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- 1. Electromagnetic Field Theory B O T HIDÉ Υ U PSILON B OOKS
- 2. E LECTROMAGNETIC F IELD T HEORY
- 3. Electromagnetic Field Theory B O T HIDÉ Swedish Institute of Space Physics and Department of Astronomy and Space Physics Uppsala University, Sweden and School of Mathematics and Systems Engineering Växjö University, Sweden ΥU PSILON B OOKS · C OMMUNA AB · U PPSALA · S WEDEN
- 4. Also available E LECTROMAGNETIC F IELD T HEORY E XERCISES by Tobia Carozzi, Anders Eriksson, Bengt Lundborg, Bo Thidé and Mattias Waldenvik Freely downloadable from www.plasma.uu.se/CEDThis book was typeset in LTEX 2ε (based on TEX 3.14159 and Web2C 7.4.2) on an AHP Visualize 9000⁄360 workstation running HP-UX 11.11.Copyright c 1997, 1998, 1999, 2000, 2001, 2002, 2003 and 2004 byBo ThidéUppsala, SwedenAll rights reserved.Electromagnetic Field TheoryISBN X-XXX-XXXXX-X
- 5. Downloaded from http://www.plasma.uu.se/CED/Book Version released 19th June 2004 at 21:47.PrefaceThe current book is an outgrowth of the lecture notes that I prepared for the four-creditcourse Electrodynamics that was introduced in the Uppsala University curriculum in1992, to become the ﬁve-credit course Classical Electrodynamics in 1997. To someextent, parts of these notes were based on lecture notes prepared, in Swedish, byB ENGT L UNDBORG who created, developed and taught the earlier, two-credit courseElectromagnetic Radiation at our faculty. Intended primarily as a textbook for physics students at the advanced undergradu-ate or beginning graduate level, it is hoped that the present book may be useful forresearch workers too. It provides a thorough treatment of the theory of electrodynam-ics, mainly from a classical ﬁeld theoretical point of view, and includes such thingsas formal electrostatics and magnetostatics and their uniﬁcation into electrodynam-ics, the electromagnetic potentials, gauge transformations, covariant formulation ofclassical electrodynamics, force, momentum and energy of the electromagnetic ﬁeld,radiation and scattering phenomena, electromagnetic waves and their propagation invacuum and in media, and covariant Lagrangian/Hamiltonian ﬁeld theoretical meth-ods for electromagnetic ﬁelds, particles and interactions. The aim has been to writea book that can serve both as an advanced text in Classical Electrodynamics and as apreparation for studies in Quantum Electrodynamics and related subjects. In an attempt to encourage participation by other scientists and students in theauthoring of this book, and to ensure its quality and scope to make it useful in higheruniversity education anywhere in the world, it was produced within a World-WideWeb (WWW) project. This turned out to be a rather successful move. By making anelectronic version of the book freely down-loadable on the net, comments have beenonly received from fellow Internet physicists around the world and from WWW ‘hit’statistics it seems that the book serves as a frequently used Internet resource. Thisway it is hoped that it will be particularly useful for students and researchers workingunder ﬁnancial or other circumstances that make it diﬃcult to procure a printed copyof the book. Thanks are due not only to Bengt Lundborg for providing the inspiration to writethis book, but also to professor C HRISTER WAHLBERG and professor G ÖRAN FÄLDT,Uppsala University, and professor YAKOV I STOMIN, Lebedev Institute, Moscow, forinteresting discussions on electrodynamics and relativity in general and on this book inparticular. Comments from former graduate students M ATTIAS WALDENVIK, T OBIAC AROZZI and ROGER K ARLSSON as well as A NDERS E RIKSSON, all at the SwedishInstitute of Space Physics in Uppsala and who all have participated in the teaching, vii
- 6. P REFACE on the material covered in the course and in this book are gratefully acknowledged. Thanks are also due to my long-term space physics colleague H ELMUT KOPKA of the Max-Planck-Institut für Aeronomie, Lindau, Germany, who not only taught me about the practical aspects of the of high-power radio wave transmitters and trans- mission lines, but also about the more delicate aspects of typesetting a book in TEX and LTEX. I am particularly indebted to Academician professor V ITALIY L AZAREV- A ICH G INZBURG , 2003 Nobel Laureate in Physics, for his many fascinating and very elucidating lectures, comments and historical footnotes on electromagnetic radiation while cruising on the Volga river at our joint Russian-Swedish summer schools during the 1990s and for numerous private discussions. 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. Uppsala, Sweden B O T HIDÉ January, 2004viii Version released 19th June 2004 at 21:47. Downloaded from http://www.plasma.uu.se/CED/Book
- 7. Downloaded from http://www.plasma.uu.se/CED/Book Version released 19th June 2004 at 21:47.ContentsPreface viiContents ixList of Figures xiii1 Classical Electrodynamics 1 1.1 Electrostatics 2 1.1.1 Coulomb’s law 2 1.1.2 The electrostatic ﬁeld 3 1.2 Magnetostatics 6 1.2.1 Ampère’s law 6 1.2.2 The magnetostatic ﬁeld 7 1.3 Electrodynamics 9 1.3.1 Equation of continuity for electric charge 9 1.3.2 Maxwell’s displacement current 10 1.3.3 Electromotive force 10 1.3.4 Faraday’s law of induction 11 1.3.5 Maxwell’s microscopic equations 14 1.3.6 Maxwell’s macroscopic equations 14 1.4 Electromagnetic duality 15 1.5 Bibliography 222 Electromagnetic Waves 25 2.1 The wave equations 26 2.1.1 The wave equation for E 26 2.1.2 The wave equation for B 26 2.1.3 The time-independent wave equation for E 27 2.2 Plane waves 30 2.2.1 Telegrapher’s equation 31 2.2.2 Waves in conductive media 32 2.3 Observables and averages 33 ix
