The paradox of a field of a moving locked charge (confined in a closed space) is considered and solved with the use of the integral Maxwell equations. While known formulas obtained for instantaneous fields of charges moving along straight and curved lines are fully correct, measurable quantities are average electric and magnetic fields of locked charges. It is shown that the average electric field of locked charges does not depend on their motion. The average electric field of protons moving in nuclei coincides with that of protons being at rest and having the same spatial distribution of the charge density. The electric field of a twisted electron is equivalent to the field of a centroid with immobile charges which spatial distribution is defined by the wave function of the twisted electron.
The energy-impulse flow of electromagnetic field, produced by the arbitrary moving charged particle, through the sphere of large radius is calculated. The result formulae are used in problem of radiation produced by hyperbolically moving particle and the problem of the synchrotron radiation energy loss. The energy-impulse radiated by a moving charged particle is shown to be not an invariant 4-tensor, for this reason calculation methods based on change of reference frame are not correct.
We discuss the seminal article in which Le Bellac and Lévy-Leblond have identified two Galilean limits of electromagnetism [1], and its modern implications. Recent works have shed a new light on the choice of gauge conditions in classical electromagnetism. We discuss various applications and experiments, such as in quantum mechanics, superconductivity, electrodynamics of continuous media, etc. Much of the current technology, where waves are not taken into account, is actually based on Galilean electromagnetism
Recently, Feigel has predicted a new effect in magnetoelectric media. The theoretical evaluation of this effect requires a careful analysis of a dynamics of the moving magnetoelectric medium and, in particular, the derivation of the energy-momentum of the electromagnetic field in such a medium. Then, one can proceed with the study of the wave propagation in this medium and derive the mechanical quantities such as the energy, the momentum, and their fluxes and the corresponding forces. In this paper, we develop a consistent general-relativistic variational approach to the moving dielectric and …
In the general setting of the problem, the explicit compact formulae are derived for the ponderomotive forces in the macroscopic electrodynamics of moving media in the Minkowski and Abraham approaches. Taking account of the Minkowski constitutive equations and making use of a special representation for the Abraham energy-momentum tensor enable one to obtain a compact expression for the Abraham force in the case of arbitrary dependence of the medium velocity on spatial coordinates and the time and for nonstationary external electromagnetic field. We term the difference between the ponderomotive…
The force due to electromagnetic induction on a test charge is calculated in different reference frames. The Faraday-Lenz Law and different formulae for the fields of a uniformly moving charge are used. The classical Heaviside formula for the electric field of a moving charge predicts that, for the particular spatial configuration considered, the inductive force vanishes in the frame in which the magnet is in motion and the test charge at rest. In contrast, consistent results, in different frames, are given by the recently derived formulae of relativistic classical electrodynamics.
The popular demonstration involving a permanent magnet falling through a conducting pipe is treated as an axially symmetric boundary value problem. Specifically, Maxwell's equations are solved for an axially symmetric magnet moving coaxially inside an infinitely long, conducting cylindrical shell of arbitrary thickness at nonrelativistic speeds. Analytic solutions for the fields are developed and used to derive the resulting drag force acting on the magnet in integral form. This treatment represents a significant improvement over existing models which idealize the problem as a point dipole mov…
The classical theory of electrodynamics is built upon Maxwell's equations and the concepts of electromagnetic field, force, energy and momentum, which are intimately tied together by Poynting's theorem and the Lorentz force law. Whereas Maxwell's macroscopic equations relate the electric and magnetic fields to their material sources (i.e., charge, current, polarization and magnetization), Poynting's theorem governs the flow of electromagnetic energy and its exchange between fields and material media, while the Lorentz law regulates the back-and-forth transfer of momentum between the media and …
We discuss the seminal article in which Le Bellac and Levy-Leblond have identified two Galilean limits of electromagnetism, and its modern implications. We use their results to point out some confusion in the literature and in the teaching of special relativity and electromagnetism. For instance, it is not widely recognized that there exist two well defined non-relativistic limits, so that researchers and teachers are likely to utilize an incoherent mixture of both. Recent works have shed a new light on the choice of gauge conditions in classical electromagnetism. We retrieve Le Bellac-Levy-Le…
