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Displacement current

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Physical quantity in electromagnetism

This article is about electric displacement current and is not to be confused withmagnetic displacement current.

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

The idea was conceived byJames Clerk Maxwell in his 1861 paperOn Physical Lines of Force, Part III in connection with the displacement of electric particles in adielectric medium. Maxwell added displacement current to theelectric current term inAmpère's circuital law. In his 1865 paperA Dynamical Theory of the Electromagnetic Field Maxwell used this amended version of Ampère's circuital law to derive theelectromagnetic 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 ofelectromagnetic waves.

Explanation

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Theelectric displacement fieldD is defined as:

$\mathbf {D} =\varepsilon _{0}\mathbf {E} +\mathbf {P} \ ,$

where:

ε₀ is thepermittivity of free space;
E is theelectric field intensity; and
P is thepolarization of the medium.

Differentiating this equation with respect to time defines thedisplacement current density (J_D), which therefore has two components in adielectric^[1] (see also "current density#Displacement current"):

$\mathbf {J} _{\mathrm {D} }=\varepsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}+{\frac {\partial \mathbf {P} }{\partial t}}\,.$

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 namedisplacement current to the first term by itself.^[2]

The second term on the right hand side, called polarization current density, comes from the change inpolarization of the individual molecules of the dielectric material. Polarization results when, under the influence of an appliedelectric 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 polarizationP. A changing state of polarization corresponds to charge movement and so is equivalent to a current, hence the term "polarization current". Thus,

$I_{\mathrm {D} }=\iint _{S}\mathbf {J} _{\mathrm {D} }\cdot \operatorname {d} \!\mathbf {S} =\iint _{S}{\frac {\partial \mathbf {D} }{\partial t}}\cdot \operatorname {d} \!\mathbf {S} ={\frac {\partial }{\partial t}}\iint _{S}\mathbf {D} \cdot \operatorname {d} \!\mathbf {S} ={\frac {\partial \Phi _{\mathrm {D} }}{\partial t}}\,.$

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 ofP was simply to change therelative permittivityε_r in the relationD =ε₀ε_rE.

The modern justification of displacement current is explained below.

Isotropic dielectric case

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

$\mathbf {D} =\varepsilon \,\mathbf {E} ~,$

where thepermittivity $\varepsilon =\varepsilon _{0}\,\varepsilon _{\mathrm {r} }$ is the product of:

ε₀, thepermittivity of free space, or theelectric constant; and
ε_r, therelative permittivity of the dielectric.

In the equation above, the use ofε accounts forthe polarization (if any) of the dielectric material.

Thescalar value of displacement current may also be expressed in terms ofelectric flux:

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

The forms in terms ofscalarε are correct only for linearisotropic materials. For linear non-isotropic materials,ε becomes amatrix; even more generally,ε may be replaced by atensor, which may depend upon the electric field itself, or may exhibit frequency dependence (hencedispersion).

For a linear isotropic dielectric, the polarizationP is given by:

$\mathbf {P} =\varepsilon _{0}\chi _{\mathrm {e} }\,\mathbf {E} =\varepsilon _{0}(\varepsilon _{\mathrm {r} }-1)\,\mathbf {E} ~,$

whereχ_e is known as thesusceptibility of the dielectric to electric fields. Note that

$\varepsilon =\varepsilon _{\mathrm {r} }\,\varepsilon _{0}=\left(1+\chi _{\mathrm {e} }\right)\,\varepsilon _{0}~.$

Necessity

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

Generalizing Ampère's circuital law

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Current in capacitors

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An example illustrating the need for the displacement current arises in connection withcapacitors with no medium between the plates. Consider the charging capacitor in the figure.

An electrically charging capacitor with an imaginary cylindrical surface surrounding the left-hand plate. Right-hand surfaceR lies in the space between the plates and left-hand surfaceL lies to the left of the left plate. No conduction current enters cylinder surfaceR, while currentI leaves through surfaceL. Consistency of Ampère's law requires a displacement currentI_D =I to flow across surfaceR.

