Displacement current: Wikis

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Electromagnetism
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Electricity · Magnetism
Electrodynamics
Free space · Lorentz force law · emf · Electromagnetic induction · Faraday’s law · Lenz's law · Displacement current · Maxwell's equations · EM field · Electromagnetic radiation · Liénard-Wiechert Potential · Maxwell tensor · Eddy current

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 density, and it has an associated magnetic field just as actual currents do. However it is not an electric current of moving charges, but a time-varying electric field. In materials, there is also a contribution from the slight motion of charges bound in atoms, dielectric polarization.

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. 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.

Contents

Explanation

The electric displacement field is defined as:

 \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, 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 in material media and in free space. It doesn't necessarily involve any actual movement of charge, but it does have an associated magnetic field, just as does a current due to charge motion. 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. Polarization results when the charges in molecules move a little under the influence of an applied electric field. 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.

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.

Why displacement current is necessary

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

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To obtain the correct magnetic field

An example illustrating the need for the displacement current arises in connection with capacitors with no medium between the plates (in free space). Consider the charging capacitor in the figure. The capacitor is in a circuit that transfers charge (on a wire external to the capacitor) from the left plate to the right plate, charging the capacitor and increasing the electric field between its plates. The same current enters the right plate (say I ) as leaves the left plate. Although current is flowing through the capacitor, 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. The explanation is that a displacement current ID flows in the vacuum, and this current produces the magnetic field in the region between the plates according to Ampère's law:[3][4]

An electrically charging capacitor with an imaginary cylindrical surface surrounding the left-hand plate. Right-hand surface R lies in the space between the plates and left-hand surface L lies to the left of the left plate. No conduction current enters cylinder surface R, while current I leaves through surface L. Consistency of Ampère's law requires a displacement current ID = I to flow across surface R.
\oint_C \mathbf{B}\ \boldsymbol{ \cdot}\ \mathrm{d}\boldsymbol{\ell} = \mu_0 I_D \ .

where

  • \oint_C is the closed line integral around some closed curve C.
  • \mathbf{B} is the magnetic field in tesla.
  • \boldsymbol{ \cdot}\ is the vector dot product.
  • \mathrm{d}\boldsymbol{\ell} is an infinitesimal element (differential) of the curve C (that is. a vector with magnitude equal to the length of the infinitesimal line element, and direction given by the tangent to the curve C).
  • \mu_0 \!\ is the magnetic constant also called the permeability of free space.
  • I_D \!\ is the net displacement current that links the curve C.

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_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 flow in the imaginary cylindrical surface shown surrounding the left plate. A current, say I, passes outward through the left surface L of the cylinder, but no conduction current (no transport of real charges) enters the right surface R. Notice that the electric field between the plates E increases as the capacitor charges. That is, in a manner described by Gauss's law, assuming no dielectric between the plates:

 Q(t) =\varepsilon_0 \oint_{\mathcal S} d \mathbf{\mathcal S} \ \boldsymbol{ \cdot} \ \boldsymbol{ E} (t) \ ,

where S 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, differentiation provides:[3]

 \frac {dQ}{dt} = \mathit I =\varepsilon_0 \oint_{\mathcal S} d \mathbf{\mathcal S} \ \boldsymbol{ \cdot} \ \frac {\partial \boldsymbol {E} }{\partial t } \approx -{ S}\ \varepsilon_0 \frac {\partial E}{\partial t} \ ,

where the sign is negative because charge leaves this plate (the charge is decreasing), and where S is the area of the face R. The electric field at face L is zero because the field due to charge on the right-hand plate balances that due to the equal but opposite charge on the left-hand plate. Under the assumption of a uniform electric field distribution inside the capacitor, the displacement current density JD is found by dividing by the area of the surface:

 J_D = \frac{I_D}{ S}= -\frac{I}{ S}= \varepsilon_0 \frac {\partial E}{\partial t} = \frac {\partial D}{\partial t} \ ,

where I is the current leaving the cylindrical surface (which must equal −ID as the two currents sum to zero) and JD is the flow of charge per unit area into the cylindrical surface through the face R.

Example showing two surfaces S1 and S2 that share the same bounding contour ∂S. However, S1 is pierced by conduction current, while S2 is pierced by displacement current.