- 8. C ONTENTS 2.4 Bibliography 34 3 Electromagnetic Potentials 35 3.1 The electrostatic scalar potential 35 3.2 The magnetostatic vector potential 36 3.3 The electrodynamic potentials 36 3.3.1 Lorenz-Lorentz gauge 38 3.3.2 Coulomb gauge 42 3.3.3 Gauge transformations 42 3.4 Bibliography 45 4 Relativistic Electrodynamics 47 4.1 The special theory of relativity 47 4.1.1 The Lorentz transformation 48 4.1.2 Lorentz space 49 4.1.3 Minkowski space 54 4.2 Covariant classical mechanics 57 4.3 Covariant classical electrodynamics 58 4.3.1 The four-potential 58 4.3.2 The Liénard-Wiechert potentials 59 4.3.3 The electromagnetic ﬁeld tensor 61 4.4 Bibliography 64 5 Electromagnetic Fields and Particles 67 5.1 Charged particles in an electromagnetic ﬁeld 67 5.1.1 Covariant equations of motion 67 5.2 Covariant ﬁeld theory 73 5.2.1 Lagrange-Hamilton formalism for ﬁelds and interactions 73 5.3 Bibliography 81 6 Electromagnetic Fields and Matter 83 6.1 Electric polarisation and displacement 83 6.1.1 Electric multipole moments 83 6.2 Magnetisation and the magnetising ﬁeld 86 6.3 Energy and momentum 88 6.3.1 The energy theorem in Maxwell’s theory 88 6.3.2 The momentum theorem in Maxwell’s theory 89 6.4 Bibliography 91 7 Electromagnetic Fields from Arbitrary Source Distributions 93 7.1 The magnetic ﬁeld 95 7.2 The electric ﬁeld 96 7.3 The radiation ﬁelds 99x Version released 19th June 2004 at 21:47. Downloaded from http://www.plasma.uu.se/CED/Book
- 9. 7.4 Radiated energy 101 7.4.1 Monochromatic signals 101 7.4.2 Finite bandwidth signals 102 7.5 Bibliography 1038 Electromagnetic Radiation and Radiating Systems 105 8.1 Radiation from extended sources 105 8.1.1 Radiation from a one-dimensional current distribution 106 8.1.2 Radiation from a two-dimensional current distribution 108 8.2 Multipole radiation 112 8.2.1 The Hertz potential 112 8.2.2 Electric dipole radiation 115 8.2.3 Magnetic dipole radiation 117 8.2.4 Electric quadrupole radiation 118 8.3 Radiation from a localised charge in arbitrary motion 119 8.3.1 The Liénard-Wiechert potentials 120 8.3.2 Radiation from an accelerated point charge 122 8.3.3 Bremsstrahlung 133 8.3.4 Cyclotron and synchrotron radiation 138 8.3.5 Radiation from charges moving in matter 145 8.4 Bibliography 152F Formulae 155 F.1 The electromagnetic ﬁeld 155 F.1.1 Maxwell’s equations 155 F.1.2 Fields and potentials 155 F.1.3 Force and energy 156 F.2 Electromagnetic radiation 156 F.2.1 Relationship between the ﬁeld vectors in a plane wave 156 F.2.2 The far ﬁelds from an extended source distribution 156 F.2.3 The far ﬁelds from an electric dipole 156 F.2.4 The far ﬁelds from a magnetic dipole 157 F.2.5 The far ﬁelds from an electric quadrupole 157 F.2.6 The ﬁelds from a point charge in arbitrary motion 157 F.3 Special relativity 157 F.3.1 Metric tensor 157 F.3.2 Covariant and contravariant four-vectors 157 F.3.3 Lorentz transformation of a four-vector 158 F.3.4 Invariant line element 158 F.3.5 Four-velocity 158 F.3.6 Four-momentum 158 F.3.7 Four-current density 158Downloaded from http://www.plasma.uu.se/CED/Book Version released 19th June 2004 at 21:47. xi
- 10. C ONTENTS F.3.8 Four-potential 158 F.3.9 Field tensor 158 F.4 Vector relations 159 F.4.1 Spherical polar coordinates 159 F.4.2 Vector formulae 160 F.5 Bibliography 161 M Mathematical Methods 163 M.1 Scalars, vectors and tensors 163 M.1.1 Vectors 163 M.1.2 Fields 165 M.1.3 Vector algebra 171 M.1.4 Vector analysis 174 M.2 Analytical mechanics 180 M.2.1 Lagrange’s equations 180 M.2.2 Hamilton’s equations 180 M.3 Bibliography 181 Index 183xii Version released 19th June 2004 at 21:47. Downloaded from http://www.plasma.uu.se/CED/Book
- 11. Downloaded from http://www.plasma.uu.se/CED/Book Version released 19th June 2004 at 21:47.List of Figures1.1 Coulomb interaction between two electric charges 31.2 Coulomb interaction for a distribution of electric charges 51.3 Ampère interaction 71.4 Moving loop in a varying B ﬁeld 124.1 Relative motion of two inertial systems 484.2 Rotation in a 2D Euclidean space 554.3 Minkowski diagram 565.1 Linear one-dimensional mass chain 747.1 Radiation in the far zone 1008.1 Linear antenna 1068.2 Electric dipole geometry 1078.3 Loop antenna 1098.4 Multipole radiation geometry 1138.5 Electric dipole geometry 1168.6 Radiation from a moving charge in vacuum 1208.7 An accelerated charge in vacuum 1228.8 Angular distribution of radiation during bremsstrahlung 1348.9 Location of radiation during bremsstrahlung 1358.10 Radiation from a charge in circular motion 1398.11 Synchrotron radiation lobe width 1418.12 The perpendicular ﬁeld of a moving charge 1448.13 Electron-electron scattering 1468.14 ˇ Vavilov-Cerenkov cone 150M.1 Tetrahedron-like volume element of matter 168 xiii
- 12. To the memory of professorL EV M IKHAILOVICH E RUKHIMOV (1936–1997) dear friend, great physicist, poet and a truly remarkable man.
- 13. Downloaded from http://www.plasma.uu.se/CED/Book Version released 19th June 2004 at 21:47. CHAPTER 1Classical ElectrodynamicsClassical 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 objects andquantum corrections must be included. However, the limiting processes used willyield results which are correct on small as well as large macroscopic scales. It took the genius of JAMES C LERK M AXWELL to unify electricity and magnet-ism into a super-theory, electromagnetism or classical electrodynamics (CED), and torealise that optics is a subﬁeld of this super-theory. Early in the 20th century, No-bel laureate H ENDRIK A NTOON L ORENTZ took the electrodynamics theory further tothe microscopic scale and also laid the foundation for the special theory of relativ-ity, formulated by Nobel laureate A LBERT E INSTEIN in 1905. In the 1930s PAULA. M. D IRAC expanded electrodynamics to a more symmetric form, including mag-netic as well as electric charges. With his relativistic quantum mechanics, he alsopaved the way for the development of quantum electrodynamics (QED) for whichR ICHARD P. F EYNMAN, J ULIAN S CHWINGER, and S IN -I TIRO T OMONAGA in 1965 re-ceived their Nobel prizes. Around the same time, physicists such as Nobel laureatesS HELDON G LASHOW, A BDUS S ALAM, and S TEVEN W EINBERG managed to unifyelectrodynamics with the weak interaction theory to yet another super-theory, elec-troweak theory. The modern theory of strong interactions, quantum chromodynamics(QCD), is inﬂuenced by QED. In this chapter we start with the force interactions in classical electrostatics andclassical magnetostatics and introduce the static electric and magnetic ﬁelds and ﬁnd 1
