Poynting Theorem Research Articles

Equations of Motion in the Theory of Relativistic Vector Fields

Within the framework of the theory of relativistic vector fields, the covariant expressions are presented for the equations of motion of the matter and the field. These expressions can be written either in terms of the field tensors, that is, the fields’ strengths and solenoidal vectors, or in terms the four-potentials, that is, the fields’ scalar and vector potentials. This state of things is due to the fact that the Lagrange function initially implied the complementarity of description in terms of the strengths and the field potentials. It is shown that the equation for the fields, obtained by taking the covariant derivative in the equation for the metric, has a deeper meaning than the ordinary equation of motion of the matter, found with the help of the principle of least action. In particular, the above-mentioned equation for the fields leads to the generalized Poynting theorem, and after integration over the volume it allows us to introduce for consideration the integral vector as a measure of the energy and the fields’ energy fluxes, associated with a system of particles and fields.

Equations of Motion in the Theory of Relativistic Vector Fields

Within the framework of the theory of relativistic vector fields, the covariant expressions are presented for the equations of motion of the matter and the field. These expressions can be written either in terms of the field tensors, that is, the fields’ strengths and solenoidal vectors, or in terms the four-potentials, that is, the fields’ scalar and vector potentials. This state of things is due to the fact that the Lagrange function initially implied the complementarity of description in terms of the strengths and the field potentials. It is shown that the equation for the fields, obtained by taking the covariant derivative in the equation for the metric, has a deeper meaning than the ordinary equation of motion of the matter, found with the help of the principle of least action. In particular, the above-mentioned equation for the fields leads to the generalized Poynting theorem, and after integration over the volume it allows us to introduce for consideration the integral vector as a measure of the energy and the fields’ energy fluxes, associated with a system of particles and fields.

Iterative formulation for the skin effect on cylindrical conductors based on the Feynman's approach

AbstractOnly qualitative aspects on the skin effect in conductors are properly described in the technical literature on power systems applications. Although the qualitative analysis is an important issue, it is not sufficient for a full approach and proper understanding of this electromagnetic phenomenon in power systems, more specifically in power transmission systems. The present work intends to provide a quantitative analysis of the skin effect on conductors based exclusively on the Maxwell Equations, whereby both physical and mathematical treatments of the phenomenon are properly analysed. This approach contrasts with the traditional one that uses a complex formulation based on both the differential Maxwell equations and potential concepts. Finally, a discussion on the application of the Poynting theorem on the external surface of the cylindrical conductor is presented in order to evaluate not only the resistance, but also the inductance as a function of the frequency, which is not properly discussed in the technical literature on power systems.

Quaternionic representation of electromagnetism for material media

In this study, the electromagnetic fields are developed in the presence of both the electric and magnetic induction fields by quaternion algebra. In this sense, the polarization and magnetization effects, which are valid in the material media, gain much importance. Quaternions are one of the most convenient tools for representing electromagnetism with regard to having non-commutative but associative algebraic division ring. By defining the quaternion induction field, the quaternion source term has been obtained in basic and elegant notation for the first time. In addition, one type of Poynting theorem, named as the Minkowski form, has been presented including the permittivity and permeability constants by quaternions.

Stored electromagnetic field energies in general materials

The most general expressions of the stored energies for time-harmonic electromagnetic fields are derived from the time-domain Poynting theorem, and are valuable in characterizing the energy storage and transport properties of complex media. A new energy conservation law for the time-harmonic electromagnetic fields, which involves the derived general expressions of the stored energies, is introduced. In contrast to the well-established Poynting theorem for time-harmonic fields, the real part of the new energy conservation law gives an equation for the sum of stored electric and magnetic field energies; the imaginary part involves an equation related to the difference between the dissipated electric and magnetic field energies. In a lossless isotropic and homogeneous medium, the new energy conservation law has a clear physical implication: the stored electromagnetic field energy of a radiating system enclosed by a surface is equal to the total field energy inside the surface subtracted by the field energy flowing out of the surface.

