The Lorentz force law and Maxwell's equations in symmetric form

This paper deals with the effects of electromagnetic forces on the orbital motion of a spacecraft. The electrostatic charge which a spacecraft generates on its surface in the Earth's magnetic field will be subject to a perturbative Lorentz force. A model incorporating all Lorentz forces as a function of orbital elements has been developed on the basis of magnetic and electric fields. This Lorentz force can be used to modify or perturb the spacecraft's orbits. Lagrange's planetary equations in the Gauss variational form are derived using the Lorentz force as a perturbation to a Keplerian orbit. Our approach incorporates orbital inclination and the true anomaly. The numerical results of Lagrange's planetary equations for some operational satellites show that the perturbation in the orbital elements of the spacecraft is a second order perturbation for a certain value of charge. The effect of the Lorentz force due to its magnetic component is three times that of the Lorentz force due to its electric component. In addition, the numerical results confirm that the strong effects are due to the Lorentz force in a polar orbit, which is consistent with realistic physical phenomena that occur in polar orbits. The results confirm that the magnitude of the Lorentz force depends on the amount of charge. This means that we can use artificial charging to create a force to control the attitude and orbital motion of a spacecraft.

The field generated by a time‐harmonic pure irrotational current distributed in a finite region of free space is derived analytically. Only an electric field is produced and is nonzero even merely within the finite interior of the current distribution, though the problem space itself is infinite. To consider the obtained field solution nonphysical according to our current physical concepts, however, lacks evidence or reasoning. It satisfies the original first‐order Maxwell's equations (MEs) and the related continuity conditions exactly, hence cannot be rejected mathematically. It is also shown that a conceptual denial of such a different kind of solution based on physical interpretations of other known solutions of the same MEs is logically not acceptable. A direct experiment study is necessary to conclude convincingly whether or not the above solution is physical. The completeness problem of MEs and their consistency with the Lorentz force law and the equation of motion are discussed. Free space MEs are incomplete or underdetermined due to the absence of a curl equation for the current density. If the solution is nonphysical, ME‐based electromagnetic (EM) theory is not self‐consistent. On the other hand, if it is physical, this type of inconsistency within the frame of the ME‐based EM theory disappears, yet it is inconsistent with the Lorentz force law and the equation of motion in this case.

This work is devoted to discussing spinning particles’ motion in the surrounding dynamic phantom AdS Black hole (BH) with phantom scalar field k. The primary goal of this research is to differentiate between the effects of electric and magnetic charges on dynamic phantom AdS BH. The study analyzes the dependence of the innermost stable circular orbit (ISCO), responsible for interactions between charged particles and the external magnetic field, and observes the circular orbits of particles revolving around the compact object. It is also observed that as electric and magnetic charges increase, the ISCO radius decreases. Additionally, stability of orbit, Keplerian frequency, and harmonic oscillations of charged particles are also discussed. We also find the condition, electric and magnetic charges of the dynamic phantom AdS BH effect the energy of the center of mass with magnetized, electrically charged, and neutral particles due to the collision. Lastly, we discuss the accretion process and obtain generalized equations for the speed of sound cs2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$c^2_{s}$$\\end{document} sec, 4-velocity u(r), energy density ρ(r)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\rho (r)$$\\end{document} and mass (M˙\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\dot{M}}$$\\end{document}). We also plot these physical parameters against constant values of mass, electric and magnetic charges, and distinct values of state parameter ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\omega $$\\end{document} to study the accretion process. We observe that radial velocity and energy density of the fluid remain positive and negative, and the rate of change of mass is increased and decreased for stiff, dust, quintessence, and phantom-like fluid, respectively.

Experimental and theoretical researches performed by the author (period: 1969-present) showed that magnetic spinor particles (magnetic charges) are real structural components of atoms and substance and immediate sources of all magnetic fields and magnetic manifestations in Nature. Magnetic charges, that constitute together with electrical charges atomic shells got of title: magnetons and antimagnetons (respectively with charges g− and g+ under fundamental condition: g = e). Furthermore, in addition to participating in structures of atomic shells, the magnetic charges exist in potential and even in real zones conduction of solids. The magnetic and electric spinor particles in atoms and substance exist in form such of the spinor associations as the magnetic and electric bispinors. Under influence of external magnetic field in conductor being implemented polarization of magnetic bispinors with formation of magnetic dipoles. Internal fields magnetic dipoles is directed against an external magnetic field and is a simple explanation of such a physical manifestation as diamagnetism. During the rotation of these dipoles in the conductor are formed of the vortex magnetic dipole fields, which and create an electromotive force, i.e. are responsible for the electromagnetic induction. Author proposed new magneto-electric technology called the magnetoelectric induction, as the result of which the permanent current of magnetic charges is created. However, such currents may be implemented solely in the superconductors. In the article presented the main reasons which more hundred year old hinder recognition and technical use of the real magnetic charges and their currents what considerably slow down the global scientific and technological progress.

