Maxwell Displacement Research Articles

Reply to J D Jackson's `Maxwell's displacement current revisited'

The points of difference between J D Jackson's `Maxwell's displacement current revisited' (1999 Eur. J. Phys. 20 495-9) and J Roche's earlier article `The present status of Maxwell's displacement current' (1998 Eur. J. Phys. 19 155-66)are carefully examined with the intention of further clarifying the issues involved.

Classical Electrodynamics without the Lorentz Condition: Extracting Energy from the Vacuum

It is shown that if the Lorentz condition is discarded, the Maxwell–Heaviside field equations become the Lehnert equations, indicating the presence of charge density and current density in the vacuum. The Lehnert equations are a subset of the O(3) Yang–Mills field equations. Charge and current density in the vacuum are defined straightforwardly in terms of the vector potential and scalar potential, and are conceptually similar to Maxwell's displacement current, which also occurs in the classical vacuum. A demonstration is made of the existence of a time dependent classical vacuum polarization which appears if the Lorentz condition is discarded. Vacuum charge and current appear phenomenologically in the Lehnert equations but fundamentally in the O(3) Yang–Mills theory of classical electrodynamics. The latter also allows for the possibility of the existence of vacuum topological magnetic charge density and topological magnetic current density. Both O(3) and Lehnert equations are superior to the Maxwell–Heaviside equations in being able to describe phenomena not amenable to the latter. In theory, devices can be made to extract the energy associated with vacuum charge and current.

DERIVATION OF THE LEHNERT FIELD EQUATIONS FROM GAUGE THEORY IN VACUUM: SPACE CHARGE AND CURRENT

It is shown that the Lehnert field equations in vacuum, with concomitant space charge and current, can be derived straightforwardly from standard gauge theory applied in vacuum, using the concept of covariant derivative and Feynman's universal influence. The Lehnert and Proca field equations are shown to be inter-related through the well-known de Broglie theorem, in which the photon mass can be interpreted as finite. These ideas go some way towards addressing the inconsistency inherent in Maxwell's famous displacement current, which has no concomitant vacuum space charge.

Maxwell's displacement current revisited

A careful analysis is made of the example of a charging capacitor to show the meaning and significance (`reality') of Maxwell's displacement current.

The present status of Maxwell's displacement current

Physics literature on the displacement current from Maxwell to the present day is reviewed with the intention of clarifying the concept. Two major traditions of interpretation are identified, one deriving from Maxwell which maintains that the displacement current is electromagnetically equivalent to an electric current and one deriving from Lorentz which denies that there is any displacement current. The article attempts to resolve various outstanding ambiguities and it concludes with an assessment of the present status of both traditions of interpretation. The appropriateness or otherwise of introducing the concept of a 'displacement current' to undergraduates is also discussed briefly.

Gravitational mass of magnetostatic field

Maxwell's displacement current equation is interpreted in the light of recent work to show that static magnetic field in free space should have a colocated and contemporaneous mass that is neither embodied in, nor can be anticipated from, the mass-energy relation. Thus magnetostatic field in the universe represents an ‘invisible’ mass. Some consequences are discussed.

Some Considerations on Maxwell's Displacement Current

Some Considerations on Maxwell's Displacement Current

Universal low-temperature ac conductivity of macroscopically disordered nonmetals.

This paper discusses a macroscopic model for ac conduction in electronically or ionically conducting disordered solids. The model considers ac conduction in an inhomogeneous solid that is characterized by a spatially randomly varying thermally activated (frequency-independent) conductivity. Discretizing Maxwell's equations leads to an equivalent electrical circuit that is a simple-cubic lattice where each pair of nodes are linked by a resistor and a capacitor in parallel. The values of the resistors are determined by the local resistivity while the capacitors are all equal, given by the infinite-frequency dielectric constant. It is shown that the capacitor currents are Maxwell's displacement currents. Assuming uncorrelated resistances, the model is solved analytically at low temperatures in the effective-medium approximation (EMA) and in a naive percolation-path approximation. Both approximations predict similar universal ac responses as T\ensuremath{\rightarrow}0, where the macroscopic frequency-dependent conductivity becomes independent of the activation-energy probability distribution. The universality represents an unusual type of regularity appearing in the extreme disorder limit. The universality prediction is tested by computer simulations of 200\ifmmode\times\else\texttimes\fi{}200 lattices in two dimensions and of 50\ifmmode\times\else\texttimes\fi{}50\ifmmode\times\else\texttimes\fi{}50 lattices in three dimensions. The computer simulations show that the EMA works very well in two dimensions in the whole temperature range studied; in particular, the low-temperature universality prediction is confirmed. In three dimensions the universality prediction is confirmed as well.

