Electromagnetic Field Theory - Nikola Tesla
ElectromagneticField TheoryB O T HIDÉϒU PSILON B OOKS
E LECTROMAGNETIC F IELD T HEORY
ElectromagneticField TheoryB O T HIDÉSwedish Institute of Space PhysicsUppsala, SwedenandDepartment of Astronomy and Space PhysicsUppsala University, SwedenandLOIS Space CentreSchool of Mathematics and Systems EngineeringVäxjö University, SwedenϒU PSILON B OOKS · U PPSALA · S WEDEN
Also availableE LECTROMAGNETIC F IELD T HEORYE XERCISESbyTobia Carozzi, Anders Eriksson, Bengt Lundborg,Bo Thidé and Mattias WaldenvikFreely downloadable fromwww.plasma.uu.se/CEDThis book was typeset in LATEX 2ε (based on TEX 3.141592 and Web2C 7.4.4) on an HP Visualize 9000 3600 workstation running HP-UX 11.11.Copyright 1997–2008 byBo ThidéUppsala, SwedenAll rights reserved.Electromagnetic Field TheoryISBN X-XXX-XXXXX-X
To the memory of professorL EV M IKHAILOVICH E RUKHIMOV (1936–1997)dear friend, great physicist, poetand a truly remarkable man.
Downloaded from http://www.plasma.uu.se/CED/BookVersion released 8th June 2008 at 23:04.C ONTENTSContentsList of FiguresixxiiiPrefacexv1Classical Electrodynamics1.1 Electrostatics1.1.1 Coulomb’s law1.1.2 The electrostatic field1.2 Magnetostatics1.2.1 Ampère’s law1.2.2 The magnetostatic field1.3 Electrodynamics1.3.1 Equation of continuity for electric charge1.3.2 Maxwell’s displacement current1.3.3 Electromotive force1.3.4 Faraday’s law of induction1.3.5 Maxwell’s microscopic equations1.3.6 Maxwell’s macroscopic equations1.4 Electromagnetic duality1.5 Bibliography1.6 Examples122366791010111215151618202Electromagnetic Waves2.1 The wave equations2.1.1 The wave equation for E2.1.2 The wave equation for B2.1.3 The time-independent wave equation for E2526262727ix
Contents2.22.32.42.5x303132333536Plane waves2.2.1 Telegrapher’s equation2.2.2 Waves in conductive mediaObservables and averagesBibliographyExample3Electromagnetic Potentials3.1 The electrostatic scalar potential3.2 The magnetostatic vector potential3.3 The electrodynamic potentials3.4 Gauge transformations3.5 Gauge conditions3.5.1 Lorenz-Lorentz gauge3.5.2 Coulomb gauge3.5.3 Velocity gauge3.6 Bibliography3.7 Examples39394040414243474949514Electromagnetic Fields and Matter4.1 Electric polarisation and displacement4.1.1 Electric multipole moments4.2 Magnetisation and the magnetising field4.3 Energy and momentum4.3.1 The energy theorem in Maxwell’s theory4.3.2 The momentum theorem in Maxwell’s theory4.4 Bibliography4.5 Example5353535658585962635Electromagnetic Fields from Arbitrary Source Distributions5.1 The magnetic field5.2 The electric field5.3 The radiation fields5.4 Radiated energy5.4.1 Monochromatic signals5.4.2 Finite bandwidth signals5.5 Bibliography65676971747475766Electromagnetic Radiation and Radiating Systems6.1 Radiation from an extended source volume at rest6.1.1 Radiation from a one-dimensional current distribution777778Version released 8th June 2008 at 23:04.Downloaded from http://www.plasma.uu.se/CED/Book
6.26.36.46.56.1.2 Radiation from a two-dimensional current distributionRadiation from a localised source volume at rest6.2.1 The Hertz potential6.2.2 Electric dipole radiation6.2.3 Magnetic dipole radiation6.2.4 Electric quadrupole radiationRadiation from a localised charge in arbitrary motion6.3.1 The Liénard-Wiechert potentials6.3.2 Radiation from an accelerated point charge6.3.3 Bremsstrahlung6.3.4 Cyclotron and synchrotron radiation6.3.5 Radiation from charges moving in 1161231247Relativistic Electrodynamics7.1 The special theory of relativity7.1.1 The Lorentz transformation7.1.2 Lorentz space7.1.3 Minkowski space7.2 Covariant classical mechanics7.3 Covariant classical electrodynamics7.3.1 The four-potential7.3.2 The Liénard-Wiechert potentials7.3.3 The electromagnetic field tensor7.4 tromagnetic Fields and Particles8.1 Charged particles in an electromagnetic field8.1.1 Covariant equations of motion8.2 Covariant field theory8.2.1 Lagrange-Hamilton formalism for fields and interactions8.3 Bibliography8.4 Example155155155161162169171FFormulæF.1 The electromagnetic fieldF.1.1 Maxwell’s equationsF.1.2 Fields and potentialsF.1.3 Force and energyF.2 Electromagnetic radiation173173173174174174Downloaded from http://www.plasma.uu.se/CED/BookVersion released 8th June 2008 at 23:04.xi
