EivCONTENTS125Radiationforsmallvelocities1278.5.3BremsstrahlungExample8.3Bremsstrahlungforlowspeedsandshortac130celerationtimes1328.5.4Cyclotronandsynchrotronradiation134Cyclotronradiation134Synchrotron radiation137Radiation in the general case137Virtual photons1398.5.5Radiation from charges moving inmatter142Vavilov-Cerenkov radiation147Bibliography149FFormulae149F.1The Electromagnetic Field149F.1.1Maxwell's equations149Constitutiverelations149F.1.2Fields and potentials149Vector and scalar potentials150Lorentz'gauge condition in vacuum150F.1.3Force and energy150Poynting's vector150Maxwell's stress tensorF.2150Electromagnetic RadiationF.2.1Relationshipbetweenthefieldvectorsinaplanewave150F.2.2150The far fields from an extended source distributionF.2.3150The far fields from an electric dipoleF.2.4151ThefarfieldsfromamagneticdipoleF.2.5151Thefarfields from an electricquadrupoleF.2.6151The fields from a point charge in arbitrary motionF.2.7151The fields from a point charge in uniform motionF.3152Special RelativityF.3.1152MetrictensorF.3.2152Covariant and contravariant four-vectors152F.3.3Lorentztransformationof afour-vectorF.3.4152Invariantlineelement152F.3.5Four-velocity153F.3.6Four-momentumF.3.7153Four-current densityF.3.8153Four-potentialDraft version released 13th November 2000 at 22:01.Downloaded fromhttp://ww.pla.se/CED/Book①由由
“main” 2000/11/13 page iv iv CONTENTS Radiation for small velocities . . . . . . . . . . . . . 125 8.5.3 Bremsstrahlung . . . . . . . . . . . . . . . . . . . . . 127 Example 8.3 Bremsstrahlung for low speeds and short acceleration times . . . . . . . . . . . . . . . . 130 8.5.4 Cyclotron and synchrotron radiation . . . . . . . . . . 132 Cyclotron radiation . . . . . . . . . . . . . . . . . . . 134 Synchrotron radiation . . . . . . . . . . . . . . . . . . 134 Radiation in the general case . . . . . . . . . . . . . . 137 Virtual photons . . . . . . . . . . . . . . . . . . . . . 137 8.5.5 Radiation from charges moving in matter . . . . . . . 139 Vavilov-Cerenk ˇ ov radiation . . . . . . . . . . . . . . 142 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 F Formulae 149 F.1 The Electromagnetic Field . . . . . . . . . . . . . . . . . . . 149 F.1.1 Maxwell’s equations . . . . . . . . . . . . . . . . . . 149 Constitutive relations . . . . . . . . . . . . . . . . . . 149 F.1.2 Fields and potentials . . . . . . . . . . . . . . . . . . 149 Vector and scalar potentials . . . . . . . . . . . . . . 149 Lorentz’ gauge condition in vacuum . . . . . . . . . . 150 F.1.3 Force and energy . . . . . . . . . . . . . . . . . . . . 150 Poynting’s vector . . . . . . . . . . . . . . . . . . . . 150 Maxwell’s stress tensor . . . . . . . . . . . . . . . . . 150 F.2 Electromagnetic Radiation . . . . . . . . . . . . . . . . . . . 150 F.2.1 Relationship between the field vectors in a plane wave 150 F.2.2 The far fields from an extended source distribution . . 150 F.2.3 The far fields from an electric dipole . . . . . . . . . . 150 F.2.4 The far fields from a magnetic dipole . . . . . . . . . 151 F.2.5 The far fields from an electric quadrupole . . . . . . . 151 F.2.6 The fields from a point charge in arbitrary motion . . . 151 F.2.7 The fields from a point charge in uniform motion . . . 151 F.3 Special Relativity . . . . . . . . . . . . . . . . . . . . . . . . 152 F.3.1 Metric tensor . . . . . . . . . . . . . . . . . . . . . . 152 F.3.2 Covariant and contravariant four-vectors . . . . . . . . 152 F.3.3 Lorentz transformation of a four-vector . . . . . . . . 152 F.3.4 Invariant line element . . . . . . . . . . . . . . . . . . 152 F.3.5 Four-velocity . . . . . . . . . . . . . . . . . . . . . . 152 F.3.6 Four-momentum . . . . . . . . . . . . . . . . . . . . 153 F.3.7 Four-current density . . . . . . . . . . . . . . . . . . 153 F.3.8 Four-potential . . . . . . . . . . . . . . . . . . . . . . 153 Downloaded from http://www.plasma.uu.se/CED/Book Draft version released 13th November 2000 at 22:01
