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A note on the uniqueness of the Poynting vector and of the Maxwell tensor
Abstract
Many alternatives to the classical Poynting vector P have been proposed, but they lack a property of locality that makes P special and makes it the right choice to represent the energy flux. We give a geometrical proof of this uniqueness. Similar considerations apply to the Maxwell stress tensor.
... The discussion of local magnetic pressure is still on going [25] [26] [27] and should be the aim of further work. The VWP is built to account for magnetic force at each node from which the global quantities can be deduced, while the Maxwell tensor gives the momentum flux across any surface, closed or not [25]. ...
... The discussion of local magnetic pressure is still on going [25] [26] [27] and should be the aim of further work. The VWP is built to account for magnetic force at each node from which the global quantities can be deduced, while the Maxwell tensor gives the momentum flux across any surface, closed or not [25]. Historically the Maxwell Tensor has been used by electrical machine designers to accurately compute global quantities such as electromagnetic torque, which is the moment of the global magnetic force applied on a cylinder surrounding the rotor on its axis of rotation. ...
This paper presents a comparison of several methods to compute the magnetic forces experienced by the stator teeth of electrical machines. In particular, the comparison focuses on the virtual work principle (VWP)-based nodal forces and the Maxwell tensor (MT) applied on different surfaces. The VWP is set as the reference. The magnetic field is computed either with finite element analysis or with the semi-analytical subdomain method (SDM). First, the magnetic saturation in iron cores is neglected (linear B-H curve). Then, the saturation effect is discussed in a second part. Homogeneous media are considered and all simulations are performed in 2-D. The link between the slot's magnetic flux and the tangential force harmonics is also highlighted. The comparison is performed on the stator of a surface-mounted permanent-magnet synchronous machine. While the different methods disagree on the local distribution of the magnetic forces at the stator surface, they give similar results concerning the integrated forces per tooth, referred as lumped forces. This conclusion is mitigated for saturated cases: the time harmonics are correctly computed with any of the presented lumped force methods but the amplitude of each harmonic is different between methods. Nonetheless, the use of the SDM remains accurate with MT in the air gap even with saturation for design and diagnostic of electromagnetic noise in electrical machines. However, for more accurate studies based on the local magnetic pressure, the VWP is strongly recommended.
... Very recently, the long-lasting discussion (not to say controversy) on the Maxwell stress as well as magnetic and electric body forces in continua gained momentum again. The interested reader is in this context referred to the contributions[49][50][51][52][53]. ...
For a long time, the search for magneto-electric materials concentrated on multi-ferroics and hard-matter composites. By contrast, rather recently the exploitation of strain-mediated magneto-electric (ME) coupling in soft composites was proposed. The basic idea behind this approach is to combine the magneto- and electro-mechanical responses of composites consisting of a soft matrix carrying magnetic inclusions. Despite that such composites are straightforward to manufacture and have cheap constituents, they did not gain much attention up to now. In this contribution, we demonstrate that ME coupling induced by finite deformations could be of significant magnitude. Our approach relies on shape effects as a special non-local phenomenon in magneto- and electro-elasticity. Based on that we characterize an up to now overlooked ME coupling mechanism which purely relies on these shape effects in soft-matter-based magnetic and electric media. While soft magnetic media are commonly realized as composites, the coupling effect to be highlighted exists independently of the origin of a body's magnetic and electric properties. We show that the magnitude of the effect is indeed significant and, among ellipsoidal bodies, most pronounced for those of spherical to moderately prolate shape. Finite-element simulations are performed to assess the quality of the analytical predictions.
This paper will analyze how the energy flux of Poynting’s vector is compared to the power flow in electrical engineering, where the power, instead, is defined by voltages and currents. There are alternatives to Poynting’s energy flux vector that agree more with circuit theory methods such that the energy flow is in the current conductor and not in the insulation surrounding it. One such basic formulation would only consist of the total current density and the voltage potential, but it would need an alternative theorem for energy transfer. Another formulation proposed by Slepian would instead still agree with Poynting’s energy transfer theorem, but it needs to add the power of alternating magnetic vector potential. The alternatives to Poynting’s vector may better illustrate the energy flow in electrical engineering, but two things could be considered in their generality. First, since they are expressed by potentials, they are gauge invariant and depend on the definition of the potentials. Second, Poynting’s vector is used to formulate the electromagnetic momentum, and any alternative energy flow vectors would not. These two notes are of minor importance in electrical engineering, and the alternatives could be used as good alternatives for describing power flow. The main purpose of this paper is to bridge the differences between the physical theory of energy flux and the methods in electrical power engineering. This could simplify the use of energy flux and Poynting’s vector in engineering problems.