ContentsF.3F.4F.5xiiF.2.1 Relationship between the field vectors in a plane waveF.2.2 The far fields from an extended source distributionF.2.3 The far fields from an electric dipoleF.2.4 The far fields from a magnetic dipoleF.2.5 The far fields from an electric quadrupoleF.2.6 The fields from a point charge in arbitrary motionSpecial relativityF.3.1 Metric tensorF.3.2 Covariant and contravariant four-vectorsF.3.3 Lorentz transformation of a four-vectorF.3.4 Invariant line elementF.3.5 Four-velocityF.3.6 Four-momentumF.3.7 Four-current densityF.3.8 Four-potentialF.3.9 Field tensorVector relationsF.4.1 Spherical polar coordinatesF.4.2 Vector 176176177177177177177178178180M Mathematical MethodsM.1 Scalars, vectors and tensorsM.1.1 VectorsM.1.2 FieldsM.1.3 Vector algebraM.1.4 Vector analysisM.2 Analytical mechanicsM.2.1 Lagrange’s equationsM.2.2 Hamilton’s equationsM.3 ExamplesM.4 203Version released 8th June 2008 at 23:04.Downloaded from http://www.plasma.uu.se/CED/Book
Downloaded from http://www.plasma.uu.se/CED/BookVersion released 8th June 2008 at 23:04.L IST OF F IGURES1.11.21.31.4Coulomb interaction between two electric chargesCoulomb interaction for a distribution of electric chargesAmpère interactionMoving loop in a varying B field357135.1Radiation in the far .14Linear antennaElectric dipole antenna geometryLoop antennaMultipole radiation geometryElectric dipole geometryRadiation from a moving charge in vacuumAn accelerated charge in vacuumAngular distribution of radiation during bremsstrahlungLocation of radiation during bremsstrahlungRadiation from a charge in circular motionSynchrotron radiation lobe widthThe perpendicular electric field of a moving chargeElectron-electron scatteringVavilov-Čerenkov lative motion of two inertial systemsRotation in a 2D Euclidean spaceMinkowski diagram1351411428.1Linear one-dimensional mass chain162M.1 Tetrahedron-like volume element of matter194xiii
Downloaded from http://www.plasma.uu.se/CED/BookVersion released 8th June 2008 at 23:04.P REFACEThis book is the result of a more than thirty-five year long love affair. In theautumn of 1972, I took my first advanced course in electrodynamics at the Department of Theoretical Physics, Uppsala University. A year later, I joined theresearch group there and took on the task of helping the late professor P ER O LOFF RÖMAN, who one year later become my Ph.D. thesis advisor, with the preparation of a new version of his lecture notes on the Theory of Electricity. Thesetwo things opened up my eyes for the beauty and intricacy of electrodynamics,already at the classical level, and I fell in love with it. Ever since that time, I haveon and off had reason to return to electrodynamics, both in my studies, researchand the teaching of a course in advanced electrodynamics at Uppsala Universitysome twenty odd years after I experienced the first encounter with this subject.The current version of the book is an outgrowth of the lecture notes that I prepared for the four-credit course Electrodynamics that was introduced in the Uppsala University curriculum in 1992, to become the five-credit course ClassicalElectrodynamics in 1997. To some extent, parts of these notes were based on lecture notes prepared, in Swedish, by my friend and colleague B ENGT L UNDBORG,who created, developed and taught the earlier, two-credit course ElectromagneticRadiation at our faculty.Intended primarily as a textbook for physics students at the advanced undergraduate or beginning graduate level, it is hoped that the present book may beuseful for research workers too. It provides a thorough treatment of the theoryof electrodynamics, mainly from a classical field theoretical point of view, andincludes such things as formal electrostatics and magnetostatics and their unification into electrodynamics, the electromagnetic potentials, gauge transformations, covariant formulation of classical electrodynamics, force, momentum andenergy of the electromagnetic field, radiation and scattering phenomena, electromagnetic waves and their propagation in vacuum and in media, and covariantLagrangian/Hamiltonian field theoretical methods for electromagnetic fields, particles and interactions. The aim has been to write a book that can serve both asan advanced text in Classical Electrodynamics and as a preparation for studies inQuantum Electrodynamics and related subjects.In an attempt to encourage participation by other scientists and students inthe authoring of this book, and to ensure its quality and scope to make it usefulxv