The capacitor is in a circuit that causes equal and opposite charges to appear on the left plate and the right plate, charging the capacitor and increasing the electric field between its plates. No actual charge is transported through the vacuum between its plates. Nonetheless, a magnetic field exists between the plates as though a current were present there as well. One explanation is that adisplacement currentI_D "flows" in the vacuum, and this current produces the magnetic field in the region between the plates according toAmpère's law:^[3]^[4]

$\oint _{C}\mathbf {B} \cdot \operatorname {d} \!{\boldsymbol {\ell }}=\mu _{0}I_{\mathrm {D} }~,$

where

$\oint _{C}$ is the closedline integral around some closed curveC;
$\mathbf {B}$ is themagnetic field measured inteslas;
$\operatorname {\cdot } ~$ is the vectordot product;
$\mathrm {d} {\boldsymbol {\ell }}$ is aninfinitesimal vector line element along the curveC, that is, a vector with magnitude equal to the length element ofC, and direction given by the tangent to the curveC;
$\mu _{0}\,$ is themagnetic constant, also called the permeability of free space; and
$I_{\mathrm {D} }\,$ is the net displacement current that passes through a small surface bounded by the curveC.

The magnetic field between the plates is the same as that outside the plates, so the displacement current must be the same as the conduction current in the wires, that is,

$I_{\mathrm {D} }=I\,,$

which extends the notion of current beyond a mere transport of charge.

Next, this displacement current is related to the charging of the capacitor. Consider the current in the imaginary cylindrical surface shown surrounding the left plate. A current, sayI, passes outward through the left surfaceL of the cylinder, but no conduction current (no transport of real charges) crosses the right surfaceR. Notice that the electric fieldE between the plates increases as the capacitor charges. That is, in a manner described byGauss's law, assuming no dielectric between the plates:

$Q(t)=\varepsilon _{0}\oint _{S}\mathbf {E} (t)\cdot \operatorname {d} \!\mathbf {S} \,,$

whereS refers to the imaginary cylindrical surface. Assuming a parallel plate capacitor with uniform electric field, and neglecting fringing effects around the edges of the plates, according tocharge conservation equation

$I=-{\frac {\mathrm {d} Q}{\mathrm {d} t}}=-\varepsilon _{0}\oint _{S}{\frac {\partial \mathbf {E} }{\partial t}}\cdot \operatorname {d} \!\mathbf {S} =S\,\varepsilon _{0}{\Biggl .}{\frac {\partial \mathbf {E} }{\partial t}}{\Biggr |}_{R}~,$

where the first term has a negative sign because charge leaves surfaceL (the charge is decreasing), the last term has a positive sign because unit vector of surfaceR is from left to right while the direction of electric field is from right to left,S is the area of the surfaceR. The electric field at surfaceL is zero because surfaceL is in the outside of the capacitor. Under the assumption of a uniform electric field distribution inside the capacitor, the displacement current densityJ_D is found by dividing by the area of the surface:

$\mathbf {J} _{\mathrm {D} }={\frac {\mathbf {I} _{\mathrm {D} }}{S}}={\frac {\mathbf {I} }{S}}=\varepsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}={\frac {\partial \mathbf {D} }{\partial t}}~,$

whereI is the current leaving the cylindrical surface (which must equalI_D) andJ_D is the flow of charge per unit area into the cylindrical surface through the faceR.

Combining these results, the magnetic field is found using the integral form ofAmpère's law with an arbitrary choice of contour provided the displacement current density term is added to the conduction current density (the Ampère-Maxwell equation):^[5]

$\oint _{\partial S}\mathbf {B} \cdot \operatorname {d} \!{\boldsymbol {\ell }}=\mu _{0}\int _{S}\left(\mathbf {J} +\epsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}\right)\cdot \operatorname {d} \!\mathbf {S} \,.$

This equation says that the integral of the magnetic fieldB around the edge⁠ $\partial S$ ⁠ of a surfaceS is equal to the integrated currentJ through any surface with the same edge, plus the displacement current term⁠ $\varepsilon _{0}\partial \mathbf {E} /\partial t$ ⁠ through whichever surface.

Example showing two surfacesS₁ andS₂ that share the same bounding contour∂S. However,S₁ is pierced by conduction current, whileS₂ is pierced by displacement current. SurfaceS₂ is closed under the capacitor plate.

