Displacement current

In physical materials (as opposed to vacuum), there is also a contribution from the slight motion of charges bound in atoms, called dielectric polarization.

The idea was conceived by James Clerk Maxwell in his 1861 paper On Physical Lines of Force, Part III in connection with the displacement of electric particles in a dielectric medium.

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 of electromagnetic waves.

The first term on the right hand side is present in material media and in free space.

Polarization results when, under the influence of an applied electric field, the charges in molecules have moved from a position of exact cancellation.

Maxwell made no special treatment of the vacuum, treating it as a material medium.

The forms in terms of scalar ε are correct only for linear isotropic materials.

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

An example illustrating the need for the displacement current arises in connection with capacitors with no medium between the plates.

Consider the current 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) crosses the right surface R. Notice that the electric field E between the plates increases as the capacitor charges.

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) 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):[5]

Any surface S1 that intersects the wire has current I passing through it so Ampère's law gives the correct magnetic field.

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.

⁠ is necessary as a second source term which gives the correct magnetic field when the surface of integration passes between the capacitor plates.

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

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

An identical wave equation can be found for the electric field by taking the curl:

Maxwell's displacement current was postulated in part III of his 1861 paper 'On Physical Lines of Force'.

[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 between Ampère'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.

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 ...

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 constant E−2 → 4πεrε0 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:

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.