Field Generated by an Irrotational Current

Maxwell's equations, Lorentz's equations of motion, and Dirac's equations are investigated on a cosmological background. Maxwell's equations are conform invariant and the equations of motion can easily be made conform invariant. Then, without introducing any new assumptions, Dirac's equations are seen to be conform invariant. In open universes the solutions of Maxwell's and Dirac's equations are the same as in flat Minkowski space. The behavior of these equations is different in closed universes by reasons of topology. In closed universes both Maxwell's and Dirac's equations provide new eigenvalue problems. Dirac's equations exhibit a marked difference between an elliptic and a spherical closed universe.

Study on propagation of cylindrical electromagnetic waves in various inhomogeneous and nonlinear media is of fundamental importance, which can be described by the cylindrical nonlinear Maxwell's equations. In recent years, finding exact solutions for these equations has emerged as a popular research topic. The exact solutions play an irreplaceable role in understanding and predicting physical phenomena, and developing numerical calculation methods, and so on. However, due to the extreme complexity of nonlinear partial differential equations, exact solutions of the cylindrical Maxwell's equations were only able to be obtained in a nonlinear and nondispersive medium whose dielectric function is an exponential function in previous researches. Actually, there is no general method at present which can exactly solve arbitrary cylindrical nonlinear Maxwell's equations. Therefore, finding physically admissible solutions meeting certain particular condition for the cylindrical nonlinear Maxwell's equations might be feasible. In this paper, we discuss the traveling wave solutions which are very important in electromagnetic theory, especially in solitary wave theory. To our knowledge, research on obtaining traveling wave solutions of the cylindrical nonlinear Maxwell's equations is still lacking. The main conclusions in this paper are listed as follows. Firstly, we introduce the cylindrical nonlinear Maxwell's equations mentioned in some previous publications, which can describe cylindrical electromagnetic waves propagation in inhomogeneous nonlinear and nondispersive media. In this paper, we focus on the nondispersive media with arbitrary nonlinearity and power-law inhomogeneity. Secondly, we point out that the electric field component E of the model has no plane traveling wave solutions E=g(r-kt), after theoretical analysis and study. Then generalized traveling wave solutions in form of E=g(lnr-kt) for the electric field component are obtained by finding correct variable substitution and solving second-order nonlinear ordinary differential equation.Finally, we provide two examples to show the physical meanings of our generalized traveling wave solutions. We find that the transmitting speeds of vibrations vary with different points of the electric field. Actually, the transmitting speed of the vibration of a certain point closer to the cylinder center is lower. As a result, we observed a physical phenomenon similar to that of self-steepening. Our work can be used to analyze the electromagnetic properties of ferroelectric materials and new materials. Theoretically, it can also provide an approach to studying the cylindrical nonlinear Maxwell's equations.

We modify the Einstein-Schrodinger theory to include a cosmological constant $\Lambda_z$ which multiplies the symmetric metric, and we show how the theory can be easily coupled to additional fields. The cosmological constant $\Lambda_z$ is assumed to be nearly cancelled by Schrodinger's cosmological constant $\Lambda_b$ which multiplies the nonsymmetric fundamental tensor, such that the total $\Lambda=\Lambda_z+\Lambda_b$ matches measurement. The resulting theory becomes exactly Einstein-Maxwell theory in the limit as $|\Lambda_z|\to\infty$. For $|\Lambda_z|\sim 1/(Planck length)^2$ the field equations match the ordinary Einstein and Maxwell equations except for extra terms which are $<10^{-16}$ of the usual terms for worst-case field strengths and rates-of-change accessible to measurement. Additional fields can be included in the Lagrangian, and these fields may couple to the symmetric metric and the electromagnetic vector potential, just as in Einstein-Maxwell theory. The ordinary Lorentz force equation is obtained by taking the divergence of the Einstein equations when sources are included. The Einstein-Infeld-Hoffmann (EIH) equations of motion match the equations of motion for Einstein-Maxwell theory to Newtonian/Coulombian order, which proves the existence of a Lorentz force without requiring sources. This fixes a problem of the original Einstein-Schrodinger theory, which failed to predict a Lorentz force. An exact charged solution matches the Reissner-Nordstrom solution except for additional terms which are $\sim 10^{-66}$ of the usual terms for worst-case radii accessible to measurement. An exact electromagnetic plane-wave solution is identical to its counterpart in Einstein-Maxwell theory.

