DOCSLIB.ORG

18 Gauge Fields, Coordinate Systems, and Geometric Phases

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
18 Gauge Fields, Coordinate Systems, and Geometric Phases 18 Gauge felds, coordinate systems, and geometric phases 18.1 Maxwell’s equations Maxwell said 1 ∂B! Faraday’s law E! = (18.1) ∇× − c ∂t 1 ∂D! Ampère’s law H! = !j + (18.2) ∇× c ! ∂t " Coulomb’s law D! = ρ (18.3) ∇ no magnetic monopoles B! =0 (18.4) ∇ and it was light [43, 44]. Together with Newton’s laws of classical mechanics and the Lorentz force !v F! = q E! + B! (18.5) c × # $ that was all what was known fundamentally about physics by the end of the 19th century. Maxwell originally didn’t write his equations in the modern notation using vectors we are familiar with. This was introduced later by Oliver Heaviside [45] and Josiah Willard Gibbs, making the equations much easier to read (and evaluate). Instead, Maxwell used 20 scalar felds, inspired by the then-popular fuid mechanics. Also his equations contain explicitly the vector potential A!, which can be eliminated from his equations. The equations for the 16 parameters above above include the relations B! = µ0 H! + λM! , (18.6) % & D! = %0E! + λP,! (18.7) introducing the macroscopic magnetization M! and electric polarization P! with the mag- netic permeability µ0 and the dielectric constant %0. The factor λ denotes a dimension- 281 18 Gauge felds, coordinate systems, and geometric phases Table 18.1: Defnitions of µ0, %0, and λ in different unit systems Unit system µ0 %0 λ 7 2 Système Internationale 4π 10− T /(A/m) 1/(µ0c )1 × Gauß 1 14π Heaviside-Lorentz 1 11 esu 1/c2 14π emu 1 1/c2 4π less constant depending on the unit system chosen.1 We are using Heaviside-Lorentz units (named after Oliver Heaviside and Hendrik Antoon Lorentz), equivalent to setting µ0 = %0 = λ =1. This makes the structure of Maxwell’s equations particularly easy to understand.2 18.1.1 The scalar and vector potentials In the following, we set the magnetization M! =0and the polarization P! =0. That is, we express felds E! and B! in terms of all charges and currents on a microscopic level. In this form, Maxwell’s equations are often named «in vacuum» (which is wrong or at least misleading) or «microscopic» in contrast to the «macroscopic» equations above involving M! and P! . 1Setting λ =4π (Gauß, esu, emu) scales electromagnetic quantities by an integration over the solid angle. The factor 4π then appears explicitly in Coulomb’s law (18.3), however drops out of calculations involving planar geometries, for example parallel charged plates. 2Maxwell’s equations are equations each establishing a certain proportionality. Historically, different pro- portionality constants have emerged, leading to different electromagnetic unit systems. Among these are the Système Internationale d’Unités, the Gaussian unit system, and the Heaviside-Lorentz system, see Table 18.1 for the corresponding values of the constants. Using these constants, we can express the Coulomb force between two point-like charged particles as λ q1q2 F = 2 (18.8) 4π#0 r and the force between two infnitely thin wires of length $ at distance r as λµ0 $ F = 2I1I2 . (18.9) 4π r Maxwell’s equations then have the form 1 1 ∂B% 1 1 ∂D% √#0 E% = , √µ0 H% = λ%j + , D% = λρ, B% =0. ∇× − √µ0 c ∂t ∇× √#0 c ! ∂t " ∇ ∇ Everything else one needs to know about electromagnetic units can be found in the appendix of John David Jackson’s book on electromagnetism. 282 18.1 Maxwell’s equations With Stokes’ theorem in differential form, ( A!)=0, (18.10) ∇· ∇× ( φ)=0 (18.11) ∇× ∇· for any suffciently well-behaved vector feld A! = A!