EvF.3.9153Field tensor153F.4VectorRelations154F.4.1Spherical polar coordinates154Basevectors154Directedlineelement154Solid angle element .154Directed area element154Volume element154F.4.2Vectorformulae.154General relations156Special relations157Integral relations157Bibliography148Appendices159M MathematicalMethods159M.1Scalars,VectorsandTensors159M.1.1 Vectors159Radius vectorM.1.2 Fields161161Scalar fields161Vectorfields162Tensor fields164ExampleM.1Tensorsin3Dspace167M.1.3Vectoralgebra167Scalarproduct167Example M.2 Inner products in complex vector spaceExampleM.3Scalarproduct,normandmetric inLorentz168space168Example M.4Metric ingeneral relativity169Dyadicproduct170Vector product170M.1.4Vectoranalysis170The del operator171Example M.5Thefour-del operator in Lorentz space172The gradientExample M.6 Gradients of scalar functions of relative dis-172tances in3D173ThedivergenceDraft vered13thNoveber2000at22:01.Downloadedfromhttp://www.plase/CED/Book①由由
“main” 2000/11/13 page v v F.3.9 Field tensor . . . . . . . . . . . . . . . . . . . . . . . 153 F.4 Vector Relations . . . . . . . . . . . . . . . . . . . . . . . . . 153 F.4.1 Spherical polar coordinates . . . . . . . . . . . . . . . 154 Base vectors . . . . . . . . . . . . . . . . . . . . . . 154 Directed line element . . . . . . . . . . . . . . . . . . 154 Solid angle element . . . . . . . . . . . . . . . . . . . 154 Directed area element . . . . . . . . . . . . . . . . . 154 Volume element . . . . . . . . . . . . . . . . . . . . 154 F.4.2 Vector formulae . . . . . . . . . . . . . . . . . . . . . 154 General relations . . . . . . . . . . . . . . . . . . . . 154 Special relations . . . . . . . . . . . . . . . . . . . . 156 Integral relations . . . . . . . . . . . . . . . . . . . . 157 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 Appendices 148 M Mathematical Methods 159 M.1 Scalars, Vectors and Tensors . . . . . . . . . . . . . . . . . . 159 M.1.1 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . 159 Radius vector . . . . . . . . . . . . . . . . . . . . . . 159 M.1.2 Fields . . . . . . . . . . . . . . . . . . . . . . . . . . 161 Scalar fields . . . . . . . . . . . . . . . . . . . . . . . 161 Vector fields . . . . . . . . . . . . . . . . . . . . . . 161 Tensor fields . . . . . . . . . . . . . . . . . . . . . . 162 Example M.1 Tensors in 3D space . . . . . . . . . . . . 164 M.1.3 Vector algebra . . . . . . . . . . . . . . . . . . . . . 167 Scalar product . . . . . . . . . . . . . . . . . . . . . 167 Example M.2 Inner products in complex vector space . . . 167 Example M.3 Scalar product, norm and metric in Lorentz space . . . . . . . . . . . . . . . . . . . . . 168 Example M.4 Metric in general relativity . . . . . . . . . 168 Dyadic product . . . . . . . . . . . . . . . . . . . . . 169 Vector product . . . . . . . . . . . . . . . . . . . . . 170 M.1.4 Vector analysis . . . . . . . . . . . . . . . . . . . . . 170 The del operator . . . . . . . . . . . . . . . . . . . . 170 Example M.5 The four-del operator in Lorentz space . . . 171 The gradient . . . . . . . . . . . . . . . . . . . . . . 172 Example M.6 Gradients of scalar functions of relative distances in 3D . . . . . . . . . . . . . . . . . . 172 The divergence . . . . . . . . . . . . . . . . . . . . . 173 Draft version released 13th November 2000 at 22:01. Downloaded from http://www.plasma.uu.se/CED/Book