PyCharge is a computational electrodynamics Python simulator that can calculate the electromagnetic fields and potentials generated by moving point charges and can self-consistently simulate dipoles modeled as Lorentz oscillators. To calculate the total fields and potentials along a discretized spatial grid at a specified time, PyCharge computes the retarded time of the point charges at each grid point, which are subsequently used to compute the analytical solutions to Maxwell's equations for each point charge. The Lorentz oscillators are driven by the electric field in the system and PyCharge…
The differential form of the Maxwell's equations was first derived based on an assumption that the media are stationary, which is the foundation for describing the electro-magnetic coupling behavior of a system. For a general case in which the medium has a time-dependent volume, shape and boundary and may move at an arbitrary velocity field v(r,t) and along a general trajectory, we derived the Maxwell's equations for a mechano-driven slow-moving media system directly starting from the integral forms of four physics laws, which should be accurate enough for describing the coupling among mechano…
It is common in the literature on classical electrodynamics and relativity theory that the transformation rules for the basic electrodynamic quantities are derived from the pre-assumption that the equations of electrodynamics are covariant against these---unknown---transformation rules. There are several problems to be raised concerning these derivations. This is, however, not our main concern in this paper. Even if these derivations are regarded as unquestionable, they leave open the following fundamental question: Are the so-obtained transformation rules indeed identical with the true transf…
The concept of gauge invariance in classical electrodynamics assumes tacitly that Maxwell's equations have unique solutions. By calculating the electromagnetic field of a moving particle both in Lorenz and in Coulomb gauge and directly from the field equations we obtain, however, contradicting solutions. We conclude that the tacit assumption of uniqueness is not justified. The reason for this failure is traced back to the inhomogeneous wave equations which connect the propagating fields and their sources at the same time.
The current density of a moving electric dipole is expressed as the sum of polarization and magnetization currents. The magnetic field due to the latter current is that of a magnetic dipole moment that is consistent with the relativistic transformations of the polarization and magnetization of macroscopic electrodynamics.
The aim of this paper is to investigate the fundamental problems of retardation and radiation caused by non-uniformly moving point sources using the theories of electrodynamics and elastodynamics. This paper investigates and compares the retarded electromagnetic fields and the retarded elastodynamic fields. For the non-uniform motion of a general point source, the Liénard-Wiechert type potentials and the radiation and nonradiation fields are derived for a point charge and for a point force.
We study the radiation reaction acting on an accelerating charge moving in noncommutative spacetime and obtain an expression for it. Radiation reaction, due to a nonrelativistic point charge, is found to receive a small noncommutative correction term. The Abraham-Lorentz equation for a point charge in noncommutative spacetime suffers from the pre-acceleration and the runaway problems. We explore as an application the radiation reaction experienced by a charge which undergoes harmonic oscillations in a noncommutative plane.
Although relativistic electrodynamics is more than 100 year old, there is one neglected topic in its presentation and application: relativistic transformations of electromagnetic integrals. Whereas in theoretical and applied electrodynamics electric and magnetic fields are mainly expressed in terms of integrals over charge and current distributions, relativistic transformations are traditionally applied to point charges and elementary currents. The purpose of this paper is to show that relativistic transformations can be easily applied to electromagnetic integrals and that relativistic transfo…
We discuss a signal theorem for charged particle detectors where the finite propagation time of the electromagnetic waves produced by a moving charge cannot be neglected. While the original Ramo-Shockley theorem and related extensions are all based on electrostatic or quasi-electrostatic approximations, the theorem presented in this report is based on the full extent of Maxwell's equations and does account for all electrodynamic effects. It is therefore applicable to all devices that detect fields and radiation from charged particles.