- 14. 1. C LASSICAL E LECTRODYNAMICS 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 dynamic connection between electricity and magnetism and how the two can be uniﬁed into one ‘super- theory’, classical electrodynamics, described by one system of coupled dynamic ﬁeld equations—the Maxwell equations. At the end of the chapter we study Dirac’s symmetrised form of Maxwell’s equa- tions by introducing (hypothetical) magnetic charges and magnetic currents into the theory. While not identiﬁed unambiguously in experiments yet, magnetic charges and currents make the theory much more appealing for instance by allowing for duality transformations in a most natural way. 1.1 Electrostatics The theory which describes physical phenomena related to the interaction between stationary electric charges or charge distributions in space with stationary boundaries is called electrostatics. For a long time electrostatics, under the name electricity, was considered an independent physical theory of its own, alongside other physical theories such as magnetism, mechanics, optics and thermodynamics. 1 1.1.1 Coulomb’s law It has been found experimentally that in classical electrostatics the interaction between stationary, electrically charged bodies can be described in terms of a mechanical force. Let us consider the simple case described by Figure 1.1 on page 3. Let F denote the force acting on a electrically charged particle with charge q located at x, due to the presence of a charge q located at x . According to Coulomb’s law this force is, in vacuum, given by the expression qq x − x qq 1 qq 1 F(x) = =− = (1.1) 4πε0 |x − x |3 4πε0 |x − x | 4πε0 |x − x | where in the last step Formula (F.71) on page 161 was used. In SI units, which we shall use throughout, the force F is measured in Newton (N), the electric charges q and 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) is the 1 The physicist and philosopher Pierre Duhem (1861–1916) once wrote: ‘The whole theory of electrostatics constitutes a group of abstract ideas and general propos- itions, 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 of a French physicist and his taste for clarity, simplicity and order. . . .’2 Version released 19th June 2004 at 21:47. Downloaded from http://www.plasma.uu.se/CED/Book
- 15. Electrostatics q x−x x q x O F IGURE 1.1: Coulomb’s law describes how a static electric charge q, located at a point x relative to the origin O, experiences an electrostatic force from a static electric charge q located at x .vacuum permittivity and c ≈ 2.9979 × 108 m/s is the speed of light in vacuum. In CGSunits ε0 = 1/(4π) and the force is measured in dyne, electric charge in statcoulomb,and length in centimetres (cm).1.1.2 The electrostatic ﬁeldInstead of describing the electrostatic interaction in terms of a ‘force action at a dis-tance’, it turns out that it is for most purposes more useful to introduce the concept ofa ﬁeld and to describe the electrostatic interaction in terms of a static vectorial electricﬁeld Estat deﬁned by the limiting process def F Estat ≡ lim (1.2) q→0 qwhere F is the electrostatic force, as deﬁned in Equation (1.1) on the preceding page,from a net 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 does notdistort the ﬁeld set up by q , the expression for Estat does not depend explicitly on qbut only on the charge q and the relative radius vector x − x . This means that we cansay that any net electric charge produces an electric ﬁeld in the space that surroundsit, regardless of the existence of a second charge anywhere in this space. 2 2 In the preface to the ﬁrst edition of the ﬁrst volume of his book A Treatise on Electricity and Magnet-ism, 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 the mathematicians saw centres of force attracting at a distance: Faraday saw a medium where they saw nothing but distance: Faraday sought the seat of the phenomena in real actionsDownloaded from http://www.plasma.uu.se/CED/Book Version released 19th June 2004 at 21:47. 3
- 16. 1. C LASSICAL E LECTRODYNAMICS Using (1.1) and Equation (1.2) on the preceding page, and Formula (F.70) on page 160, we ﬁnd that the electrostatic ﬁeld Estat at the ﬁeld point x (also known as the observation point), due to a ﬁeld-producing electric charge q at the source point x , is given by q x−x q 1 q 1 Estat (x) = =− = (1.3) 4πε0 |x − x | 3 4πε0 |x − x | 4πε0 |x − x | In the presence of several ﬁeld producing discrete electric charges q i , located at the points xi , 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 ﬁeld 1 x − xi Estat (x) = 4πε0 ∑ qi 3 (1.4) i x − xi If the discrete electric charges are small and numerous enough, we introduce the electric charge density ρ, measured in C/m3 in SI units, located at x within a volume V of limited extent and replace summation with integration over this volume. This allows us to describe the total ﬁeld as 1 x−x 1 1 Estat (x) = d3x ρ(x ) 3 = − 4πε d3x ρ(x ) 4πε0 V |x − x | 0 V |x − x | (1.5) 1 3 ρ(x ) =− dx 4πε0 V |x − x | where we used Formula (F.70) on page 160 and the fact that ρ(x ) does not depend on the unprimed (ﬁeld point) coordinates on which operates. We emphasise that under the assumption of linear superposition, Equation (1.5) above is valid for an arbitrary distribution of electric charges, including discrete charges, in which case ρ is expressed in terms of Dirac delta distributions: ρ(x ) = ∑ qi δ(x − xi ) (1.6) i as illustrated in Figure 1.2 on the facing page. Inserting this expression into expres- sion (1.5) above we recover expression (1.4). Taking the divergence of the general Estat expression for an arbitrary electric charge distribution, Equation (1.5) above, and using the representation of the Dirac going on in the medium, they were satisﬁed that they had found it in a power of action at a distance impressed on the electric ﬂuids.’ 3 In fact, vacuum exhibits a quantum mechanical nonlinearity due to vacuum polarisation eﬀects mani- festing themselves in the momentary creation and annihilation of electron-positron pairs, but classically this nonlinearity is negligible.4 Version released 19th June 2004 at 21:47. Downloaded from http://www.plasma.uu.se/CED/Book