Prefacein higher university education anywhere in the world, it was produced within aWorld-Wide Web (WWW) project. This turned out to be a rather successful move.By making an electronic version of the book freely down-loadable on the net,comments have been received from fellow Internet physicists around the worldand from WWW ‘hit’ statistics it seems that the book serves as a frequently usedInternet resource.1 This way it is hoped that it will be particularly useful forstudents and researchers working under financial or other circumstances that makeit difficult to procure a printed copy of the book.Thanks are due not only to Bengt Lundborg for providing the inspiration towrite this book, but also to professor C HRISTER WAHLBERG and professor G ÖRANFÄLDT, Uppsala University, and professor YAKOV I STOMIN, Lebedev Institute,Moscow, for interesting discussions on electrodynamics and relativity in generaland on this book in particular. Comments from former graduate students M ATTIASWALDENVIK, TOBIA C AROZZI and ROGER K ARLSSON as well as A NDERS E RIKS SON , all at the Swedish Institute of Space Physics in Uppsala and who all haveparticipated in the teaching on the material covered in the course and in this bookare gratefully acknowledged. Thanks are also due to my long-term space physicscolleague H ELMUT KOPKA of the Max-Planck-Institut für Aeronomie, Lindau,Germany, who not only taught me about the practical aspects of high-power radiowave transmitters and transmission lines, but also about the more delicate aspectsof typesetting a book in TEX and LATEX. I am particularly indebted to Academicianprofessor V ITALIY L AZAREVICH G INZBURG, 2003 Nobel Laureate in Physics, forhis many fascinating and very elucidating lectures, comments and historical noteson electromagnetic radiation and cosmic electrodynamics while cruising on theVolga river at our joint Russian-Swedish summer schools during the 1990s, andfor numerous private discussions over the years.Finally, I would like to thank all students and Internet users who have downloaded and commented on the book during its life on the World-Wide Web.I dedicate this book to my son M ATTIAS, my daughter K AROLINA, myhigh-school physics teacher, S TAFFAN RÖSBY, and to my fellow members of theC APELLA P EDAGOGICA U PSALIENSIS.Uppsala, SwedenJune, 20081 AtxviB O T HIDÉwww.physics.irfu.se/ btthe time of publication of this edition, more than 600 000 downloads have been recorded.Version released 8th June 2008 at 23:04.Downloaded from http://www.plasma.uu.se/CED/Book
Downloaded from http://www.plasma.uu.se/CED/BookVersion released 8th June 2008 at 23:04.1C LASSICAL E LECTRODYNAMICSClassical electrodynamics deals with electric and magnetic fields and interactionscaused by macroscopic distributions of electric charges and currents. This meansthat the concepts of localised electric charges and currents assume the validity ofcertain mathematical limiting processes in which it is considered possible for thecharge and current distributions to be localised in infinitesimally small volumes ofspace. Clearly, this is in contradiction to electromagnetism on a truly microscopicscale, where charges and currents have to be treated as spatially extended objectsand quantum corrections must be included. However, the limiting processes usedwill yield results which are correct on small as well as large macroscopic scales.It took the genius of JAMES C LERK M AXWELL to consistently unify electricityand magnetism into a super-theory, electromagnetism or classical electrodynamics (CED), and to realise that optics is a subfield of this super-theory. Early inthe 20th century, H ENDRIK A NTOON L ORENTZ took the electrodynamics theoryfurther to the microscopic scale and also laid the foundation for the special theory of relativity, formulated by A LBERT E INSTEIN in 1905. In the 1930s PAULA. M. D IRAC expanded electrodynamics to a more symmetric form, includingmagnetic as well as electric charges. With his relativistic quantum mechanics,he also paved the way for the development of quantum electrodynamics (QED)for which R ICHARD P. F EYNMAN, J ULIAN S CHWINGER, and S IN -I TIRO TOMON AGA in 1965 received their Nobel prizes in physics. Around the same time, physicists such as S HELDON G LASHOW, A BDUS S ALAM, and S TEVEN W EINBERG wereable to unify electrodynamics the weak interaction theory to yet another supertheory, electroweak theory, an achievement which rendered them the Nobel prizein physics 1979. The modern theory of strong interactions, quantum chromodynamics (QCD), is influenced by QED.In this chapter we start with the force interactions in classical electrostatics1