Maxwell's electrodynamics postulates the finite propagation speed of electromagnetic (EM) action and the notion of EM fields, but it only satisfies the requirement of the covariance in Minkowski metric (Lorentz invariance). Darwin's force law of moving charges, which originates from Maxwell's field theory complies the Lorentz invariance as well. Poincaré's principle stating that physical laws can be formulated with identical meaning on different geometries suggest, that the retarded EM interaction of moving charges might be covariant even in Euclidean geometry (Galilean invariance). Keeping th…
In this paper, we give the covariant formulation of second gradient electrodynamics, which is a generalized electrodynamics of second order including derivatives of higher order. The relativistic form of the field equations, the energy-momentum tensor and the Lorentz force density are presented. For an electric point charge, the generalized Lienard-Wiechert potentials and the corresponding electromagnetic field strength tensor are given as retarded integral expressions. Explicit formulas for the electromagnetic potential vector and electromagnetic field strength tensor of a uniformly moving po…
The radiation reaction problem for an electric charge moving in flat space-time of three dimensions is discussed. The divergences stemming from the pointness of the particle are studied. A consistent regularization procedure is proposed, which exploits the Poincaré invariance of the theory. Effective equation of motion of radiating charge in an external electromagnetic field is obtained via the consideration of energy-momentum and angular momentum conservation. This equation includes the effect of the particle's own field. The radiation reaction is determined by the Lorentz force of point-like…
Generation of electromagnetic fields by moving charges is a fascinating topic where the tight connection between classical electrodynamics and special relativity becomes particularly apparent. One can gain direct insight into the fascinating structure of such fields by visualizing the electric field lines. However, the calculation of electric field lines for arbitrarily moving charges is far from trivial. Here, we derive an equation for the director that points from the retarded position of a moving charge towards a specific field line position, which allows for a simple construction of these …
We propose a modification of Maxwell's macroscopic fundamental set of equations in vacuum in order to clarify Faraday's law of induction. Using this procedure, the Lorentz force is no longer separate from Maxwell's equations. The Lorentz transformations are shown to be related to the convective derivative, which is introduced in the electrodynamics of moving bodies. The new formulation is in complete agreement with the actual set of Maxwell's equations for bodies at rest, the only novel feature is a new kind of electromotive force. Heinrich Hertz was the first to propose a similar electrodynam…
The classical electromagnetic friction of a charged particle moving with prescribed constant velocity parallel to a planar imperfectly conducting surface is reinvestigated. As a concrete example, the Drude model is used to describe the conductor. The transverse electric and transverse magnetic contributions have very different character both in the low velocity (nonrelativistic) and high velocity (ultrarelativistic) regimes. Both numerical and analytical results are given. Most remarkably, the transverse magnetic contribution to the friction has a maximum for $|\mathbf{v}|<c$, and persists in …
We consider a self-action problem for an electric charge arbitrarily moving in flat spacetime of three dimensions. Its electromagnetic field satisfies the Maxwell equations in Minkowski space of three dimensions. In this space electromagnetic waves propagate not just at a speed of light, but also at all speeds smaller than or equal to the speed of light. The massive particle may "fill" its own field, which acts on it just like an external one. The radiation reaction is determined by Lorentz force of point-like charge acting upon itself plus non-local term which provides finiteness of the self-…
Using the canonical formalism, Compagno and Persico [J. Phys. A: Math. Gen. 35 (2002) 3629--45] have calculated the 'radiation-reaction' force on a uniform spherical charge moving rigidly, slowly and slightly from its position at the time when the transverse electric field is assumed to vanish. This force is shown to result in the same time-averaged self-force as that which has been obtained by different means for the test charge of a Bohr--Rosenfeld field-measurement procedure and which Compagno and Persico claimed to be incorrect.