Combining these results, the magnetic field is found using the integral form of Ampère'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} \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 displacement current term ε0E / ∂t through the surface.[6] Applying the Ampère-Maxwell equation to surface S1 we find:

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

However, applying this law to surface S2, which is bounded by exactly the same curve \partial S , but lies between the plates, provides:

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

Any surface that intersects the wire has current I passing through it so Ampère's law gives the correct magnetic field. Also, any surface bounded by the same loop but passing between the capacitor's plates has no charge transport flowing through it, but the ε0E / ∂t term provides a second source for the magnetic field besides charge conduction current. 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 field B found above.

To obtain consistency between Ampère's law and current continuity

In a more mathematical vein, the same results can be obtained from the underlying differential equations. Consider for simplicity a non-magnetic medium where the relative magnetic permeability is unity, and the complication of magnetization current is absent. [7] 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, contradicting the continuity equation. (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:[8][9]

 \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.[10] In the particular situation where there is no polarization (P=0); which occurs in free space, for example; the displacement current is:[11]

 \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:[12]

-\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 the electric field can be expressed in the general form:

 \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.[13] 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:

 \boldsymbol{D} = \varepsilon \boldsymbol{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 polarization of the dielectric.

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

 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 time dependence (dispersion).

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

\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:

\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.[14] 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 between Ampère's 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 concluded, using Newton's equation for the speed of sound (Lines of Force, Part III, equation (132)), that “light consists of transverse undulations in the same medium that is the cause of electric and magnetic phenomena.”

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 if h is the displacement, R the electromotive force, and E a coefficient depending on the nature of the dielectric:
R = -4\pi \mathrm E^2 h \ ;

and if r 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.—Part III – The theory of molecular vortices applied to statical electricity , pp. 14–15

With some change of symbols (and units): r → J, R → −E and the material constant E−24π εrε0 these equations take the familiar form:

J = \frac{d}{dt} \frac {1}{4 \pi \mathrm E^2} \mathit E = \frac{d}{dt} \varepsilon_r\varepsilon_0 \mathit E = \frac{d}{dt} \mathit 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. (It should be noted that 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. See Nahin[15], Stepin[16], and other historical references in the reference list.)

So it was that displacement current became associated with capacitors. Once Maxwell's sea of molecular vortices had been abandoned in the 20th century (along with the aether), an interpretation of displacement current evolved that treated free space explicitly, allowing a separation of free space from material media, unlike Maxwell's original concept.[17] The modern displacement current can be derived in connection with ideal capacitors in free space by relating the magnetic field to current using the equation I = C ∂V/∂t, where the charging current is I = ∂Q/∂t, 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 polarization of a dielectric.
  2. The displacement current that is associated with magnetization and wireless telegraphy (that is, with electromagnetic waves). In this case the electric field term E will have a zero divergence and will be compatible with the time varying electric field term in Faraday's law of induction.
  3. The virtual displacement current that is associated with maintaining a magnetic field in a charging or discharging ideal capacitor in free space despite the solenoidal nature of Ampère's Circuital Law.

(1) and (3), are connected with cable telegraphy and involve a non-zero divergence for 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 (1) and (3) above through capacitor theory.[18] Kirchhoff never used the concept of displacement current. Instead, he manipulated the non-zero divergent E of Gauss's law with the zero-divergent, time-varying E of Faraday's law as if they were one and the same thing.