A plea for the introduction, in advanced electromagnetics courses, of some basic differential geometric notions: covectors, differential forms, Hodge operators. The main advantages of this evolution should be felt in computational electromagnetism. It may also shed some new light on the concept of material isotropy.
As Oliver Heaviside observed, “However mysterious energy (and its flux) may be in some of its theoretical aspects, there must
be something in it, because it is convertible into dollars, the ultimate official measure of value.”22 The ability to track electromagnetic energy is helpful in designing antennas with low reactive field energy. Such antennas
would be highly efficient with a low Q and broad bandwidth — ideal for UWB-SP applications. Electromagnetic energy localization
is thus a valuable technique for UWB-SP antenna design.
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When two magnets are stuck together, where do magnetic forces operate and which formulas should one apply to compute them? Such frequently asked questions do not find immediate answers in the literature on forces, mainly because the force field is obtained, by the Virtual Power Principle (VPP), as a (mathematical, vector-valued) distribution, not as a plain vector field, which would be more convenient for practical computation. We intend to show how to represent this single distribution by two vector fields, one of them borne by the bulk of the matter, the other one localized at material interfaces where discontinuities of permeability, of magnetization, etc., do occur. Suitable extensions of the classical Maxwell tensor play an important role in the computation (by the so-called “pillbox trick”) of this interface vector field.
Poynting’s choice for the energy flow vector of the electromagnetic field has certain unattractive physical features. In order to eliminate such features Hines proposed an alternative choice. Here we show that Hines’s choice does not lead to Larmor’s result for the rate of radiation by an accelerated nonrelativistic charge.
The gauge presented here, which we call the Poincaré gauge, is a generalization of the well-known expressions phi = -r.E0 and A = 1/2 B0×r for the scalar and vector potentials which describe static, uniform electric and magnetic fields. This gauge provides a direct method for calculating a vector potential for any given static or dynamic magnetic field. After we establish the validity and generality of this gauge, we use it to produce a simple and unambiguous method of computing the flux linking an arbitrary knotted and twisted closed circuit. The magnetic flux linking the curve bounding a Möbius band is computed as a simple example. Arguments are then presented that physics students should have the opportunity of learning early in their curriculum modern geometric approaches to physics. (The language of exterior calculus may be as important to future physics as vector calculus was to the past.) Finally, an appendix illustrates how the Poincaré gauge (and others) may be derived from Poincaré's lemma relating exact and closed exterior differential forms.
Momentum conservation and the validity of the center-of-mass law are examined for systems made up of electrostatic charges and magnets, in terms of the requirements of special relativity theory. The approach used is that of a quasimicroscopic electromagnetic theory, in which the interaction of the field with material bodies is described by using models for these bodies which involve charge and current densities. It is shown that if small ``electromagnetic mass'' terms are neglected, both conservation of momentum and the center-of-mass law hold in all cases. In some cases a ``hidden momentum'' contained in stationary matter plays an important role, as pointed out recently by Shockley and James [W. Shockley and R. P. James, Phys. Rev. Lett. 18, 876 (1967).] In such cases the center-of-mass law can fail in nonrelativistic theory. This is illustrated by the discussion of a special model. Another case in which a ``hidden momentum'' required by relativity theory is important is the explanation of the null result of the Trouton-Noble experiment. This is discussed in Sec. V.
It is shown in a very simple manner how a momentum density may be associated with an electromagnetic field.
An alternative choice for the energy flow vector of the electromagnetic field is proposed by discarding a term ∇×(phi H) from the Poynting vector. This new choice for the energy flow vector is in harmony with our ordinary intuitions, and yields Larmor's radiation formula for an accelerated charge.
nature of this paper renders its adequate abstraction difficult. The principle of conservation of energy, when applied to a theory such as Maxwell’s, which postulates the definite localisation of energy, takes a more special form, viz., that of the continuity of energy. Its general nature is discussed.
Even in static fields, where there is no observable energy flow, Poynting vector momentum must be considered to avoid an apparent violation of the angular-momentum law. This often-neglected aspect of the Poynting vector, is illustrated in an easily calculated example. Two other simple and rigorously solvable pedagogical examples illustrate the role of the Poynting vector in defining the energy flow in static fields.
