DOCSLIB.ORG

Electromagnetic Force and Torque in Lorentz and Einstein-Laub

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Electromagnetic Force and Torque in Lorentz and Einstein-Laub Electromagnetic force and torque in Lorentz and Einstein-Laub formulations Masud Mansuripur College of Optical Sciences, The University of Arizona, Tucson, Arizona 85721 [Published in Optical Trapping and Optical Micromanipulation XI, K. Dholakia and G.C. Spalding, eds., Proceedings of SPIE Vol. 9164, 91640B~1-16 (2014). DOI: 10.1117/12.2060554] The Lorentz force law of classical electrodynamics requires the introduction of hidden energy and hidden momentum in situations where an electric field acts on a magnetic material. In contrast, the Einstein-Laub formulation does not invoke hidden entities. The total force and torque exerted by electromagnetic fields on a given object are independent of whether the force and torque densities are 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 throughout material media. 1. Introduction. 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 the Lorentz force law. Whereas Maxwell’s macroscopic equations relate the electric and magnetic fields to their material sources (i.e., charge, current, polarization and magnetization), 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. As it turns out, an alternative force law, first proposed in 1908 by Einstein and Laub, exists that is consistent with Maxwell’s macroscopic 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 magnetic material, the Einstein-Laub formulation of EM force and torque does not invoke hidden entities under such circumstances. Moreover, the total force and the total torque exerted by EM fields on any given object turn out to be independent of whether force and torque densities are evaluated using the Lorentz law or obtained in accordance with the Einstein-Laub formulas. Hidden entities aside, the two formulations differ only in their predicted force and torque distributions throughout material media. Such differences in distribution are occasionally measurable, and could serve as a guide in deciding which formulation, if either, corresponds to physical reality. It has been suggested that magnetic dipoles cannot exist in classical physics. A famous argument in support of this assertion is given in Niels Bohr’s PhD dissertation (1911).1 The argument is now known as the Bohr-van Leeuwen theorem. Feynman also mentions this point briefly in his Lectures on Physics.2 We believe this is a consequence of accepting Maxwell’s so- called microscopic equations as the fundamental laws of electrodynamics, in which case Ampèrian current loops become the standard model for magnetic dipoles; it is, in fact, the existence and properties of such stable current loops that spell trouble for magnetism in classical physics. If, instead, one adopts Maxwell’s macroscopic equations as fundamental (i.e., equations that admit, in addition to electric charge and current, both electric and magnetic dipoles as basic building blocks of matter), then classical physics will have incorporated electric and magnetic dipoles without much ado—simply by fiat. One will have to make a substantial adjustment in his/her view of Maxwell’s macroscopic equations in order to arrive at such a conclusion. Once the adjustment is made, however, classical electrodynamics will be enriched and strengthened. Elsewhere we have shown the self-consistency of this viewpoint as well as its consistency with the conservation laws and with special relativity.3-8 It is from this perspective that the Einstein- Laub force and torque equations (rather than the Lorentz force law) can be considered as foundational postulates of classical electrodynamics. 1 In this view, polarization ( , ) and magnetization ( , ) are interpreted as the densities of electric and magnetic dipoles at different locations in space-time, with the implicit assumption that individual dipoles are point푷-dipoles.풓 푡 That way, one can푴 fill풓 푡 every region of space, no matter how small, with as few or as many dipoles as one may desire. The density of each kind of dipole can thus be an arbitrary function of the position coordinate, . 2. Maxwell’s macroscopic equations. We begin our examination풓 of Maxwell’s macroscopic 9 equations with his first equation, also known as Gauss’s law, namely, ( , ) = ( , ). (1) Since, by definition, free 휵 ∙ 푫 풓 푡 휌 풓 푡 ( , ) = ( , ) + ( , ), (2) where is the permittivity of푫 free풓 푡space,휀 0Eq푬 .