newton's equation for the speed of sound (equation 132), that light consists of transverse undulations in the same medium that is the cause of electric and magnetic phenomena.

It is now believed that displacement current does not exist as a real current (movement of charge). It is defined as a quantity proportional to the time derivative of the electric field, and it is deemed to be able to exist in pure vacuum. The present day concept of displacement current therefore simply refers to the fact that a changing electric field has an associated magnetic field.

[edit] See also