- 17. Electrostatics q x − xi x qi V xi O F IGURE 1.2: Coulomb’s law for a distribution of individual charges xi localised within a volume V of limited extent.delta distribution, Formula (F.73) on page 161, we ﬁnd that 1 x−x · Estat (x) = · d3x ρ(x ) 4πε0 V |x − x |3 1 1 =− d3x ρ(x ) · 4πε0 V |x − x | (1.7) 1 1 =− d3x ρ(x ) 2 4πε0 V |x − x | 1 ρ(x) = d3x ρ(x ) δ(x − x ) = ε0 V ε0which is the diﬀerential form of Gauss’s law of electrostatics. Since, according to Formula (F.62) on page 160, × [ α(x)] ≡ 0 for any 3D R3scalar ﬁeld α(x), we immediately ﬁnd that in electrostatics 1 ρ(x ) × Estat (x) = − × d3x =0 (1.8) 4πε0 V |x − x |i.e., that Estat is an irrotational ﬁeld. To summarise, electrostatics can be described in terms of two vector partial diﬀer-ential equations ρ(x) · Estat (x) = (1.9a) ε0 × Estat (x) = 0 (1.9b)representing four scalar partial diﬀerential equations.Downloaded from http://www.plasma.uu.se/CED/Book Version released 19th June 2004 at 21:47. 5
- 18. 1. C LASSICAL E LECTRODYNAMICS 1.2 Magnetostatics While electrostatics deals with static electric charges, magnetostatics deals with sta- tionary electric currents, i.e., electric charges moving with constant speeds, and the interaction between these currents. Here we shall discuss this theory in some detail. 1.2.1 Ampère’s law Experiments on the interaction between two small loops of electric current have shown that they interact via a mechanical force, much the same way that electric charges interact. In Figure 1.3 on the facing page, let F denote such a force acting on a small loop C, with tangential line element dl, located at x and carrying a current I in the direction of dl, due to the presence of a small loop C , with tangential line element dl , located at x and carrying a current I in the direction of dl . According to Ampère’s law this force is, in vacuum, given by the expression µ0 II (x − x ) F(x) = dl × dl × 4π C C |x − x |3 (1.10) µ0 II 1 =− dl × dl × 4π C C |x − x | In SI units, µ0 = 4π × 10−7 ≈ 1.2566 × 10−6 H/m is the vacuum permeability. From the deﬁnition of ε0 and µ0 (in SI units) we observe that 107 1 ε 0 µ0 = 2 (F/m) × 4π × 10−7 (H/m) = 2 (s2 /m2 ) (1.11) 4πc c which is a most useful relation. At ﬁrst glance, Equation (1.10) above may appear unsymmetric in terms of the loops and therefore to be a force law which is in contradiction with Newton’s third law. However, by applying the vector triple product ‘bac-cab’ Formula (F.51) on page 160, we can rewrite (1.10) as µ0 II 1 F(x) = − dl dl · 4π C C |x − x | (1.12) µ0 II x−x − dl ·dl 4π C C |x − x |3 Since the integrand in the ﬁrst integral is an exact diﬀerential, this integral vanishes and we can rewrite the force expression, Equation (1.10) above, in the following sym- metric way µ0 II x−x F(x) = − dl · dl (1.13) 4π C C |x − x |3 which clearly exhibits the expected symmetry in terms of loops C and C .6 Version released 19th June 2004 at 21:47. Downloaded from http://www.plasma.uu.se/CED/Book
- 19. Magnetostatics J C dl x−x dl x C J x O F IGURE 1.3: Ampère’s law describes how a small loop C, carrying a static electric current I through its tangential line element dl located at x, experiences a magnetostatic force from a small loop C , carrying a static electric current I through the tangential line element dl located at x . The loops can have arbitrary shapes as long as they are simple and closed.1.2.2 The magnetostatic ﬁeldIn analogy with the electrostatic case, we may attribute the magnetostatic interactionto a static vectorial magnetic ﬁeld Bstat . It turns out that the elemental Bstat can bedeﬁned as def µ0 I x−x dBstat (x) ≡ dl × (1.14) 4π |x − x |3which expresses the small element dBstat (x) of the static magnetic ﬁeld set up at theﬁeld point x by a small line element dl of stationary current I at the source pointx . The SI unit for the magnetic ﬁeld, sometimes called the magnetic ﬂux density ormagnetic induction, is Tesla (T). If we generalise expression (1.14) to an integrated steady state electric currentdensity j(x), measured in A/m2 in SI units, we obtain Biot-Savart’s law: µ0 x−x µ0 1 Bstat (x) = d3x j(x ) × =− d3x j(x ) × 4π V |x − x |3 4π V |x − x | µ0 j(x ) = × d3x 4π V |x − x | (1.15)where we used Formula (F.70) on page 160, Formula (F.57) on page 160, and thefact that j(x ) does not depend on the unprimed coordinates on which operates.Comparing Equation (1.5) on page 4 with Equation (1.15), we see that there exists aDownloaded from http://www.plasma.uu.se/CED/Book Version released 19th June 2004 at 21:47. 7
- 20. 1. C LASSICAL E LECTRODYNAMICS close analogy between the expressions for Estat and Bstat but that they diﬀer in their vectorial characteristics. With this deﬁnition of Bstat , Equation (1.10) on page 6 may we written F(x) = I dl × Bstat (x) (1.16) C In order to assess the properties of Bstat , we determine its divergence and curl. Taking the divergence of both sides of Equation (1.15) on the preceding page and utilising Formula (F.63) on page 160, we obtain µ0 j(x ) · Bstat (x) = · × d3x =0 (1.17) 4π V |x − x | since, according to Formula (F.63) on page 160, · ( × a) vanishes for any vector ﬁeld a(x). Applying the operator ‘bac-cab’ rule, Formula (F.64) on page 160, the curl of Equation (1.15) on the preceding page can be written µ0 j(x ) × Bstat (x) = × × d3x = 4π V |x − x | µ0 1 µ0 1 =− d3x j(x ) 2 + d3x [j(x ) · ] 4π V |x − x | 4π V |x − x | (1.18) In the ﬁrst of the two integrals on the right hand side, we use the representation of the Dirac delta function given in Formula (F.73) on page 161, and integrate the second one by parts, by utilising Formula (F.56) on page 160 as follows: 1 d3x [j(x ) · ] V |x − x | ∂ 1 1 = xk ˆ d3x · j(x ) − d3x · j(x ) V ∂xk |x − x | V |x − x | ∂ 1 1 = xk ˆ dS · j(x ) − d3x · j(x ) S ∂xk |x − x | V |x − x | (1.19) Then we note that the ﬁrst integral in the result, obtained by applying Gauss’s theorem, vanishes when integrated over a large sphere far away from the localised source j(x ), and that the second integral vanishes because · j = 0 for stationary currents (no charge accumulation in space). The net result is simply × Bstat (x) = µ0 d3x j(x )δ(x − x ) = µ0 j(x) (1.20) V8 Version released 19th June 2004 at 21:47. Downloaded from http://www.plasma.uu.se/CED/Book