풓(1)푡 may푷 be풓 written푡 as = 휀0 . (3) ( , ) ( , ) Therefore, one is justified in휀 0treating휵 ∙ 푬 휌free − 휵 ∙ 푷 as the bound-charge-density associated with electric dipoles. This is also exactly what has traditionally been taken to be the bound bound charge-density of electric dipoles.2,9−,10휵 ∙Aside푷 풓 푡 from the fact that we assume 휌 ( , ) 풓can푡 represent any arbitrary vector function of space-time, it is clear that the divergence of this function produces a bound charge-density, in the same way that one has always as푷sumed풓 푡 that polarization should. This does not mean that the bound charge-density is an arbitrary function of space-time; it is the divergence of another (arbitrary) function, but the bound charge-density is not arbitrary. Together with the corresponding bound-current-density, ( , ) = / (introduced below), they satisfy the charge-current continuity equation, + / = 0. bound Next we examine Maxwell’s second (macroscopic) equation, also known푱 as 풓the푡 Maxwell휕푷 휕푡- Ampère equation:9 휵 ∙ 푱 휕휌 휕푡 × = + / . (4) This can be written as 휵 푯 퐽free 휕푫 휕푡 × = ( + / + × ) + (1/ ) / , (5) −1 2 where, by definition, 0 free 휵 푩 휇 푱 휕푷 휕푡 휇0 휵 푴 푐 휕푬 휕푡 ( , ) = ( , ) + ( , ). (6) Here is the permeability푩 풓of 푡free space휇0푯 풓and푡 =푴1풓 푡 is the speed of light in vacuum. / × The two contributions0 to the bound current-density are seen0 0to be and . Of course the휇 charge-current continuity equation is satisfied푐 ⁄ �because휇 휀 of the way / −1 and 0 are both related to the same (albeit arbitrary) function ( , ), and also휕푷 because휕푡 휇 (휵 ×푴 ) is always zero. Again, ( , ) may be an arbitrary function of the space-time, but휕푷 its휕푡 contribution−휵 ∙ 푷 to bound electrical charge (zero) and bound electrical 푷current풓 푡 ( × ) satisfy휵 ∙ the휵 charge푴 - current continuity equation.푴 풓 푡 −1 0 The above arguments reveal that, once one accepts the 휇functions휵 푴 ( , ) and ( , ) appearing in the macroscopic equations as representing arbitrary distributions of electric and magnetic dipole densities, there will be no need to introduce additional 푷bound풓 푡 charges푴 풓and푡 currents. The vector functions ( , ) and ( , ), precisely because of the way that they appear 푷 풓 푡 푴 풓 푡 2 in Maxwell’s macroscopic equations, give rise to the same bound-charge and bound-current densities as have always been associated with polarization and magnetization. Maxwell’s third equation, × = , also known as Faraday’s law,9 does not require any reinterpretation, nor does the fourth equation, = 0. Together with Eqs.(3) and (5), these equations relate 휵 =푬 − 휕푩⁄휕푡 and = + / + × to ( , ), the electric field at an arbitrary location ( , ), and휵 ∙to푩 ( , ), the local, instantaneous−1 total free total free 0 value of the magnetic induction,휌 which휌 is −related휵 ∙ 푷 to the푱 -field 푱via Eq휕.(6)푷 .휕푡 Alternatively,휇 휵 푴 one may푬 풓 write푡 the macroscopic equations in the following,풓 푡 equally valid푩 풓 way:푡 퐻 = , (7) × = ( + / ) + / , 휀0휵 ∙ 푬 휌free − 휵 ∙ 푷 (8) × = 휵 푯 푱free 휕푷 휕푡 휀0휕푬 ,휕푡 (9) = 휵 푬 − 휕푴⁄휕푡 − 휇0 휕. 푯⁄휕푡 (10) Clearly, the above equations 0now relate the total electric-charge and electric-current ( ) 휇 휵( ∙)푯 −휵 ∙ 푴 densities = and = + / as well as the total magnetic-charge ( ) ( ) and magnetice -current densities e = and = / to ( , ) and ( , ). 휌total 휌free − 휵 ∙ 푷 푱total 푱free 휕푷 휕푡 Neither set of equations is superiorm to the other. Whilem electric dipoles in both cases are represented by electric charges 휌andtotal currents,−휵 ∙ 푴magnetic푱 totaldipoles휕 푴appear휕푡 as푬 Ampèrian풓 푡 electric푯 풓 푡 - current-loops in the first, and as Gilbertian magnetic charges and currents in the second case. 3. Poynting’s theorem. Maxwell’s macroscopic equations yield two different, albeit closely related, versions of the Poynting theorem.2,9-11 The first version is obtained by dot-multiplying and into the Maxwell-Ampère and Faraday equations, respectively, then subtracting one equation from the other. We will have 푬 푯 ( × ) + (½ + ½ ) + + + = 0. (11) 휕 휕푷 휕푴 According휵 ∙ 푬 to 푯the above휕푡 휀vers0푬 ∙ion푬 of the휇0푯 theorem,∙ 푯 푬 the∙ � 푱freePoynting휕푡 � vector푯 ∙ 휕푡is = × , the stored energy-density in the fields is = ½ + ½ , and the rates of EM energy 1 exchange between fields, on the one hand, and free currents and electric and magnetic푺 푬 dipoles푯 , 1 0 0 on the other hand, are , ℰ , and 휀 푬 ∙ 푬 , 휇respectively.