1. Classical Electrodynamicsand classical magnetostatics and introduce the static electric and magnetic fieldsto find two uncoupled systems of equations for them. Then we see how the conservation of electric charge and its relation to electric current leads to the dynamicconnection between electricity and magnetism and how the two can be unifiedinto one ‘super-theory’, classical electrodynamics, described by one system ofeight coupled dynamic field equations—the Maxwell equations.At the end of this chapter we study Dirac’s symmetrised form of Maxwell’sequations by introducing (hypothetical) magnetic charges and magnetic currentsinto the theory. While not identified unambiguously in experiments yet, magnetic charges and currents make the theory much more appealing, for instance byallowing for duality transformations in a most natural way.1.1 ElectrostaticsThe theory which describes physical phenomena related to the interaction between stationary electric charges or charge distributions in a finite space whichhas stationary boundaries is called electrostatics. For a long time, electrostatics,under the name electricity, was considered an independent physical theory of itsown, alongside other physical theories such as magnetism, mechanics, optics andthermodynamics.11.1.1 Coulomb’s lawIt has been found experimentally that in classical electrostatics the interactionbetween stationary, electrically charged bodies can be described in terms of amechanical force. Let us consider the simple case described by figure 1.1 onpage 3. Let F denote the force acting on an electrically charged particle withcharge q located at x, due to the presence of a charge q′ located at x′ . Accordingto Coulomb’s law this force is, in vacuum, given by the expression qq′qq′ ′11qq′ x x′ (1.1) F(x) 4πε0 x x′ 34πε04πε0 x x′ x x′ 1 Thephysicist and philosopher P IERRE D UHEM (1861–1916) once wrote:‘The whole theory of electrostatics constitutes a group of abstract ideas and general propositions, formulated in the clear and concise language of geometry and algebra, and connected with one another by the rules of strict logic. This whole fully satisfies the reason ofa French physicist and his taste for clarity, simplicity and order. . . .’2Version released 8th June 2008 at 23:04.Downloaded from http://www.plasma.uu.se/CED/Book
Electrostaticsqx x′xq′x′OF IGURE 1 .1 : Coulomb’s law describes how a static electric charge q, located ata point x relative to the origin O, experiences an electrostatic force from a staticelectric charge q′ located at x′ .where in the last step formula (F.71) on page 179 was used. In SI units, which weshall use throughout, the force F is measured in Newton (N), the electric charges qand q′ in Coulomb (C) [ Ampère-seconds (As)], and the length x x′ in metres(m). The constant ε0 107 /(4πc2 ) 8.8542 10 12 Farad per metre (F/m) isthe vacuum permittivity and c 2.9979 108 m/s is the speed of light in vacuum.In CGS units ε0 1/(4π) and the force is measured in dyne, electric charge instatcoulomb, and length in centimetres (cm).1.1.2 The electrostatic fieldInstead of describing the electrostatic interaction in terms of a ‘force action at adistance’, it turns out that it is for most purposes more useful to introduce theconcept of a field and to describe the electrostatic interaction in terms of a staticvectorial electric field Estat defined by the limiting processdefFq 0 qEstat lim(1.2)where F is the electrostatic force, as defined in equation (1.1) on page 2, from anet electric charge q′ on the test particle with a small electric net electric chargeq. Since the purpose of the limiting process is to assure that the test charge q doesnot distort the field set up by q′ , the expression for Estat does not depend explicitlyon q but only on the charge q′ and the relative radius vector x x′ . This meansthat we can say that any net electric charge produces an electric field in the spaceDownloaded from http://www.plasma.uu.se/CED/BookVersion released 8th June 2008 at 23:04.3