- 21. Electrodynamics1.3 ElectrodynamicsAs we saw in the previous sections, the laws of electrostatics and magnetostatics canbe summarised in two pairs of time-independent, uncoupled vector partial diﬀerentialequations, namely the equations of classical electrostatics ρ(x) · Estat (x) = (1.21a) ε0 × Estat (x) = 0 (1.21b)and the equations of classical magnetostatics · Bstat (x) = 0 (1.22a) stat ×B (x) = µ0 j(x) (1.22b)Since there is nothing a priori which connects Estat directly with Bstat , we must con-sider classical electrostatics and classical magnetostatics as two independent theories. However, when we include time-dependence, these theories are uniﬁed into onetheory, classical electrodynamics. This uniﬁcation of the theories of electricity andmagnetism is motivated by two empirically established facts: 1. Electric charge is a conserved quantity and electric current is a transport of electric charge. This fact manifests itself in the equation of continuity and, as a consequence, in Maxwell’s displacement current. 2. A change in the magnetic ﬂux through a loop will induce an EMF electric ﬁeld in the loop. This is the celebrated Faraday’s law of induction.1.3.1 Equation of continuity for electric chargeLet j(t, x) denote the time-dependent electric current density. In the simplest caseit can be deﬁned as j = vρ where v is the velocity of the electric charge dens-ity ρ. In general, j has to be deﬁned in statistical mechanical terms as j(t, x) =∑α qα d3v v fα (t, x, v) where fα (t, x, v) is the (normalised) distribution function forparticle species α with electric charge qα . The electric charge conservation law can be formulated in the equation of con-tinuity ∂ρ(t, x) + · j(t, x) = 0 (1.23) ∂twhich states that the time rate of change of electric charge ρ(t, x) is balanced by adivergence in the electric current density j(t, x).Downloaded from http://www.plasma.uu.se/CED/Book Version released 19th June 2004 at 21:47. 9
- 22. 1. C LASSICAL E LECTRODYNAMICS 1.3.2 Maxwell’s displacement current We recall from the derivation of Equation (1.20) on page 8 that there we used the fact that in magnetostatics · j(x) = 0. In the case of non-stationary sources and ﬁelds, we must, in accordance with the continuity Equation (1.23) on the preceding page, set · j(t, x) = −∂ρ(t, x)/∂t. Doing so, and formally repeating the steps in the derivation of Equation (1.20) on page 8, we would obtain the formal result µ0 ∂ 1 × B(t, x) = µ0 d3x j(t, x )δ(x − x ) + d3x ρ(t, x ) V 4π ∂t V |x − x | ∂ = µ0 j(t, x) + µ0 ε0 E(t, x) ∂t (1.24) where, in the last step, we have assumed that a generalisation of Equation (1.5) on page 4 to time-varying ﬁelds allows us to make the identiﬁcation4 1 ∂ 1 ∂ 1 1 d3x ρ(t, x ) = − d3x ρ(t, x ) 4πε0 ∂t V |x − x | ∂t 4πε0 V |x − x | ∂ 1 ρ(t, x ) ∂ = − d3x = E(t, x) ∂t 4πε0 V |x − x | ∂t (1.25) The result is Maxwell’s source equation for the B ﬁeld ∂ × B(t, x) = µ0 j(t, x) + ε0 E(t, x) (1.26) ∂t where the last term ∂ε0 E(t, x)/∂t is the famous displacement current. This term was introduced, in a stroke of genius, by Maxwell[8] in order to make the right hand side of this equation divergence free when j(t, x) is assumed to represent the density of the total electric current, which can be split up in ‘ordinary’ conduction currents, polar- isation currents and magnetisation currents. The displacement current is an extra term which behaves like a current density ﬂowing in vacuum. As we shall see later, its existence has far-reaching physical consequences as it predicts the existence of elec- tromagnetic radiation that can carry energy and momentum over very long distances, even in vacuum. 1.3.3 Electromotive force If an electric ﬁeld E(t, x) is applied to a conducting medium, a current density j(t, x) will be produced in this medium. There exist also hydrodynamical and chemical processes which can create currents. Under certain physical conditions, and for certain 4 Later, we will need to consider this generalisation and formal identiﬁcation further.10 Version released 19th June 2004 at 21:47. Downloaded from http://www.plasma.uu.se/CED/Book
- 23. Electrodynamicsmaterials, one can sometimes assume a linear relationship between the electric currentdensity j and E, called Ohm’s law: j(t, x) = σE(t, x) (1.27)where σ is the electric conductivity (S/m). In the most general cases, for instance inan anisotropic conductor, σ is a tensor. We can view Ohm’s law, Equation (1.27) above, as the ﬁrst term in a Taylor ex-pansion of the law j[E(t, x)]. This general law incorporates non-linear eﬀects such asfrequency mixing. Examples of media which are highly non-linear are semiconduct-ors and plasma. We draw the attention to the fact that even in cases when the linearrelation between E and j is a good approximation, we still have to use Ohm’s law withcare. The conductivity σ is, in general, time-dependent (temporal dispersive media)but then it is often the case that Equation (1.27) is valid for each individual Fouriercomponent of the ﬁeld. If the current is caused by an applied electric ﬁeld E(t, x), this electric ﬁeld willexert work on the charges in the medium and, unless the medium is super-conducting,there will be some energy loss. The rate at which this energy is expended is j · Eper unit volume. If E is irrotational (conservative), j will decay away with time.Stationary currents therefore require that an electric ﬁeld which corresponds to anelectromotive force (EMF) is present. In the presence of such a ﬁeld EEMF , Ohm’slaw, Equation (1.27) above, takes the form j = σ(Estat + EEMF ) (1.28)The electromotive force is deﬁned as E= dl · (Estat + EEMF ) (1.29) Cwhere dl is a tangential line element of the closed loop C.1.3.4 Faraday’s law of inductionIn Subsection 1.1.2 we derived the diﬀerential equations for the electrostatic ﬁeld. Inparticular, on page 5 we derived Equation (1.8) which states that × Estat (x) = 0 andthus that Estat is a conservative ﬁeld (it can be expressed as a gradient of a scalar ﬁeld).This implies that the closed line integral of Estat in Equation (1.29) above vanishes andthat this equation becomes E= dl · EEMF (1.30) C It has been established experimentally that a nonconservative EMF ﬁeld is pro-duced in a closed circuit C if the magnetic ﬂux through this circuit varies with time.Downloaded from http://www.plasma.uu.se/CED/Book Version released 19th June 2004 at 21:47. 11
