General Classical Electrodynamics

Abstract

Maxwell’s Classical Electrodynamics (MCED) suffers several inconsistencies: (1) the Lorentz force law of MCED violates Newton’s Third Law of Motion (N3LM) in case of stationary and divergent or convergent current distributions; (2) the general Jefimenko electric field solution of MCED shows two longitudinal far fields that are not waves; (3) the ratio of the electrodynamic energy-momentum of a charged sphere in uniform motion has an incorrect factor of four-thirds. A consistent General Classical Electrodynamics (GCED) is presented that is based on Whittaker’s reciprocal force law that satisfies N3LM. The Whittaker force is expressed as a scalar magnetic field force, added to the Lorentz force. GCED is consistent only if it is assumed that the electric potential velocity in vacuum, a, is much greater than c; GCED reduces to MCED, in case we assume a equals c. Longitudinal electromagnetic waves and superluminal longitudinal electric potential waves are predicted. This theory has been verified by seemingly unrelated experiments, such as the detection of superluminal Coulomb fields and longitudinal Ampère forces, and has a wide range of electrical engineering applications.

Keywords: Classical Electrodynamics, Longitudinal Ampère Force, Scalar Fields, Longitudinal Electric Waves, Superluminal Velocity, Energy Conversion.

1. Introduction

An alternative to Maxwell’s Classical Electrodynamics (MCED) theory is presented, called General Classical Electrodynamics (GCED), that is free of inconsistencies. For the development of this theory we make use of the fundamental theorem of vector algebra. The proof of this fundamental theorem is based on the three dimensional delta function and the sifting property of this function. The fundamental theorem of vector algebra is as follows: a vector function can be decomposed into two unique vector functions, a longitudinal part and a transverse part. The lowercase subindexes l and t will have the meaning of longitudinal and transverse in this paper. The longitudinal vector function is curl free, and the transverse vector function is divergence free. We assume that the function is well behaved, that is, zero at infinite distance.

The charge conservation law, also called charge continuity, is true for all types of charge-current density distributions. The physics of current free charge density distributions is called Electrostatics. The physics of stationary current density distributions is called General Magnetostatics (GMS). A special case of GMS are divergence free current distributions, and this is widely called Magnetostatics (MS) in the scientific educational literature. In case of Magnetostatics, the charge density distribution has to be static as well, such that the electric field and the magnetic field are both static.

The Maxwell-Lorentz force law satisfies Newton’s third law of motion (N3LM) in case of Electrostatics and Magnetostatics, however, this force law violates N3LM in case of General Magnetostatics. A violation of N3LM means that momentum is not conserved by GMS systems, for which there is no experimental evidence. This remarkable inconsistency in classical physics is rarely mentioned in the scientific educational literature.

This is not the only problematic aspect of MCED. Jefimenko’s electric field expression that is derived from MCED theory shows two longitudinal electric field terms that do not interact by induction with other fields, therefore these electric fields cannot be field waves and nevertheless these electric fields fall off in magnitude by distance, as far fields, which is inconsistent. A third inconsistency is the problematic four-thirds factor in the ratio of the electric energy and the electromagnetic momentum of a charged sphere. In the next sections we describe these related inconsistencies of MCED theory in more detail, and how to resolve them.

(Sections 2 and 3.1 to 3.3 — General Magnetostatics and Whittaker’s force law, the generalisation of Faraday induction to the scalar magnetic field, Special Classical Electrodynamics and the far-field inconsistency of Jefimenko’s solution — are omitted for length; the complete text is at the source.)

3.4. The wave types

These wave equations describe the Transverse Electromagnetic (TEM) wave and two types of longitudinal electric waves. One type of longitudinal electric wave is expressed only in terms of the electric charge potential, so it is not induced by electric currents. It will be called a potential wave. The second type of longitudinal electric wave is associated with the curl free electric current potential, and it will be called a Longitudinal Electromagnetic (LEM) wave.

