BABEL OF UNITS THE EVOLUTION OF UNITS SYSTEMS IN CLASSICAL ELECTROMAGNETISM Neal J. Carron Rock West Solutions, Inc. Santa Barbara, CA 93101 [email protected] May 21, 2015 1 BABEL OF UNITS CONTENTS 1 INTRODUCTION ..................................................................................................................... 3 2 UNITS vs DIMENSIONS .......................................................................................................... 5 3 THE PROBLEM ........................................................................................................................ 6 4 OVERVIEW .............................................................................................................................. 8 4.1 An Instructive Imaginary History; The Case of Gravity ................................................... 8 4.2 The Real (but very short) History of Electricity and Magnetism ...................................... 9 4.3 History of Development of Units Systems ...................................................................... 14 5 GENERAL RELATIONS IN ALL SYSTEMS ....................................................................... 19 5.1 A Relation Between ke and km ......................................................................................... 20 5.2 The Magnetic Field ......................................................................................................... 21 5.3 Maxwell’s Equations in Arbitrary Units ......................................................................... 23 5.4 Constitutive Relations ..................................................................................................... 25 6 THE VARIOUS SYSTEMS .................................................................................................... 28 6.1 The Electromagnetic Units System ................................................................................. 28 6.1.1 The unit of magnetic flux density B ......................................................................................... 31 6.1.2 The unit of magnetic field H .................................................................................................... 32 6.1.3 Constitutive relations ............................................................................................................... 32 6.1.4 Other dynamical quantities ...................................................................................................... 33 6.1.5 Maxwell’s Equations in Electromagnetic Units ....................................................................... 34 6.2 The Electrostatic Units System ....................................................................................... 35 6.2.1 Maxwell’s Equations in Electrostatic Units ............................................................................. 37 6.3 The Gaussian System ...................................................................................................... 38 6.3.1 Maxwell’s Equations in Gaussian Units .................................................................................. 39 6.4 The Variant-Gaussian System ......................................................................................... 41 6.4.1 Maxwell’s Equations in Variant-Gaussian Units ..................................................................... 42 6.5 “Rationalizing” a Unit System ........................................................................................ 44 6.6 The Heaviside-Lorentz System ....................................................................................... 45 6.6.1 Maxwell’s Equations in Heaviside-Lorentz Units ................................................................... 46 6.7 The SI System ................................................................................................................. 46 6.7.1 Constitutive relations ............................................................................................................... 49 6.7.2 Maxwell’s Equations in SI ....................................................................................................... 49 6.7.3 Other dynamical quantities ...................................................................................................... 50 6.8 Physical Properties of the Electron ................................................................................. 51 7 A MAGNETIC EXAMPLE ..................................................................................................... 52 8 UNITS USED IN SOME TEXTBOOKS................................................................................. 53 9 APPENDIX A. Gauss' Absolute Measure of Magnetic Field ............................................... 54 10 APPENDIX B. Applying Conversion Relations .................................................................... 57 11 APPENDIX C. A link between electricity and magnetism before Oersted ........................... 63 12 REFERENCES ......................................................................................................................... 64 2 1 INTRODUCTION Few things are more confusing than the various sets of units employed in classical electromagnetism1. Five systems (of very many2) in common use are: • The International System of Units (SI) • Gaussian • Variant-Gaussian • Heaviside-Lorentz (rationalized Gaussian), and • Various sets of “Natural” units. The International System of Units, universally abbreviated SI (from the French Le Système International d’Unités), is the modern metric system of measurement. It applies to all of physics, and is now officially preferred. In electromagnetic context it used to be called MKS or MKSA, or, more precisely, Rationalized MKSA. Gaussian is commonly used in electromagnetism, especially in fundamental discussions. It has replaced pure esu or emu. SI and Gaussian are by far the most common systems today. Variant-Gaussian3 is sometimes preferred to Gaussian because of the size of its current unit. It has no standard name, and is sometimes confusingly also called Gaussian. Heaviside-Lorentz and natural units ( = c = 1) are commonly used in Quantum Electrodynamics, less so in classical electrodynamics4. Other natural systems are often employed (e.g. Atomic Units in atomic structure studies). Natural units systems are not really for classical electromagnetism, and are not discussed here. A good discussion is in Klauber [Kl13]. See also Tomilin [To99]. For most purposes one need understand only the two major systems, Gaussian and SI. The surest way to understand the Gaussian is to understand the two earlier systems of which it is an amalgam: pure Electrostatic Units (esu), and pure Electromagnetic Units (emu). Each of these latter two systems is essentially never used anymore. However the letters “esu” are sometimes still loosely used when referring to electric quantities in Gaussian units, such as electric field (1 esu = 3×104 V/m) or charge (electron charge = 4.803×10−10 esu). 1 One is the units in radiometry and photometry. Nit, stilb, Talbot, radiant exitance, Lumen, candela, lux, Lambert, phot, sterance, apostilb, and the Bougie-Hectomètre-Carrè are among the quantities that occur. See the IR Handbook, Section 1.1. 2 A few examples: Sommerfeld [So52] discusses a five dimension system in which the magnetic pole is the fifth basic quantity, but does not use it. The Russian Kalantaroff proposed a system with four basic dimensions: length, time, charge, and magnetic flux [Ka29. So52]. In that system mass would be expressed in terms of his four basic dimensions; see [Ki62]. Ampere himself used his own system, called Electrodynamic Units. Cohn (1900, 1927) introduced another system [So52]. Mendelson [Me15] discusses one early suggestion. Many others, some bizarre, have been proposed. 3 This system appears to have no official name. We simply call it Variant-Gaussian. 4 One of the earliest uses of units with c = 1 is in Einstein’s 1934 lecture, well before quantum field theory matured [To07]. 3 Here we recount the development of the six systems: Electromagnetic, Electrostatic, Gaussian, Variant-Gaussian, Heaviside-Lorentz, and SI, and come to appreciate why there is more than one system in the first place. Handbooks and texts provide conversion factors and some technical discussion, but not always a good understanding of how so many unit systems came to be (with good reason; it’s a long story). Several texts provide good discussions (Jackson, Panofsky & Phillips, Stratton, Sommerfeld, etc.). Jackson’s treatment is straightforward and, to the present writer, the clearest of textbook discussions. He suggests Birge [Bi34, Bi35a,b]; these three papers are indeed excellent sources of understanding, emphasizing the arbitrariness in the number of base dimensions and the choice of units. They were written during a time (1930’s) of vehement discussions about which system should be adopted internationally. See also Page [Pa70a]. The article by Silsbee [Si62] is a thorough comparative and historical treatment of the various systems, although it belabors the different conceptual approaches of theorists and experimentalists. It is surprising that a thing ostensibly as simple as units can be so troublesome. What should be simple rules have turned out to be very confusing. The subject of units in electromagnetism has confounded a huge number of scientists for more than 180 years. Untold papers have been written advocating this or that system. References cited in the three papers by