(!x, t) and scalar feld φ = φ(!x, t),we can use the divergence-freeness of the magnetic feld (18.4) and Faraday’s law (18.1)to express E! and B! through B! = A,! (18.12) ∇× 1 ∂ E! + A! =0 (18.13) ∇× c ∂t # $ 1 ∂ E! = A! + φ . (18.14) ⇒ − c ∂t ∇ # $ We replace E! and B! accordingly in Ampère’s law (18.2) and obtain 1 ∂ 1 ∂ 1 A! φ A! = !j. (18.15) ∇× ∇× − c ∂t −∇ − c ∂t c % & # $ Using the identity A! = A! ( ) A! (18.16) ∇× ∇× ∇ ∇ − ∇·∇ % & % & ∆ and the d’Alembert operator ' () * 1 ∂2 := ∆ (18.17) ! c2 ∂t2 − we obtain 1 ∂ 1 A! + φ + A! = !j. (18.18) ! ∇ c ∂t ∇ c # $ Replacing E! from Eq. (18.14) in Coulomb’s law (18.3) yields 1 ∂ φ A! = ρ (18.19) ∇ −∇ − c ∂t # $ 1 ∂ ∆φ A! = ρ (18.20) ⇒− − c ∂t∇ 1 ∂ 1 ∂ φ φ + A! = ρ, (18.21) ⇒ ! − c ∂t c ∂t ∇ # $ where in the last line we just added and subtracted (1/c2)∂2φ/∂t2. We note that the equations (18.18) and (18.21) are exactly equivalent to Maxwell’s equations: Four frst- order differential equations are replaced by two second-order equations. Typeset June 8, 2021 283 18 Gauge felds, coordinate systems, and geometric phases 18.2 Gauge invariance The felds A! and φ are not uniquely determined by these equations: Shifting with Λ= Λ(!x, t) according to A! A! + Λ=: A!", (18.22) → ∇ 1 ∂ φ φ Λ=: φ", (18.23) → − c ∂t doesn’t change the physical felds B! and E! : B! = A! A!" = A! + Λ ∇× →∇× ∇× ∇ (18.24) = A! = B,! % & ∇× 1 ∂ 1 ∂ 1 ∂ 1 ∂ E! = φ A! φ" A!" = φ Λ A! + Λ −∇ − c ∂t → −∇ − c ∂t −∇ − c ∂t − c ∂t ∇ # $ (18.25) 1 ∂ % & = φ A! = E.! −∇ − c ∂t This is called a gauge degree of freedom: The felds A! and φ are defned only up to a gradient of a (possibly space- and time-dependent) scalar feld Λ, and any particular choice of Λ (called «fxing the gauge») leads to exactly the same physics. Therefore any theory involving electromagnetic felds must be formulated in a form which respects this gauge invariance. Including the Lorentz force into Lagrange’s equations ∂ d ∂ L L =0 (18.26) ∂xi − dt ∂x˙ i where ∂ /∂xi are the components of the canonical force and pi = ∂ /∂x˙ i are the com- L L ponents of the canonical momentum leads to e p! = m!x˙ + A! (18.27) c for the canonical momentum of a particle with charge e. We stress that the canonical momentum p! , being just a variable conjugate to the position !x explicitly depends on the gauge chosen. Thus in order to write the kinetic momentum in the Lagrange function in a gauge invariant form, we need to substitute it according to e m!x˙ p! A,! (18.28) → − c a substitution we will meet again in quantum mechanics, where in real-space represen- tation p! =(h¯/i) is the operator entering the canonical commutation relation [!x,p ! ]= ∇ i¯h. 284 18.2 Gauge invariance 18.2.1 Common gauges The Lorenz gauge This is named after the danish physicist Ludvig Valentin Lorenz, not to be confused with Hendrik Lorentz mentioned above. It reads 1 ∂ φ + A! =0, (18.29) c ∂t ∇ turning Maxwell’s equations into a set of inhomogeneous wave equations, particularly popular in the calculation of time dependent electromagnetic felds. This does not fx the gauge completely, as any scalar ψ = ψ(!x, t) satisfying a homogeneous wave equation ! ψ =0 (18.30) can still be used to shift A! A! + ψ and φ φ (1/c)∂ψ/∂t without changing the → ∇ → − physics. We note that the Lorenz gauge is Lorentz invariant. The Coulomb gauge Here we set A! =0. (18.31) ∇ Eq. (18.21) then reduces to Poisson’s equation, ∆φ = ρ, (18.32) − the solution of which is the instantaneous Coulomb potential.3 This also allows us to write the vector potential for a static uniform magnetic feld (a very common experi- mental setup in solid-state physics) as 1 A! = !x B.! (18.33) −2 × Two remarks: (a) The Coulomb gauge is not Lorentz invariant. (b) A! diverges. The temporal gauge In this gauge, the scalar feld itself φ =0. (18.34) Used by Hermann Weyl who discovered the principle of gauge invariance in quantum mechanics in 1918, quite some years before Schrödinger’s equation was established. 