Maxwell equations

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.

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].

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.

A new solution to Maxwell's differential equations is proposed. A new approach for writing solutions to these equations under consideration uses quaternions. The equations are written as a kind of generalization of the Cauchy-Riemann equations and have a form of partial differential equation of first order. The Green's function was found for direct (without potentials) solutions of Maxwell's equations. To calculate Green's function, we use factorization of the d'Alembert operator and the fact that Green's function for the d'Alembert operator is known. Three examples of determining the electromagnetic field strength were considered. This is an example of finding the strength of the electromagnetic field created by the charge q moving with constant speed v along the axis x1. The example of finding the electric field strength created by a uniformly charged thin rod at a point perpendicular to the rod at a distance R from the rod. The example of finding the electric field strength created by a dipole at a point located perpendicular to the middle of the dipole at a distance R from the middle of the dipole.

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.

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

For over 100 years after the publication of Maxwell's equations in 1865, essentially all solution techniques for electromagnetic fields and waves were based on Fourier‐domain concepts, assuminga prioria time‐harmonic (sinusoidal steady‐state) field variation and possibly the existence of a particular Green's function or a set of spatial modes. In 1966, Kane Yee's seminal paper introduced a complete paradigm shift in how to solve Maxwell's equations, reporting a field evolution‐in‐time technique that subsequently evolved into the finite‐difference time‐domain (FDTD) method. In the decades since the publication of Yee's paper, there has been an explosion of interest in FDTD and related grid‐based time‐marched solutions of Maxwell's equations among scientists and engineers. During this period, FDTD modeling has evolved to an advanced stage enabling large‐scale simulations of full‐wave time‐domain electromagnetic wave interactions with volumetrically complex structures over large frequency ranges, spatial scales, and timescales. Currently, FDTD modeling spans the electromagnetic spectrum from ultralow frequencies to visible light. FDTD modeling is routinely conducted as an invaluable virtual laboratory bench in scientific inquiry and exploration in electrodynamics; as an integral part of the electromagnetic engineering design and optimization process; and as a powerful forward solver in imaging and sensing inverse problems. This article reviews the technical basis of the key features of FDTD solution techniques for Maxwell's equations and provides 18 modeling examples spanning the electromagnetic spectrum to illustrate the power, flexibility, and robust nature of FDTD computational electrodynamics simulations.

The finite-difference time domain (FDTD) method is a numerical technique based on the finite difference concept. It is employed to solve Maxwell's equations for the electric and magnetic field distributions in both the time and spatial domains. The FDTD method utilizes the central difference approximation to discretize two of Maxwell's curl equations, namely, Faraday's and Ampere's laws, in both the time and spatial domains, and then solve the resulting equations numerically to derive the electric and magnetic field distributions at each time step and spatial point using the explicit leap-frog scheme. The FDTD solution, thus derived, is second-order accurate, although the difference formulation is first order, and is stable if the time step size is chosen to satisfy the special criterion.

THROUGHOUT the nineteenth century there were intensive efforts to develop mechanical models of electromagnetic phenomena based on the hypothesis of a linear aether continuum pervading all space1. One of the most interesting models was the gyrostatic aether proposed by MacCullagh in 1837, a medium characterized by resistance to rotation but not to compression or distortion, and designed to secure a dynamical foundation for Fresnel's wave theory of light2. This proposal lay dormant until taken up again many years later by Fitzgerald (1880), followed by Kelvin (1892) and Larmor (1894), who assimilated it to Maxwell's field equations. Larmor pictured the electron as a nucleus of rotational strain in the gyrostatic aether, thereby partially anticipating Volterra's3 dislocation singularities which may exist in an elastic continuum. Although Larmor's electron as it stands is incapable of motion, it marked a distinct advance over alternative conceptions, including that of Lorentz (1892), which insisted on an inexplicable dichotomy between the aether and ordinary matter. This supposed dichotomy was responsible for many paradoxes and, in spite of Larmor's suggestion, prevented a rapprochement with the new relativity physics in 1905. An aether with mechanical properties completely imaging electromagnetic phenomena, and containing mobile singularities imaging material particles, would be perfectly compatible with the Special Theory of Relativity. It would, however, go beyond that theory in also providing a rational theory of matter. The purpose of this communication is to introduce the gyrostatic aether on a fresh mathematical basis, capable of assimilating charges and currents as well as fields, and thereby fulfilling the primary condition of any acceptable aether hypothesis. An extended development will be presented elsewhere.