As depicted in the figure to the right, the current crossing surfaceS₁ is entirely conduction current. Applying the Ampère-Maxwell equation to surfaceS₁ yields:

$B={\frac {\mu _{0}I}{2\pi r}}~.$

However, the current crossing surfaceS₂ is entirely displacement current. Applying this law to surfaceS₂, which is bounded by exactly the same curve⁠ $\partial S$ ⁠, but lies between the plates, produces:

$B={\frac {\mu _{0}I_{\mathrm {D} }}{2\pi r}}~.$

Any surfaceS₁ that intersects the wire has currentI passing through it soAmpère's law gives the correct magnetic field. However a second surfaceS₂ bounded by the same edge⁠ $\partial S$ ⁠ could be drawn passing between the capacitor plates, therefore having no current passing through it. Without the displacement current term Ampere's law would give zero magnetic field for this surface. Therefore, without the displacement current term Ampere's law gives inconsistent results, the magnetic field would depend on the surface chosen for integration. Thus the displacement current term⁠ $\varepsilon _{0}\partial \mathbf {E} /\partial t$ ⁠ is necessary as a second source term which gives the correct magnetic field when the surface of integration passes between the capacitor plates. Because the current is increasing the charge on the capacitor's plates, the electric field between the plates is increasing, and the rate of change of electric field gives the correct value for the fieldB found above.

Mathematical formulation

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In a more mathematical vein, the same results can be obtained from the underlying differential equations. Consider for simplicity a non-magnetic medium where therelative magnetic permeability is unity, and the complication ofmagnetization current (bound current) is absent, so that $\mathbf {M} =0$ and $\mathbf {J} =\mathbf {J} _{\mathrm {f} }$ .The current leaving a volume must equal the rate of decrease of charge in a volume. In differential form thiscontinuity equation becomes:

$\nabla \cdot \mathbf {J} _{\mathrm {f} }=-{\frac {\partial \rho _{\mathrm {f} }}{\partial t}}\,,$

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:

$\nabla \times \mathbf {B} =\mu _{0}\mathbf {J} _{\mathrm {f} }\,,$

which implies that the divergence of the current term vanishes, contradicting the continuity equation. (Vanishing of thedivergence is a result of themathematical identity that states the divergence of acurl is always zero.) This conflict is removed by addition of the displacement current, as then:^[6]^[7]

$\nabla \times \mathbf {B} =\mu _{0}\left(\mathbf {J} +\varepsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}\right)=\mu _{0}\left(\mathbf {J} _{\mathrm {f} }+{\frac {\partial \mathbf {D} }{\partial t}}\right)\,,$

and

$\nabla \cdot \left(\nabla \times \mathbf {B} \right)=0=\mu _{0}\left(\nabla \cdot \mathbf {J} _{\mathrm {f} }+{\frac {\partial }{\partial t}}\nabla \cdot \mathbf {D} \right)\,,$

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

$\nabla \cdot \mathbf {D} =\rho _{\mathrm {f} }\,.$

Wave propagation

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The added displacement current also leads to wave propagation by taking the curl of the equation for magnetic field.^[8]

$\mathbf {J} _{\mathrm {D} }=\epsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}\,.$

Substituting this form forJ intoAmpère's law, and assuming there is no bound or free current density contributing toJ:

$\nabla \times \mathbf {B} =\mu _{0}\mathbf {J} _{\mathrm {D} }\,,$

with the result:

$\nabla \times \left(\nabla \times \mathbf {B} \right)=\mu _{0}\epsilon _{0}{\frac {\partial }{\partial t}}\nabla \times \mathbf {E} \,.$

However, $\nabla \times \mathbf {E} =-{\frac {\partial }{\partial t}}\mathbf {B} \,,$

leading to thewave equation:^[9] $-\nabla \times \left(\nabla \times \mathbf {B} \right)=\nabla ^{2}\mathbf {B} =\mu _{0}\epsilon _{0}{\frac {\partial ^{2}}{\partial t^{2}}}\mathbf {B} ={\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}\mathbf {B} \,,$

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

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

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 thecurl:

$\nabla \times \left(\nabla \times \mathbf {E} \right)=-{\frac {\partial }{\partial t}}\nabla \times \mathbf {B} =-\mu _{0}{\frac {\partial }{\partial t}}\left(\mathbf {J} +\epsilon _{0}{\frac {\partial }{\partial t}}\mathbf {E} \right)\,.$