The Generalized Poynting Theorem for the General Field and Solution of the 4/3 Problem

The generalized Poynting theorem is applied to the equilibrium system of particles, both inside and outside the system. The particles are bound to each other by means of the electromagnetic and gravitational fields, acceleration field and pressure field. As a result, the correlation is found between the acceleration field coefficient, the pressure field coefficient, the gravitational constant and the vacuum permittivity. This correlation also depends on the ratio of the charge density to the mass density of the particles inside the sphere. Due to the correlation between the given field coefficients the 4/3 problem is solved and the expression for the relativistic energy of the system is refined.

Comparison between Lambert and Maxwell Approaches in the Modelling of Microwave Heating of Liquid Foods

Microwave heating of liquid foods in laminar flow through a circular tube has been modeled. In particular, skim milk as a Newtonian fluid and apple sauce and tomato sauce as non-Newtonian fluids have been considered. The temperature profiles have been obtained solving the motion and energy equations in transient regime and Maxwell’s equations in the frequency domain. Numerical resolution of Finite Element Method has been implemented in Comsol Multiphysics. The generation term due to the microwave heating has been evaluated according both to Lambert’s law and Poynting theorem. Finally, a comparison between the two methods has been made in order to check to what extent the results obtained with the simpler Lambert’s law approximation are comparable with those deriving from the exact solution of Maxwell equations. Dielectric properties are considered to be temperature dependent.

Analytical Computation for AC Resistance and Reactance of Electric Machine Windings in Ferromagnetic Slots

Under the effects of alternating current (AC), the resistance and reactance of electric machine windings are different from their static state. As electric machine windings usually comprise many conductors, it makes the modeling and analysis very cumbersome using the finite element method. This paper presents an analytical method for computing AC resistance and reactance of electric machine windings placed in ferromagnetic slots, which is more convenient and efficient, and could be extended to the slots with different complex structures. First, the slot is divided into a number of layers based on the concept of discretization. Then, the magnetic field governing equation is established, and the distributions of magnetic field intensity and the current density of conductors under AC conditions are solved. By Poynting's theorem, the complex power flowing into the conductors is calculated, and then the effective resistance and reactance of the windings are obtained. Finally, the deduced analytical solutions are compared with the finite element method, and the results show that they have a good agreement for both single-layer winding and the double-layer windings with current phase difference. The presented analytical model is useful for understanding the AC effects and optimization of electric machine windings placed in ferromagnetic slots.

The involvement of vaginal microbiome in threatened premature delivery

This investigation shows that an active emitting electromagnetic (EM) Dirichlet source (i.e., with axial polarization of the electric field) in a homogeneous non-dissipative/non-absorptive medium placed near a perfectly conducting boundary can render total invisibility (i.e. zero extinction cross-section or efficiency) in addition to a radiation force cancellation on its surface. Based upon the Poynting theorem, the mathematical expression for the extinction, radiation and amplification cross-sections (or efficiencies) are derived using the partial-wave series expansion method in cylindrical coordinates. Moreover, the analysis is extended to compute the self-induced EM radiation force on the active source, resulting from the waves reflected by the boundary. The numerical results predict the generation of a zero extinction efficiency, achieving total invisibility, in addition to a radiation force cancellation which depend on the source size, the distance from the boundary and the associated EM mode order of the active source. Furthermore, an attractive EM pushing force on the active source directed toward the boundary or a repulsive pulling one pointing away from it can arise accordingly. The numerical predictions and computational results find potential applications in the design and development of EM cloaking devices, invisibility and stealth technologies.

Lagrangian dynamics approach for the derivation of the energy densities of electromagnetic fields in some typical metamaterials with dispersion and loss

The Lagrangians and dissipation functions are proposed for use in the electrodynamics of the double-negative and chiral metamaterials with finite loss. The double-negative metamaterial considered here is the ‘wires and split-rings’ periodic structure, while the chiral metamaterial is the ‘single-resonance helical resonators’ array. For either system, application of Legendre transformation leads to a Hamiltonian density identical to the energy density obtained in our previous work based on the Poynting theorem and the mechanism of the power loss. This coincidence implies the correctness of the energy density formulas we obtained before. The Lagrangian description and Hamiltonian formulation can be further developed for exploring the properties of the elementary excitations or quasiparticles in dispersive metamaterials due to light–matter interactions.

A Design Approach for High-Q FBARs With a Dual-Step Frame.