References

  1. ^ John D Jackson (1999). Classical Electrodynamics (3rd Edition ed.). Wiley. p. 238. ISBN 047130932X.  
  2. ^ For example, see David J Griffiths (1999). Introduction to Electrodynamics (3rd Edition ed.). Pearson/Addison Wesley. p. 323. ISBN 013805326X.   and Tai L Chow (2006). Introduction to Electromagnetic Theory. Jones & Bartlett. p. 204. ISBN 0763738271. http://books.google.com/books?id=dpnpMhw1zo8C&pg=PA153&dq=isbn=0763738271#PPA204,M1.  
  3. ^ a b Stuart B. Palmer, Mircea S. Rogalski (1996). Advanced University Physics. Taylor & Francis. p. 214. ISBN 2884490655. http://books.google.com/books?id=TF6Igz5lJLgC&pg=PP1&dq=Physics+inauthor:%22Stuart+B+Palmer%22&lr=&as_brr=0&as_pt=ALLTYPES#PPA214,M1.  
  4. ^ Raymond A. Serway, John W. Jewett (2006). Principles of Physics. Thomson Brooks/Cole. p. 807. ISBN 053449143X. http://books.google.com/books?id=1DZz341Pp50C&pg=PA807.  
  5. ^ from Feynman, Richard P.; Robert Leighton, Matthew Sands (1963). The Feynman Lectures on Physics, Vol. 2. Massachusetts, USA: Addison-Wesley. pp. 18–4. ISBN 0201021161.  
  6. ^ 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.
  7. ^ 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.
  8. ^ Raymond Bonnett, Shane Cloude (1995). An Introduction to Electromagnetic Wave Propagation and Antennas. Taylor & Francis. p. 16. ISBN 1857282418. http://books.google.com/books?id=gME9zlyG304C&pg=PA16&dq=wave+%22displacement+current%22&lr=&as_brr=0#PPA16,M1.  
  9. ^ JC Slater and NH Frank (1969). Electromagnetism (Reprint of 1947 edition ed.). Courier Dover Publications. p. 84. ISBN 0486622630. http://books.google.com/books?id=GYsphnFwUuUC&pg=PA83&dq=displacement+%22ampere%27s+law%22&lr=&as_brr=0#PPA84,M1.  
  10. ^ JC Slater and NH Frank. Electromagnetism (op. cit. ed.). p. 91. ISBN 0486622630. http://books.google.com/books?id=GYsphnFwUuUC&pg=PA83&dq=displacement+%22ampere%27s+law%22&lr=&as_brr=0#PPA91,M1.  
  11. ^ Wave propagation occurs in materials as well as in free space; the intention here is just to keep things simple.
  12. ^ J Billingham, A C King (2006). Wave Motion. Cambridge University Press. p. 182. ISBN 0521634504. http://books.google.com/books?id=bNePaHM20LQC&pg=PA179&dq=wave+%22displacement+current%22&lr=&as_brr=0#PPA181,M1.  
  13. ^ 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.
  14. ^ Daniel M. Siegel (2003). Innovation in Maxwell's Electromagnetic Theory. Cambridge University Press. p. 85. ISBN 0521533295. http://books.google.com/books?id=AbQq85U8K0gC&pg=PA123&dq=Kirchhoff+displacement+current&lr=&as_brr=0#PPA85,M1.  
  15. ^ Paul J. Nahin (2002). Oliver Heaviside: The Life, Work, and Times of an Electrical Genius of the Victorian Age. Johns Hopkins University Press. p. 109. ISBN 0801869099. http://books.google.com/books?id=e9wEntQmA0IC&pg=PA109&dq=history+Maxwell+symmetry+of+field+equations&lr=&as_brr=0&as_pt=ALLTYPES.  
  16. ^ Vyacheslav Stepin (2002). Theoretical Knowledge. Springer. p. 202. ISBN 1402030452. http://books.google.com/books?id=4LEns8rzBOEC&pg=PA202&dq=history+Maxwell+symmetry+of+field+equations&lr=&as_brr=0&as_pt=ALLTYPES.  
  17. ^ The final disengagement of "vacuum" from real media occurred with the international agreement to use the material-unrelated terms electric constant and magnetic constant to replace the seemingly material-related terms permittivity of vacuum and permeability of vacuum. These constants have defined (not measured) values that refer to free space, which is viewed as an unattainable idealization; not as a real, observable medium, not equivalent even to a quantum vacuum.
  18. ^ Daniel M. Siegel (2003). Innovation in Maxwell's Electromagnetic Theory. Cambridge University Press. p. 123. ISBN 0521533295. http://books.google.com/books?id=AbQq85U8K0gC&pg=PA123&dq=Kirchhoff+displacement+current&lr=&as_brr=0#PPA123,M1.  

Maxwell's papers

Further reading

  • AM Bork Maxwell, Displacement Current, and Symmetry (1963)
  • AM Bork Maxwell and the Electromagnetic Wave Equation (1967)

See also


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