In electrical engineering, electric charges usually move at speeds substantially slower than the speed of light. Therefore, relativistic mechanics and the Lorentz transformation are rarely applied; instead, the much simpler Newtonian mechanics and the Galilean transformation are used. This approximation is hoped to yield useful results. In this article, the exact solution of Maxwell's equations for arbitrarily moving point charges is used to demonstrate that this approximation is not suitable for point charges, even if they move extremely slowly. Nevertheless, in electrical engineering, great practical demand exists for a functioning and consistent electrodynamic model for non-relativistic point charges. This article presents such a model and demonstrates that it ensures the universal constancy of the speed of light for all receiving antennas, while satisfying the Galilean principle of relativity regarding the point charges, and being compatible with Newtonian mechanics and the classical Newtonian conservation of momentum. The framework can describe electromagnetic waves as well as all static and quasistatic effects with excellent quality if they are generated by non-relativistic point charges. The framework is based on the exact solution of Maxwell's equations for arbitrarily moving point charges and a simplified Lorentz force law, which has been calibrated so that the force for slow velocities matches Ampère's original force law and thus fulfills the predictions of magnetostatics and quasistatics.

This article addresses whether the Lorentz force formula, as used in electrostatics and magnetostatics, is also applicable to non-relativistic electrodynamics. To answer this question, the article uses the Liénard-Wiechert potentials to calculate the general analytical solution of Maxwell's equations for arbitrarily moving, non-relativistic point charges. It is then shown that the obtained solution only produces the correct force of a direct current on a moving test charge under illogical assumptions. This problem does not arise from the Maxwell equations but is probably related to the fact that the Lorentz force is already contained in Maxwell's equations due to Maxwell's addition. The Lorentz force should therefore not be added a second time in the form of an explicit supplementary formula. Instead, it is reasonable to integrate an old hypothesis by Carl Friedrich Gauss from 1835 into Maxwell's electrodynamics. This integration turns Maxwell's electrodynamics into Weber-Maxwell electrodynamics and resolves the contradictions.

Weber-Maxwell electrodynamics is a modernized, compressed, cleansed and, in many respects, advantageous representation of classical electrodynamics that results from the Liénard-Wiechert potentials. In the non-relativistic domain, it is compatible with both Maxwell's electrodynamics and Weber electrodynamics. It is suitable for all electrical engineering tasks, ranging from electrical machines to radar and high-frequency technologies. Weber-Maxwell electrodynamics also simplifies access to quantum physics and other areas of modern physics, such as optics and atomic physics. Particular advantages of Weber-Maxwell electrodynamics are its simple and fast computability in computer calculations and, as it is based on point charges, in the simulation of plasmas. The latter is particularly important for fusion research. Moreover, Weber-Maxwell electrodynamics is also highly suited to academic and post-primary education, as it allows an easy comprehension of both magnetism and electromagnetic waves. Due to the novelty of Weber-Maxwell electrodynamics, there are currently no articles that summarize its most important aspects. The present article aims to achieve this.

Weber-Maxwell electrodynamics combines classical Weber electrodynamics and Maxwell's equations, including all four field equations and the Lorentz force, into a single de facto equivalent three-dimensional wave equation. From classical Weber electrodynamics, Weber-Maxwell electrodynamics inherits properties in which the concept of the magnetic field is unnecessary, and Newton's third law is satisfied under all circumstances. From Maxwell's electrodynamics, Weber-Maxwell electrodynamics inherits the ability to be compatible with electromagnetic waves. This article shows that in Weber-Maxwell electrodynamics, all conservation laws are satisfied, and that electromagnetic waves in an isolated system do not possess energy and momentum, but only mediate them between particles of matter. Furthermore, the article shows that the modern formulation of Weber electrodynamics is clearly superior to standard electrodynamics in electrical engineering, because it not only eliminates the internal contradictions, but also represents considerable simplification and compression.

Maxwell's equations from the 19th century and the almost equally old Lorentz force equation provide the theoretical basis of all of electrical engineering and consequently the foundation for the majority of all modern technologies. For point charges, this system of partial differential equations reduces to the Weber-Maxwell wave equation. In this article, it is shown that this wave equation can be solved analytically for arbitrarily moving point charges, including accelerated charges, and that it is possible to present an analytical solution in the style of Coulomb's law. This finding demonstrates that classical electrodynamics, with all of its numerous wave phenomena, can also be represented without differential equations. This is surprising and unexpected. Moreover, this work enables the development of a novel class of electromagnetic field solvers that are highly superior in terms of speed and quality to existing solvers based on finite-difference time-domain methods, the method of moments, or finite element methods. In the future, this solution will make it possible to simulate any electromagnetic task in interactive form at previously unattainable quality, even on low-performance computers.