- 24. 1. C LASSICAL E LECTRODYNAMICS dS v B(x) v C dl B(x) F IGURE 1.4: A loop C which moves with velocity v in a spatially varying magnetic ﬁeld B(x) will sense a varying magnetic ﬂux during the motion. This is formulated in Faraday’s law which, in Maxwell’s generalised form, reads d E(t, x) = dl · E(t, x) = − Φm (t, x) C dt (1.31) d ∂ =− dS · B(t, x) = − dS · B(t, x) dt S S ∂t where Φm is the magnetic ﬂux and S is the surface encircled by C which can be in- terpreted as a generic stationary ‘loop’ and not necessarily as a conducting circuit. Application of Stokes’ theorem on this integral equation, transforms it into the diﬀer- ential equation ∂ × E(t, x) = − B(t, x) (1.32) ∂t which is valid for arbitrary variations in the ﬁelds and constitutes the Maxwell equa- tion which explicitly connects electricity with magnetism. Any change of the magnetic ﬂux Φm will induce an EMF. Let us therefore consider the case, illustrated if Figure 1.4, that the ‘loop’ is moved in such a way that it links12 Version released 19th June 2004 at 21:47. Downloaded from http://www.plasma.uu.se/CED/Book
- 25. Electrodynamicsa magnetic ﬁeld which varies during the movement. The convective derivative isevaluated according to the well-known operator formula d ∂ = +v· (1.33) dt ∂twhich follows immediately from the rules of diﬀerentiation of an arbitrary diﬀeren-tiable function f (t, x(t)). Applying this rule to Faraday’s law, Equation (1.31) on thepreceding page, we obtain d ∂B E(t, x) = − dS · B = − dS · − dS · (v · )B (1.34) dt S S ∂t S During spatial diﬀerentiation v is to be considered as constant, and Equation (1.17)on page 8 holds also for time-varying ﬁelds: · B(t, x) = 0 (1.35)(it is one of Maxwell’s equations) so that, according to Formula (F.59) on page 160, × (B × v) = (v · )B (1.36)allowing us to rewrite Equation (1.34) in the following way: d E(t, x) = dl · EEMF = − dS · B C dt S (1.37) ∂B =− dS · − dS · × (B × v) S ∂t SWith Stokes’ theorem applied to the last integral, we ﬁnally get ∂B E(t, x) = dl · EEMF = − dS · − dl · (B × v) (1.38) C S ∂t Cor, rearranging the terms, ∂B dl · (EEMF − v × B) = − dS · (1.39) C S ∂twhere EEMF is the ﬁeld which is induced in the ‘loop’, i.e., in the moving system. Theuse of Stokes’ theorem ‘backwards’ on Equation (1.39) above yields ∂B × (EEMF − v × B) = − (1.40) ∂tIn the ﬁxed system, an observer measures the electric ﬁeld E = EEMF − v × B (1.41)Downloaded from http://www.plasma.uu.se/CED/Book Version released 19th June 2004 at 21:47. 13
- 26. 1. C LASSICAL E LECTRODYNAMICS Hence, a moving observer measures the following Lorentz force on a charge q qEEMF = qE + q(v × B) (1.42) corresponding to an ‘eﬀective’ electric ﬁeld in the ‘loop’ (moving observer) EEMF = E + v × B (1.43) Hence, we can conclude that for a stationary observer, the Maxwell equation ∂B ×E=− (1.44) ∂t is indeed valid even if the ‘loop’ is moving. 1.3.5 Maxwell’s microscopic equations We are now able to collect the results from the above considerations and formulate the equations of classical electrodynamics valid for arbitrary variations in time and space of the coupled electric and magnetic ﬁelds E(t, x) and B(t, x). The equations are ρ ·E= (1.45a) ε0 ∂B ×E=− (1.45b) ∂t ·B=0 (1.45c) ∂E × B = ε 0 µ0 + µ0 j(t, x) (1.45d) ∂t In these equations ρ(t, x) represents the total, possibly both time and space depend- ent, electric charge, i.e., free as well as induced (polarisation) charges, and j(t, x) represents the total, possibly both time and space dependent, electric current, i.e., conduction currents (motion of free charges) as well as all atomistic (polarisation, magnetisation) currents. As they stand, the equations therefore incorporate the clas- sical interaction between all electric charges and currents in the system and are called Maxwell’s microscopic equations. Another name often used for them is the Maxwell- Lorentz equations. Together with the appropriate constitutive relations, which relate ρ and j to the ﬁelds, and the initial and boundary conditions pertinent to the physical situation at hand, they form a system of well-posed partial diﬀerential equations which completely determine E and B. 1.3.6 Maxwell’s macroscopic equations The microscopic ﬁeld equations (1.45) provide a correct classical picture for arbitrary ﬁeld and source distributions, including both microscopic and macroscopic scales.14 Version released 19th June 2004 at 21:47. Downloaded from http://www.plasma.uu.se/CED/Book
- 27. Electromagnetic dualityHowever, for macroscopic substances it is sometimes convenient to introduce newderived ﬁelds which represent the electric and magnetic ﬁelds in which, in an averagesense, the material properties of the substances are already included. These ﬁelds arethe electric displacement D and the magnetising ﬁeld H. In the most general case,these derived ﬁelds are complicated nonlocal, nonlinear functionals of the primaryﬁelds E and B: D = D[t, x; E, B] (1.46a) H = H[t, x; E, B] (1.46b)Under certain conditions, for instance for very low ﬁeld strengths, we may assumethat the response of a substance to the ﬁelds is linear so that D = εE (1.47) −1 H=µ B (1.48)i.e., that the derived ﬁelds are linearly proportional to the primary ﬁelds and that theelectric displacement (magnetising ﬁeld) is only dependent on the electric (magnetic)ﬁeld. The ﬁeld equations expressed in terms of the derived ﬁeld quantities D and H are · D = ρ(t, x) (1.49a) ∂B ×E=− (1.49b) ∂t ·B=0 (1.49c) ∂D ×H= + j(t, x) (1.49d) ∂tand are called Maxwell’s macroscopic equations. We will study them in more detailin Chapter 6.1.4 Electromagnetic dualityIf we look more closely at the microscopic Maxwell equations (1.45), we see thatthey exhibit a certain, albeit not a complete, symmetry. Let us follow Dirac and makethe ad hoc assumption that there exist magnetic monopoles represented by a magneticcharge density, which we denote by ρm = ρm (t, x), and a magnetic current density,which we denote by jm = jm (t, x). With these new quantities included in the theory,and with the electric charge density denoted ρe and the electric current density denotedje , the Maxwell equations will be symmetrised into the following four coupled, vector,Downloaded from http://www.plasma.uu.se/CED/Book Version released 19th June 2004 at 21:47. 15