Measurement of quasistatic Maxwell's displacement current.

We have used a superconducting quantum interference device to measure all components of the small magnetic field inside a coaxial capacitor that is being charged at a low frequency (695 Hz). The expected result is that ${B}_{z}$ and ${B}_{\ensuremath{\rho}}$ should be 0 and that ${B}_{\ensuremath{\varphi}}$ should increase linearly from the end of the capacitor. This prediction depends upon Maxwell's displacement current ${\mathrm{J}}_{D}$ being strictly equal to (1/4\ensuremath{\pi})\ensuremath{\partial}D/\ensuremath{\partial}t in the annular region between the electrodes. Anomalous fields would arise if ${J}_{D}$ twisted a little in going from one electrode to the other. We find no evidence for such a twist. ${B}_{\ensuremath{\varphi}}$ is as predicted; ${B}_{\ensuremath{\rho}}$ and ${B}_{z}$=0 everywhere to within a few percent of ${B}_{\ensuremath{\varphi}}$. Our results show that the displacement current flows within \ensuremath{\Delta}\ensuremath{\varphi}=22\ifmmode^\circ\else\textdegree\fi{} of the radial direction. We have also searched for a B field that is in phase with the charge (rather than the current) and for fields which vary as the second harmonic of the charging frequency. The former search is interpreted to show that in a traveling TEM wave B is perpendicular to E to within 0.8\ifmmode\times\else\texttimes\fi{}${10}^{\mathrm{\ensuremath{-}}8}$ rad. The latter search is loosely connected to nonlinear electromagnetic effects such as those provided by axions.

Erwin Schrödinger's views on gravitational physics during his last years at the University of Vienna and some research ensuing from it

The author, who was Schrodinger's assistant during his last years in Vienna, gives an account of Schrodinger's views and activities during that time which lead him to a different approach to research on the relations between gravitation and quantum phenomena. Various features of past research are outlined in nontechnical terms. A heuristic argument is presented for the role of the zero-point energy of massive particles in counteracting gravitational collapse and the formation of horizons. Arguments are presented for the view that progress in describing extreme gravitational phenomena can be achieved by the new outlook obtained from the introduction of the analog of Maxwell's vacuum displacement term with a quasiconstant parameter, rather than from renormalization of special processes, even if this is successful. The results can be expected to be in accord with Schrodinger's conjectures.

Measuring the unmeasurable

After more than a century, the reality of Maxwell's displacement current has been demonstrated by direct measurement. So, too, but less conclusively, has been the Aharonov/Bohm effect.

Correction to "Maxwell's Displacement Current"

Correction to "Maxwell's Displacement Current"

Maxwell's Displacement Current

Maxwell's Displacement Current

A Remark on Maxwell's Displacement Current

One usually derives Ampere's law by taking the curl of the magnetic field given by the law of Biot and Savartfor a volume distribution of currents J. In the derivation, one drops a term that contains ∇ · J, in order to treat stationary processes. It is shown that if that term is kept one gets, for ∇ × H, a term which is the usual conduction current J plus a second term similar to Maxwell's displacement current, present even outside the current distribution J. That extra term is δD0/δt, where D0 = ε0E0 and E0 is the instantaneous Coulomb field produced by the charges in question.

