DOCSLIB.ORG

Undergraduate Lecture Notes in Physics Undergraduate Lecture Notes in Physics (ULNP) Publishes Authoritative Texts Covering Topics Throughout Pure and Applied Physics

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Undergraduate Lecture Notes in Physics Undergraduate Lecture Notes in Physics (ULNP) Publishes Authoritative Texts Covering Topics Throughout Pure and Applied Physics Undergraduate Lecture Notes in Physics Undergraduate Lecture Notes in Physics (ULNP) publishes authoritative texts covering topics throughout pure and applied physics. Each title in the series is suitable as a basis for undergraduate instruction, typically containing practice problems, worked examples, chapter summaries, and suggestions for further reading. ULNP titles must provide at least one of the following: • An exceptionally clear and concise treatment of a standard undergraduate subject. • A solid undergraduate-level introduction to a graduate, advanced, or non-standard subject. • A novel perspective or an unusual approach to teaching a subject. ULNP especially encourages new, original, and idiosyncratic approaches to physics teaching at the undergraduate level. The purpose of ULNP is to provide intriguing, absorbing books that will continue to be the reader’s preferred reference throughout their academic career. Series editors Neil Ashby University of Colorado, Boulder, CO, USA William Brantley Department of Physics, Furman University, Greenville, SC, USA Matthew Deady Physics Program, Bard College, Annandale-on-Hudson, NY, USA Michael Fowler Department of Physics, University of Virginia, Charlottesville, VA, USA Morten Hjorth-Jensen Department of Physics, University of Oslo, Oslo, Norway Michael Inglis SUNY Suffolk County Community College, Long Island, NY, USA Heinz Klose Humboldt University, Oldenburg, Niedersachsen, Germany Helmy Sherif Department of Physics, University of Alberta, Edmonton, AB, Canada More information about this series at http://www.springer.com/series/8917 Francesco Lacava Classical Electrodynamics From Image Charges to the Photon Mass and Magnetic Monopoles 123 Francesco Lacava Università degli studi di Roma “La Sapienza” Rome Italy ISSN 2192-4791 ISSN 2192-4805 (electronic) Undergraduate Lecture Notes in Physics ISBN 978-3-319-39473-2 ISBN 978-3-319-39474-9 (eBook) DOI 10.1007/978-3-319-39474-9 Library of Congress Control Number: 2016943959 © Springer International Publishing Switzerland 2016 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG Switzerland To the memory of my parents Biagio and Sara and my brother Walter Preface In the undergraduate program of Electricity and Magnetism emphasis is given to the introduction of fundamental laws and to their applications. Many interesting and intriguing subjects can be presented only shortly or are postponed to graduate courses on Electrodynamics. In the last years I examined some of these topics as supplementary material for the course on Electromagnetism for the M.Sc. students in Physics at the University Sapienza in Roma and for a series of special lectures. This small book collects the notes from these lectures. The aim is to offer to the readers some interesting study cases useful for a deeper understanding of the Electrodynamics and also to present some classical methods to solve difficult problems. Furthermore, two chapters are devoted to the Electrodynamics in relativistic form needed to understand the link between the electric and magnetic fields. In the two final chapters two relevant experimental issues are examined. This introduces the readers to the experimental work to confirm a law or a theory. References of classical books on Electricity and Magnetism are provided so that the students get familiar with books that they will meet in further studies. In some chapters the worked out problems extend the text material. Chapter 1 is a fast survey of the topics usually taught in the course of Electromagnetism. It can be useful as a reference while reading this book and it also gives the opportunity to focus on some concepts as the electromagnetic potential and the gauge transformations. The expansion in terms of multipoles for the potential of a system of charges is examined in Chap. 2. Problems with solutions are proposed. Chapter 3 introduces the elegant method of image charges in vacuum. In Chap. 4 the method is extended to problems with dielectrics. This last argument is rarely presented in textbooks. In both chapters examples are examined and many prob- lems with solutions are proposed. Analytic complex functions can be used to find the solutions for the electric field in two-dimensional problems. After a general introduction of the method, Chap. 5 discusses some examples. In the Appendix to the chapter the solutions for vii viii Preface two-dimensional problems are derived by solving the Laplace equation with boundary conditions. Chapter 6 aims at introducing the relativistic transformations of the electric and magnetic fields by analysing the force on a point charge moving parallel to an infinite wire carrying a current. The equations of motion are formally the same in the laboratory and in the rest frame of the charge but the forces acting on the charge are seen as different in the two frames. This example introduces the transformations of the fields in special relativity. In Chap. 7 a short historic introduction mentions the difficulties of the classical physics at the end of the 19th century in explaining some phenomena observed in Electrodynamics. The problem of invariance in the Minkowsky spacetime