푯 ∙ 푯 To obtain the second version of Poynting’s theorem, we dot-multiply and into the free Maxwell-Ampère and 푬Faraday∙ 푱 equations,푬 ∙ 휕푷⁄휕푡 respectively,푯 ∙ 휕푴⁄ then휕푡 subtract one from the other to find11 푬 푩 ( × ) + (½ + ½ ) + + + × = 0. (12) −1 휕 −1 휕푷 −1 Here휵 ∙ 휇 0the푬 Poynting푩 휕푡 vector휀0푬 is∙ 푬 = 휇0 푩 ×∙ 푩 , the푬 ∙ stored�푱free energy휕푡 휇-density0 휵 푴in� the fields is = ½ + ½ , and the rates− 1of EM energy exchange between the fields and the 2 0 material media are −1 , 푺 , and휇 푬 푩 × . 2 0 0 ℰ There휀 푬 are∙ 푬 obvious휇 푩differences∙ 푩 between the−1 above versions of Poynting’s theorem. The free 0 magnetic field energy푬 ∙ is푱 stored푬 ∙ 휕in푷 the⁄휕푡 -field푬 i∙n휇 the휵 first푴 version, and in the -field in the second.
Load more

Recommended publications
  • Harvard Physics Circle Lecture 14: Magnetism, Biot-Savart, Ampere’S Law
    Harvard Physics Circle Lecture 14: Magnetism, Biot-Savart, Ampere’s Law Atınç Çağan Şengül January 30th, 2021 1 Theory 1.1 Magnetic Fields We are dealing with the same problem of how charged particles interact with each other. We have a group of source charges and a test charge that moves under the influence of these source charges. Unlike electrostatics, however, the source charges are in motion. One of the simplest experiments one can do to gain insight on how magnetism works is observing two parallel wires that have currents flowing through them. The force causing this attraction and repulsion is not electrostatic since the wires are neutral. Even if they were not neutral, flipping the wires would not flip the direction of the force as we see in the experiment. Magnetic fields are what is responsible for this phenomenon. A stationary charge produces only and electric field E~ around it, while a moving charge creates a magnetic field B~ . We will first study the force acting on a charge under the influence of an ambient magnetic field, before we delve into how moving charges generate such magnetic fields. 1 1.2 The Lorentz Force Law For a particle with charge q moving with velocity ~v in a magnetic field B~ , the force acting on the particle by the magnetic field is given by, F~ = q(~v × B~ ): (1) This is known as the Lorentz force law. Just like F = ma, this law is based on experiments rather than being derived. Notice that unlike the electrostatic version of this law where the force is parallel to the electric field (F~ = qE~ ), here, the force is perpendicular to both the velocity of the particle and the magnetic field.
    [Show full text]
  • On the History of the Radiation Reaction1 Kirk T
    On the History of the Radiation Reaction1 Kirk T. McDonald Joseph Henry Laboratories, Princeton University, Princeton, NJ 08544 (May 6, 2017; updated March 18, 2020) 1 Introduction Apparently, Kepler considered the pointing of comets’ tails away from the Sun as evidence for radiation pressure of light [2].2 Following Newton’s third law (see p. 83 of [3]), one might suppose there to be a reaction of the comet back on the incident light. However, this theme lay largely dormant until Poincar´e (1891) [37, 41] and Planck (1896) [46] discussed the effect of “radiation damping” on an oscillating electric charge that emits electromagnetic radiation. Already in 1892, Lorentz [38] had considered the self force on an extended, accelerated charge e, finding that for low velocity v this force has the approximate form (in Gaussian units, where c is the speed of light in vacuum), independent of the radius of the charge, 3e2 d2v 2e2v¨ F = = . (v c). (1) self 3c3 dt2 3c3 Lorentz made no connection at the time between this force and radiation, which connection rather was first made by Planck [46], who considered that there should be a damping force on an accelerated charge in reaction to its radiation, and by a clever transformation arrived at a “radiation-damping” force identical to eq. (1). Today, Lorentz is often credited with identifying eq. (1) as the “radiation-reaction force”, and the contribution of Planck is seldom acknowledged. This note attempts to review the history of thoughts on the “radiation reaction”, which seems to be in conflict with the brief discussions in many papers and “textbooks”.3 2 What is “Radiation”? The “radiation reaction” would seem to be a reaction to “radiation”, but the concept of “radiation” is remarkably poorly defined in the literature.
