Potentials of a uniformly moving point charge in the Coulomb gauge
V. Hnizdo
备注：
To appear in European Journal of Physics. A delta-function identity for a second-order partial derivative of the Coulomb potential of a point charge is now used for the regularization of the rhs of the Poisson equation for the difference between the vector potentials in the Coulomb and Lorenz gauges
The Coulomb-gauge vector potential of a uniformly moving point charge is obtained by calculating the gauge function for the transformation between the Lorenz and Coulomb gauges. The expression obtained for the difference between the vector potentials in the two gauges is shown to satisfy a Poisson equation to which the inhomogeneous wave equation for this quantity can be reduced. The right-hand side of the Poisson equation involves an important but easily overlooked delta-function term that arises from a second-order partial derivative of the Coulomb potential of a point charge.
We provide exact expressions for the electrostatic energy of uniformly-charged prolate and oblate spheroidal shells. We find that uniformly-charged prolate spheroids of eccentricity greater than 0.9 have lower Coulomb energy than a sphere of the same area. For the volume-constrained case, we find that a sphere has the highest Coulomb energy among all spheroidal shells. Further, we derive the change in the Coulomb energy of a uniformly-charged shell due to small, area-conserving perturbations on the spherical shape. Our perturbation calculations show that buckling-type deformations on a sphere …
The formula for the frequency shift of a plane-polarized light wave reflected from a uniformly moving mirror is derived directly from the constant light speed postulate and the basic principles of wave optics. Unlike the original derivation by Einstein, our derivation does not involve Lorentz transformations. As such, it lies within the scope of a first course in special relativity.
We developed a formula for the law of reflection of a plane-polarized light beam from an inclined flat mirror in uniform rectilinear motion by a direct application of the Huygens-Fresnel principle. Applying the obtained formula and the postulates of special relativity to describe a specific thought experiment, we showed that the moving mirror must be contracted along the direction of its motion by the usual Lorentz factor. The result emphasizes the reality of Lorentz contraction by showing that the contraction is a direct consequence of the first and second postulates of special relativity, an…
The purpose of this paper is to investigate the fundamental problem of the non-uniform subsonic motion of a point force and line forces in an unbounded, homogeneous, isotropic medium in analogy to the electromagnetic Liénard-Wiechert potentials. The exact closed-form solutions of the displacement and elastic fields produced by the point force and line forces are calculated. The displacement fields can be identified with the elastodynamic Liénard-Wiechert tensor potentials. For a non-uniformly moving point force, we decompose the elastic fields into a radiation part and a non-radiation part. We…
The second-order partial derivatives of the Coulomb potential of a point charge can be regularized using the Coulomb potential of a charge of the oblate spheroidal shape that a moving rest-frame-spherical charge acquires by the Lorentz contraction. This `physical' regularization is shown to be fully equivalent to the standard delta-function identity involving these derivatives.
We consider non-stationary free and forced transverse oscillation of an infinite taut string on the Winkler foundation subjected to a discrete mass-spring system non-uniformly moving at a given sub-critical speed. The speed of the mass-spring system is assumed to be a slowly time-varying function. To describe a non-vanishing free oscillation we use an analytic approach based on the method of stationary phase and the method of multiple scales. The moving oscillator is characterized by a partial frequency, which can be greater or less than the cut-off frequency. Accordingly, a sub-critical unifo…
The refraction of a light ray by a homogeneous, isotropic and non-dispersive transparent material half-space in uniform rectilinear motion is investigated theoretically. The approach is an amalgamation of the original Fermat's principle and the fact that an isotropic optical medium at rest becomes optically anisotropic in a frame where the medium is moving at a constant velocity. Two cases of motion are considered: a) the material half-space is moving parallel to the interface; b) the material half-space is moving perpendicular to the interface. In each case, a detailed analysis of the obtaine…
We investigate the coupling of the electromagnetic sources (charge and current densities) to the scalar and vector potentials in classical electrodynamics, using Green function techniques. As is well known, the scalar potential shows an action-at-a-distance behavior in Coulomb gauge. The conundrum generated by the instantaneous interaction has intrigued physicists for a long time. Starting from the differential equations that couple the sources to the potentials, we here show in a concise derivation, using the retarded Green function, how the instantaneous interaction cancels in the calculatio…
In 19th century Maxwell derived Maxwell equations from the knowledge of three experimental physical laws: the Coulomb's law, the Ampere's force law and Faraday's law of induction. However, theoretical basis for Ampere's force law and Faraday's law remains unknown to this day. Furthermore, the Lorentz force is considered as experimental phenomena, the theoretical foundation of this force is still unknown. To answer these fundamental theoretical questions, we derive Lienard Wiechert potentials, Maxwell's equations and Lorentz force from two simple postulates: (a) when all charges are at rest the…
In this work, Stroh's formalism is endowed with causal properties on the basis of an analysis of the radiation condition in the Green tensor of the elastodynamic wave equation. The modified formalism is applied to dislocations moving uniformly in an anisotropic medium. In practice, accounting for causality amounts to a simple analytic continuation procedure whereby to the dislocation velocity is added an infinitesimal positive imaginary part. This device allows for a straightforward computation of velocity-dependent field expressions that are valid whatever the dislocation velocity ---includin…
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This work introduces original explicit solutions for the elastic fields radiated by non-uniformly moving, straight, screw or edge dislocations in an isotropic medium, in the form of time-integral representations in which acceleration-dependent contributions are explicitly separated out. These solutions are obtained by applying an isotropic regularization procedure to distributional expressions of the elastodynamic fields built on the Green tensor of the Navier equation. The obtained regularized field expressions are singularity-free, and depend on the dislocation density rather than on the pla…
We present an elementary analysis of the effects on light reflected from a uniformly moving mirror by using the photon picture of light and the conservation laws for energy and momentum of the system photon-mirror. Such a dynamical approach to the problem seems very suitable for introductory physics courses, not requiring any previous knowledge in wave optics or special relativity.
We extend the method of Huygens' construction in a uniformly moving optical medium [Am. J. Phys. 72, 934 (2004)] to the case when the medium is dispersive in its rest frame of reference. The first-order Huygens' construction analysis of the light drag in a transversely moving dispersive slab is in agreement with the results of the experiment by Jones (1975) and with the Player-Rogers formula for the downstream deflection of the beam. The derivation purports that the original Huygens' principle remains valid in non-stationary situations if it is modified to include the relativistic effects on t…
The electric field of a uniformly accelerated charge shows a plane of discontinuity, where the field extending only on one side of the plane, terminates abruptly on the plane with a finite value. This indicates a non-zero divergence of the electric field in a source-free region, implying a violation of Gauss law. In order to make the field compliant with Maxwell's equations everywhere, an additional field component, proportional to a $δ$-function at the plane of discontinuity, is required. Such a "$δ$-field" might be the electromagnetic field of the charge, moving with a uniform velocity appro…
A Galilean invariant description of the retarded electric interaction of two moving charges is carried out, considering the EM action to propagate at the speed of light. Through the altering of the Liénard-Wiechert potential (LWP) a new retarded, electric potential called retarded Coulomb potential (RCP) is ordered to the moving charges. RCP depends on the velocities both of the interacting charges. The force law of the electric interaction of moving charges is determined by means of the second order approximation of RCP. The force law obtained is the Galilean invariant Webers force law, surpr…