Initially we assume that the values of these phase velocities are independent constants, and that is why we introduced the new constants for the polarizability of vacuum and the associated velocity. The additional theoretical prediction of the LEM wave and the potential wave are testable, as before Maxwell’s TEM wave prediction was tested by Hertz. Three power- and force laws can be derived that are associated with the potential wave, the LEM wave and the TEM wave. Similar energy flux vectors to Poynting’s vector for the TEM wave can be defined for the potential wave and for the LEM wave. For very small values of the vacuum polarizability, the potential wave contribution to momentum change becomes very small as well, and yet the potential wave contribution to power flux might be substantial.

3.5. The Lorentz premise and the Whittaker premise

In case of general magnetostatic currents, the density of the scalar magnetic field force must equal the current density times the scalar magnetic field, according to Whittaker’s force law that satisfies N3LM. Hence, the factor in the GCED force theorem for general magnetostatics must be approximately equal to one in order to fulfill N3LM. We conclude that the electric potential velocity must generally be much greater than the speed of light, and this is the crux of GCED theory. We will call the assumption that the potential velocity is much greater than the speed of light the Whittaker premise, after E. T. Whittaker. This premise should not be confused with the Coulomb gauge condition. GCED in the Whittaker premise is called GCED-WP.

The assumption that the potential velocity equals the speed of light will be called the Lorentz premise, after H. A. Lorentz. This premise should not be confused with the Lorenz gauge condition. GCED in the Lorentz premise (GCED-LP) is equivalent with the inconsistent MCED theory, and for this reason the Lorentz premise must be false in general. This is proven as follows: under the Lorentz premise, the scalar field condition implies that the superimposed scalar field exists as a free field wave that is not sourced by any charge current density distribution. If it is not sourced by anything, it simply does not exist, then we can set it to zero. It is easy to verify that GCED-LP reduces to MCED by setting both scalar fields to zero.

The scalar fields equation is a physical condition for physical potentials and physical scalar fields, since the Whittaker premise is true. We have shown that MCED, and in particular the confusing indeterminacy of the potentials in the context of MCED, can be avoided by replacing the Lorentz premise with the Whittaker premise. In Sec. 4.1 we refer to the experiments that verify the Whittaker premise and falsify the Lorentz premise.

The potentials of GCED-WP are determinate and physical, such that a unique solution of charge-current potentials describes the physics of a particular charge- and current distribution. With the potential velocity much greater than the speed of light, the scalar and vector potentials are solutions of decoupled inhomogeneous wave equations, whose solutions are retarded potentials. From these potentials we identify three near field terms, that fall off in magnitude by the square of the distance, and six far field terms of the potential, LEM and TEM waves, that fall off in magnitude by the distance, so GCED-WP is far field consistent.

Beside Electrostatics and General Magnetostatics, we define two other types of restricted behavior. A charge-current distribution is Quasi Dynamic (QD) if it is assumed that the potential velocity is infinite. A charge-current distribution is Quasi Static (QS) if it is assumed that both the potential velocity and the speed of light are infinite. For QD distributions, the retardation of the Coulomb field and the scalar potential is unnoticed. The length of the circuit is much smaller than the wavelength of the far potential wave, such that detection of a far potential gradient is impossible. QD is often referred to as instantaneous action at a distance.

3.7. The four-thirds problem

The electrostatic energy and the electromagnetic momentum of an electron with charge distributed on the surface of a sphere with the classical electron radius, moving with constant speed, show the following inconsistency in MCED: the mass inferred from the momentum is four-thirds of the mass inferred from the energy, which is known as the four-thirds problem in the context of MCED.

David E. Rutherford offered a solution to this problem. Firstly, Rutherford proves that the electrostatic energy of the electron is twice the usual value, since the work rate in order to charge the electron sphere with its charge is equal to the time derivative of the product of charge and potential, and not just to the potential times the time derivative of the charge. If we consider only the net charge potential, the GCED power density is the gradient of the potential contracted with the current plus the time derivative of the potential times the charge density. The volume integral of the first term equals the potential times the rate of change of charge; the volume integral of the second equals the charge times the rate of change of potential; and therefore the energy flow of charging the electron sphere equals the time derivative of the product, so the corrected energy also follows from GCED. This shows that the four-thirds problem is in fact a two-thirds problem: an electromagnetic momentum of one third of the mass times the velocity is missing.