A Hertzian dipole is a small antenna that emits radio waves. In this article, the interaction of a Hertzian dipole with the field reflected from the environment is analyzed. The study shows that a single Hertzian dipole can behave like a quantum object: in particular, it can interfere with itself at a double slit, owing to a ponderomotive force. Therefore, it cannot reach every location behind a double slit with the same probability, and the probability distribution for the location of an impact on a screen behind the double slit may show a clear interference pattern. This effect disappears if one of the slits is closed, or if the system of the Hertzian dipole, double slit and standing wave is disturbed by external influences. Both particle interference and the collapse of the wave function are considered to be effects that cannot be explained by classical physics. However, Hertzian dipoles can be tangible objects moving on classical trajectories, and their location and velocity can be measured simultaneously at any time. Thus, the Hertzian dipole provides an example of how central features of quantum mechanics can be explained by means of classical physics.

Owing to the principle of relativity, the present state of knowledge explicitly allows Maxwell's equations to be solved not only in the rest frame of an electromagnetic transmitter but also directly in the rest frame of the receiver without use of the Lorentz transformation and the Lorentz force. Recently, such a calculation was first performed for the Hertzian dipole. The analysis of the resulting formula breaks new scientific ground and indicates that Maxwell's equations predict that electromagnetic waves in vacuum propagate at the speed of light, notably for each receiver, even when these receivers have relative velocities with respect to each other. Although this paradoxical phenomenon was expected, the finding that Maxwell's equations nevertheless predict a classical Doppler effect was unexpected and indicates inconsistent or not yet fully understood aspects of canonical Lorentz-Einstein electrodynamics consisting of Maxwell's equations, Lorentz force and Lorentz transformation.

Aiming to bypass the Lorentz force, this study analyzes Maxwell’s equations from the perspective of a receiver at rest. This approach is necessary because experimental results suggest that the general validity of the Lorentz force might be questionable in non-stationary cases. Calculations in the receiver’s rest frame are complicated and, thus, are rarely performed. In particular, the most important case is missing: namely, the solution of a Hertzian dipole moving in the rest frame of the receiver. The present article addresses this knowledge gap. First, this work demonstrates how the inhomogeneous wave equation can be derived and generically solved in the rest frame of the receiver. Subsequently, the solution for two uniformly moving point charges is derived, and the close connection between Maxwell’s equations and Weber electrodynamics is highlighted. The gained insights are then applied to compute the far-field solution of a moving Hertzian dipole in the receiver’s rest frame. The resulting solution is analyzed, and an explanation is presented regarding why an invariant and symmetric wave equation is possible for Weber electrodynamics and why the invariance could be the consequence of a quantum effect.

The predictions of Maxwell’s equations depend on the reference frame in which they are solved. If one solves Maxwell’s equations in the rest frame of the transmitter, which is the common approach, one obtains Lorentz-Einstein electrodynamics by adding the special theory of relativity. Here, for formal reasons, no information velocities greater than the speed of light in vacuum are possible. If, however, one solves Maxwell’s equations rigorously in the rest frame of the receiver, one comes to a field-theoretical generalization of Weber electrodynamics, which differs from Lorentz-Einstein electrodynamics. Although Einstein’s postulates are also fulfilled in this Weber-Maxwell electrodynamics, in a specifically designed experimental setup of two mutually stationary and very distant antennas, electromagnetic waves may travel at velocities that exceed the speed of light in vacuum. This effect, previously predicted only theoretically, has now been experimentally investigated and confirmed. This finding indicates that Lorentz-Einstein electrodynamics is incorrect and that Maxwell’s equations should instead be interpreted in terms of Weber electrodynamics. As a subsidiary result, these findings can enable remarkable new technologies, such as a highly compact method for radio direction finding (RDF).

One of the basic assumptions of physics is that information cannot be transferred more rapidly than light. If one equates information with data, then information transfer is a process in which data are transmitted between two remote locations. According to this basic principle, the transmission process should require an appropriate amount of time so that the distance between these locations divided by the needed time does not exceed the speed of light in a vacuum. However, it is technologically feasible to transmit a voice message, music, or any bit sequence at a low bit rate between locations separated by hundreds of meters, with almost no loss of time. Herein, we demonstrate that this behavior most likely does not contradict special relativity, and we explain this phenomenon based on a model of a superconducting cable.

The magnetic force acts exclusively perpendicular to the direction of motion of a test charge, whereas the electric force does not depend on the velocity of the charge. This article provides experimental evidence that, in addition to these two forces, there is a third electromagnetic force that (i) is proportional to the velocity of the test charge and (ii) acts parallel to the direction of motion rather than perpendicular. This force cannot be explained by the Maxwell equations and the Lorentz force, since it is mathematically incompatible with this framework. However, this force is compatible with Weber electrodynamics and Ampère’s original force law, as this older form of electrodynamics not only predicts the existence of such a force but also makes it possible to accurately calculate the strength of this force.