is examined. The formulas of Electrodynamics are written in covariant form. The electromagnetic tensor is introduced and the Maxwell equations in covariant form are given. Chapter 8 presents a lecture by Feynman on the capacitor at high frequency. The effects produced by iterative corrections due to the induction law and to the dis- placement current are considered. For very high frequency of the applied voltage, the capacitor becomes a resonant cavity. This is a very interesting example for the students. The students are encouraged to refer to the Feynman lectures for further comments and for other arguments. The energy and momentum conservation in the presence of an electromagnetic field are considered in Chap. 9. The Poynting’s vector is introduced and some simple applications to the resistor, to the capacitor and to the solenoid are presented. The transfer of energy in an electric circuit in terms of the flux of the Poynting’s vector is also examined. Then the Maxwell stress tensor is introduced. Some problems with solutions complete the chapter. The Feynman paradox or paradox of the angular momentum is very intriguing. It is very useful to understand the dynamics of the electromagnetic field. Chapter 10 presents the paradox with comments. An original example of a rotating charged system in a damped magnetic field is discussed. The need to test the dependence on the inverse square of the distance for the Coulomb’s law was evident when the law was stated. The story of these tests is presented in Chap. 11. The most sensitive method, based on the Faraday’s cage, was introduced by Cavendish and was used until the half of last century. After that time the test was interpreted in terms of a test on a non-null mass of the photon. The theory is shortly presented and experiments and limits are reported. Chapter 12 introduces the problem of the magnetic monopoles. In a paper Dirac showed that the electric charge is quantized if a magnetic monopole exists in the Universe. The Dirac’s relation is derived. The properties of a magnetic monopole crossing the matter are presented. Experiments to search the magnetic monopoles and their results are mentioned. In the Appendix the general formulas of the differential operators used in Electrodynamics are derived for orthogonal systems of coordinates and the expressions for spherical and cylindrical coordinates are given. Preface ix I wish to thank Professors L. Angelani, M. Calvetti, A. Ghigo, S. Petrarca, and F. Piacentini for useful suggestions. A special thank is due to Professor M. Testa for helpful discussions and encouragement. I am grateful to Dr. E. De Lucia for reviewing the English version of this book and to Dr. L. Lamagna for reading and commenting this work. Rome, Italy Francesco Lacava May 2016 Contents 1 Classical Electrodynamics: A Short Review.................. 1 1.1 Coulomb’s Law and the First Maxwell Equation . 1 1.2 Charge Conservation and Continuity Equation . 4 1.3 Absence of Magnetic Charges in Nature and the Second Maxwell Equation . 4 1.4 Laplace’s Laws and the Steady Fourth Maxwell Equation . 5 1.5 Faraday’s Law and the Third Maxwell Equation . 6 1.6 Displacement Current and the Fourth Maxwell Equation . 7 1.7 Maxwell Equations in Vacuum . 7 1.8 Maxwell Equations in Matter. 7 1.9 Electrodynamic Potentials and Gauge Transformations . 9 1.10 Electromagnetic Waves. 13 2 Multipole Expansion of the Electrostatic Potential ............. 17 2.1 The Potential of the Electric Dipole . 17 2.2 Interaction of the Dipole with an Electric Field . 18 2.3 Multipole Expansion for the Potential of a Distribution of Point Charges . 19 2.4 Properties of the Electric Dipole Moment . 22 2.5 The Quadrupole Tensor .
Load more

Recommended publications
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [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]
  • 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.
    [Show full text]
  • On Electromagnetic Duality
    On Electromagnetic Duality Thomas B. Mieling Faculty of Physics, University of Vienna Boltzmanngasse 5, 1090 Vienna, Austria (Dated: November 14, 2018) CONTENTS III. DUAL TENSORS I. Introduction 1 This section introduces the notion of dual ten- sors, whose proper treatment requires some famil- II. Conventions 1 iarity with exterior algebra. Readers unfamiliar with this topic may skip section III A, since sec- III. Dual Tensors 1 tion III B suces for the calculations in the rest of A. The Hodge Dual 1 the paper, except parts of the appendix. B. The Complex Dual of Two-Forms 2 IV. The Free Maxwell-Field 2 A. The Hodge Dual V. General Duality Transformations 3 For an oriented vector space V of dimension VI. Coupled Maxwell-Fields 3 n with a metric tensor, the Hodge star opera- tor provides an isomorphism between the k-vectors VII. Applications 4 n − k-vectors. In this section, we discuss the gen- eral denition in Euclidean and Lorentzian vector VIII. Conclusion 5 spaces and give formulae for concrete calculations in (pseudo-)orthonormal frames. A. Notes on Exterior Algebra 6 The metric tensor g on V induces a metric tensor on ^kV (denoted by the same symbol) by bilinear B. Collection of Proofs 7 extension of the map References 8 g(v1 ^ · · · ^ vk; w1 ^ · · · ^ wk) = det(g(vi; wj)): If the metric on V is non-degenerate, so is g on I. INTRODUCTION ^kV . Let w be a k-vector. For every n − k-vector In abelian gauge theories whose action does not v the exterior product v ^ w is proportional to depend on the gauge elds themselves, but only the unit n-vector Ω, since ^nV is one-dimensional.