Maxwell's equations within a dielectric and/or a magnetic medium were first developed macroscopically and must be complemented by constitutive relations to obtain solutions. These relations connect D with E (and with B in optically active material) and H with B (and again with E in optically active material). The atomic and molecular theories of quantum mechanics allow a microscopic approach to derive these constitutive relations where the macroscopic electric and magnetic fields are averages of the microscopic fields e and b. In classical electromagnetic theory Lorentz [1] originally showed how to derive the Maxwell's macroscopic equations from electron theory using microscopic fields obeying the Maxwell's equations in vacuo but coupled to electronic and ionic sources. There are two distinct steps in this procedure. The first introduces microscopic polarization fields, both electric amd magnetic, p and m from which microscopic electric displacement vector field d and auxiliary magnetic field h are simply constructed. The resulting equations for the microscopic fields e, b, d, and h are called the atomic field equations. The second step is the statistical one where the macroscopic fields E, B, D, and H are defined as averages of the microscopic fields and these macroscopic fields are then shown to obey the phenomenological macroscopic Maxwell's equations. A historical appraisal may be found in the recent book by de Groot [2].

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.

It is well known that there is no direct one-to-one correspondence between the electromagnetic theory based on the physical laws and that based on the Maxwell's differential equations. For example, in order to derive the boundary conditions from the Maxwell's differential equations, one assumes that some integral identities derived from them are valid even when the field components (or material parameters) are discontinuous. This assumption violates, in a sense, the completeness of the theory of electromagnetism based on the Maxwell's differential equations. We will prove that if one postulates that the Maxwell's equations are valid in the sense of distributions, then this incompleteness will be removed and the boundary conditions will appear implicitly in the basic differential equations.

In the first part of this work, we introduce a monochromatic solution to the scalar wave equation in free space, defined by a superposition of monochromatic nondiffracting half Bessel-lattice optical fields, which is determined by two scalar functions; one is defined on frequency space, and the other is a complete integral to the eikonal equation in free space. We obtain expressions for the geometrical wavefronts, the caustic region, and the Poynting vector. We highlight that this solution is stable under small perturbations because it is characterized by a caustic of the hyperbolic umbilical type. In the second part, we introduce the corresponding solution to the Maxwell equations in free space.

In this paper, the solution to long standing problem of deriving Maxwell’s equations and Lorentz force from first principles, i.e., from Coulomb’s law, is presented. This problem was studied by many authors throughout history but it was never satisfactorily solved, and it was never solved for charges in arbitrary motion. In this paper, relativistically correct Liénard–Wiechert potentials for charges in arbitrary motion and Maxwell equations are both derived directly from Coulomb’s law by careful mathematical analysis of the moment just before the charge in motion stops. In the second part of this paper, the electrodynamic energy conservation principle is derived directly from Coulomb’s law by using similar approach. From this energy conservation principle the Lorentz force is derived. To make these derivations possible, the generalized Helmholtz theorem was derived along with two novel vector identities. The special relativity was not used in our derivations, and the results show that electromagnetism as a whole is not the consequence of special relativity, but it is rather the consequence of time retardation.

