The Energy-Momentum Tensor of Electromagnetic Fields in Matter
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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. -
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. -
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. -
On the Covariant Representation of Integral Equations of the Electromagnetic Field
On the covariant representation of integral equations of the electromagnetic field Sergey G. Fedosin PO box 614088, Sviazeva str. 22-79, Perm, Perm Krai, Russia E-mail: [email protected] Gauss integral theorems for electric and magnetic fields, Faraday’s law of electromagnetic induction, magnetic field circulation theorem, theorems on the flux and circulation of vector potential, which are valid in curved spacetime, are presented in a covariant form. Covariant formulas for magnetic and electric fluxes, for electromotive force and circulation of the vector potential are provided. In particular, the electromotive force is expressed by a line integral over a closed curve, while in the integral, in addition to the vortex electric field strength, a determinant of the metric tensor also appears. Similarly, the magnetic flux is expressed by a surface integral from the product of magnetic field induction by the determinant of the metric tensor. A new physical quantity is introduced – the integral scalar potential, the rate of change of which over time determines the flux of vector potential through a closed surface. It is shown that the commonly used four-dimensional Kelvin-Stokes theorem does not allow one to deduce fully the integral laws of the electromagnetic field and in the covariant notation requires the addition of determinant of the metric tensor, besides the validity of the Kelvin-Stokes theorem is limited to the cases when determinant of metric tensor and the contour area are independent from time. This disadvantage is not present in the approach that uses the divergence theorem and equation for the dual electromagnetic field tensor. -
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. -
Electromagnetic Particles Guy Michel Stephan
Electromagnetic particles Guy Michel Stephan To cite this version: Guy Michel Stephan. Electromagnetic particles. 2017. hal-01446417 HAL Id: hal-01446417 https://hal.archives-ouvertes.fr/hal-01446417 Preprint submitted on 2 Feb 2017 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Electromagnetic particles. G.M. St´ephan 26, Chemin de Quo Vadis 22730, Tregastel email : [email protected] 25 janvier 2017 R´esum´e The covariant derivative of the 4-components electromagnetic potential in a flat Minkowski space- time is split into its antisymmetric and symmetric parts. While the former is well known to describe the electromagnetic field, we show that the latter describes the associated particles. When symmetry principles are applied to the invariants in operations of the Poincar´egroup, one finds equations which describe the structure of the particles. Both parts of the tensor unify the concept of matter-wave duality. Charge and mass are shown to be associated to the potential. 1 Introduction. When Born and Infeld published their work[1] on the foundations of non linear electromagnetism in the year 1934, their aim was to describe the electron and more generally the physical world from a purely electromagnetic theory. -
Electromagnetic Forces and Internal Stresses in Dielectric Media
PHYSICAL REVIEW E 85, 046606 (2012) Electromagnetic forces and internal stresses in dielectric media Winston Frias* and Andrei I. Smolyakov Department of Physics and Engineering Physics, University of Saskatchewan, 116 Science Place Saskatoon, SK S7N 5E2, Canada (Received 16 August 2011; revised manuscript received 26 January 2012; published 23 April 2012) The macroscopic electromagnetic force on dielectric bodies and the related problem of the momentum conservation are discussed. It is argued that different forms of the momentum conservation and, respectively, different forms of the force density, correspond to the different ordering in the macroscopic averaging procedure. Different averaging procedures and averaging length scale assumptions result in expressions for the force density with vastly different force density profiles which can potentially be detected experimentally by measuring the profiles of the internal stresses in the medium. The expressions for the Helmholtz force is generalized for the dissipative case. It is shown that the net (integrated) force on the body in vacuum is the same for Lorentz and Helmholtz expressions in all configuration. The case of a semi-infinite medium is analyzed and it is shown that explicit assumptions on the boundary conditions at infinity remove ambiguity in the force on the semi-infinite dielectric. DOI: 10.1103/PhysRevE.85.046606 PACS number(s): 03.50.De I. INTRODUCTION This article is organized as follows. In Sec. II, the problem of the momentum exchange and the force is explored from Electromagnetic forces on charged particles result in an the microscopic point of view. In Sec. III, the Abraham and overall force imparted on a body that is called radiation Minkowski expressions for the momentum and the stress (ponderomotive) pressure. -
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. -
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 -
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. -
Formulation of Energy Momentum Tensor for Generalized Fields of Dyons
Formulation of Energy Momentum Tensor for Generalized fields of dyons Gaurav Karnatak∗, P. S. Bisht∗, O. P. S. Negi∗ November 26, 2016 ∗Department of Physics Kumaun University S. S. J. Campus Almora-263601(Uttarakhand) India Email- [email protected] [email protected] [email protected] Abstract The energy momentum tensor of generalized fields of dyons and energy momentum conservation laws of dyons has been developed in simple, compact and consistent manner. We have obtained the Maxwell’s field theory of energy momentum tensor of dyons (electric and magnetic) of electromagnetic field, Poynting vector and Poynting theorem for generalized fields of dyons in a simple, unique and consistent way. Keywords: Dyons, Electromagnetic fields, Energy-momentum tensor, Poynting vector. 1 Introduction The concept of the energy-momentum tensor in classical field theory has a long history, especially in Ein- stein’s theory of gravity [1]. The energy-momentum tensor combines the densities and flux densities of energy and momentum of the fields into one single object. The problem of giving a concise definition of this object able to provide the physically correct answer under all circumstances, for an arbitrary Lagrangian field theory on an arbitrary space-time background, has puzzled physicists for decades. The classical field theory with space-time translation invariance has a conserved energy-momentum tensor [1, 2]. The classical Lagrangian in constructing the energy-momentum tensor, like the canonical energy-momentum tensor was noticed long ago. The question is based on the Noethern theorem, according to which a field theory with space-time translation invariance has a conserved energy-momentum tensor. -
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.