    [Show full text]
  • The Lorentz Force
    CLASSICAL CONCEPT REVIEW 14 The Lorentz Force We can find empirically that a particle with mass m and electric charge q in an elec- tric field E experiences a force FE given by FE = q E LF-1 It is apparent from Equation LF-1 that, if q is a positive charge (e.g., a proton), FE is parallel to, that is, in the direction of E and if q is a negative charge (e.g., an electron), FE is antiparallel to, that is, opposite to the direction of E (see Figure LF-1). A posi- tive charge moving parallel to E or a negative charge moving antiparallel to E is, in the absence of other forces of significance, accelerated according to Newton’s second law: q F q E m a a E LF-2 E = = 1 = m Equation LF-2 is, of course, not relativistically correct. The relativistically correct force is given by d g mu u2 -3 2 du u2 -3 2 FE = q E = = m 1 - = m 1 - a LF-3 dt c2 > dt c2 > 1 2 a b a b 3 Classically, for example, suppose a proton initially moving at v0 = 10 m s enters a region of uniform electric field of magnitude E = 500 V m antiparallel to the direction of E (see Figure LF-2a). How far does it travel before coming (instanta> - neously) to rest? From Equation LF-2 the acceleration slowing the proton> is q 1.60 * 10-19 C 500 V m a = - E = - = -4.79 * 1010 m s2 m 1.67 * 10-27 kg 1 2 1 > 2 E > The distance Dx traveled by the proton until it comes to rest with vf 0 is given by FE • –q +q • FE 2 2 3 2 vf - v0 0 - 10 m s Dx = = 2a 2 4.79 1010 m s2 - 1* > 2 1 > 2 Dx 1.04 10-5 m 1.04 10-3 cm Ϸ 0.01 mm = * = * LF-1 A positively charged particle in an electric field experiences a If the same proton is injected into the field perpendicular to E (or at some angle force in the direction of the field.
    [Show full text]
  • The Lorentz Law of Force and Its Connections to Hidden Momentum
    The Lorentz force law and its connections to hidden momentum, the Einstein-Laub force, and the Aharonov-Casher effect Masud Mansuripur College of Optical Sciences, The University of Arizona, Tucson, Arizona 85721 [Published in IEEE Transactions on Magnetics, Vol. 50, No. 4, 1300110, pp1-10 (2014)] Abstract. The Lorentz force of classical electrodynamics, when applied to magnetic materials, gives rise to hidden energy and hidden momentum. Removing the contributions of hidden entities from the Poynting vector, from the electromagnetic momentum density, and from the Lorentz force and torque densities simplifies the equations of the classical theory. In particular, the reduced expression of the electromagnetic force-density becomes very similar (but not identical) to the Einstein-Laub expression for the force exerted by electric and magnetic fields on a distribution of charge, current, polarization and magnetization. Examples reveal the similarities and differences among various equations that describe the force and torque exerted by electromagnetic fields on material media. An important example of the simplifications afforded by the Einstein-Laub formula is provided by a magnetic dipole moving in a static electric field and exhibiting the Aharonov-Casher effect. 1. Introduction. The classical theory of electrodynamics is based on Maxwell’s equations and the Lorentz force law [1-4]. In their microscopic version, Maxwell’s equations relate the electromagnetic (EM) fields, ( , ) and ( , ), to the spatio-temporal distribution of electric charge and current densities, ( , ) and ( , ). In any closed system consisting of an arbitrary distribution of charge and 푬current,풓 푡 Maxwell’s푩 풓 푡 equations uniquely determine the field distributions provided that the휌 sources,풓 푡 푱 and풓 푡 , are fully specified in advance.