1. Classical Electrodynamicsthat surrounds it, regardless of the existence of a second charge anywhere in thisspace.2Using (1.1) and equation (1.2) on page 3, and formula (F.70) on page 179,we find that the electrostatic field Estat at the field point x (also known as theobservation point), due to a field-producing electric charge q′ at the source pointx′ , is given by q′ ′11q′q′ x x′ Estat (x) 4πε0 x x′ 34πε04πε0 x x′ x x′ (1.3)In the presence of several field producing discrete electric charges q′i , locatedat the points x′i , i 1, 2, 3, . . . , respectively, in an otherwise empty space, the assumption of linearity of vacuum3 allows us to superimpose their individual electrostatic fields into a total electrostatic fieldEstat (x) 14πε0 q′iix x′ix x′i(1.4)3If the discrete electric charges are small and numerous enough, we introducethe electric charge density ρ, measured in C/m3 in SI units, located at x′ withina volume V ′ of limited extent and replace summation with integration over thisvolume. This allows us to describe the total field as ZZ11x x′13 ′′ Estat (x) d3x′ ρ(x′ )dxρ(x) 4πε0 V ′4πε0 V ′ x x′ x x′ 3Zρ(x′ )1 d3x′ 4πε0 x x′ V′(1.5)where we used formula (F.70) on page 179 and the fact that ρ(x′ ) does not dependon the unprimed (field point) coordinates on which operates.2 In the preface to the first edition of the first volume of his book A Treatise on Electricity and Magnetism, first published in 1873, James Clerk Maxwell describes this in the following almost poetic manner[9]:‘For instance, Faraday, in his mind’s eye, saw lines of force traversing all space where themathematicians saw centres of force attracting at a distance: Faraday saw a medium wherethey saw nothing but distance: Faraday sought the seat of the phenomena in real actionsgoing on in the medium, they were satisfied that they had found it in a power of action ata distance impressed on the electric fluids.’3 In fact, vacuum exhibits a quantum mechanical nonlinearity due to vacuum polarisation effects manifesting themselves in the momentary creation and annihilation of ele
8 Electromagnetic Fields and Particles 155 8.1 Charged particles in an electromagnetic field 155 8.1.1 Covariant equations of motion 155 8.2 Covariant field theory 161 8.2.1 Lagrange-Hamilton formalism for fields and interactions 162 8.3 Bibliography 169 8.4 Example 171 F Formulæ 173 F.1 The
The Lost Journals of Nikola Tesla Chapter One The Secret Life Of Nikola Tesla Nikola Tesla was beyond a doubt the greatest genius of the 20th century. Our way of life today, the technology that we take for granted, is all
Tesla’s AC power system is the worldwide standard today. [6] [7] [8] 1.1.2 The Tesla Coil In 1981, Nikola Tesla invented one of his most famous devices - the Tesla coil. To be fair, electrical coils exists before Nikola Tesla. Ruhmkor coils, named after Heinrich Ruhmkor
NIKOLA TESLA AND THE NEW YORKER HOTEL. Nikola Tesla lived in rooms 3327 and 3328 of the Hotel New Yorker from 1933 to 1943. He was an ethnic Serbian born in Croatia, on 10 July 1856. Tesla invented the system of AC power that we use today, including the AC generator, AC motor, and the method of transmission of power.
Tesla coil Tesla turbine Teleforce Tesla's oscillator Tesla electric car Tesla principle Alternating current Induction motor Rotating magnetic field Wireless technology Particle beam weapon Death ray. Born 10 July 1856 Smiljan, Austrian Empire Died 7 January 1943 (age 86) New York City, New York, USA
Fig.1 Radiant energy according to Nikola Tesla At first glance, this contradicts the age-old experience of studying the electromagnetic field (according to modern concepts, any electromagnetic field has components wh
Nikola Tesla’s Contributions to Radio Developments 135 Fig. 3 – First drawing of Tesla Coil presented in his 1891 lecture. Tesla’s oscillator is relatively easily outlined, although its detailed description is extremely complex. In the following analysis we will assume that primary and secondary circuits have been adjusted to the same .
Tesla, Goodness, New Energy, Creation. 1. Introduction . This is the article I’ve prepared and submitted for presentation on . Nikola Tesla 70th Year Memorial Conference “TESLA SPIRIT AWARD BENEFIT”, January 7. th . 2013, New York. The article was not accepted
Tesla coil is, in case you don [t know. The Tesla Coil was invented by an electrical engineer named Nikola Tesla around 1891 as a device to transmit electricity wirelessly over large distances. It is an air-cored resonant transformer capable of achieving extremely high output voltages at high frequencies. A Tesla coil differs from basic iron .