The validity of Poynting's theorem and the various expressions for the energy density, linear and angular momentum densities of the electromagnetic fields are re-examined in the presence of magnetic monopoles. It is found that the well known results in conventional electrodynamics without monopoles remain largely intact, provided that both the Maxwell equations and the Lorentz force law are modified in a consistent fashion as is already done previously in the literature.

The generation of Lorentz force is limited by the local magnetic field and the relative velocity of the spacecraft. As a result of this constraint, the Lorentz force cannot completely take place of the traditional propulsion technologies. In recent years, studying the Lorentz force provided by modulating spacecraft electrostatic charge (magnetic and electric fields) has become a possible good means to control the spacecraft without fuel consumption or reduce the fuel cost. Most of the previous studies concerned the Lorentz force of the magnetic field only. In this paper, we developed a combination of Lorentz forces provided by modulating spacecraft's electrostatic charge (magnetic and electric fields) that can be used to keep the desired relative distances and orientations. We are derived nonlinear dynamic models of satellite relative motion considering J2 perturbation, and Lorentz forces to develop a mathematical model for a new technique establishing increase the level of charging in the spacecraft surface. These solutions indicate that the principal effects of the Lorentz force on a spacecraft can be used to correct the drift in the relative position of formation flying due to the effect of Second zonal harmonics, where the magnitude of charge to mass ratio is the key player for the correction of this drift. Numerical results for different examples of formation flying confirmed the capability of Lorentz force to correct the drift in relative position. These corrections depend on the value of charge to mass ratio and magnitude of the relative position.

Monopoles, duality, triality

A proof is given that no action principle exists for the classical electromagnetic field when its sources are both charged particles and magnetic monopoles unless an extra condition, not derivable from the action principle, is assumed. The extra condition is that charges never touch magnetic monopoles. The Lorentz force law predicts that a charge and a magnetic monopole approaching each other along a straight line will collide. Since the Lorentz force law can only be gotten from an action principle with the aid of this extra condition, the necessary extra condition is in contradiction to the law derived with its help. Thus there is no satisfactory action principle for the classical electromagnetic field if both charges and magnetic monopoles exist. The foregoing provides an aesthetic argument against the existence of magnetic monopoles. An action principle has then been constructed using the extra condition. It is analogous to the usual action principle for (charge-only) electromagnetic theory, but ${J}^{\ensuremath{\nu}}{A}_{\ensuremath{\nu}}$ is replaced by ${J}^{\ensuremath{\nu}}{R}_{\ensuremath{\nu}}\ensuremath{\rightarrow}{K}^{\ensuremath{\nu}}{T}_{\ensuremath{\nu}}$ where ${J}^{\ensuremath{\nu}}$ and ${K}_{\ensuremath{\nu}}$ are the 4-dimensional current and magnetic monopole densities, respectively, and ${R}_{\ensuremath{\nu}}$ and ${T}_{\ensuremath{\nu}}$ are gaugeless potentials. Electromagnetic theory is formulated in a gaugeless way in terms of these effective potentials in the Appendix.

I would like to present some new results regarding the origin of mass and distributions of electric and magnetic charges in an elementary particle. The discreteness of the electric charge distribution and its confinement results from the layered distribution, with alternating signs, of the magnetic charge. It is found that magnetic charge layers thin out towards the surface of the particle while the electric charge increases so that most of it resides on the particle’s surface. The fact of the electric and magnetic charges lying on a circle [i.e., \(r_o^2 = \left( {{{{\text{2G}}} \mathord{\left/ {\vphantom {{{\text{2G}}} {\text{c}}}} \right. \kern-\nulldelimiterspace} {\text{c}}}^4 } \right)\left( {e^2 + g^2 } \right)\)] implies that the discreteness of the magnetic charge distribution will result in the discreteness of electric charge distribution.