EviCONTENTS173Example M.7Divergence in 3D173The Laplacian...173Example M.8 The Laplacian and the Dirac delta+174The curl :...174Example M.9 The curl ofa gradient175Example M.10 The divergence of a curlM.2 Analytical Mechanics1761176M.2.1Lagrange's equations176M.2.2Hamilton's equations177BibliographyDownloaded fromhttp://www.plasma.uu.se/ceD/BookDraft version released 13th November 2000 at 22:01.田由①
“main” 2000/11/13 page vi vi CONTENTS Example M.7 Divergence in 3D . . . . . . . . . . . . . 173 The Laplacian . . . . . . . . . . . . . . . . . . . . . . 173 Example M.8 The Laplacian and the Dirac delta . . . . . 173 The curl . . . . . . . . . . . . . . . . . . . . . . . . . 174 Example M.9 The curl of a gradient . . . . . . . . . . . 174 Example M.10 The divergence of a curl . . . . . . . . . 175 M.2 Analytical Mechanics . . . . . . . . . . . . . . . . . . . . . . 176 M.2.1 Lagrange’s equations . . . . . . . . . . . . . . . . . . 176 M.2.2 Hamilton’s equations . . . . . . . . . . . . . . . . . . 176 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 Downloaded from http://www.plasma.uu.se/CED/Book Draft version released 13th November 2000 at 22:01
EList of Figures21.1Coulomb interaction51.2Ampereinteraction1.312Moving loop in a varying B field5.152Relativemotion of two inertial systems595.2Rotation in a 2D Euclidean space5.359Minkowski diagram766.1Linear one-dimensional masschain8.198Radiation in the far zone8.2112Radiation from a moving charge in vacuum8.3114Anacceleratedchargeinvacuum8.4128Angular distribution of radiation during bremsstrahlung1298.5Location of radiation duringbremsstrahlung1338.6Radiationfromachargeincircularmotion8.7135Synchrotron radiation lobe width8.8138The perpendicular field of a moving charge8.9144Vavilov-Cerenkovcone164M.1 Surface element of a material body165M.2Tetrahedron-likevolume elementof mattervi①由D
“main” 2000/11/13 page vii List of Figures 1.1 Coulomb interaction . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Ampère interaction . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Moving loop in a varying B field . . . . . . . . . . . . . . . . 12 5.1 Relative motion of two inertial systems . . . . . . . . . . . . 52 5.2 Rotation in a 2D Euclidean space . . . . . . . . . . . . . . . . 59 5.3 Minkowski diagram . . . . . . . . . . . . . . . . . . . . . . . 59 6.1 Linear one-dimensional mass chain . . . . . . . . . . . . . . . 76 8.1 Radiation in the far zone . . . . . . . . . . . . . . . . . . . . 98 8.2 Radiation from a moving charge in vacuum . . . . . . . . . . 112 8.3 An accelerated charge in vacuum . . . . . . . . . . . . . . . . 114 8.4 Angular distribution of radiation during bremsstrahlung . . . . 128 8.5 Location of radiation during bremsstrahlung . . . . . . . . . . 129 8.6 Radiation from a charge in circular motion . . . . . . . . . . . 133 8.7 Synchrotron radiation lobe width . . . . . . . . . . . . . . . . 135 8.8 The perpendicular field of a moving charge . . . . . . . . . . 138 8.9 Vavilov-Cerenk ˇ ov cone . . . . . . . . . . . . . . . . . . . . . 144 M.1 Surface element of a material body . . . . . . . . . . . . . . . 164 M.2 Tetrahedron-like volume element of matter . . . . . . . . . . . 165 vii
EaceThis book is the result of a twenty-five year long love affair. In 1972, I tookmy first advanced course in electrodynamics at the Theoretical Physics depart-ment, Uppsala University. Shortly thereafter, I joined the research group thereand took on the task of helping my supervisor, professor PER-OLOF FRO-MAN, with the preparation of a new version of his lecture notes on ElectricityTheory. These two things opened up my eyes for the beauty and intricacy ofelectrodynamics, already at the classical level, and Ifell in love with it.Eversincethattime,Ihaveoffandonhadreasontoreturntoelectro-dynamics,bothinmy studies,researchandteaching,andthecurrentbookistheresultof