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In electromagnetism, displacement current is a quantity that is defined in terms of the rate of change of electric displacement field. Displacement current has the units of electric current and it has an associated magnetic field.

The idea was conceived by Maxwell in his 1861 paper On Physical Lines of Force 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 an historical landmark in physics by virtue of uniting electricity, magnetism and optics into one single unified theory.

In more recent times, displacement current has been extended to apply to the vacuum space between the plates of a capacitor in order to maintain the solenoidal nature of Ampère's Circuital Law

Contents

Explanation

The electric displacement field is defined as:

\boldsymbol{D} = \varepsilon_0 \boldsymbol{E} + \boldsymbol{P}\ .

Differentiating this equation with respect to time defines the displacement current, which therefore has two components in a dielectric:1

\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 everywhere, even in a vacuum. It doesn't involve any actual movement of charge, but it nevertheless has an associated magnetic field, as if it were an actual current. Some authors apply the name displacement current to only this contribution.2

The second term on the right hand side is associated with the polarization of the individual molecules of the dielectric material. Even though charges cannot flow freely in a dielectric, the charges in molecules can move a little under the influence of an electric field. The positive and negative charges in molecules separate, causing an increase in the state of polarization P. This changing state of polarization is equivalent to a current.

This polarization is the displacement current as it was originally conceived by Maxwell. Maxwell had no conception of the modern concept of displacement current associated with the first term on the right hand side of the above equations. For Maxwell, D and P were equivalent.

The modern concept of displacement current is justified as explained below.

Why it is necessary

Example showing need for displacement current
See also: Capacitance#Capacitance and 'displacement current'

Some implications of the displacement current follow, which agree with experimental observation, and with the requirements of logical consistency for the theory of electromagnetism.

To obtain the correct magnetic field

An example illustrating the need for the displacement current arises in connection with capacitors that have a pure vacuum between the plates. The explanation is demonstrated by the example3 shown in the diagram at right. The diagram shows a capacitor being charged by current I flowing through a wire, which creates a magnetic field B around it. The magnetic field is found from the integral form of Ampère's law with the displacement current term added (the Ampère-Maxwell equation):

\oint_{\partial S} \boldsymbol{B} \cdot d\boldsymbol{\ell} = \mu_0 \int_S (\boldsymbol{J} + \epsilon_0 \frac {\partial \boldsymbol{E}}{\partial t}) \cdot d\boldsymbol{S} \,

This equation says that the integral of the magnetic field B around a loop ∂S is equal to the integrated current J through any surface spanning the loop, plus the current ε0 ∂E / ∂t through the surface.4 Any surface that intersects the wire, such as S1, has current I passing through it so Ampère's law law gives the correct magnetic field:

B = \frac {\mu_0 I}{2 \pi r}\,

But surface S2 bounded by the same loop but passing between the capacitor's plates has no current flowing through it, so without the displacement current term Ampère's law gives:

B = 0\,

So without the displacement current term Ampère's law fails; it gives the wrong answer for the magnetic field when applied to some surfaces. The ε0 ∂E / ∂t term provides a second source for the magnetic field besides current. Since the current is increasing the charge on the capacitor's plates, the electric field between the plates is increasing, so the rate of change of electric field through the surface S2 is positive, and its magnitude gives the correct value for the field B found above.

To obtain conservation of charge

Consider for simplicity a non-magnetic medium where the relative magnetic permeability is unity, and the complication of magnetization current is absent. 5 The the current leaving a volume must equal the rate of decrease of charge in a volume. In differential form this continuity equation becomes:

\nabla \boldsymbol{\cdot J_f} = -\frac {\partial \rho_f}{\partial t} \ ,

where the left side is the divergence of the free current density and the right side is the rate of decrease of the free charge density. However, Ampère's law in its original form states:

\boldsymbol{ \nabla \times B} = \mu_0 \boldsymbol J_f \ ,

which implies that the divergence of the current term vanishes. (Vanishing of the divergence is a result of the mathematical identity that states the divergence of a curl is always zero.) This conflict is removed by addition of the displacement current, as then:67

\boldsymbol{ \nabla \times B} = \mu_0 \left(\boldsymbol J +\varepsilon_0 \frac {\partial \boldsymbol E}{\partial t}\right) = \mu_0 \left( \boldsymbol J_f +\frac {\partial \boldsymbol D}{\partial t}\right) \ ,

and

\boldsymbol{ \nabla \cdot } \left( \boldsymbol{\nabla \times B}\right ) = 0 = \mu_0 \left( \nabla \cdot \boldsymbol J_f +\frac {\partial }{\partial t} \boldsymbol {\nabla \cdot D } \right ) \ ,

which is in agreement with the continuity equation because of Gauss's law:

\boldsymbol {\nabla \cdot D} = \rho_f \ .