A space containing electric currents may be regarded as a field where energy is transformed at certain points into the electric and magnetic kinds by means of batteries, dynamos, thermoelectric actions, and so on, while in other parts of the field this energy is again transformed into heat, work done by electromagnetic forces, or any form of energy yielded by currents. Formerly a current was regarded as something travelling along a conductor, attention being chiefly directed to the conductor, and the energy which appeared at any part of the circuit, if considered at all, was supposed to be conveyed thither through the conductor by the current. But the existence of induced currents and of electromagnetic actions at a distance from a primary circuit from which they draw their energy, has led us, under the guidance of Faraday and Maxwell, to look upon the medium surrounding the conductor as playing a very important part in the development of the phenomena. If we believe in the continuity of the motion of energy, that is, if we believe that when it disappears at one point and reappears at another it must have passed through the intervening space, we are forced to conclude that the surrounding medium contains at least a part of the energy, and that it is capable of transferring it from point to point. Upon this basis Maxwell has investigated what energy is contained in the medium, and he has given expressions which assign to each part of the field a quantity of energy depending on the electromotive and magnetic intensities and on the nature of the matter at that part in regard to its specific inductive capacity and magnetic permeability. These expressions account, as far as we know, for the whole energy. According to Maxwell’s theory, currents consist essentially in a certain distribution of energy in and around a conductor, accompanied by transformation and consequent movement of energy through the field.
In an effort to clarify the role of surface charges on the conductors of
elementary electric circuits and the electric fields in the space around
them, we present a quantitative analysis of (two-dimensional) circular
current loops. It is also noted that, in general, lines of Poynting flux
lie in the equipotential surfaces of quasistatic systems.
A recently described “paradox” in classical electromagnetictheory is resolved by recognizing the fact that even a staticelectromagnetic field may have angular momentum. A charged particle, emitted from a source in a magnetic field, acquires a readily calculable angular momentum under the influence of the Lorentz force. The angular momentum of the static field after the emission of the particle can be independently calculated, and differs from its previous value by an amount which is exactly the negative of the angular momentum acquired by the particle. If the magnetic field is subsequently turned off, the field angular momentum which disappears is found to be exactly accounted for by changes in mechanical angular momenta. These results, although hardly surprising, provide a very clear illustration of the reality of field angular momentum, especially since the geometry is so simple that exact calculations can readily be made.
1. The remarkable experimental work of late years has inaugurated a new era in the development of the Faraday-Maxwellian theory of the ether, considered as the primary medium concerned in electrical phenomena—electric, magnetic, and electromagnetic. Maxwell’s theory is no longer entirely a paper theory, bristling with unproved possibilities. The reality of electromagnetic waves has been thoroughly demonstrated by the experiments of Hertz and Lodge, Fitzgerald and Trouton, J. J. Thomson, and others; and it appears to follow that, although Maxwell’s theory may not be fully correct, even as regards the ether (as it is certainly not fully comprehensive as regards material bodies), yet the true theory must be one of the same type, and may probably be merely an extended form of Maxwell’s. No excuse is therefore now needed for investigations tending to exhibit and elucidate this theory, or to extend it, even though they be of a very abstract nature. Every part of so important a theory deserves to be thoroughly examined, if only to see what is in it, and to take note of its unintelligible parts, with a view to their future explanation or elimination.
The non-existence of divergenceless supplements to Poynting's vector and the localization of energy and momentum densities follow from demonstration that conventional electrodynamics uniquely predicts the operationally-defined localized densities conceptually measurable by utilizing idealized limits of pulsed test-current distributions.
The making of the two-dimensional printed circuit type models of current-carrying conducting systems and the use of these models for demonstrating the electric fields of current-carrying conductors is described. The models are produced by drawing the systems under consideration on glass plates using a transparent conducting ink. The electric lines of force inside and outside the elements of these models are demonstrated with the aid of grass seeds strewed upon them. © 1962, American Association of Physics Teachers. All rights reserved.
Although Poynting's theorem receives general acceptance in the treatment of electromagnetic energy, an alternative theorem, Macdonald's, has equal claim to validity at the present state of our knowledge. In a comparison of many results derived from the two, Macdonald's theorem exhibits sufficient superiority to warrant its consideration more generally in electromagnetics.