    [Show full text]
  • True Energy-Momentum Tensors Are Unique. Electrodynamics Spin
    True energy-momentum tensors are unique. Electrodynamics spin tensor is not zero Radi I. Khrapko∗ Moscow Aviation Institute, 125993, Moscow, Russia Abstract A true energy-momentum tensor is unique and does not admit an addition of a term. The true electrodynamics’ energy-momentum tensor is the Maxwell-Minkowski tensor. It cannot be got with the Lagrange formalism. The canonical energy-momentum and spin tensors are out of all relation to the physical reality. The true electrodynamics’ spin tensor is not equal to a zero. So, electrodynamics’ ponderomotive action comprises a force from the Maxwell stress tensor and a torque from the spin tensor. A gauge non-invariant expression for the spin tensor is presented and is used for a con- sideration of a circularly polarized standing wave. A circularly polarized light beam carries a spin angular momentum and an orbital angular momentum. So, we double the beam’s angular momentum. The Beth’s experiment is considered. PACS: 03.05.De Keywords: classical spin; electrodynamics; Beth’s experiment; torque 1 An outlook on the standard electrodynamics The main points in the electrodynamics are an electric current four-vector density jα and elec- tromagnetic field which is described by a covariant skew-symmetric electromagnetic field tensor αβ αµ βν Fµν , or by a contravariant tensor F = Fµν g g (α, ν, . = 0, 1, 2, 3, or = t, x, y, z). The four-vector density jα comprises a charge density and an electric current three-vector density: jα = j0 = ρ, ji (i, k, . =1, 2, 3, or = x, y, z). gαµ = g00 =1, gij = δij .
    [Show full text]
  • The Procedure of Finding the Stress-Energy Tensor And
    Advanced Studies in Theoretical Physics Vol. 8, 2014, no. 18, 771 - 779 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/astp.2014.47101 The Procedure of Finding the Stress-Energy Tensor and Equations of Vector Field of Any Form Sergey G. Fedosin Sviazeva Str. 22-79, Perm, 614088 Perm region, Russian Federation Copyright © 2014 Sergey G. Fedosin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract A method allowing us to introduce into the Lagrangian the terms, which characterize an arbitrary vector field of a system, is described. As a result of applying the principle of least action it becomes possible to find all the main characteristics of this field, including its energy and momentum, field equations, force of interaction with the matter. Keywords: Four-potential; pressure field; acceleration field; field equations. 1. Introduction The field concept is widely used not only in the gravitation theory but also in other physical theories. Further we will consider the properties of vector fields in four-dimensional space. The main characteristic of the electromagnetic vector field is the 4-potential A , A , where c is the speed of light, and A denote the scalar and c vector potentials, respectively. If the system contains a set of particles, each of which generates its own potential, then the potentials and A of the system of particles depend mainly on the general system parameters – the dimensions of the system, the total charge, etc.
    [Show full text]
  • CIUFOLINI & WHEELER: Gravitation and Inertia Is
    COPYRIGHT NOTICE: CIUFOLINI & WHEELER: Gravitation and Inertia is published by Princeton University Press and copyrighted, (c) 1996, by Princeton University Press. All rights reserved. This text may be used and shared in accordance with the fair-use provisions of US copyright law, and it may be archived and redistributed in electronic form, provided that this entire notice is carried and provided that Princeton University Press is notified and no fee is charged for access. Archiving, redistribution, or republication of this text on other terms, in any medium, requires the consent of Princeton University Press. For COURSE PACK PERMISSIONS, refer to entry on previous menu. For more information, send e-mail to [email protected] The Bartlett Press, Inc. ciufolin 4:00 p.m. 6 iii 1995 · · 2 Einstein Geometrodynamics If Einstein gave us a geometric account of motion and gravity, if according to his 1915 and still-standard geometrodynamics spacetime tells mass how to move and mass tells spacetime how to curve, then his message requires mathematical tools to describe position and motion, curvature and the action of mass on curvature. The tools (see the mathematical appendix) will open the doorways to the basic ideas—equivalence principle, geometric structure, field equation, equation of motion, equation of geodesic deviation—and these ideas will open the doorways to more mathematical tools—exact solutions of Einstein’s geometrodynamics field equation, equations of conservation of source, and the principle that the boundary of a boundary is zero. The final topics in this chapter—black holes, singularities, and gravitational waves— round out the interplay of mathematics and physics that is such a central feature of Einstein’s geometrodynamics.