3A contradiction to Einstein’s postulate of a fnite speed of light c, refecting the fact that the scalar po- tential itself is not a physical object. Typeset June 8, 2021 285 18 Gauge felds, coordinate systems, and geometric phases The axial gauge A very special choice, A3 =0. (18.35) Used in particle physics to avoid «ghost felds». 18.3 Nonrelativistic quantum mechanics In this section we discuss the Hamiltonian for a single nonrelativistic scalar particle of mass m, charge e. For vanishing electromagnetic felds, its Hamiltonian and the Schrödinger equation of motion simply reads 2 1 h¯ H = , (18.36) 2m i ∇ # $ ∂ HΨ=i¯h Ψ. (18.37) ∂t A fundamental principle of quantum mechanics states that the properties of a physi- cal object cannot be determined by measuring its wavefunction (which is impossible), but rather only by determining expectation values of operators expressed in a certain representation.4 That is equivalent to the freedom to choose an arbitrary phase for the wavefunction: This phase cannot be expressed as the expectation value of an operator, thus cannot be measured, hence its choice is arbitrary and doesn’t affect the physics of the system. So if we set ie/(hc¯ )Λ Ψ ΨΛ := e Ψ, (18.38) → Schrödinger’s equation should not be changed. However taking the gradient of the shifted wavefunction shifts the momentum operator and the time derivative h¯ h¯ e + Λ, (18.39) i ∇→ i ∇ c∇ ∂ ∂ e ∂ i¯h i¯h Λ (18.40) ∂t → ∂t − c ∂t and therefore does change the form of Eq. (18.37) except for the trivial case of a constant, single-valued global phase.
Load more

Recommended publications
  • Simple Circuit Theory and the Solution of Two Electricity Problems from The
    Simple circuit theory and the solution of two electricity problems from the Victorian Age A C Tort ∗ Departamento de F´ısica Te´orica - Instituto de F´ısica Universidade Federal do Rio de Janeiro Caixa Postal 68.528; CEP 21941-972 Rio de Janeiro, Brazil May 22, 2018 Abstract Two problems from the Victorian Age, the subdivision of light and the determination of the leakage point in an undersea telegraphic cable are discussed and suggested as a concrete illustrations of the relationships between textbook physics and the real world. Ohm’s law and simple algebra are the only tools we need to discuss them in the classroom. arXiv:0811.0954v1 [physics.pop-ph] 6 Nov 2008 ∗e-mail: [email protected]. 1 1 Introduction Some time ago, the present author had the opportunity of reading Paul J. Nahin’s [1] fascinating biog- raphy of the Victorian physicist and electrician Oliver Heaviside (1850-1925). Heaviside’s scientific life unrolls against a background of theoretical and technical challenges that the scientific and technological developments fostered by the Industrial Revolution presented to engineers and physicists of those times. It is a time where electromagnetic theory as formulated by James Clerk Maxwell (1831-1879) was un- derstood by only a small group of men, Lodge, FitzGerald and Heaviside, among others, that had the mathematical sophistication and imagination to grasp the meaning and take part in the great Maxwellian synthesis. Almost all of the electrical engineers, or electricians as they were called at the time, considered themselves as “practical men”, which effectively meant that most of them had a working knowledge of the electromagnetic phenomena spiced up with bits of electrical theory, to wit, Ohm’s law and the Joule effect.