    [Show full text]
  • The Lorentz Force and the Radiation Pressure of Light
    The Lorentz Force and the Radiation Pressure of Light Tony Rothman∗ and Stephen Boughn† Princeton University, Princeton NJ 08544 Haverford College, Haverford PA 19041 LATEX-ed October 23, 2018 Abstract In order to make plausible the idea that light exerts a pressure on matter, some introductory physics texts consider the force exerted by an electromag- netic wave on an electron. The argument as presented is both mathematically incorrect and has several serious conceptual difficulties without obvious resolu- tion at the classical, yet alone introductory, level. We discuss these difficulties and propose an alternative demonstration. PACS: 1.40.Gm, 1.55.+b, 03.50De Keywords: Electromagnetic waves, radiation pressure, Lorentz force, light 1 The Freshman Argument arXiv:0807.1310v5 [physics.class-ph] 21 Nov 2008 The interaction of light and matter plays a central role, not only in physics itself, but in any freshman electricity and magnetism course. To develop this topic most courses introduce the Lorentz force law, which gives the electromagnetic force act- ing on a charged particle, and later devote some discussion to Maxwell’s equations. Students are then persuaded that Maxwell’s equations admit wave solutions which ∗[email protected] †[email protected] 1 Lorentz... 2 travel at the speed of light, thus establishing the connection between light and elec- tromagnetic waves. At this point we unequivocally state that electromagnetic waves carry momentum in the direction of propagation via the Poynting flux and that light therefore exerts a radiation pressure on matter. This is not controversial: Maxwell himself in his Treatise on Electricity and Magnetism[1] recognized that light should manifest a radiation pressure, but his demonstration is not immediately transparent to modern readers.
    [Show full text]
  • Lecture 18 Relativity & Electromagnetism
    Electromagnetism - Lecture 18 Relativity & Electromagnetism • Special Relativity • Current & Potential Four Vectors • Lorentz Transformations of E and B • Electromagnetic Field Tensor • Lorentz Invariance of Maxwell's Equations 1 Special Relativity in One Slide Space-time is a four-vector: xµ = [ct; x] Four-vectors have Lorentz transformations between two frames with uniform relative velocity v: 0 0 x = γ(x − βct) ct = γ(ct − βx) where β = v=c and γ = 1= 1 − β2 The factor γ leads to timepdilation and length contraction Products of four-vectors are Lorentz invariants: µ 2 2 2 2 02 0 2 2 x xµ = c t − jxj = c t − jx j = s The maximum possible speed is c where β ! 1, γ ! 1. Electromagnetism predicts waves that travel at c in a vacuum! The laws of Electromagnetism should be Lorentz invariant 2 Charge and Current Under a Lorentz transformation a static charge q at rest becomes a charge moving with velocity v. This is a current! A static charge density ρ becomes a current density J N.B. Charge is conserved by a Lorentz transformation The charge/current four-vector is: dxµ J µ = ρ = [cρ, J] dt The full Lorentz transformation is: 0 0 v J = γ(J − vρ) ρ = γ(ρ − J ) x x c2 x Note that the γ factor can be understood as a length contraction or time dilation affecting the charge and current densities 3 Notes: Diagrams: 4 Electrostatic & Vector Potentials A reminder that: 1 ρ µ0 J V = dτ A = dτ 4π0 ZV r 4π ZV r A static charge density ρ is a source of an electrostatic potential V A current density J is a source of a magnetic vector potential A Under a
    [Show full text]
  • The Energy-Momentum Tensor of Electromagnetic Fields in Matter
    The energy-momentum tensor of electromagnetic fields in matter Rodrigo Medina1, ∗ and J Stephany2, † 1Centro de F´ısica, Instituto Venezolano de Investigaciones Cient´ıficas, Apartado 20632 Caracas 1020-A, Venezuela. 2Departamento de F´ısica, Secci´on de Fen´omenos Opticos,´ Universidad Sim´on Bol´ıvar,Apartado Postal 89000, Caracas 1080-A, Venezuela. In this paper we present a complete resolution of the Abraham-Minkowski controversy about the energy-momentum tensor of the electromagnetic field in matter. This is done by introducing in our approach several new aspects which invalidate previous discussions. These are: 1)We show that for polarized matter the center of mass theorem is no longer valid in its usual form. A contribution related to microscopic spin should be considered. This allows to disregard the influencing argument done by Balacz in favor of Abraham’s momentum which discuss the motion of the center of mass of light interacting with a dielectric block. 2)The electromagnetic dipolar energy density −P · E contributes to the inertia of matter and should be incorporated covariantly to the the energy- momentum tensor of matter. This implies that there is also an electromagnetic component in matter’s momentum density given by P × B which accounts for the diference between Abraham and Minkowski’s momentum densities. The variation of this contribution explains the results of G. B. Walker, D. G. Lahoz and G. Walker’s experiment which until now was the only undisputed support to Abraham’s force. 3)A careful averaging of microscopic Lorentz’s force results in the µ 1 µ αβ unambiguos expression fd = 2 Dαβ ∂ F for the force density which the field exerts on matter.