    [Show full text]
  • On the Local Extension of Killing Vector Fields in Electrovacuum Spacetimes
    On the local extension of Killing vector fields in electrovacuum spacetimes Elena Giorgi May 28, 2019 Abstract We revisit the problem of extension of a Killing vector field in a spacetime which is solution to the Einstein-Maxwell equation. This extension has been proved to be unique in the case of a Killing vector field which is normal to a bifurcate horizon in [9]. Here we generalize the extension of the vector field to a strong null convex domain in an electrovacuum spacetime, inspired by the same technique used in [6] in the setting of Ricci flat manifolds. We also prove a result concerning non-extendibility: we show that one can find local, stationary electrovacuum extension of a Kerr-Newman solution in a full neighborhood of a point of the horizon (that is not on the bifurcation sphere) which admits no extension of the Hawking vector field. This generalizes the construction in [6] to the electrovacuum case. 1 Introduction In (pseudo)-Riemannian manifolds, Killing vector fields are generators of local isometries. They therefore play a major role in the description of the symmetries of the manifold. In addition, in Mathematical General Relativity the identification of Killing vector fields in spacetimes veryfing the Einstein equation is closely related to the no-hair theorem for black holes. Neverthless, the problem of extension of Killing vector fields in (pseudo)-Riemannian manifolds can be stated independently of its applications to General relativity (see Section 1.1 of Introduction for a summary of the relation with the problem of rigidity of black holes). The main purpose of the present paper is to address some aspects of the following problem.
    [Show full text]
  • An Introduction to Relativistic Electrodynamics Part I: Calculus with 4-Vectors and 4-Dyadics
    An introduction to relativistic electrodynamics Part I: Calculus with 4-vectors and 4-dyadics Jonas Larsson¦ Department of Physics, Ume˚aUniversity, SE-901 87 Ume˚a,Sweden Karl Larsson: Department of Mathematics and Mathematical Statistics, Ume˚aUniversity, SE-901 87 Ume˚a,Sweden (Dated: October 22, 2018) The conventional way of introducing relativity when teaching electrodynamics is to leave Gibbs’ vector calculus for a more general tensor calculus. This sudden change of formalism can be quite problematic for the students and we therefore in this two-part paper consider alternate approaches. In this Part I we use a simplified tensor formalism with 4-vectors and 4-dyadics (i.e., second order tensors built by 4-vectors) but with no tensors of higher order than two. This allows for notations in good contact with the coordinate-free Gibbs’ vector calculus that the students already master. Thus we use boldface notations for 4-vectors and 4-dyadics without coordinates and index algebra to formulate Lorentz transformations, Maxwell’s equations, the equation of the motion of charged particles and the stress-energy conservation law. By first working with this simplified tensor for- malism the students will get better prepared to learn the standard tensor calculus needed in more advanced courses. I. INTRODUCTION electromagnetic field tensor happens to be just of order two is it possible to postpone the use of index algebra to In a first course in electrodynamics, on the level of say more advanced courses. Griffith’s textbook1, the mathematical formalism is es- Instead, since the students are confident with vector sentially Gibbs’ vector calculus with one exception; the calculus, it is beneficial to keep in contact with that for- chapter on relativistic electrodynamics.
    [Show full text]
  • Quaternion Electromagnetism and the Relation with Two-Spinor Formalism
    universe Article Quaternion Electromagnetism and the Relation with Two-Spinor Formalism In Ki Hong 1 and Choong Sun Kim 1,2,* 1 Department of Phyics and IPAP, Yonsei University, Seoul 03722, Korea; [email protected] 2 Institute of High Energy Physics, Dongshin University, Naju 58245, Korea * Correspondence: [email protected] Received: 26 April 2019; Accepted: 28 May 2019; Published: 3 June 2019 Abstract: By using complex quaternion, which is the system of quaternion representation extended to complex numbers, we show that the laws of electromagnetism can be expressed much more simply and concisely. We also derive the quaternion representation of rotations and boosts from the spinor representation of Lorentz group. It is suggested that the imaginary “i” should be attached to the spatial coordinates, and observe that the complex conjugate of quaternion representation is exactly equal to parity inversion of all physical quantities in the quaternion. We also show that using quaternion is directly linked to the two-spinor formalism. Finally, we discuss meanings of quaternion, octonion and sedenion in physics as n-fold rotation. Keywords: quaternion; electromagnetism; representation theory; Cayley-Dickson algebra; special relativity; twistor theory 1. Introduction There are several papers claiming that the quaternion or the octonion can be used to describe the laws of classical electromagnetism in a simpler way [1–6]. However, they are mainly limited to describing Maxwell equations. Furthermore, the meaning of quaternion and the reasons electromagnetic laws can be concisely described by them have not been well discussed up to now. Here, we list more diverse quaternion representations of the relations in electromagnetism than previously known and we introduce a new simpler notation to express quaternions.
    [Show full text]