    [Show full text]
  • The Concept of Field in the History of Electromagnetism
    The concept of field in the history of electromagnetism Giovanni Miano Department of Electrical Engineering University of Naples Federico II ET2011-XXVII Riunione Annuale dei Ricercatori di Elettrotecnica Bologna 16-17 giugno 2011 Celebration of the 150th Birthday of Maxwell’s Equations 150 years ago (on March 1861) a young Maxwell (30 years old) published the first part of the paper On physical lines of force in which he wrote down the equations that, by bringing together the physics of electricity and magnetism, laid the foundations for electromagnetism and modern physics. Statue of Maxwell with its dog Toby. Plaque on E-side of the statue. Edinburgh, George Street. Talk Outline ! A brief survey of the birth of the electromagnetism: a long and intriguing story ! A rapid comparison of Weber’s electrodynamics and Maxwell’s theory: “direct action at distance” and “field theory” General References E. T. Wittaker, Theories of Aether and Electricity, Longam, Green and Co., London, 1910. O. Darrigol, Electrodynamics from Ampère to Einste in, Oxford University Press, 2000. O. M. Bucci, The Genesis of Maxwell’s Equations, in “History of Wireless”, T. K. Sarkar et al. Eds., Wiley-Interscience, 2006. Magnetism and Electricity In 1600 Gilbert published the “De Magnete, Magneticisque Corporibus, et de Magno Magnete Tellure” (On the Magnet and Magnetic Bodies, and on That Great Magnet the Earth). ! The Earth is magnetic ()*+(,-.*, Magnesia ad Sipylum) and this is why a compass points north. ! In a quite large class of bodies (glass, sulphur, …) the friction induces the same effect observed in the amber (!"#$%&'(, Elektron). Gilbert gave to it the name “electricus”.
    [Show full text]
  • HISTORICAL SURVEY SOME PIONEERS of the APPLICATIONS of FRACTIONAL CALCULUS Duarte Valério 1, José Tenreiro Machado 2, Virginia
    HISTORICAL SURVEY SOME PIONEERS OF THE APPLICATIONS OF FRACTIONAL CALCULUS Duarte Val´erio 1,Jos´e Tenreiro Machado 2, Virginia Kiryakova 3 Abstract In the last decades fractional calculus (FC) became an area of intensive research and development. This paper goes back and recalls important pio- neers that started to apply FC to scientific and engineering problems during the nineteenth and twentieth centuries. Those we present are, in alphabet- ical order: Niels Abel, Kenneth and Robert Cole, Andrew Gemant, Andrey N. Gerasimov, Oliver Heaviside, Paul L´evy, Rashid Sh. Nigmatullin, Yuri N. Rabotnov, George Scott Blair. MSC 2010 : Primary 26A33; Secondary 01A55, 01A60, 34A08 Key Words and Phrases: fractional calculus, applications, pioneers, Abel, Cole, Gemant, Gerasimov, Heaviside, L´evy, Nigmatullin, Rabotnov, Scott Blair 1. Introduction In 1695 Gottfried Leibniz asked Guillaume l’Hˆopital if the (integer) order of derivatives and integrals could be extended. Was it possible if the order was some irrational, fractional or complex number? “Dream commands life” and this idea motivated many mathematicians, physicists and engineers to develop the concept of fractional calculus (FC). Dur- ing four centuries many famous mathematicians contributed to the theo- retical development of FC. We can list (in alphabetical order) some im- portant researchers since 1695 (see details at [1, 2, 3], and posters at http://www.math.bas.bg/∼fcaa): c 2014 Diogenes Co., Sofia pp. 552–578 , DOI: 10.2478/s13540-014-0185-1 SOME PIONEERS OF THE APPLICATIONS . 553 • Abel, Niels Henrik (5 August 1802 - 6 April 1829), Norwegian math- ematician • Al-Bassam, M. A. (20th century), mathematician of Iraqi origin • Cole, Kenneth (1900 - 1984) and Robert (1914 - 1990), American physicists • Cossar, James (d.