Secondly, Rutherford derives the electromagnetic momentum by means of an electromagnetic momentum density expression that includes the electric field times the scalar magnetic field alongside the usual cross product of the electric field with the vector magnetic field; the volume integral of the additional term is equal to the missing momentum. The time derivative of this momentum density expression is exactly the last term of the GCED force theorem when we apply the Whittaker premise, so the corrected momentum follows also from GCED-WP.

Now the two masses agree without the factor of four-thirds, and this means that the famous mass-energy equation can be derived from the non-relativistic GCED-WP theory by evaluating the static energy and the electromagnetic momentum of a charged sphere in uniform motion. The consistent GCED-WP theory does not suffer the four-thirds problem either.

4. Review of CED experiments

The development of GCED-WP was motivated by Nikola Tesla’s remarkable achievements in electrical engineering. Tesla described his long distance electric energy transport system as transmission of longitudinal electric waves, conducted by a single wire or the natural media, including the aether. Mainstream physics predicts that longitudinal electric waves exist as sound waves conducted by material media only. GCED-WP predicts luminal longitudinal electromagnetic waves, and superluminal electric potential waves in vacuum as well, hence, for the first time in history Tesla’s observation of a longitudinal aether sound wave is supported by an exact theory. The characteristic quarter wave length distribution, across the unwound wire length of the secondary coil of Tesla’s transformer device, is hard to explain by conventional electrodynamics theory, where GCED-WP explains this wave type naturally as a LEM wave. A most general review is required for a wide range of CED experiments that may verify or falsify the new aspects of GCED-WP theory.

4.1. The superluminal Coulomb field

SCED and GCED-WP predict that the Coulomb field is superluminal. Superluminal evanescent tunneling of fields has been reported. Usually such effects are explained as quantum effects, however, GCED-WP explains such effects as a Coulomb near field with superluminal speed. The authors of the electron-beam study experimentally proved that the Coulomb near field of a uniformly moving electron beam is rigidly carried by the beam itself, which is further described as follows: the Coulomb near field travels with velocity much greater than the speed of light. It is impossible to explain these results by means of MCED, since Jefimenko’s electric field expression predicts a field retardation time interval given by the distance divided by the speed of light for both the electric near field and the electric far field.

4.2. General Magnetostatic force experiments

The historic Magnetostatic force experiments carried out by Ampère, Gauss, Weber and other famous scientists are examples of open circuit currents. It is certain that batteries or very large capacitor banks were used as electric current sources and current sinks that typically showed time varying charge densities and divergent currents at the current source and sink interface, however, delivered a steady voltage and a stationary current.

Specific General Magnetostatic experiments such as Ampère’s historic hairpin experiment demonstrate the existence of the longitudinal Ampère force. In particular, Stefan Marinov and Genady Nikolaev published on the results of several GMS force experiments that prove the existence of longitudinal Ampère forces. Nikolaev suggested a classical explanation for the Aharonov-Bohm effect: it is a longitudinal Ampère force, acting on the free electrons that pass through a double slit and pass a shielded solenoid on both sides of the solenoid. Such a force does not deflect the free electrons, and it slightly decelerates (delays) or accelerates (advances) the electrons, depending on which side the electrons pass the solenoid, which explains classically the observed phase shift in the interference pattern.

An excellent example of General Magnetostatics is Marinov’s stationary current motor that works as claimed, as observed by Phipps and others. The Marinov motor configuration is very similar to the Aharonov-Bohm experiment: two stationary electric currents, conducted by a metallic rotor ring, pass a solenoid at both sides, such that only a longitudinal Ampère force can explain the ring rotation. Marinov referred to Whittaker’s force law and Newton’s third law of motion in order to explain the ring rotation. Wesley’s theory of the Marinov motor is incorrect, because the relation between current and moving charge has been applied incorrectly for the movement of net electric charge in the conducting ring and the net electric current in the conducting ring.

4.3. Nikolaev induction

Experiments have yet to be done to verify or falsify Nikolaev’s induction law. According to this law, primary sinusoidal divergent currents induce a secondary sinusoidal divergent electric field and similar secondary currents, depending on the resistance in the secondary circuits, such that the secondary current is 90 degrees or more out of phase with the primary currents.