Overview Having shown in Chapter 5 that the magnetic force must exist, we will now study the various properties of the magnetic field and show how it can be calculated for an arbitrary (steady) current distribution. The Lorentz force gives the total force on a charged particle as F = q E + q v × B . The results from the previous chapter give us the form of the magnetic field due to a long straight wire. This form leads to Ampere’s law , which relates the line integral of the magnetic field to the current enclosed by the integration loop. It turns out that Ampere’s law holds for a wire of any shape. When supplemented with a term involving changing electric fields, this law becomes one of Maxwell’s equations (as we will see in Chapter 9). The sources of magnetic fields are currents, in contrast with the sources of electric fields, which are charges; there are no isolated magnetic charges, or monopoles . This statement is another of Maxwell’s equations. As in the electric case, the magnetic field can be obtained from a potential, but it is now a vector potential ; its curl gives the magnetic field. The Biot–Savart law allows us to calculate (in principle) the magnetic field due to any steady current distribution. One distribution that comes up often is that of a solenoid (a coil of wire), whose field is (essentially) constant inside and zero outside. This field is consistent with an Ampere’s-law calculation of the discontinuity of B across a sheet of current. By considering various special cases, we derive the Lorentz transformations of the electric and magnetic fields. The electric (or magnetic) field in one frame depends on both the electric and magnetic fields in another frame. The Hall effect arises from the q v × B part of the Lorentz force. This effect allows us, for the first time, to determine the sign of the charge carriers in a current.

9 - Computational Electromagnetics: The Finite-Difference Time-Domain Method

An electromagnetism module is under development in the commercial software LS-DYNA in order to perform coupled mechanical/thermal/electromagnetic simulations. The main part of the module is a so-called eddy current solver. This module allows us to introduce source electrical currents into solid conductors, and to compute the associated magnetic fields, electric fields, induced currents, Lorentz forces, Joule heating and so forth. The Maxwell equations are solved using a Finite Element Method (FEM) for the solid conductors coupled with a Boundary Element Method (BEM) for the surrounding air (or insulators). Both the FEM and the BEM are based on discrete differential forms (Nedelec-like elements). The use of the BEM for the air allows to handle complex 3D geometries with multiply connected conductors, very small air gaps, and motion of the conductors, due, in particular, to the electromagnetic forces. The variable of the BEM is a so-called surface current allowing the connection of the model to external current sources by simple dirichlet constraints, as well as direct computations of the self and mutual inductances of the system. In order to handle the large and dense BEM matrices, a domain decomposition is performed and low rank approximations are done on the off-diagonal blocks of the resulting block matrices. The diagonal blocks are used as an efficient preconditioner when solving the BEM part of the system. The singularities arising in the computation of the self and neighbor matrix elements of the Galerkin BEM are taken into account using a method based on Duffy transforms. The BEM method will be presented as well as benchmarks and real life application examples.

Simply constructing linear combinations of the electric and magnetic fields yields symmetric forms for the basic relationships in electromagnetism. The resultant description contains magnetic monopoles, electric charges and an arbitrary parameter ?.This approach allows educators to introduce several interesting topics at the introductory college level without resorting to more advanced techniques.

It has previously been shown that complex-valued electromagnetic fields can substantially increase the symmetry of Maxwell’s Equations (MEs). They are consistent with known experimental findings in classical electrodynamics and result in some interesting predictions. For example, complex MEs predict the existence of magnetic monopoles that would have escaped detection in past experimental searches for them. This paper extends the basic complex-valued MEs for use inside matter. The increased symmetry of the extended MEs is demonstrated by an electromagnetic duality transformation analogous to that of the standard MEs and a fundamentally new type of duality transform. A derived wave equation unexpectedly shows that the imaginary-valued portion of waves inside of matter propagates without attenuation or reduced speed. Demonstrating the existence of the imaginary-valued field components predicted by this theory could have substantial implications for understanding physical and biological phenomena. To illustrate this, ways in which imaginary-valued field components would contribute to existing electromagnetic field theories of consciousness are described. The ability of complex-valued fields to account for disparate phenomena (failure of past experimental searches to find magnetic monopoles; several poorly understood features of subjective time and memory) increases the probability of their existence.