    [Show full text]
  • A Solution of the Interpretation Problem of Lorentz Transformations
    Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 30 July 2020 doi:10.20944/preprints202007.0705.v1 Article A Solution of the Interpretation Problem of Lorentz Transformations Grit Kalies* HTW University of Applied Sciences Dresden; 1 Friedrich-List-Platz, D-01069 Dresden, [email protected] * Correspondence: [email protected], Tel.: +49-351-462-2552 Abstract: For more than one hundred years, scientists dispute the correct interpretation of Lorentz transformations within the framework of the special theory of relativity of Albert Einstein. On the one hand, the changes in length, time and mass with increasing velocity are interpreted as apparent due to the observer dependence within special relativity. On the other hand, real changes are described corresponding to the experimental evidence of mass increase in particle accelerators or of clock delay. This ambiguity is accompanied by an ongoing controversy about valid Lorentz-transformed thermodynamic quantities such as entropy, pressure and temperature. In this paper is shown that the interpretation problem of the Lorentz transformations is genuinely anchored within the postulates of special relativity and can be solved on the basis of the thermodynamic approach of matter-energy equivalence, i.e. an energetic distinction between matter and mass. It is suggested that the velocity-dependent changes in state quantities are real in each case, in full agreement with the experimental evidence. Keywords: interpretation problem; Lorentz transformation; special relativity; thermodynamics; potential energy; space; time; entropy; non-mechanistic ether theory © 2020 by the author(s). Distributed under a Creative Commons CC BY license. Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 30 July 2020 doi:10.20944/preprints202007.0705.v1 2 of 25 1.
    [Show full text]
  • Oliver Heaviside (1850-1925) - Physical Mathematician D
    Oliver Heaviside (1850-1925) - Physical Mathematician D. A. EDGE. AFIMA The Mary Erskine School, Edinburgh 4m-; JL Mathematics is of two kinds, Rigorous and Physical. The former is Narrow: the latter Downloaded from Bold and Broad. To have to stop to formulate rigorous demonstrations would put a stop to most physico-mathematical enquiries. Am I to refuse to eat because I do not fully understand the mechanism of digestion? Oliver Heaviside http://teamat.oxfordjournals.org/ Preface Details Too little attention is paid on educational courses in Oliver Heaviside was the youngest of four sons of mathematics and science to the personalities of the Thomas Heaviside, then at 55 King Street, Camden great innovators. When teaching at Paisley College Town, London. His mother was formerly Rachel of Technology I was very impressed with the tape- West, who had been a governess in the Spottiswoode slide facilities in the library and I thought that here famjly and whose sister was the wife of Sir Charles was an ideal way of presenting biographical Wheatstone, the telegraphist and prolific inventor. material. Not only would well-prepared material be Thomas Heaviside was a skilful wood-engraver who at University of Iowa Libraries/Serials Acquisitions on June 17, 2015 instructive but also it would be entertaining. moved to London from Teesside with his wife and I was inspired to produce a tape-slide presentation young sons in about 1849. Oliver was born on May on Oliver Heaviside in particular by an excellent 18th, 1850. radio talk by Dr. D. M. A. Mercer.1 The essay below It has been suggested2 that the Heaviside family is along the lines of my script and is presented with a moved to London because the development of view to interesting others in Heaviside and perhaps photography was making wood-engravers redun- in the idea mentioned at the outset dant and there were better prospects of employment Summary in the capital.