4.4. LEM waves

Wesley and Monstein published a paper on the transmission of a potential wave, by means of a pulsating surface charge on a centrally fed ball antenna. They assumed divergent currents are not present in such an antenna, however, this suggests a violation of charge conservation. We suggest that a centrally fed ball antenna conducts curl free divergent currents that induce mainly LEM waves, the scalar potential field being negligible, and that Wesley and Monstein actually observed LEM waves instead of potential waves. They tested and confirmed the longitudinal polarity of the received electric field.

Ignatiev and Leus used a similar ball shaped antenna to send wireless longitudinal electric waves with a wavelength of 2.5 km. They measured a phase difference between the wireless signal and an optical glass fiber control signal — the two signals are synchronous at the sender location — at a 0.5 km distance from the sender location. They concluded from the measured phase shift that the wireless signal is faster than the optical glass fiber signal, and that the wireless signal has a phase velocity of 1.12 times the speed of light. However, we assume that the discrepancy of 0.12 is due to an incorrect interpretation of the data, for instance, the optical glass fiber control signal has a phase velocity slower than the speed of light — in most cases it is 200,000 km per second, depending on the refractive index of the glass fiber. Combining the results from the experiments by Wesley, Monstein, Ignatiev and Leus, we conclude that the results verify the existence of the LEM wave that has luminal speed in air.

A very efficient quasi-superconducting Single Wire Electric Power System (SWEP) has been tested, that meets the same power requirements for standard 50/60 Hz three-phase AC power lines. The SWEP system transmits a high frequency high voltage signal that is sent and received by tuned and synchronized Tesla transformers. Floating SWEP applications are described that do not require grounding of the SWEP system, therefore it is reasonable to assume the SWEP system is based on the LEM wave concept. Single wire transmission systems that transport TEM wave energy are ground return systems that always require ground connections, such that the electric field is perpendicular to the wire and ground.

4.5. The scalar potential field and potential waves

The scalar potential field only exists as a far field, so observable effects of this type of field are not similar to near field force interaction, but occur through the emission and reception of potential wave energy. In order to induce observable scalar potential fields, one needs to induce high electric charge potentials with very high frequencies. High scalar potential fields may be induced by means of collective tunneling of many electrons through a potential energy barrier, since the tunneling of electrons is practically instantaneous. The potential wave energy flux vector also depends on the magnitude of electric fields, which can be optimized as well. Negative Resistance Oscillators (NROs) and Negative Conductance Oscillators (NCOs) may be suitable senders and receivers of continuous potential waves; such oscillators induce the highest electric potential frequencies, and are either based on avalanche ionization effects or collective quantum tunneling effects, or both.

Podkletnov’s impulse gravity generator emits a far field signal pulse with velocity of at least 64 times the speed of light, over a distance of 1211 meter. The wireless pulse is generated by means of a high voltage discharge, a maximum of 2 million volt, from a superconducting flat surface electrode to another non-superconducting electrode. The emitted pulse travels into the direction longitudinal, that is parallel, to the electronic discharge direction. TEM wave radiation, transverse to the direction of discharge, was not detected. Podkletnov concluded that the longitudinal direction signal is not a TEM wave, nor a beam of mass particles. We assume that Podkletnov’s impulse gravity device generates potential waves; the measured signal speed of at least 64 times the speed of light agrees with the GCED-WP prediction of a superluminal potential wave phase velocity. Secondly, Podkletnov expects that the superluminal signal frequency matches the tunneling frequency of the discharged electrons; during the discharge pulse, many electrons tunnel collectively through many superconducting layers before leaving the superconductor. This implies that a high scalar potential field is induced, since electron tunneling is an almost instantaneous electronic effect. The electrodynamic nature of the impulse gravity force has not yet been fully investigated by Podkletnov and Modanese. This recently discovered force could also be a novel impulse dia-electric force, that is repulsive for a wide range of materials, similar to the repulsive diamagnetic force in a strong magnetic field.

The reception of potential waves is most likely the reverse process of the transmission of potential waves, and requires the fastest movement of electrons, such as quantum tunneling through an energy barrier. The hypothesis that potential waves may stimulate quantum tunneling of electrons will be called the potential wave electric effect, similar to the photo-electric effect of electron emissions by a metal surface that is exposed to TEM waves.