- 28. 1. C LASSICAL E LECTRODYNAMICS partial diﬀerential equations: ρe ·E= (1.50a) ε0 ∂B ×E=− − µ 0 jm (1.50b) ∂t · B = µ 0 ρm (1.50c) ∂E × B = ε 0 µ0 + µ 0 je (1.50d) ∂t We shall call these equations Dirac’s symmetrised Maxwell equations or the electro- magnetodynamic equations. Taking the divergence of (1.50b), we ﬁnd that ∂ · ( × E) = − ( · B) − µ0 · jm ≡ 0 (1.51) ∂t where we used the fact that, according to Formula (F.63) on page 160, the divergence of a curl always vanishes. Using (1.50c) to rewrite this relation, we obtain the equation of continuity for magnetic monopoles ∂ρm + · jm = 0 (1.52) ∂t which has the same form as that for the electric monopoles (electric charges) and currents, Equation (1.23) on page 9. We notice that the new Equations (1.50) exhibit the following symmetry (recall that ε0 µ0 = 1/c2 ): E → cB (1.53a) cB → −E (1.53b) e m cρ → ρ (1.53c) m e ρ → −cρ (1.53d) e m cj → j (1.53e) m e j → −cj (1.53f) which is a particular case (θ = π/2) of the general duality transformation (depicted by the Hodge star operator) E = E cos θ + cB sin θ (1.54a) c B = −E sin θ + cB cos θ (1.54b) e e m c ρ = cρ cos θ + ρ sin θ (1.54c) m e m ρ = −cρ sin θ + ρ cos θ (1.54d) e e m c j = cj cos θ + j sin θ (1.54e) m e m j = −cj sin θ + j cos θ (1.54f)16 Version released 19th June 2004 at 21:47. Downloaded from http://www.plasma.uu.se/CED/Book
- 29. Electromagnetic dualitywhich leaves the symmetrised Maxwell equations, and hence the physics they describe(often referred to as electromagnetodynamics), invariant. Since E and j e are (true orpolar) vectors, B a pseudovector (axial vector), ρe a (true) scalar, then ρm and θ, whichbehaves as a mixing angle in a two-dimensional ‘charge space’, must be pseudoscalarsand jm a pseudovector. FARADAY ’ S LAW AS A CONSEQUENCE OF CONSERVATION OF MAGNETIC CHARGE E XAMPLE 1.1Postulate 1.1 (Indestructibility of magnetic charge). Magnetic charge exists and is indes-tructible in the same way that electric charge exists and is indestructible. In other words wepostulate that there exists an equation of continuity for magnetic charges: ∂ρm (t, x) + · jm (t, x) = 0 ∂t Use this postulate and Dirac’s symmetrised form of Maxwell’s equations to deriveFaraday’s law. The assumption of the existence of magnetic charges suggests a Coulomb-like law for mag-netic ﬁelds: µ0 x−x µ0 1 Bstat (x) = d3x ρm (x ) =− d3x ρm (x ) 4π V |x − x |3 4π V |x − x | (1.55) µ0 ρm (x ) =− d3x 4π V |x − x |[cf. Equation (1.5) on page 4 for Estat ] and, if magnetic currents exist, a Biot-Savart-like law forelectric ﬁelds [cf. Equation (1.15) on page 7 for Bstat ]: µ0 x−x µ0 1 Estat (x) = − d3x jm (x ) × = d3x jm (x ) × 4π V |x − x |3 4π V |x − x | (1.56) µ0 jm (x ) =− × d3x 4π V |x − x |Taking the curl of the latter and using the operator ‘bac-cab’ rule, Formula (F.59) on page 160,we ﬁnd that µ0 jm (x ) × Estat (x) = − × × d3x = 4π V |x − x | (1.57) µ0 1 µ0 1 =− d3x jm (x ) 2 + d3x [jm (x ) · ] 4π V |x − x | 4π V |x − x |Comparing with Equation (1.18) on page 8 for Estat and the evaluation of the integrals there, weobtain × Estat (x) = −µ0 d3x jm (x ) δ(x − x ) = −µ0 jm (x) (1.58) V We assume that Formula (1.56) above is valid also for time-varying magnetic currents.Then, with the use of the representation of the Dirac delta function, Equation (F.73) on page 161,the equation of continuity for magnetic charge, Equation (1.52) on the preceding page, andthe assumption of the generalisation of Equation (1.55) to time-dependent magnetic chargeDownloaded from http://www.plasma.uu.se/CED/Book Version released 19th June 2004 at 21:47. 17
- 30. 1. C LASSICAL E LECTRODYNAMICS distributions, we obtain, formally, µ0 ∂ 1 × E(t, x) = −µ0 d3x jm (t, x )δ(x − x ) − d3x ρm (t, x ) V 4π ∂t V |x − x | ∂ = −µ0 jm (t, x) − B(t, x) ∂t (1.59) [cf. Equation (1.24) on page 10] which we recognise as Equation (1.50b) on page 16. A transformation of this electromagnetodynamic result by rotating into the ‘electric realm’ of charge space, thereby letting jm tend to zero, yields the electrodynamic Equation (1.50b) on page 16, i.e., the Faraday law in the ordinary Maxwell equations. This process also provides an alternative interpretation of the term ∂B/∂t as a magnetic displacement current, dual to the electric displacement current [cf. Equation (1.26) on page 10]. By postulating the indestructibility of a hypothetical magnetic charge, we have thereby been able to replace Faraday’s experimental results on electromotive forces and induction in loops as a foundation for the Maxwell equations by a more appealing one. E ND OF EXAMPLE 1.1 E XAMPLE 1.2 D UALITY OF THE ELECTROMAGNETODYNAMIC EQUATIONS Show that the symmetric, electromagnetodynamic form of Maxwell’s equations (Dirac’s symmetrised Maxwell equations), Equations (1.50) on page 16, are invariant under the duality transformation (1.54). Explicit application of the transformation yields ρe · E= cos θ + cµ0 ρm sin θ · (E cos θ + cB sin θ) = ε0 (1.60) 1 1 ρe = ρe cos θ + ρm sin θ = ε0 c ε0 ∂ B ∂ 1 × E+ = × (E cos θ + cB sin θ) + − E sin θ + B cos θ ∂t ∂t c ∂B 1 ∂E = −µ0 jm cos θ − cos θ + cµ0 je sin θ + sin θ ∂t c ∂t (1.61) 1 ∂E ∂B − sin θ + cos θ = −µ0 jm cos θ + cµ0 je sin θ c ∂t ∂t = −µ0 (−cje sin θ + jm cos θ) = −µ0 jm 1 ρe · B= · (− E sin θ + B cos θ) = − sin θ + µ0 ρm cos θ c cε0 (1.62) = µ0 (−cρe sin θ + ρm cos θ) = µ0 ρm18 Version released 19th June 2004 at 21:47. Downloaded from http://www.plasma.uu.se/CED/Book
- 31. Electromagnetic duality 1 ∂ E 1 1 ∂ × B− = × (− E sin θ + B cos θ) − 2 (E cos θ + cB sin θ) c2 ∂t c c ∂t 1 1 ∂B 1 ∂E = µ0 jm sin θ + cos θ + µ0 je cos θ + 2 cos θ c c ∂t c ∂t (1.63) 1 ∂E 1 ∂B − 2 cos θ − sin θ c ∂t c ∂t 1 m = µ0 j sin θ + je cos θ = µ0 je c QED E ND OF EXAMPLE 1.2 D IRAC ’ S SYMMETRISED M AXWELL EQUATIONS FOR A FIXED MIXING ANGLE E XAMPLE 1.3 Show that for a ﬁxed mixing angle θ such that ρm = cρe tan θ (1.64a) m e j = cj tan θ (1.64b)the symmetrised Maxwell equations reduce to the usual Maxwell equations. Explicit application of the ﬁxed mixing angle conditions on the duality transformation(1.54) on page 16 yields 1 1 ρe = ρe cos θ + ρm sin θ = ρe cos θ + cρe tan θ sin θ c c (1.65a) 1 e 2 e 2 1 e = (ρ cos θ + ρ sin θ) = ρ cos θ cos θ ρm = −cρe sin θ + cρe tan θ cos θ = −cρe sin θ + cρe sin θ = 0 (1.65b) 1 1 e je = je cos θ + je tan θ sin θ = (je cos2 θ + je sin2 θ) = j (1.65c) cos θ cos θ jm = −cje sin θ + cje tan θ cos θ = −cje sin θ + cje sin θ = 0 (1.65d)Hence, a ﬁxed mixing angle, or, equivalently, a ﬁxed ratio between the electric and magneticcharges/currents, ‘hides’ the magnetic monopole inﬂuence (ρ m and jm ) on the dynamic equa-tions. We notice that the inverse of the transformation given by Equation (1.54) on page 16 yields E = E cos θ − c B sin θ (1.66)This means that ·E= · E cos θ − c · B sin θ (1.67)Furthermore, from the expressions for the transformed charges and currents above, we ﬁnd that ρe 1 ρe · E= = (1.68) ε0 cos θ ε0andDownloaded from http://www.plasma.uu.se/CED/Book Version released 19th June 2004 at 21:47. 19