    [Show full text]
  • Lorentz-Violating Electrostatics and Magnetostatics
    Publications 10-1-2004 Lorentz-Violating Electrostatics and Magnetostatics Quentin G. Bailey Indiana University - Bloomington, [email protected] V. Alan Kostelecký Indiana University - Bloomington Follow this and additional works at: https://commons.erau.edu/publication Part of the Electromagnetics and Photonics Commons, and the Physics Commons Scholarly Commons Citation Bailey, Q. G., & Kostelecký, V. A. (2004). Lorentz-Violating Electrostatics and Magnetostatics. Physical Review D, 70(7). https://doi.org/10.1103/PhysRevD.70.076006 This Article is brought to you for free and open access by Scholarly Commons. It has been accepted for inclusion in Publications by an authorized administrator of Scholarly Commons. For more information, please contact [email protected]. Lorentz-Violating Electrostatics and Magnetostatics Quentin G. Bailey and V. Alan Kosteleck´y Physics Department, Indiana University, Bloomington, IN 47405, U.S.A. (Dated: IUHET 472, July 2004) The static limit of Lorentz-violating electrodynamics in vacuum and in media is investigated. Features of the general solutions include the need for unconventional boundary conditions and the mixing of electrostatic and magnetostatic effects. Explicit solutions are provided for some simple cases. Electromagnetostatics experiments show promise for improving existing sensitivities to parity- odd coefficients for Lorentz violation in the photon sector. I. INTRODUCTION electrodynamics in Minkowski spacetime, coupled to an arbitrary 4-current source. There exists a substantial Since its inception, relativity and its underlying theoretical literature discussing the electrodynamics limit Lorentz symmetry have been intimately linked to clas- of the SME [19], but the stationary limit remains unex- sical electrodynamics. A century after Einstein, high- plored to date. We begin by providing some general in- sensitivity experiments based on electromagnetic phe- formation about this theory, including some aspects as- nomena remain popular as tests of relativity.
    [Show full text]
  • 6.013 Electromagnetics and Applications, Chapter 5
    Chapter 5: Electromagnetic Forces 5.1 Forces on free charges and currents 5.1.1 Lorentz force equation and introduction to force The Lorentz force equation (1.2.1) fully characterizes electromagnetic forces on stationary and moving charges. Despite the simplicity of this equation, it is highly accurate and essential to the understanding of all electrical phenomena because these phenomena are observable only as a result of forces on charges. Sometimes these forces drive motors or other actuators, and sometimes they drive electrons through materials that are heated, illuminated, or undergoing other physical or chemical changes. These forces also drive the currents essential to all electronic circuits and devices. When the electromagnetic fields and the location and motion of free charges are known, the calculation of the forces acting on those charges is straightforward and is explained in Sections 5.1.2 and 5.1.3. When these charges and currents are confined within conductors instead of being isolated in vacuum, the approaches introduced in Section 5.2 can usually be used. Finally, when the charges and charge motion of interest are bound within stationary atoms or spinning charged particles, the Kelvin force density expressions developed in Section 5.3 must be added. The problem usually lies beyond the scope of this text when the force-producing electromagnetic fields are not given but are determined by those same charges on which the forces are acting (e.g., plasma physics), and when the velocities are relativistic. The simplest case involves the forces arising from known electromagnetic fields acting on free charges in vacuum.