Maxwell's displacement current and his theory of light

Every physics student knows Maxwell added a term to Ampere's law. The original law related the magnetic force to the conduction current. Maxwell's new term is the time rate of change of displacement. The resultant version of the law thus relates the magnetic force to the total current, the sum of the conduction and displacement currents.* Yet, though physicists are correctly taught that this addition was Maxwell's, they are usually misinformed about the reasons which led him to make it. Two explanations are particularly common. The first is that Maxwell changed Ampere's law because of an aesthetic desire to give the electromagnetic equations a greater symmetry. The addition of the time rate of change of displacement brings the law into closer analogy with Faraday's law, which relates the time rate of change of magnetic induction to the electric force.** The trouble with this explanation is that Maxwell's equations, as first collected together, exhibit no symmetry between Ampere's and Faraday's laws. This is because Faraday's law does not appear. Instead, there are equations for the magnetic induction and electric force separately, in terms of the vector and scalar potentials [1]. The second explanation commonly offered is that Maxwell compared Ampere's law in its original form with the equation for the continuity of charge, and realized the two were inconsistent*** [2]. Now Maxwell did compare these equations in 1856, as we shall see later, but he did not draw the conclusion that they were incompatible and that consequently Ampere's law must be altered. He did not alter Ampere's law for another five years.

Contributions to the integration of the magnetoelasticity equations

This paper considers the magnetoelasticity equations for a homogeneous and isotropic elastic medium which satisfies the condition εμ e = ε 0 μ e0 . It is assumed that Maxwell's displacement currents are neglected. The equations are written in dimensionless variables. The linearization of the system of equations leads to the obtaining of a single equation for the current density. As a practical application, Cauchy's problem is solved for the perfectly conductive infinite plane in the presence of a homogeneous magnetic field. It is shown that the wave propagation takes place without dispersion. In the final part of the paper Cauchy's problem is solved for the elastic half-plane subjected to a normal pressure on the boundary. The solution is obtained by a continuous superposition of plane waves. ]It is demonstrated that in both cases Cauchy's problem is correctly set.

An experimental problem in the atmospheric-electrical synopsis: The separation of conduction carrent and Maxwell's displacement current in measurements of the air-earth current

In order to separate the different influences causing the atmospheric-electrical variations, a synoptical method of research has been applied. The measurement of the air-earth current, needed in this method, is commonly disturbed by the variations of the atmospheric-electrical field which produce a displacement current in the “antenna.” This difficulty can be avoided by an arrangement proposed by H. W. Kasemir, which enables us to separate the displacement current and the conduction current. The “method of adaptation” is described. A few examples show that by employing this method all variations of the air-earth current can be measured correctly.

Rayleigh's Law of Extinction and the Quantum Hypothesis

THE bearing of Rayleigh's law of gaseous extinction on some of the fundamental aspects of radiation theory does not seem to have been sufficiently emphasised in recent reports and publications on modern molecular physics. The coefficient of attenuation κ of radiation of wave-length λ travelling through a gas containing n0 molecules per unit volume was given by Rayleigh1 so long ago as 1871 in the form κ=84, π 3(µ20 – I)2λ−4/n0, µ0 being the refractive index of the gas. It is of importance to notice that the law in question is one of the most fundamental results of molecular dynamics, its final expression being an invariant with respect to the theories of the æther or of the molecule employed2, while in its derivation there is no need to draw on resources outside classical dynamics and continuous energy-flow. From the point of view of elementary electromagnetic theory, the above expression κ for is very easily derived along lines suggested in a problem set in part ii. of the “Mathematical Tripos”3; use is made of the conventional electrical doublet set into forced vibrations by a train of electromagnetic waves; by making use of the radiation formula for accelerated charges and Poynting's theorem, the flow of energy from the doublet is easily calculated in terms of the amplitude of vibration; the oscillations of the doublet contribute a term to Maxwell's displacement current, enabling the amplitude to be expressed in terms of the refractive index of the gas; by considering the depletion of energy from the original beam as a result of this scattering, and eliminating the amplitude, the above expression for κ is easily obtained. In a recent paper, Natanson4 has subjected the derivation of Rayleigh's law to minute criticism on the grounds of the classical electromagnetic theory, allowing for a damping term arising from the mechanical reaction due to radiation, and taking into special consideration the summation of the aggregate radiation from the random distribution of doublets which are supposed to constitute the molecules of the gas; the final result is a vindication of the above expression for the coefficient of attenuation to a very high order of accuracy. It may be noticed in passing that the same electromagnetic system forms the basis of Planck's5 theory of black-body radiation, the interpretation of experiment in this case, however, necessitating the hypothesis of discontinuous energy-flow, or the emission of energy by quanta.