    [Show full text]
  • 13. Maxwell's Equations and EM Waves. Hunt (1991), Chaps 5 & 6 A
    13. Maxwell's Equations and EM Waves. Hunt (1991), Chaps 5 & 6 A. The Energy of an Electromagnetic Field. • 1880s revision of Maxwell: Guiding principle = concept of energy flow. • Evidence for energy flow through seemingly empty space: ! induced currents ! air-core transformers, condensers. • But: Where is this energy located? Two equivalent expressions for electromagnetic energy of steady current: ½A J ½µH2 • A = vector potential, J = current • µ = permeability, H = magnetic force. density. • Suggests energy located outside • Suggests energy located in conductor. conductor in magnetic field. 13. Maxwell's Equations and EM Waves. Hunt (1991), Chaps 5 & 6 A. The Energy of an Electromagnetic Field. • 1880s revision of Maxwell: Guiding principle = concept of energy flow. • Evidence for energy flow through seemingly empty space: ! induced currents ! air-core transformers, condensers. • But: Where is this energy located? Two equivalent expressions for electrostatic energy: ½qψ# ½εE2 • q = charge, ψ = electric potential. • ε = permittivity, E = electric force. • Suggests energy located in charged • Suggests energy located outside object. charged object in electric field. • Maxwell: Treated potentials A, ψ as fundamental quantities. Poynting's account of energy flux • Research project (1884): "How does the energy about an electric current pass from point to point -- that is, by what paths and acording to what law does it travel from the part of the circuit where it is first recognizable as electric and John Poynting magnetic, to the parts where it is changed into heat and other forms?" (1852-1914) • Solution: The energy flux at each point in space is encoded in a vector S given by S = E × H.
    [Show full text]
  • Oliver Heaviside 1850-1925
    Oliver Heaviside 1850-1925 The words inductance, capacitance, and impedance were given to us by the great English physicist and engineer, Oliver Heaviside, who also pioneered in the use of Laplace and Fourier transforms in the analysis of electric circuits, first suggested the existence of an ionized atmospheric layer (now called the ionosphere) that can reflect radio waves, and is said to have predicted the increase of mass of a charge moving at great speeds before Einstein formulated his theory of relativity. He not only coined the word impedance but introduced its concept to the solution of ac circuits. Heaviside was born in London, the youngest of four sons of Thomas Heaviside, an engraver and watercolorist, and Rachel Elizabeth West, a sister-in-law of the famous physicist Sir Charles Wheatstone. Young Oliver's schooling ended when he was 16, but he trained himself at home in languages, mathematics, and the natural sciences. He became a telegraph operator in 1870, but in 1874 he was forced to retire because of increasing deafness. From then until his death he led a hermit like existence, devoting himself to investigations of electrical phenomena and publishing such works as Electrical Papers in 1892 and a three-volume treatise, Electromagnetic Theory (1893 – 1912). His free and original use of mathematics was decades ahead of his time and evoked controversy with his contemporaries. Nevertheless, his fame spread, and because of his great store of knowledge and the scientific help he generously extended to all who sought it, his home became known as The Inexhaustible Cavity. He was elected Fellow of the Royal Society in 1891 and received an honorary doctorate from the University of Gottingen.
