Maxwell Equations In Form Research Articles

Complex Symmetric Formulation of Maxwell Equations for Fields and Potentials

Maxwell equations have two types of asymmetries between the electric and magnetic fields. The first asymmetry is the inhomogeneity induced by the absence of magnetic charge sources. The second asymmetry is due to parity. We show how both asymmetries are naturally resolved under an alternative formulation of Maxwell equations for fields or potentials that uses a compact complex vector operator representation. The developed complex symmetric operator formalism can be easily applied to performing the continuity equation, the field wave equations, the Maxwell equations for potentials, the gauge transformations, and the 4-momentum representation; in general, the developed formalism constitutes a simple way of unfolding the Maxwell theory. Finally, we provide insights for extending the presented analysis within the context of (i) bicomplex numbers and tessarine algebra; and (ii) Lp-spaces in nonlinear Maxwell equations.

Mean-Field Formulation of Maxwell Equations to Model Electrically Inhomogeneous and Isotropic Media

Maxwell equations were originally designed to describe classic electromagnetic phenomena in any type of medium. In particular, to describe electromagnetic phenomena under the quasistatic electric approximation in media that are electrically inhomogeneous and isotropic, such as for example when there are strong spatial variations of conductivity, the formalism must be adapted according to the problem considered. We review here two approaches to this problem, first a “microscopic” model, where the spatial variations of conductivity and permittivity are explicitly taken into account. In a second “macroscopic” model, these spatial variations are taken on average by using a mean-field formulation of Maxwell equations. Both of these models can describe the electromagnetic behavior of inhomogeneous media. We illustrate this formalism to describe the electric behavior of biological media, such as brain tissue, which is typically very inhomogeneous. We show that the theory predicts that for the typical frequency range of biological phenomena (lower than about 1000 Hz), the inhomogeneous nature of the medium has a determinant influence.

Fast modal method for crossed grating computation, combining finite formulation of Maxwell equations with polynomial approximated constitutive relations

We present a modal method for the fast analysis of 2D-layered gratings. It combines exact discrete formulations of Maxwell equations in 2D space with polynomial approximations of the constitutive equations, and provides a sparse formulation of the eigenvalue equations. In specific cases, the use of sparse matrices allows us to calculate the electromagnetic response while solving only a small fraction of the eigenmodes. This significantly increases computational speed up to 100×, as shown on numerical examples of both dielectric and metallic subwavelength gratings.

Three‐dimensional eddy current analysis in unbounded domains by a DEM‐BEM formulation

PurposeTo propose a novel 3D hybrid approach, based on a discrete formulation of Maxwell equations (the cell method – CM), suitable for solving eddy current problems in unbounded domains.Design/methodology/approachField equations for magnetodynamics are expressed directly in algebraic form thanks to the CM. The eddy current problem inside bulk conductors is formulated in terms of discrete modified vector potential, whereas magnetic scalar potential is used in order to model the free space. The CM is coupled to the boundary element method by using a surface boundary operator, which maps the surface magnetic fluxes to the surface magnetic scalar potentials. This leads to a unique set of linear equations to be solved in terms of discrete potentials. The eddy currents in bulk conductors are then obtained from discrete potentials.FindingsIt is shown that formulation of hybrid approaches can be simplified by expressing field equations directly in algebraic form without need of weighted residual techniques. An original strategy, based on Green's formula for the magnetic scalar potential, is proposed in order to couple conducting parts to the exterior domain.Research limitations/implicationsConducting bodies with multiply connected parts cannot be modelled by the proposed approach, since it is based on the magnetic scalar potential. The resulting global matrix is partially dense and non‐symmetric; therefore, standard iterative solvers such as GMRES have to be used.Practical implicationsThe proposed approach can be suitably used for analyzing eddy current problems involving models with high degree of complexity, large air domains and moving parts. These are typical of induction heating processes.Originality/valueThis paper proposes a new 3D hybrid approach, based on a discrete formulation of Maxwell equations. A novel coupling strategy relying on integral electromagnetic variables, i.e. magnetic fluxes and magnetic scalar potentials, is devised in order to solve uniquely for eddy currents inside conducting bodies.

A surface integral formulation of Maxwell equations for topologically complex conducting domains

A general and effective method is presented to numerically solve the electric field integral equation (EFIE) for topologically complex conducting domains by the finite element method. A new technique is proposed to decompose the surface current density into a solenoidal part and a nonsolenoidal remainder to avoid the low frequency breakdown. The surface current density field is approximated through div-conforming (facet) elements. The solenoidal part is represented through the null space of the discrete approximation D of the surface divergence operator in the subspace spanned by the facet elements, whereas the nonsolenoidal remainder is represented through its complement. The basis functions of the null space and its complement are evaluated, respectively, by the null and pseudo-inverse of the matrix D. The completeness of the null-pinv basis functions is studied. Unlike the loop-star and loop-tree basis functions, the null-pinv basis functions allow to deal with topologically complex conducting domains in a general and readily applicable way. A topological interpretation of the "null-pinv" decomposition is given and a general and simple method to evaluate the null and pseudo-inverse of D is proposed. The computational complexity of the proposed method is discussed.

Analysis of the performance of a multi-stage pulsed linear induction launcher

In this paper the performance of a multistage pulsed linear induction launcher (PLIL) is analyzed. Attention is focused on the phenomena occurring during the transition between two adjacent stages and on the effects of the coaxial deviation on the motion of the sleeve. The possibility of contacts between the sleeve and the flyway tube and their dynamics is investigated. The analysis has been performed by means of an integral formulation of Maxwell equations that results in an equivalent time varying network. Mechanical coupling has been taken into account assuming a rigid sleeve with three degrees of freedom; the first two are related to the motion of the mass center and the third one is related to rotation around it. Thermal coupling has been considered assuming that the ohmic losses are dissipated adiabatically in the stator windings and in the sleeve. The model has been tested by comparison with results obtained by other researchers in the analysis of a single stage pulsed induction launcher. The analysis of a capacitor-driven five stage launcher has been performed. As the speed of the barrel increases the armature capture effect becomes more evident; the number of contacts with the flyway tube increases and the impacts happen at higher velocity.

The group of external transformations of electromagnetic field variables

A quaternion formulation of Maxwell equations in a medium is given. Adding gradient terms to these equations, they become invariant under the transformations of field variables forming a representation of the group S0(3, 3) — the group of external transformations of electromagnetic field variables. A discussion of possible physical applications of the generalized equations of macroscopic electrodynamics is given.

The eriksenlike form of the foldy-wouthuysen transformation for the photon

The Foldy-Wouthuysen transformation for free or interacting photons is obtained in an Eriksenlike form from a (6× 6) real formulation of Maxwell equations.

Sur l'analogie formelle des équations de Maxwell et des équations de Dirac à masse nulle

A formulation of Maxwell equations, analogous to Moses formulation 1), but using true Dirac matrices, is considered. It is shown how the transformation of the wave function ensuring the invariance of Maxwell theory for the Lorentz group is related to the transformation of a Dirac spinor.