- 32. 1. C LASSICAL E LECTRODYNAMICS · B = µ 0 ρm = 0 (1.69) so that 1 ρe ρe ·E= cos θ − 0 = (1.70) cos θ ε0 ε0 and so on for the other equations. QED E ND OF EXAMPLE 1.3 The invariance of Dirac’s symmetrised Maxwell equations under the similarity transformation means that the amount of magnetic monopole density ρ m is irrelevant for the physics as long as the ratio ρm /ρe = tan θ is kept constant. So whether we assume that the particles are only electrically charged or have also a magnetic charge with a given, ﬁxed ratio between the two types of charges is a matter of convention, as long as we assume that this fraction is the same for all particles. Such particles are referred to as dyons [14]. By varying the mixing angle θ we can change the fraction of magnetic monopoles at will without changing the laws of electrodynamics. For θ = 0 we recover the usual Maxwell electrodynamics as we know it.5 E XAMPLE 1.4 C OMPLEX FIELD SIX - VECTOR FORMALISM The complex ﬁeld six-vector G(t, x) = E(t, x) + icB(t, x) (1.71) where E, B ∈ R3 and hence G ∈ C3 , has a number of interesting properties: 1. The inner product of G with itself G · G = (E + icB) · (E + icB) = E 2 − c2 B2 + 2icE · B (1.72) is conserved. I.e., E 2 − c2 B2 = Const (1.73a) E · B = Const (1.73b) as we shall see later. 2. The inner product of G with the complex conjugate of itself G · G∗ = (E + icB) · (E − icB) = E 2 + c2 B2 (1.74) 5 As Julian Schwinger (1918–1994) put it [15]: ‘. . . there are strong theoretical reasons to believe that magnetic charge exists in nature, and may have played an important role in the development of the universe. Searches for magnetic charge continue at the present time, emphasizing that electromagnetism is very far from being a closed object’.20 Version released 19th June 2004 at 21:47. Downloaded from http://www.plasma.uu.se/CED/Book
- 33. Electromagnetic duality is proportional to the electromagnetic ﬁeld energy. 3. As with any vector, the cross product of G with itself vanishes: G × G = (E + icB) × (E + icB) = E × E − c2 B × B + ic(E × B) + ic(B × E) (1.75) = 0 + 0 + ic(E × B) − ic(E × B) = 0 4. The cross product of G with the complex conjugate of itself G × G∗ = (E + icB) × (E − icB) = E × E + c2 B × B − ic(E × B) + ic(B × E) (1.76) = 0 + 0 − ic(E × B) − ic(E × B) = −2ic(E × B) is proportional to the electromagnetic power ﬂux. E ND OF EXAMPLE 1.4 D UALITY EXPRESSED IN THE COMPLEX FIELD SIX - VECTOR E XAMPLE 1.5 Expressed in the complex ﬁeld vector, introduced in Example 1.4 on the facing page, theduality transformation Equations (1.54) on page 16 become G = E + ic B = E cos θ + cB sin θ − iE sin θ + icB cos θ (1.77) = E(cos θ − i sin θ) + icB(cos θ − i sin θ) = e−iθ (E + icB) = e−iθ Gfrom which it is easy to see that 2 G · G∗ = G = e−iθ G · eiθ G∗ = |G|2 (1.78)while G · G = e−2iθ G · G (1.79) Furthermore, assuming that θ = θ(t, x), we see that the spatial and temporal diﬀerentiationof G leads to ∂ G ∂t G ≡ = −i(∂t θ)e−iθ G + e−iθ ∂t G (1.80a) ∂t ∂ · G ≡ · G = −ie−iθ θ · G + e−iθ · G (1.80b) −iθ −iθ ∂× G≡ × G = −ie θ×G+e ×G (1.80c)which means that ∂t G transforms as G itself only if θ is time-independent, and that · Gand × G transform as G itself only if θ is space-independent. E ND OF EXAMPLE 1.5Downloaded from http://www.plasma.uu.se/CED/Book Version released 19th June 2004 at 21:47. 21
- 34. 1. C LASSICAL E LECTRODYNAMICS 1.5 Bibliography [1] T. W. BARRETT AND D. M. G RIMES, Advanced Electromagnetism. Foundations, Theory and Applications, World Scientiﬁc Publishing Co., Singapore, 1995, ISBN 981-02-2095- 2. [2] R. B ECKER, Electromagnetic Fields and Interactions, Dover Publications, Inc., New York, NY, 1982, ISBN 0-486-64290-9. [3] W. G REINER, Classical Electrodynamics, Springer-Verlag, New York, Berlin, Heidelberg, 1996, ISBN 0-387-94799-X. [4] E. H ALLÉN, Electromagnetic Theory, Chapman & Hall, Ltd., London, 1962. [5] J. D. JACKSON, Classical Electrodynamics, third ed., John Wiley & Sons, Inc., New York, NY . . . , 1999, ISBN 0-471-30932-X. [6] L. D. L ANDAU AND E. M. L IFSHITZ, The Classical Theory of Fields, fourth revised Eng- lish ed., vol. 2 of Course of Theoretical Physics, Pergamon Press, Ltd., Oxford . . . , 1975, ISBN 0-08-025072-6. [7] F. E. L OW, Classical Field Theory, John Wiley & Sons, Inc., New York, NY . . . , 1997, ISBN 0-471-59551-9. [8] J. C. M AXWELL, A dynamical theory of the electromagnetic ﬁeld, Royal Society Trans- actions, 155 (1864). [9] J. C. M AXWELL, A Treatise on Electricity and Magnetism, third ed., vol. 1, Dover Pub- lications, Inc., New York, NY, 1954, ISBN 0-486-60636-8. [10] J. C. M AXWELL, A Treatise on Electricity and Magnetism, third ed., vol. 2, Dover Pub- lications, Inc., New York, NY, 1954, ISBN 0-486-60637-8. [11] D. B. M ELROSE AND R. C. M C P HEDRAN, Electromagnetic Processes in Dispersive Me- dia, Cambridge University Press, Cambridge . . . , 1991, ISBN 0-521-41025-8. [12] W. K. H. PANOFSKY AND M. P HILLIPS, Classical Electricity and Magnetism, second ed., Addison-Wesley Publishing Company, Inc., Reading, MA . . . , 1962, ISBN 0-201-05702- 6. [13] F. ROHRLICH, Classical Charged Particles, Perseus Books Publishing, L.L.C., Reading, MA . . . , 1990, ISBN 0-201-48300-9. [14] J. S CHWINGER, A magnetic model of matter, Science, 165 (1969), pp. 757–761. [15] J. S CHWINGER , L. L. D E R AAD , J R ., K. A. M ILTON , AND W. T SAI, Classical Electro- dynamics, Perseus Books, Reading, MA, 1998, ISBN 0-7382-0056-5. [16] J. A. S TRATTON, Electromagnetic Theory, McGraw-Hill Book Company, Inc., New York, NY and London, 1953, ISBN 07-062150-0.22 Version released 19th June 2004 at 21:47. Downloaded from http://www.plasma.uu.se/CED/Book
- 35. Bibliography[17] J. VANDERLINDE, Classical Electromagnetic Theory, John Wiley & Sons, Inc., New York, Chichester, Brisbane, Toronto, and Singapore, 1993, ISBN 0-471-57269-1.Downloaded from http://www.plasma.uu.se/CED/Book Version released 19th June 2004 at 21:47. 23

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