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Displacement current
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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 in connection with the displacement of electric particles in a dielectric medium.
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Encyclopedia
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 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.
Displacement current applies not only to material media, but to free space as well.
Explanation
The electric displacement field is defined as:
Differentiating this equation with respect to time defines the displacement current, which therefore has two components in a dielectric:
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.
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 er in the relation D = ere0 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.
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:
where:
- is the closed line integral around some closed curve C.
- is the magnetic field in tesla.
- is the vector dot product.
- 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).
- is the magnetic constant also called the permeability of free space.
- 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,
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:
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:
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:
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.
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):
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 e0 ?E / ?t through the surface.
Applying the Ampère-Maxwell equation to surface S1 we find:
However, applying this law to surface S2, which is bounded by exactly the same curve , but lies between the plates, provides:
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 e0 ?E / ?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.
The current leaving a volume must equal the rate of decrease of charge in a volume. In differential form this continuity equation becomes:
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:
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:
and
which is in agreement with the continuity equation because of Gauss's law:
To obtain wave propagation
The added displacement current also leads to wave propagation by taking the curl of the equation for magnetic field. In the particular situation where there is no polarization (P=0); which occurs in free space, for example; the displacement current is:
Substituting this form for J into Ampère's law, and assuming there is no bound or free current density contributing to J :
with the result:
However,
leading to the wave equation:
where use is made of the vector identity that holds for any vector field V(r, t):
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:
If J, P and ? are zero (as in free space), the result is:
It should be noted that the electric field can be expressed in the general form:
where f is the electric potential (which can be chosen to satisfy Poisson's equation) and A is a vector potential. The ?f 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, ?f does not contribute to ?×E.
Simplifications
In the case of a very simple dielectric material the constitutive relation holds:
where the permittivity e = e0 er,
In this equation the use of e, accounts for
the polarization of the dielectric.
The scalar value of displacement current may also be expressed in terms of electric flux:
The forms in terms of e are correct only for linear isotropic materials. More generally e 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:
where ?e is known as the electric susceptibility of the dielectric. Note that:
History and interpretation
Maxwell's displacement current was postulated in part III of his 1861 paper ''. Few topics in modern physics have caused as much confusion and misunderstanding as that of displacement current. 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):
He is careful to point out the treatment is one of analogy:
In part III, in relation to displacement current, he says
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:
With some change of symbols (and units): r ? J, R ? -E and the material constant E-2 ? 4p ere0 these equations take the familiar form:
-
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, Stepin, 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. 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.
The displacement current that is associated with polarization of a dielectric.
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.
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. 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.
Maxwell's papers
- Maxwell's paper of 1855
- Maxwell's paper of 1861
- Maxwell's paper of 1864
Further reading
- Maxwell, Displacement Current, and Symmetry (1963)
- Maxwell and the Electromagnetic Wave Equation (1967)
See also
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