    [Show full text]
  • The Daniell Cell, Ohm's Law and the Emergence of the International System of Units
    The Daniell Cell, Ohm’s Law and the Emergence of the International System of Units Joel S. Jayson∗ Brooklyn, NY (Dated: December 24, 2015) Telegraphy originated in the 1830s and 40s and flourished in the following decades, but with a patchwork of electrical standards. Electromotive force was for the most part measured in units of the predominant Daniell cell. Each company had their own resistance standard. In 1862 the British Association for the Advancement of Science formed a committee to address this situation. By 1873 they had given definition to the electromagnetic system of units (emu) and defined the practical units of the ohm as 109 emu units of resistance and the volt as 108 emu units of electromotive force. These recommendations were ratified and expanded upon in a series of international congresses held between 1881 and 1904. A proposal by Giovanni Giorgi in 1901 took advantage of a coincidence between the conversion of the units of energy in the emu system (the erg) and in the practical system (the joule) in that the same conversion factor existed between the cgs based emu system and a theretofore undefined MKS system. By introducing another unit, X (where X could be any of the practical electrical units), Giorgi demonstrated that a self consistent MKSX system was tenable without the need for multiplying factors. Ultimately the ampere was selected as the fourth unit. It took nearly 60 years, but in 1960 Giorgi’s proposal was incorporated as the core of the newly inaugurated International System of Units (SI). This article surveys the physics, physicists and events that contributed to those developments.
    [Show full text]
  • The Beauty of Equations
    Proceedings of Bridges 2013: Mathematics, Music, Art, Architecture, Culture The Beauty of Equations Robert P. Crease Philosophy Department Stony Brook University Stony Brook, NY 11794 E-mail: [email protected] Abstract This paper discusses grounds for ascribing beauty to equations. Philosophical accounts of beauty in the context of science tend to stress one or more of three aspects of beautiful objects: depth, economy, and definitiveness. The paper considers several cases that engage these aspects in different ways, depending on whether the equation is considered as a signpost, as a stand-in for a phenomenon, as making a disciplinary transformation, or as inspiring a personal revelation. Introduction Certain equations have become more than scientific tools or educational instruments, and exert what might be called cultural force. A few equations have acquired celebrity status in that people recognize them while knowing next to nothing about them, while other equations attract fascination and awe because they are thought to harbor deep significance. For these reasons, equations are frequently encountered outside science and mathematics, and even turn up in novels, plays, and films, having effectively acquired the status of cultural touchstones. Sometimes what the public encounters in the form of an equation is simply an ordinary thought cloaked in mathematical dress. Even the form of certain equations can be culturally influential, as when instructions for behavior are presented as equations. Several recent books have broached the subject of the meaning, greatness, or aesthetic value of certain important equations [1-5]. I want to discuss the last of these subjects: aesthetics. Scientists do sometimes ascribe beauty to theories or equations, and occasionally declare beauty to be an essential property of fundamental equations.
    [Show full text]
  • Celebrating 125 Years of Oliver Heaviside's 'Electromagnetic Theory'
    ISSN 1364-503X | Volume 376 | Issue 2134 | 13 December 2018 Phil. Trans. R. Soc. A | R. Soc. A Phil. Trans. The Royal Society is a self-governing Fellowship of many of the world’s most distinguished scientists drawn from all areas of science, engineering, and medicine. The Society’s fundamental purpose, as it 376 Volume has been since its foundation in 1660, is to recognise, promote, and support excellence in science and to Celebrating 125 years of Oliver Heaviside’s ‘Electromagnetic encourage the development and use of science for Theory’ the benefi t of humanity. | Issue 2134 Theme issue compiled and edited by Christopher Donaghy-Spargo and Alex Yakovlev The Society’s strategic priorities emphasise its commitment to the highest quality science, to curiosity-driven research, and to the development | and use of science for the benefi t of society. 13 December 2018 These priorities are: • Promoting science and its benefi ts • Recognising excellence in science • Supporting outstanding science • Providing scientifi c advice for policy • Fostering international and global cooperation ‘Electromagnetic Theory’ 125 years of Oliver Heaviside’s • Education and public engagement For further information on the Royal Society The Royal Society 6 – 9 Carlton House Terrace London SW1Y 5AG T +44 20 7451 2500 W royalsociety.org For further information on Philosophical Transactions of the Royal Society A T +44 20 7451 2214 E [email protected] W rsta.royalsocietypublishing.org 1364-503X(20181213)376:2134 ISBN: 978-1-78252-358-1 ISSN 1364-503X The Royal Society Registered Charity No 207043 RSTA_376_2134_Cover.indd 2-4 30/10/18 5:13 PM Oliver Heaviside’s electromagnetic theory rsta.royalsocietypublishing.org Christopher Donaghy-Spargo1 and Alex Yakovlev2 1Department of Engineering, Durham University, South Road, Durham DH1 3LE, UK Introduction 2School of Engineering, Newcastle University, Merz Court, Cite this article: Donaghy-Spargo C, Yakovlev NewcastleuponTyneNE17RU,UK A.