To obtain wave propagation

The added displacement current also leads to wave propagation by taking the curl of the equation for magnetic field.8 In the particular situation where there is no polarization (P=0), the displacement current is:

\boldsymbol{J_D} = \epsilon_0\frac { \partial \boldsymbol{E} } { \partial t }

Substituting this form for J into Ampère's law, and assuming there is no bound or free current density contributing to J :

\boldsymbol{ \nabla \times B} = \mu_0 \boldsymbol {J_D} \ ,

with the result:

\boldsymbol{ \nabla \times}\left( \boldsymbol {\nabla \times B} \right ) = \mu_0 \epsilon_0 \frac {\partial}{\partial t} \boldsymbol {\nabla \times E} \ .

However,

\boldsymbol {\nabla \times E} = -\frac{\partial }{\partial t} \boldsymbol B \ ,

leading to the wave equation:9

-\boldsymbol{ \nabla \times}\left( \boldsymbol {\nabla \times B} \right ) = \nabla^2 \boldsymbol B =\mu_0 \epsilon_0 \frac {\partial^2}{\partial t^2} \boldsymbol {B } = \frac{1}{c^2} \frac {\partial^2}{\partial t^2} \boldsymbol {B } \ ,

where use is made of the vector identity that holds for any vector field V(r, t):

\boldsymbol{\nabla \times}\left( \boldsymbol { \nabla \times V}\right ) = \boldsymbol {\nabla}\left(\boldsymbol{\nabla \cdot V}\right ) - \nabla^2 \boldsymbol V \ ,

and the fact that the divergence of the magnetic field is zero. An identical wave equation can be found for the electric field by taking the curl:

\boldsymbol {\nabla \times } \left( \boldsymbol {\nabla \times E} \right) = -\frac {\partial}{\partial t}\boldsymbol {\nabla \times } \boldsymbol{B}=-\mu_0 \frac {\partial}{\partial t} \left( \boldsymbol J + \epsilon_0\frac {\partial}{\partial t} \boldsymbol E \right) \ .

If J, P and ρ are zero (as in free space), the result is:

\nabla^2 \boldsymbol E =\mu_0 \epsilon_0 \frac {\partial^2}{\partial t^2} \boldsymbol {E } = \frac{1}{c^2} \frac {\partial^2}{\partial t^2} \boldsymbol {E } \ .

It should be noted that in general the electric field is given by,

\boldsymbol{E} = - \boldsymbol{\nabla} \varphi - \frac { \partial \boldsymbol{A} } { \partial t } \ ,

where φ is the electric potential (which can be chosen to satisfy Poisson's equation) and A is a vector potential.10 The ∇φ component on the right hand side is the Gauss's law component, and this is the component that is relevant to the conservation of charge argument above. The second term on the right-hand side is the one relevant to the electromagnetic wave equation, because it is the term that contributes to the curl of E. Because of the vector identity that says the curl of a gradient is zero, ∇φ does not contribute to ∇ × E.

Simplifications

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

\mathbf{D} = \varepsilon \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.

History and Interpretation

Maxwell's displacement current was postulated in part III of his 1861 paper 'On Physical Lines of Force'. Few topics in modern physics have caused as much confusion and misunderstanding as that of displacement current.11 This is in part due to the fact that Maxwell himself did not appear to be altogether clear on its precise physical interpretation. Maxwell conceived of the idea of displacement current in connection with tangential stress in his sea of molecular vortices.12 This is discussed in the preamble to part III in his 1861 paper. In this part, Maxwell says,

“I conceived the rotating matter to be the substance of certain cells, divided from each other by cell-walls composed of particles which are very small compared with the cells, and that it is by the motions of these particles, and their tangential action on the substance in the cells, that the rotation is communicated from one cell to another.”

Although the above quote would tend to point towards a magnetization explanation for displacement current, Maxwell eventually settled on an interpretation of displacement current in terms of linear polarization of a dielectric. Irrespective of whether he had linear displacement or rotational displacement in mind, 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.