The problem of heat transfer with temperature control including a genetic algorithm for optimization of the process is studied in this work. The problem occurs in the cancer therapy called hyperthermia in which tissue around the tumor is heated to about 40 C by a set o antennas that involve the patient, with phase orientation and directed to a spot that receives heat due the incidence of electromagnetic waves. The problem o heat transfer in biosystems employs the Pennes' equation and Maxwell's equations for determining the electric and magnetic field. A two-dimensional model is studied with the Boundary Element Method including the Dual Reciprocity Method (DRBEM). The heterogeneity of the medium can be treated with the combination of various homogeneous media by the inclusion of sub-regions in the DRBEM. It is proposed a new approximating function for DRBEM to treat the domain effects of the integral equations related to Maxwell's and Pennes' equations in steady state, reduced to nonhomogeneous Helmholtz equations. The convergence of the function is evaluated in examples with simple geometry. The model was tested for a case with tumors in different depths been the results compared with the available literature.

Maxwell's fundamental equations and the resulting wave equation for propagation are reviewed. Closed‐form solutions for transverse‐electromagnetic (TEM), transverse‐magnetic (TM), and transverse‐electric (TE) waves in guided‐wave structures are obtained from Maxwell's equations. Propagation parameters (intrinsic wave impedance, attenuation and phase change per‐unit‐length, frequency ranges of use), broad‐banding techniques, and design examples are presented. Dielectric waveguides for applications at microwave and optical frequencies are also presented with design examples and diagrams for better understanding.

In Chapter 4 we have derived perturbation theory for Maxwell's equations to find corrections to the electromagnetic state eigenfrequency ω due to small changes in the material dielectric constant. Applied to the case of systems incorporating absorbing materials, we have concluded that absorption introduces an imaginary part to the modal frequency, thus resulting in decay of the modal power in time. While this result is intuitive for resonator states localized in all spatial directions, it is somewhat not straightforward to interpret for the case of waveguides in which energy travels freely along the waveguide direction. In the case of waveguides, a more natural description of the phenomenon of energy dissipation would be in terms of a characteristic modal decay length, or, in other words, in terms of the imaginary contribution to the modal propagation constant. The Hamiltonian formulation of Maxwell's equations in the form (2.70) is an eigenvalue problem with respect to ω 2 , thus perturbation theory formalism based on this Hamiltonian form is most naturally applicable for finding frequency corrections. In the following sections we develop the Hamiltonian formulation of Maxwell's equations in terms of the modal propagation constant, which allows, naturally, perturbative formulation with respect to the modal propagation constant. Eigenstates of a waveguide in Hamiltonian formulation In what follows, we introduce the Hamiltonian formulation of Maxwell's equations for waveguides, [1] which is an eigenvalue problem with respect to the modal propagation constant β. A waveguide is considered to possess continuous translational symmetry in the longitudinal ẑ direction.

It is well known that many of the field equations from theoretical physics (e.g. Einstein field equations, Maxwell's equations, Klein-Gordon equation) can be obtained from a variational principle with a suitably chosen Lagrange density. In the case of the Einstein equations the corresponding Lagrangian is degenerate (i.e., the associated Euler-Lagrange equations are of second order whereas in general these would be of fourth order), while in the cases of the Maxwell and Klein-Gordon equations the Lagrangian usually used is not degenerate.

Maxwell's vector wave equations are solved for dielectric configurations that match the symmetry of a spherical computational domain. The electric or magnetic field components and the inverse of the dielectric profile are series expansion defined using basis functions composed of the lowest order spherical Bessel function, polar angle single index dependant Legendre polynomials and azimuthal complex exponential (BLF). The series expressions and non-traditional form of the basis functions result in an eigenvalue matrix formulation of Maxwell's equations that are relatively compact and accurately solvable on a desktop PC. The BLF matrix returns the frequencies and field profiles for steady states modes. The key steps leading to the matrix populating expressions are provided. The validity of the numerical technique is confirmed by comparing the results of computations to those published using complementary techniques.

ABSTRACTIn this article, we carry out a systematic stability analysis for the explicit Runge–Kutta scheme coupled with a discontinuous Galerkin discretization in space for solving the time‐dependent Maxwell's equations. Many so‐called RKDG methods have been developed and successfully solved Maxwell's equations. However, to our best knowledge, there are quite limited rigorous analyses of RKDG methods devoted to Maxwell's equations. This paper is our initial effort in filling this gap by establishing the stability analysis for the RKDG methods of order two, three, and four. Numerical results consistent with our analysis are presented for the 3‐D Maxwell's equations.