myownteaching of acoursein advanced electrodynamicsatUppsala University some twenty odd years after I experienced the first en-counterwiththissubject.Thebook istheoutgrowthofthelecturenotesthatIprepared for the four-credit course Electrodynamics that was introduced in theUppsalaUniversitycurriculum in1992,tobecomethefive-creditcourseClas-sical Electrodynamics in 1997. To some extent, parts ofthese notes were basedonlecturenotesprepared,inSwedish,byBENGTLUNDBORGwhocreateddevelopedandtaught the earlier,two-creditcourse ElectromagneticRadiationatourfaculty.Intended primarily as a textbook for physics students at the advanced un-dergraduate or beginninggraduate level, Ihope thebook may be useful forresearch workerstoo.Itprovidesathoroughtreatmentofthetheoryof elec-trodynamics, mainly from a classical field theoretical point of view, and in-cludes such things as electrostatics and magnetostatics and their unificationinto electrodynamics, the electromagnetic potentials, gauge transformations,covariantformulationofclassical electrodynamics,force,momentum and en-ergy of the electromagnetic field, radiation and scattering phenomena, electro-magnetic waves and their propagation invacuum and inmedia, and covariantLagrangian/Hamiltonian field theoretical methods for electromagnetic fields,particles and interactions.The aimhas been to writeabookthat can servebothasanadvancedtext inClassicalElectrodynamicsandasapreparationforstudies in Quantum Electrodynamics and related subjectsIn an attempt to encourage participation by other scientists and students inxi0由
“main” 2000/11/13 page xi Preface This book is the result of a twenty-five year long love affair. In 1972, I took my first advanced course in electrodynamics at the Theoretical Physics department, Uppsala University. Shortly thereafter, I joined the research group there and took on the task of helping my supervisor, professor PER-OLOF FRÖ- MAN, with the preparation of a new version of his lecture notes on Electricity Theory. These two 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 have off and on had reason to return to electrodynamics, both in my studies, research and teaching, and the current book is the result of my own teaching of a course in advanced electrodynamics at Uppsala University some twenty odd years after I experienced the first encounter with this subject. The book is the 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 Classical Electrodynamicsin 1997. To some extent, parts of these notes were based on lecture notes prepared, in Swedish, by BENGT LUNDBORG who created, developed and taught the earlier, two-credit course Electromagnetic Radiation at our faculty. Intended primarily as a textbook for physics students at the advanced undergraduate or beginning graduate level, I hope the book may be useful for research workers too. It provides a thorough treatment of the theory of electrodynamics, mainly from a classical field theoretical point of view, and includes such things as electrostatics and magnetostatics and their unification into electrodynamics, the electromagnetic potentials, gauge transformations, covariant formulation of classical electrodynamics, force, momentum and energy of the electromagnetic field, radiation and scattering phenomena, electromagnetic waves and their propagation in vacuum and in media, and covariant Lagrangian/Hamiltonian field theoretical methods for electromagnetic fields, particles and interactions. The aim has been to write a book that can serve both as an advanced text in Classical Electrodynamics and as a preparation for studies in Quantum Electrodynamics and related subjects. In an attempt to encourage participation by other scientists and students in xi