4.6. Evidence for natural longitudinal electric waves as energy source

The most important GCED-WP application may be the conversion of naturally existing longitudinal electric far fields into useful electricity. Nikola Tesla was convinced that such an energy conversion is possible. In particular, the conversion of natural superluminal potential waves can be of great importance, since a high frequency energetic potential wave penetrates much deeper into the earth and earth atmosphere than electromagnetic waves, because of its relatively long wavelength. We assume that a few free energy devices that were invented in the electronic age might actually function as claimed and make use of the potential wave electric effect.

Dr. T. Henry Moray’s radiant energy device converted the energy flow of natural cosmic aether waves into 50 kilowatt of useful electricity, day and night. Moray’s device did not include batteries or large capacitor banks to store energy. Moray tuned his radiant energy device into a natural high frequency signal, that we assume is a potential wave of natural origin. Dr. Harvey Fletcher, who was the co-discoverer of the elementary charge of the electron as the assistant of Nobel prize winner Dr. Millikan, signed an affidavit describing that Moray’s radiant energy receiver functioned as claimed. The most proprietary component of Moray’s device was a high voltage cold cathode tube that contained a Germanium electrode doped with impurities, called the detector tube by Moray. The high voltage high frequency electric potential at the first energy receiving stage appeared to be at over 200,000 volts. Electron tunneling and avalanche ionization effects explain the observed high frequency signal generated by Moray’s valve tube, as well as the reported negative slope in a part of the conduction characteristic of the Moray tube. Moray’s suggestion that transverse electromagnetic gamma rays are the energy source of his device is unlikely, since the cosmic gamma ray intensity at the earth surface is too weak to explain 50 kilowatt of continuous output power generated by such a small device. The conversion of mass energy into electricity as alternative explanation for Moray’s device power output was rejected by Moray himself, therefore potential wave energy conversion remains one of the few explanations.

Dr. P. N. Correa’s energy conversion system is also based on a cold cathode plasma tube, which shows an excess electric power output. Correa described an anomalous and longitudinal cathode reaction force during self-pulsed abnormal glow discharges in a cold cathode plasma tube. Correa observed the abnormal glow discharge in a negative slope current-voltage regime. The same pre-discharge glow has been observed by Podkletnov, just before the pulse discharge of his gravity impulse device. A similar excess energy result was achieved by Dr. Chernetsky by means of a self-pulsed high voltage discharge tube filled with hydrogen gas. Chernetsky’s hydrogen gas tube generated longitudinal electronic waves in the electrical circuits attached to the tube, powering several hundred watt lamps.

These self-oscillating electronic plasma tube systems are without doubt negative resistance and negative conductance oscillators, optimized for Fowler-Nordheim quantum tunneling from a cold cathode to vacuum, and optimized for avalanche effects. The self-oscillation criteria for NROs and NCOs are not fully understood even today. We suggest that this type of device can function as a powerful potential wave energy receiver, which explains its excess energy output.

5. Conclusions

N3LM describes the motion of bodies that have mass; this law does not take into account the momentum of massless electromagnetic radiation. N3LM can be replaced by the more general principle of conservation of the sum of mass- and massless momentum. It is sometimes concluded that MCED satisfies the more general principle of momentum conservation, however, MCED violates this principle as well: circuits of stationary currents do not send or receive electromagnetic radiation with massless momentum, and it was already shown that the Grassmann force law violates N3LM in case of General Magnetostatics. One might take into account infrared radiation caused by friction, emitted by magnetostatic current circuits. However, infrared radiation does not compensate for the non-reciprocal Lorentz force, since the magnitude of momentum changes due to infrared radiation usually is much smaller than the occurring Ampère forces exerted on conductors, and secondly, this radiation is usually emitted in all directions such that the contribution of momentum change due to infrared radiation emission nullifies.

The fundamental theorem of vector algebra, that holds for time dependent vector functions as well, is essential for a comprehensive treatise and derivation of a consistent classical electrodynamics. We showed that GCED-LP reduces to the inconsistent MCED, that shows indeterminate potentials. Therefore, we conclude that the Lorentz premise is false, and that the Whittaker premise is fundamentally true. GCED-WP is a consistent electrodynamics theory, naturally based on the law of charge continuity and not on divergence free currents, that solves the four-thirds problem as well. Experimental results exist that verify GCED-WP and falsify MCED. GCED-WP with the extra condition that the current divergence and the rate of change of charge density both vanish reduces to SCED, which is a consistent theory and the essence of MCED. Classical electrodynamics is closely tied to modern physics theories, such as relativity theory and quantum mechanics.