    [Show full text]
  • Energy, Momentum, and Symmetries 1 Fields 2 Complex Notation
    Energy, Momentum, and Symmetries 1 Fields The interaction of charges was described through the concept of a field. A field connects charge to a geometry of space-time. Thus charge modifies surrounding space such that this space affects another charge. While the abstraction of an interaction to include an intermedi- ate step seems irrelevant to those approaching the subject from a Newtonian viewpoint, it is a way to include relativity in the interaction. The Newtonian concept of action-at-a-distance is not consistent with standard relativity theory. Most think in terms of “force”, and in particular, force acting directly between objects. You should now think in terms of fields, i.e. a modification to space-time geometry which then affects a particle positioned at that geometric point. In fact, it is not the fields themselves, but the potentials (which are more closely aligned with energy) which will be fundamental to the dynamics of interactions. 2 Complex notation Note that a description of the EM interaction involves the time dependent Maxwell equa- tions. The time dependence can be removed by essentially employing a Fourier transform. Remember in an earlier lecture, the Fourier transformation was introduced. ∞ (ω) = 1 dt F (t) e−iωt F √2π −∞R ∞ F (t) = 1 dt (ω) eiωt √2π −∞R F Now assume that all functions have the form; F (~x, t) F (~x)eiωt → These solutions are valid for a particular frequency, ω. One can superimpose solutions of different frequencies through a weighted, inverse Fourier transform to get the time depen- dence of any function. Also note that this assumption results in the introduction of complex functions to describe measurable quantities which in fact must be real.
    [Show full text]
  • The Maxwell Stress Tensor and Electromagnetic Momentum
    Progress In Electromagnetics Research Letters, Vol. 94, 151–156, 2020 The Maxwell Stress Tensor and Electromagnetic Momentum Artice Davis1, * and Vladimir Onoochin2 Abstract—In this paper, we discuss two well-known definitions of electromagnetic momentum, ρA and 0[E × B]. We show that the former is preferable to the latter for several reasons which we will discuss. Primarily, we show in detail—and by example—that the usual manipulations used in deriving the expression 0[E×B] have a serious mathematical flaw. We follow this by presenting a succinct derivation for the former expression. We feel that the fundamental definition of electromagnetic momentum should rely upon the interaction of a single particle with the electromagnetic field. Thus, it contrasts with the definition of momentum as 0[E × B] which depends upon a (defective) integral over an entire region, usually all space. 1. INTRODUCTION The correct form of the electromagnetic energy-momentum tensor has been debated for almost a century. A large number of papers have been written [1], some of which concentrate upon a single particle interaction with the electromagnetic field in free space [2, 3]; others discuss the interaction between a field and a dielectric material upon which it impinges [4]. In works which discuss the Minkowski-Abraham controversy, the system considered is usually on the macroscopic level. On this level, singularities caused by point charges cannot be treated. We show that neglecting the effect due the properties of classical charges creates difficulties which cannot be resolved within the framework typically discussed by textbooks on classical electrodynamics [5]. Our present concept of electromagnetic field momentum density has two interpretations: (i) As 0[E × B], the scaled Poynting vector.
    [Show full text]
  • Magnetic Fields, Magnetic Forces, and Sources of Magnetic Fields
    Magnetic Fields, Magnetic Forces, and Sources of Magnetic Fields W07D2 1 Announcements Problem Set 6 Due Thursday at 9 pm .This problem set is an introduction to Experiment 1 and the two problems from W06D2. Week 8 LS1 due Mon at 8:30 am Week 8 LS2 due Mon at 8:30 am Week 8 LS3 due Wed at 8:30 am Week 8 LS4 due Wed at 8:30 am 2 Outline Magnetic Field Lorentz Force Law Magnetic Force on Current Carrying Wire Sources of Magnetic Fields Biot-Savart Law 3 Magnetic Field of the Earth North magnetic pole located in southern hemisphere http://www.youtube.com/watch?v=AtDAOxaJ4Ms 4 Magnetic Force on Moving Charges Force on ! ! positive charge B ! Fq = qvq × B vq B B q vq F F q q B q + B Force on negative charge Magnetic force is perpendicular to both velocity of the charge and magnetic field 5 CQ: Cross Product and Magnetic Force An electron is traveling up in a magnetic field that points to the right. What is the direction of the force on the electron? v 1. Up. q 2. Down. B 3. Left. q = e 4. Right. 5. Into plane of figure. 6. Out of plane of figure. 6 Group Problem: Cyclotron Motion A positively charged particle vq with charge +q and mass m is B moving with speed v in a + uniform magnetic field of + q magnitude B directed into the plane of figure will undergo circular motion. Find (1) the radius R of the orbit, (2) the period T of the motion, (3) the angular frequency ω.
    [Show full text]