Displacement current is a quantity related to a changing electric field. It is not a real current (movement of charge) in a vacuum, but it has the units of current, as movement of charge does, and has an associated magnetic field. It was postulated in 1865 by James Clerk Maxwell when formulating what are today known as Maxwell's equations.

Explanation

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The displacement current density is mathematically defined by the rate of change of the electric displacement field, D:

$\backslash mathbf\{J\}\_\backslash mathrm\{D\}\; =\; \backslash frac\{\backslash partial\; \backslash mathbf\{D\}\}\{\backslash partial\; t\}\; =\backslash varepsilon\; \backslash frac\{\backslash partial\; \backslash mathbf\{E\}\}\{\backslash partial\; t\}$

(since D = εE) and where the permittivity ε = ε₀ ε_r,

• ε_r is the relative permittivity of the dielectric and
• ε₀ is the permittivity of free space ( 8.854 E-12 Fm^-1)

In this equation the use of ε, rather than ε₀, accounts for the polarisation of the dielectric. The scalar value of displacement current may also be expressed in terms of electric flux:

$I\_\backslash mathrm\{D\}\; =\backslash varepsilon\; \backslash frac\{d\backslash Phi\_E\}\{dt\}$

The forms in terms of $\backslash varepsilon$ are only correct for linear isotropic materials, unless you consider $\backslash varepsilon$ to be a tensor, in which case they are valid for all linear materials.

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

$\backslash mathbf\{P\}\; =\; \backslash varepsilon\_0\; \backslash chi\_e\; \backslash mathbf\{E\}\; =\; \backslash varepsilon\_0\; (\backslash varepsilon\_r\; -\; 1)\; \backslash mathbf\{E\}$

where $\backslash chi\_e$ is known as the electric susceptibility of the dielectric. Note that:

$\backslash varepsilon\; =\; \backslash varepsilon\_r\; \backslash varepsilon\_0\; =\; (1+\backslash chi\_e)\backslash varepsilon\_0$

The electric displacement field is defined as:

$\backslash mathbf\{D\}\; =\; \backslash varepsilon\_0\; \backslash mathbf\{E\}\; +\; \backslash mathbf\{P\}$

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

$\backslash mathbf\{J\}\_\backslash mathrm\{D\}\; =\; \backslash varepsilon\_0\; \backslash frac\{\backslash partial\; \backslash mathbf\{E\}\}\{\backslash partial\; t\}\; +\; \backslash frac\{\backslash partial\; \backslash mathbf\{P\}\}\{\backslash partial\; t\}$

The first part is present everywhere, even in a vacuum; it does not involve any actual movement of charge, but still has an associated magnetic field, as if it were an actual current. The second part is caused by changing polarization of individual molecules. Even though charges are not free to flow in a dielectric, the changing polarization causes small movements of charge which produce a polarization current.

Mathematical necessity

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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 Ampere's Law. It was also thought (and still is) that electric charge cannot be created or destroyed. This principle is expressed mathematically with the continuity equation. Taken together, these two equations give the absurd result that the amount of electric charge at any particular place never changes. However, with the aid of Gauss's Law, it is straightforward to show that if Ampere's Law is modified so that both electric current and displacement current generate the magnetic field, this problem is resolved. For a detailed mathematical treatment, see The origin of the electromagnetic wave equation.

Interpretation

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Maxwell interpreted the displacement current as a real motion of charges, even in a vacuum, where he supposed that it corresponded to motion of dipole charges in the ether. Although this interpretation has been abandoned, Maxwell's correction to Ampere's law remains valid (a changing electric field produces a magnetic field).

With the addition of the displacement current, Maxwell concluded that light was a form of electromagnetism (see Electromagnetic wave equation and Electromagnetic waves).

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

See also

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• Electric displacement field
• Maxwell's equations
• Electromagnetic wave equation
• Permittivity

Electricity | Electrodynamics | Electromagnetism

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