The false Lorentz premise is key to understanding the second postulate of Special Relativity (SR) theory, a proposition in the following circular arguments: MCED does not predict longitudinal wave modes in vacuum, therefore vacuum cannot be a physical medium — a physical medium allows for longitudinal waves — therefore a relative motion of an observer with respect to vacuum is not possible, such that the speed of light has an absolute constant value regardless of the relative motion of light source and observer (the second SR postulate), such that the Lorentz transform has preference over the Galilei transform, which further forbids velocities higher than the constant speed of light, such that the potential velocity equals the speed of light, and this reduces GCED to MCED, and so on. These circular arguments of SR theory began with the false Lorentz premise, which is the most fundamental assumption of SR, and which reduces GCED to the inconsistent MCED. One-way TEM wave experiments prove the anisotropy of the TEM wave velocity in vacuum, and the recent gravity impulse speed measurements by E. Podkletnov are strong evidence for a superluminal wave speed of 64 times the speed of light. These experiments falsify both the Lorentz-Poincaré SR theory and Hilbert’s general relativity theory. To formulate a relativistic and consistent GCED-WP theory, a relativity principle other than Lorentz invariance should be applied anyway. Heinrich Hertz and Thomas E. Phipps showed how to cast MCED into a Galilei invariant form by simply replacing the partial time differential operator by the total time differential operator. In this way GCED-WP can be cast into a relativistic Galilei invariant GCED-WP.

The unfounded conjecture from Quantum Mechanics theory, that the linearly dependent scalar photon and longitudinal photon do not contribute to field observables, is obviously based on the false Lorentz premise, and is expressed classically as the vanishing of both scalar fields. The indeterminacy of the quantum wave function is tied to the indeterminacy of the MCED potentials. A gauge transform of the relativistic quantum wave equation is a transformation of the MCED potentials and the quantum wave function such that the phase of the transformed function differs by a constant from the phase of the original function, and such that the relativistic quantum wave equation is gauge invariant. This means the wave function is unphysical and indeterminate as well. The indeterminacy of the wave function is explained as probabilistic behavior of elementary particles, the Born rule: the particle velocity is the wave group velocity, and is not associated with wave phase velocity. Gauge invariance is also known as gauge symmetry, and is supposed to be the guiding principle of modern particle physics, however, this gauge freedom is merely the consequence of a false Lorentz premise, that reduces determinate GCED potentials to indeterminate MCED potentials. Recent scanning tunneling microscope experiments falsify Heisenberg’s uncertainty relations by close to two orders of magnitude, which is proof for the deterministic physical nature of the quantum wave function.

The determinate potentials of GCED-WP are agreeable with a determinate quantum wave function, where the phase of the wave function has physical meaning. The De Broglie-Bohm pilot wave theory comes into mind, in order to reinterpret quantum entanglement behavior as interferences of physical pilot waves. The unknown nature of Bohm’s pilot wave has been an objection against pilot wave theory, ever since Bohm and de Broglie offered this interpretation of Schrödinger’s equation solutions. Caroline H. Thompson published a paper on the universal wave function as the potential wave aether. Indeed, superluminal potential wave fields, acting as particle pilot waves, is a natural suggestion, such that the pilot wave nature is no longer ghost like and unknown. Quantum non-locality and non-causal entanglement may be confused with causal and local quasi dynamics. GCED-WP offers a classical foundation for the elementary particle-wave duality. The electromagnetic momentum and the near potential energy describe the particle mass energy and mass momentum, and the pilot potential wave particle interference describes the particle wave nature.

Although a consistent physics foundation based on determinate functions is the final destiny of modern physics, it is of greater importance that GCED-WP inspires scientists and engineers to review past classical electrodynamics experiments, which may birth a new era of science and technology with respect to telecommunication and energy conversion.

Acknowledgements

We are grateful to Samer Al Duleimi and Ernst van Den Bergh for valuable discussions.

(The 44-item reference list is omitted; the complete list is at the source.)