This paper presents a new approach for the design of film bulk acoustic resonators (FBARs) with high-quality factor ${Q}_{a}$ at antiresonant frequency ${f}_{a}$ . Lamb waves are among the main contributors to the loss of energy at ${f}_{a}$ and the deterioration of ${Q}_{a}$ . The targeted approach is an optimization process of a dual-step frame structure to alleviate the problem with Lamb waves at ${f}_{a}$ . Such a frame is a combination of two neighboring steps that have different acoustic impedances reflecting the two Lamb modes that carry the largest amount of energy. The dimensions of these steps are computed using the diffraction grating method. To do so, the dispersion curves of Lamb waves and their associated power flows are calculated using a global matrix method and the acoustic Poynting's Theorem. This results in an enhancement in ${Q}_{a}$ of nearly 82% for an FBAR with a dual-step frame in comparison with an FBAR with no frame, as well as a smoother electrical response, at frequencies above ${f}_{a}$ but below the resonant frequency ${f}_{r}$ spurious resonances are moderately increased.

Assessing the Time Dependence of Reconnection With Poynting's Theorem: MMS Observations

AbstractWe investigate the time dependence of electromagnetic‐field‐to‐plasma energy conversion in the electron diffusion region of asymmetric magnetic reconnection. To do so, we consider the terms in Poynting's theorem. In a steady state there is a perfect balance between the divergence of the electromagnetic energy flux and the conversion between electromagnetic field and particle energy . This energy balance is demonstrated with a particle‐in‐cell simulation of reconnection. We also evaluate each of the terms in Poynting's theorem during an observation of a magnetopause reconnection region by Magnetospheric Multiscale (MMS). We take the equivalence of both sides of Poynting's theorem as an indication that the errors associated with the approximation of each term with MMS data are small. We find that, for this event, balance between is only achieved for a small fraction of the energy conversion region at/near the X‐point. Magnetic energy was rapidly accumulating on either side of the current sheet at roughly 3 times the predicted energy conversion rate. Furthermore, we find that while and are observed, as is expected for reconnection, the energy accumulation is driven by the overcompensation for by . We note that due to the assumptions necessary to do this calculation, the accurate evaluation of may not be possible for every MMS‐observed reconnection event; but, if possible, this is a simple approach to determine if reconnection is or is not in a steady state.

A Novel Electromagnetic Power‐Based Characteristic Mode for Magnetodielectric Materials

AbstractIn this paper, we focus on the characteristics of electromagnetic power of characteristic modes (CMs) for magnetodielectric materials (MDM). A generalized integral form of Poynting's theorem is proposed. Based on the generalized integral form of Poynting's theorem, an explicit formulation for electromagnetic power of traditional volume integral equation‐based CM and Poggio, Miller, Chang, Harrington, Wu, and Tsai‐based CM of homogeneous MDM is presented for the first time. Furthermore, from the concept of electromagnetic power, a novel electromagnetic power‐based CM for MDM is proposed. Numerical results are given to show that the proposed CM is effective in solving scattering problem.

Electromagnetic energy within an isolated C60 molecule

By using the equation of motion for the dynamic response of the σ and π electrons in the carbon nanostructure, the classical Poynting theorem of conservation of electromagnetic energy is extended so as to be applicable to derive an exact expression for the time-averaged electromagnetic energy within an isolated C60 molecule, which is irradiated by a plane and time-harmonic electromagnetic wave. Numerical results show that the stored energy exhibits resonant enhancement near resonances of extinction cross section of the system.

Active electromagnetic invisibility cloaking and radiation force cancellation

This investigation shows that an active emitting electromagnetic (EM) Dirichlet source (i.e., with axial polarization of the electric field) in a homogeneous non-dissipative/non-absorptive medium placed near a perfectly conducting boundary can render total invisibility (i.e. zero extinction cross-section or efficiency) in addition to a radiation force cancellation on its surface. Based upon the Poynting theorem, the mathematical expression for the extinction, radiation and amplification cross-sections (or efficiencies) are derived using the partial-wave series expansion method in cylindrical coordinates. Moreover, the analysis is extended to compute the self-induced EM radiation force on the active source, resulting from the waves reflected by the boundary. The numerical results predict the generation of a zero extinction efficiency, achieving total invisibility, in addition to a radiation force cancellation which depend on the source size, the distance from the boundary and the associated EM mode order of the active source. Furthermore, an attractive EM pushing force on the active source directed toward the boundary or a repulsive pulling one pointing away from it can arise accordingly. The numerical predictions and computational results find potential applications in the design and development of EM cloaking devices, invisibility and stealth technologies.