Where the magnetization explanation was totally compatible with electromagnetic induction for the purposes of deriving the the electromagnetic wave equation, (see his 1864 paper entitled A Dynamical Theory of the Electromagnetic Field), Maxwell's linear polarization explanation diverted attention towards the electric capacitor circuit, and it is a common myth that Maxwell conceived of displacement current within the context of the electric capacitor circuit. Once displacement current had become associated with capacitors, and once Maxwell's sea of molecular vortices had been abandoned in the 20th century, a new concept of displacement current evolved which bore no physical resemblance to Maxwell's original concept. The modern displacement current can be derived in connection with vacuum capacitors using the equation Q = CV, where Q is electric charge, C is capacitance, and V is voltage. We can therefore identify three different kinds of displacement current.

(1) The displacement current that is associated with magnetization and wireless telegraphy. In this case the electric field term \mathbf{E} will have a zero divergence and will be compatible with the time varying electric field term in Faraday's law of induction.
(2) The displacement current that is associated with linear polarization of a dielectric.
(3) The virtual displacement current that is associated with maintaining the solenoidal nature of Ampère's Circuital Law in a vacuum capacitor circuit.

The latter two, (2) and (3), are connected with cable telegraphy and involve a non-zero divergence for \mathbf{E}. Interestingly, in 1857, Kirchhoff derived the cable telegraphy equation using the interrelationships between Poisson's equation and the equation of continuity which would connect to (2) and (3) above through capacitor theory.13 Kirchhoff never used the concept of displacement current. But in doing so, he manipulated the non-zero divergent \mathbf{E} of Gauss's law with the zero divergent time varying \mathbf{E} of Faraday's law as if they were one and the same thing.

References

  1. ^ John D Jackson (1999). Classical Electrodynamics, 3rd Edition, Wiley, p. 238. ISBN 047130932X. 
  2. ^ For example, see David J Griffiths (1999). Introduction to Electrodynamics, 3rd Edition, Pearson/Addison Wesley, p. 323. ISBN 013805326X.  and Tai L Chow (2006). Introduction to Electromagnetic Theory. Jones & Bartlett, p. 204. ISBN 0763738271. 
  3. ^ from Feynman, Richard P.; Robert Leighton, Matthew Sands (1963). The Feynman Lectures on Physics, Vol. 2. Massachusetts, USA: Addison-Wesley, p.18-4. ISBN 0201021161. 
  4. ^ This formulation is in terms of the B-field, rather than the H-field, which means the current J is the total current density due both to conduction and to polarization and magnetization. See Ampère's law for more detail.
  5. ^ The restriction to a non-magnetic medium can be lifted by including the magnetization current. That adds some formal complication, but does not affect the continuity equation because the divergence of the magnetization current is zero. See magnetization current.
  6. ^ Raymond Bonnett, Shane Cloude (1995). An Introduction to Electromagnetic Wave Propagation and Antennas. Taylor & Francis, p. 16. ISBN 1857282418. 
  7. ^ JC Slater and NH Frank (1969). Electromagnetism, Reprint of 1947 edition, Courier Dover Publications, p. 84. ISBN 0486622630. 
  8. ^ JC Slater and NH Frank. Electromagnetism, op. cit., p. 91. ISBN 0486622630. 
  9. ^ J Billingham, A C King (2006). Wave Motion. Cambridge University Press, p. 182. ISBN 0521634504. 
  10. ^ There is some flexibility in choice of the scalar and vector potential called gauge freedom. In the Coulomb gauge, φ satisfies Poisson's equation. In the Lorentz gauge both satisfy an inhomogeneous wave equation.
  11. ^ Daniel M. Siegel (2003). Innovation in Maxwell's Electromagnetic Theory. Cambridge University Press, p. 85. ISBN 0521533295. 
  12. ^ The idea of vortices preceded Maxwell, having been part of the thinking of Rankine, Thomson and Helmholtz. Olivier Darrigol (2003). Electrodynamics from Ampere to Einstein. Oxford University Press, p. 148. ISBN 0198505930. , HT Pledge (2007). Science Since 1500. Read Books, p. 147. ISBN 1406768723. 
  13. ^ Daniel M. Siegel (2003). Innovation in Maxwell's Electromagnetic Theory. Cambridge University Press, p. 123. ISBN 0521533295. 

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