When we move the Hall component in a uniform gradient magnetic field under the constant current, the Hall electromotive force it outputs depends only on the amount of its displacement in the magnetic field. According to this characteristic, this paper analyses the basic principles, the selection of uniform gradient magnetic field and the output of displacement expression data. This paper makes a research from the four aspects. With improvement and innovation, we try to design a sensor which can measure miniature vibration. Introduction Nowadays, as the deepening of the scientific research, we have become increasingly demanding for the accuracy of measurements. That, how to measure small vibrations conveniently and quickly, is a research aspect that receives much concern. Hall displacement sensor, compared with other displacement sensors, has features such as quick response, simple structure and low cost, and some disadvantages about the measurement as well. To achieve the effect small vibrations to be measured, we will improve and research the sensor, based on its disadvantages. Fundamental principles Hall effect When a magnetic field is present to the semiconductor that is perpendicular to the direction of the current, the electrons and holes inside the semiconductor will experience a force, which is called the Lorentz force, and the electrons and holes will accumulate on one face of the material, establishing an electric field. The force from the electric field opposes the Lorentz force. When the electric field reaches a certain value, the electric force and the Lorentz force experienced by the holes and electrons are balance. The electric field opposes the migration of further charge, so a steady electrical potential is established, making the subsequent electron and holes can pass without deflection. This is called the Hall effect. As is shown in the Fig.1. Suppose the conductor is a cuboid, whose length, width and height are a, b, and c respectively, the electricity flows in from surface cd vertically, and that the magnetic field intensity is B vertical to surface ab, the voltage between the two boards front and back is therefore Hall voltage Fig.1: Hall effec International Power, Electronics and Materials Engineering Conference (IPEMEC 2015) © 2015. The authors Published by Atlantis Press 969 According to the formula of Hall voltage U, with a fixed electric current I, as long as the magnetic field gradient is constant, the value of Hall voltage across the Hall element will show a positive liner relationship with its displacement within the Hall element. Hall element. Produce Hall element using the principle of Hall Effect. The performance curve of Hall element is as follows: The purpose is to find appropriate electric current and magnetic field, so that with a fixed electric current and constant magnetic field gradient, the Hall element will work with in interval AB in the phase diagram. The working status of Hall element is also associated with temperature. For the sake of measuring accuracy, SS495A1 Hall element is chosen, whose working temperature is between -40C and 150C with strong flexibility, matching the requirement of measurement and design. . Fig.2: Characteristic Choosing the magnetic field As is shown in the first step, in order to make the instrument reflect tiny vibration, a liner relationship between Hall voltage and displacement is in need. Thus, we need to select a magnetic field with constant gradient. This essay develops the study according to an experiment – correction verify procedure, fitting Hall voltage and displacement of Hall elements in magnetic field of various shapes, through which we find that magnetic field of bar magnet is pretty close. Consequently, we made further improvement based on this idea, and finally choose to fix two angel-shaped iron on the two sides of the bar. The specific measuring method is as follows: Fig.3 Measuring method Fitting this set of data using MATlAB, we have Fig.4:

We re-examined the four-dimensional spacetime formulation of invariance of electromagnetic fields between two inertial frames under Lorentz transformation, which predicts a pure electric (magnetic) field in one inertial frame is composed the Cartesian components of a pure both electric and magnetic fields in another inertial frame. This contradicts the Lorentz invariance condition which requires that the vector quantities in one inertial frame must have the same form in another inertial frame. In this work, we introduce a three-dimensional quasi-time vector to modify the classical four-dimensional spacetime (3+1) to a new six-dimensional spacetime (3+3) and derive spacetime metric equation and relativistic velocity. We use the classical vector transformation theory to derive expressions for Cartesian components of relativistic velocity and net electromagnetic force vectors. Considering two massive inertial frames form a closed system, we integrated the transformed relativistic velocity with the law of conservation of energy to prove that contrary to the common belief, the electromagnetic field that appears as a purely electric (magnetic) field in one massive inertial frame, it also appears as a pure electric (magnetic) field in another massive inertial frame under Lorentz transformation. As an application of the proposed six-dimensional spacetime theory, we prove Lorentz invariance of Maxwell’s equations with and without charge and current source. We also prove the scalar electromagnetic wave equations with and without charge and current source and the conservation laws of the continuity equations of current and densities of electromagnetic energy and linear and angular momentums between two massive inertial frames under Lorentz transformation.