A self-action problem for a point-like charged particle arbitrarily moving in flat space-time of six dimensions is considered. A consistent regularization procedure is proposed which relies on energy-momentum and angular momentum balance equations. Structure of the angular momentum tensor carried by the retarded "Liénard-Wiechert" field testifies that a point-like source in six dimensions possesses an internal angular momentum. Its magnitude is proportional to the square of acceleration. It is the so-called {\em rigid} relativistic particle; its motion is determinated by the higher-derivative …
A self-action problem for a pointlike charged particle arbitrarily moving in flat spacetime of three dimensions is considered. Outgoing waves carry energy-momentum and angular momentum; the radiation removes energy and angular momentum from the source which then undergoes a radiation reaction. We decompose Noether quantities carried by electromagnetic field into bound and radiative components. The bound terms are absorbed by individual particle's characteristics within the renormalization procedure. Radiative terms together with already renormalized 3-momentum and angular momentum of pointlike…
Up until now, a consistent causal theory of point charged particles (for example electrons) interacting with electromagnetic field is not known. The well-known problem is that the standard Lorentz force alone (in the case of point particles) does not lead to a theory in which momentum and energy are conserved. The need of radiation reaction force (or self-force) thus arises. The well known candidate for such force, the Lorentz-Abraham-Dirac reaction force, gives non-causal particle behavior, i.e., the particle starts to move before the arrival of external electromagnetic fields. Alternative, c…
Maxwell's equations are formulated in arbitrary moving frames by means of tetrad fields, which are interpreted as reference frames adapted to observers in space-time. We assume the existence of a general distribution of charges and currents in an inertial frame. Tetrad fields are used to project the electromagnetic fields and sources on accelerated frames. The purpose is to study several configurations of fields and observers that in the literature are understood as paradoxes. For instance, are the two situations, (i) an accelerated charge in an inertial frame, and (ii) a charge at rest in an …
A fully relativistically covariant formulation of the classical Maxwell electrodynamics of an arbitrarily-moving point charge is presented, purely in terms of gauge invariant potentials without entailing any gauge fixing. A new, relativistically covariant energy-momentum tensor for the radiation fields is derived and yields results for the angular power distribution, in full agreement with results derived in a frame-dependent manner in standard texts of classical electrodynamics. This is then used to present a full derivation, not available in standard texts, of the energy-momentum and orbital…
The classical theory of radiating point-charges is revisited: the retarded potentials, fields, and currents are defined as nonlinear generalized functions. All calculations are made in a Colombeau algebra, and the spinor representations provided by the biquaternion formulation of classical electrodynamics are used to make all four-dimensional integrations exactly and in closed-form. The total rate of energy-momentum radiated by an arbitrarily moving relativistic point-charge under the effect of its own field is shown to be rigorously equal to minus the self-interaction force due to that field.…
The complete charge-current density and field strength of an arbitrarily accelerated relativistic point-charge are explicitly calculated. The current density includes, apart from the well-established three-dimensional delta-function which is sufficient for its global conservation, additional delta-contributions depending on the second and third proper-time derivatives of the position, which are necessary for its local conservation as required by the internal consistency of classical electrodynamics which implies that local charge-conservation is an {identity}. Similarly, the field strength inc…
In 1892 H.A. Lorentz started the search for a classical equation of motion for pointlike charged particles that takes into account the radiation reaction force. This search culminated in the Lorentz-Abraham-Dirac equation of motion, which is not satisfactory since it exhibits self-acceleration causing runaway solutions. In spite of ongoing efforts for more than a century, there is yet no acceptable classical equation of motion for a pointlike charge, cf. the recent paper by Rohrlich and the comments about his proposal. So, it is still an open question how to augment continuous classical electr…
The gravitomagnetic field is the force exerted by a moving body on the basis of the intriguing interplay between geometry and dynamics which is the analog to the magnetic field of a moving charged body in electromagnetism. The existence of such a field has been demonstrated based on special relativity approach and also by special relativity plus the gravitational time dilation for two different cases, a moving infinite line and a uniformly moving point mass, respectively. We treat these two approaches when the applied cases are switched while appropriate key points are employed. Thus, we demon…
In this work we modify the wave-corpuscle mechanics for elementary charges introduced by us recently. This modification is designed to better describe electromagnetic (EM) phenomena at atomic scales. It includes a modification of the concept of the classical EM field and a new model for the elementary charge which we call a balanced charge (b-charge). A b-charge does not interact with itself electromagnetically, and every b-charge possesses its own elementary EM field. The EM energy is naturally partitioned as the interaction energy between pairs of different b-charges. We construct EM theory …
We present a mechanical analog of a quantum wave-particle duality: a vibrating string threaded through a freely moving bead or `masslet'. For small string amplitudes, the particle movement is governed by a set of non-linear dynamical equations that couple the wave field to the masslet dynamics. Under specific conditions, the particle achieves a regime of {\it transparency} in which the field and the particle's dynamics appear decoupled. In that special case, the particle conserves its momentum and a guiding wave obeying a Klein-Gordon equation, with real or imaginary mass, emerges. Similar to …
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…