    [Show full text]
  • Heaviside's Operator Calculus
    Heaviside’s Operator Calculus ©2007-2009 Ron Doerfler Dead Reckonings: Lost Art in the Mathematical Sciences http://www.myreckonings.com/wordpress May 14, 2009 An operational calculus converts derivatives and integrals to operators that act on functions, and by doing so ordinary and partial linear differential equations can be reduced to purely algebraic equations that are much easier to solve. There have been a number of operator methods created as far back as Liebniz, and some operators such as the Dirac delta function created controversy at the time among mathematicians, but no one wielded operators with as much flair and abandon over the objections of mathematicians as Oliver Heaviside, the reclusive physicist and pioneer of electromagnetic theory. The name of Oliver Heaviside (1850-1925) is not well-known to the general public today. However, it was Heaviside, for example, who developed Maxwell’s electromagnetic equations into the four vector calculus equations in two unknowns that we are familiar with today; Maxwell left them as 20 equations in 20 unknowns expressed as quaternions, a once-popular mathematical system currently experiencing a revival for fast coordinate transformations in video games. Heaviside also did important early work in long-distance telegraphy and telephony, introducing induction-loading of long cables to minimize distortion and patenting the coaxial cable. At one time the ionosphere was called the Heaviside layer after his suggestion (and that of Arthur Kennelly) that a layer of charged ions in the upper atmosphere (now just one layer of the ionosphere) would account for the puzzlingly long distances that radio waves traveled.
    [Show full text]
  • A Brief History of Electromagnetism
    A Brief History of The Development of Classical Electrodynamics Professor Steven Errede UIUC Physics 436, Fall Semester 2015 Loomis Laboratory of Physics The University of Illinois @ Urbana-Champaign 900 BC: Magnus, a Greek shepherd, walks across a field of black stones which pull the iron nails out of his sandals and the iron tip from his shepherd's staff (authenticity not guaranteed). This region becomes known as Magnesia. 600 BC: Thales of Miletos (Greece) discovered that by rubbing an 'elektron' (a hard, fossilized resin that today is known as amber) against a fur cloth, it would attract particles of straw and feathers. This strange effect remained a mystery for over 2000 years. 1269 AD: Petrus Peregrinus of Picardy, Italy, discovers that natural spherical magnets (lodestones) align needles with lines of longitude pointing between two pole positions on the stone. ca. 1600: Dr. William Gilbert (court physician to Queen Elizabeth) discovers that the earth is a giant magnet just like one of the stones of Peregrinus, explaining how compasses work. He also investigates static electricity and invents an electric fluid which is liberated by rubbing, and is credited with the first recorded use of the word 'Electric' in a report on the theory of magnetism. Gilbert's experiments subsequently led to a number of investigations by many pioneers in the development of electricity technology over the next 350 years. ca. 1620: Niccolo Cabeo discovers that electricity can be repulsive as well as attractive. 1630: Vincenzo Cascariolo, a Bolognese shoemaker, discovers fluorescence. 1638: Rene Descartes theorizes that light is a pressure wave through the second of his three types of matter of which the universe is made.
    [Show full text]