The usefulness of Poynting's theorem in magnetic turbulence

Abstract. We rewrite Poynting's theorem, already used in a previous publication Treumann and Baumjohann (2017a) to derive relations between the turbulent magnetic and electric power spectral densities, to make explicit where the mechanical contributions enter. We then make explicit use of the relativistic transformation of the turbulent electric fluctuations to obtain expressions which depend only on the magnetic and velocity fluctuations. Any electric fluctuations play just an intermediate role. Equations are constructed for the turbulent conductivity spectrum in Alfvénic and non-Alfvénic turbulence in extension of the results in the above citation. An observation-based discussion of their use in application to solar wind turbulence is given. The inertial range solar wind turbulence exhibits signs of chaos and self-organization.

Mode Matching for Optical Antennas.

The emission rate of a point dipole can be strongly increased in the presence of a well-designed optical antenna. Yet, optical antenna design is largely based on radio-frequency rules, ignoring, e.g., Ohmic losses and non-negligible field penetration in metals at optical frequencies. Here, we combine reciprocity and Poynting's theorem to derive a set of optical-frequency antenna design rules for benchmarking and optimizing the performance of optical antennas driven by single quantum emitters. Based on these findings a novel plasmonic cavity antenna design is presented exhibiting a considerably improved performance compared to a reference two-wire antenna. Our work will be useful for the design of high-performance optical antennas and nanoresonators for diverse applications ranging from quantum optics to antenna-enhanced single-emitter spectroscopy and sensing.

Force, torque, linear momentum, and angular momentum in classical electrodynamics

The classical theory of electrodynamics is built upon Maxwell's equations and the concepts of electromagnetic (EM) field, force, energy, and momentum, which are intimately tied together by Poynting's theorem and by the Lorentz force law. Whereas Maxwell's equations relate the fields to their material sources, Poynting's theorem governs the flow of EM energy and its exchange between fields and material media, while the Lorentz law regulates the back-and-forth transfer of momentum between the media and the fields. An alternative force law, first proposed by Einstein and Laub, exists that is consistent with Maxwell's equations and complies with the conservation laws as well as with the requirements of special relativity. While the Lorentz law requires the introduction of hidden energy and hidden momentum in situations where an electric field acts on a magnetized medium, the Einstein-Laub (E-L) formulation of EM force and torque does not invoke hidden entities under such circumstances. Moreover, total force/torque exerted by EM fields on any given object turns out to be independent of whether the density of force/torque is evaluated using the law of Lorentz or that of Einstein and Laub. Hidden entities aside, the two formulations differ only in their predicted force and torque distributions inside matter. Such differences in distribution are occasionally measurable, and could serve as a guide in deciding which formulation, if either, corresponds to physical reality.

Sedeonic equations of ideal fluid

In the present paper, we propose the generalized equations for an ideal fluid based on space-time algebra of sixteen-component sedeons. It is shown that the dynamics of isentropic fluid can be described by sedeonic first-order wave equation for fluid potentials. The key features of the proposed formalism are illustrated on the problem of the sound waves propagation. We consider the plane wave solution of linearized sedeonic wave equation and derive the second-order relations for the sound potential analogues to the Poynting theorem in electrodynamics. The generalization of proposed sedeonic equations for the description of viscous fluid is also discussed.

Poynting theorem in terms of beam shape coefficients and applications to axisymmetric, dark and non-dark, vortex and non-vortex, beams

Electromagnetic arbitrary shaped beams may be described by using expansions over a set of basis functions, with expansion coefficients containing sub-coefficients called beam shape coefficients which encode the structure of the beam. In this paper, the Poynting theorem is expressed in terms of these beam shape coefficients. Special cases (axisymmetric, dark and non-dark beams) are thereafter considered, as well as specific applications to paradigmatic examples, from trivial cases (plane waves and spherical waves) to the more sophisticated case of vortex beams.