Both, the kinematics and the statics of the mechanical state of a solid can be represented in the language of differential geometry. If the state is defect-free then the geometry is, in general, Riemannian. The above statements are proved for the general nonlinear case. In particular it is argued that stress function tensor and stress tensor play the role of the metric and Einstein tensor, respectively, of the stress space whereas the equilibrium conditions for forces and moments (in absence of volume forces and moments) are the Bianchi identies of the stress space. Extension to solids with defects is possible.Sowohl die Kinematik als auch die Statik des mechanischen Zustands eines Festkörpers lassen sich in der Sprache der Differentialgeometrie darstellen. Wenn der Zustand defektfrei ist, ist die Geometrie im allgemeinen eine Riemannsche Geometrie. Die obigen Behauptungen werden für den allgemeinen nichtlinearen Fall bewiesen. Insbesondere wird gezeigt, daß Spannungsfunktions-tensor und Spannungstensor die Rolle des metrischen bzw. Einsteintensors des Spannungsraumes spielen, während die Gleichgewichtsbedingungen für die Kräfte und Momente (in Abwesenheit von Volumenkräften und Momenten) die Bianchi-Identitäten des Spannungsraumes sind. Erweiterung auf Festkörper mit Defekten ist möglich.
Methods of calculating and measuring the flow of electromagnetic energy are compared and contrasted. The differences between the low-frequency and high-frequency approaches to energy flow problems are discussed and suggestions are made to ease the difficulties in the way of students and teachers faced with these apparently irreconcilable differences.
A new method for deriving the usual Poynting vector is given. This method also yields other equally valid energy flow vectors with the same or other equally valid postulated electromagnetic energy densities. Examples of such alternative Poynting vectors with their associated energy densities are given. A definition is given which distinguishes between the conduction and displacement currents in matter. It is shown that by addition of a term, the postulated energy flow widely used by electrical power engineers and based on wattmeter readings becomes a valid alternative Poynting vector.
The conditions which a valid postulated electric-energy flow must satisfy are given and are stated to be insufficient for its unique determination. The commonly used Vi energy-flow postulate is shown by examples to be not generally valid, but by adding a simple term it can be made equally valid with other valid energy-flow postulates. Various examples are given of the application of this corrected energy-flow postulate. On power systems the engineer commonly limits his use of the uncorrected Vi postulate to applications where the correcting term should have a negligible net effect. Various examples of such use are discussed.
Purpose
The paper aims at proposing a uniform and demonstrative description of two well‐known and widely used approximations of slowly time‐varying electromagnetic fields, i.e. the electro‐quasistatic and the magneto‐quasistationary approximation to Maxwell's equations.
Design/methodology/approach
Under both approximations, the orders of magnitude of the relative errors of the dominant fields are analyzed by using three characteristic time constants. These time constants are determined by considering the material properties, the characteristic length scale and the characteristic time scale.
Findings
Limiting curves which show the domains of applicability of the two approximations are retrieved from the estimation of their relative errors. The relation between the domains of validity of the electro‐quasistatic and magneto‐quasistationary approximations was found and depicted in a combined diagram.
Research limitations/implications
The study is restricted to slowly time‐varying electromagnetic fields. Heuristic and local estimates based on local material properties were used for the analysis. Rigorous estimations of the errors (e.g. also considering the field problem's topology) of the magneto‐quasistationary approximation are already known in the literature. A rigorous estimation of the error of the electro‐quasistatic approximation is, therefore, suggested for future research.
Originality/value
The combined diagram showing the domains of validity of both approximations considered here in a uniform way is novel. It gives rise to an intuitive and easily accessible understanding of their applicability.
Lorenz' work is remarkable, since it was performed in parallel with that of Maxwell. The reader will probably agree that the paternity suit must be decided in Lorenz' favour. it appears that the various authors of textbooks whi sinned against historical accuracy–the undersigned being regretfully one of them–should amend their references in future printings of their books!
Considerations on the Concept of Poynting Vector Contribution to Power Theory Development
- L S Czarnecki
L.S. Czarnecki: "Considerations on the Concept of Poynting Vector Contribution to Power Theory Development",
Sixth Int. Wkshp on Power Definitions, Milan (2003).
Domains of validity of quasistatic and quasistationary field approximations
- T Steinmetz
- S Kurz
- M Clemens
T. Steinmetz, S. Kurz, M. Clemens: "Domains of validity of quasistatic and quasistationary field approximations",
COMPEL, 30, 4 (2011), pp. 1237-47.
Considerations on the concept of Poynting vector contribution to power theory development.Sixth Int. Wkshp on Power Definitions Milan
- Czarneckils