IfJ,P, andρ are zero, the result is:

$\nabla ^{2}\mathbf {E} =\mu _{0}\epsilon _{0}{\frac {\partial ^{2}}{\partial t^{2}}}\mathbf {E} ={\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}\mathbf {E} \,.$

The electric field can be expressed in the general form:

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

whereφ is theelectric potential (which can be chosen to satisfyPoisson's equation) andA is avector potential (i.e.magnetic vector potential, not to be confused with surface area, asA is denoted elsewhere). 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 ofE. Because of the vector identity that says the curl of a gradient is zero,∇φ does not contribute to∇×E.

History and interpretation

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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.^[10] This is in part due to the fact that Maxwell used a sea of molecular vortices in his derivation, while modern textbooks operate on the basis that displacement current can exist in free space. Maxwell's derivation is unrelated to the modern day derivation for displacement current in the vacuum, which is based on consistency betweenAmpère's circuital law for the magnetic field and the continuity equation for electric charge.

Maxwell's purpose is stated by him at (Part I, p. 161):

I propose now to examine magnetic phenomena from a mechanical point of view, and to determine what tensions in, or motions of, a medium are capable of producing the mechanical phenomena observed.

He is careful to point out the treatment is one of analogy:

The author of this method of representation does not attempt to explain the origin of the observed forces by the effects due to these strains in the elastic solid, but makes use of the mathematical analogies of the two problems to assist the imagination in the study of both.

In part III, in relation to displacement current, he 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.

Clearly Maxwell was driving at magnetization even though the same introduction clearly talks about dielectric polarization.

Maxwell compared the speed of electricity measured byWilhelm Eduard Weber andRudolf Kohlrausch (193,088 miles/second) and the speed of light determined by theFizeau experiment (195,647 miles/second). Based on their same speed, he concluded that "light consists of transverse undulations in the same medium that is the cause of electric and magnetic phenomena."^[11]

But although the above quotations point towards a magnetic explanation for displacement current, for example, based upon the divergence of the above curl equation, Maxwell's explanation ultimately stressed linear polarization of dielectrics:

This displacement ... is the commencement of a current ... The amount of displacement depends on the nature of the body, and on the electromotive force so that ifh is the displacement,R the electromotive force, andE a coefficient depending on the nature of the dielectric:
$R=-4\pi \mathrm {E} ^{2}h\,;$ and ifr is the value of the electric current due to displacement $r={\frac {dh}{dt}}\,,$ These relations are independent of any theory about the mechanism of dielectrics; but when we find electromotive force producing electric displacement in a dielectric, and when we find the dielectric recovering from its state of electric displacement ... we cannot help regarding the phenomena as those of an elastic body, yielding to a pressure and recovering its form when the pressure is removed.

— On Physical Lines of Force, Part III,The theory of molecular vortices applied to statical electricity, pp. 14–15

With some change of symbols (and units) combined with the results deduced in the section§ Current in capacitors (r →J,R → −E, and the material constantE⁻² → 4πε_rε₀ these equations take the familiar form between a parallel plate capacitor with uniform electric field, and neglecting fringing effects around the edges of the plates:

$J={\frac {d}{dt}}{\frac {1}{4\pi \mathrm {E} ^{2}}}E={\frac {d}{dt}}\varepsilon _{r}\varepsilon _{0}E={\frac {d}{dt}}D\,.$

When it came to deriving the electromagnetic wave equation from displacement current in his 1865 paper 'A Dynamical Theory of the Electromagnetic Field', he got around the problem of the non-zero divergence associated with Gauss's law and dielectric displacement by eliminating the Gauss term and deriving the wave equation exclusively for the solenoidal magnetic field vector.

Maxwell's emphasis on polarization diverted attention towards the electric capacitor circuit, and led to the common belief that Maxwell conceived of displacement current so as to maintain conservation of charge in an electric capacitor circuit. There are a variety of debatable notions about Maxwell's thinking, ranging from his supposed desire to perfect the symmetry of the field equations to the desire to achieve compatibility with the continuity equation.^[12]^[13]