The Kerner-Mann fermions tunneling framework is extended to the spin particles with electric and magnetic charges in this paper. We rewrite the electromagnetic field tensor and the Lagrangian of the field corresponding to the source with electric and magnetic charges to redefine an equivalent charge. We only consider the case that the ratio of the electric charge and magnetic charge of the emission is constant and equal to the source. The result shows that when the energy conservation together with the electric charge and magnetic charge conservations are taken into account in the dynamical background space time, the emission rate agrees with the underlying unitary theory and the actual radiation spectrum of charged and magnetized fermions also derivates from the pure thermal one.

The quest for magnetic monopoles has captivated theoretical physicists due to their potential to reshape our understanding of electromagnetism. Despite extensive research, definitive mathematical proof of their existence remains elusive. This paper presents a rigorous mathematical framework supporting the existence of magnetic monopoles within the covariant formulation of classical electromagnetism and special relativity. Utilizing tensor calculus and the covariance form of Maxwell's equations, we derive expressions for magnetic charge density and magnetic current density through Lorentz transformations. Our approach emphasizes the expansion of tensorial equations, particularly focusing on the covariant derivatives of the electromagnetic field tensor and its dual. The results demonstrate the recovery of magnetic charge and current densities in Maxwell's equations, providing a theoretical foundation for the existence of magnetic monopoles. A magnetic monopole model is introduced, establishing the relationship between the electric field, the monopole’s magnetic charge, and their dependence on the observer's relative velocity as a direct consequence of the modified Gauss's law for magnetism. This model suggests the existence of dipole bosons—electric or magnetic—which function analogously to the Higgs boson. These dipole bosons are proposed to confer mass, charge, and spin to charged particles, whether they are electric or magnetic in nature.

Recent attempts to obtain the force of radiative reaction in the classical equations of motion of point charges have proceeded from two different viewpoints. Each of these has to introduce one basic assumption in addition to Maxwell's equations, namely the conservation law for the electromagnetic energy-momentum tensor in field theory and the relation between the Lorentz force and the momentum of the particle in action-at-a-distance theory. In previous field-theoretical derivations the Lorentz-Dirac equations including radiation damping are obtained only if one takes the field produced by the particle to be the retarded field; but equations of motion without the damping term are obtained if one uses half the sum of retarded and advanced fields. On the other hand, the theory of action at a distance as developed by Wheeler and Feynman was able to obtain the radiation damping using fields symmetric in time. It is noted that the need for the exclusive use of retarded fields arose only in the field-theoretical derivations for the one-particle problem. The considerations of Wheeler and Feynman on the total field due to all particles of the universe are, however, applicable to field theory in the symmetric form as well as to action-at-a-distance theory. The acceptance of their condition of "complete absorption" again leads to the radiation damping term in the equations of motion of the many-body problem. Some implications of this result are discussed.

The objective of this paper is to establish the properties of the electromagnetic wave propagation in a diversity of situations in material media with magnetic monopoles and even in the situations of entities simultaneously containing electric and magnetic charges. This analysis requires the knowledge and solutions of the \Maxwell equations in material media compatible with the existence of magnetic monopoles and the extended concepts of linear responses (conductivity, split-charge susceptibility, kinetic susceptibility, permittivity and magnetic permeability) in the case of presence of electric and magnetic charges. This analysis can facilitate insights and suggestions for electrical and optical experiments afiording a better knowledge of the materials whose behaviour can be analyzed under the consideration of the existence of entities with equivalent properties of the magnetic monopoles.

We present the main aspects of Weber's electrodynamics and of Maxwell's equations. We discuss Maxwell's point of view related to Weber's electrodynamics. We compare Weber's force with Lorentz's force. We analyse the relation between Weber's law and Maxwell's equations. Finally, we discuss some experiments performed and proposed with which we can distinguish Weber's force from Lorentz's one.

We present, design and generate a new kind of vector optical fields with linear polarization distributions modeling to electric and magnetic field lines. The geometric configurations of "electric charges" and "magnetic charges" can engineer the spatial structure and symmetry of polarizations of vector optical field, providing additional degrees of freedom assisting in controlling the field symmetry at the focus and allowing engineering of the field distribution at the focus to the specific applications.