④xiiPREFACEtheauthoring ofthisbook,and to ensure its quality and scopetomake it usefulin higher university education anywhere in the world, it was produced withina World-Wide Web (WWW) project. This turned out to be a rather successfulmove.By making an electronic version of the book freely down-loadable onthenet,Ihavenotonlyreceivedcommentson itfromfellowInternetphysicistsaround theworld,butknow,fromWWW"hit'statisticsthat atthetimeofwriting this, the book serves as a frequently used Internet resource. This wayit is my hope that it will be particularly useful for students and researchersworking under financial or other circumstances that make it difficult to procureaprintedcopyofthebook.I am grateful not only to Per-Olof Froman and Bengt Lundborg for provid-ing the inspirationfor my writing this book, but also to CHRISTER WAHLBERGat Uppsala University for interesting discussions on electrodynamics in generalandonthisbookinparticular,andtomyformergraduatestudentsMATTIAsWALDENVIKandTOBIACAROZZIaswellasANDERSERIKSSON,allattheSwedish Institute of Space Physics, Uppsala Division,and who have parti-cipated in the teaching and commented on the material covered in the courseand in this book. Thanks are also due to my long-term space physics col-leagueHELMUTKoPKAof theMax-Planck-InstitutfurAeronomie,LindaulGermany, who not only taught me about the practical aspects of the of high-powerradiowavetransmittersandtransmission lines,butalsoaboutthemoreOdelicate aspects oftypesetting a book in TeX andLATpX. I am particularlyindebtedtoAcademicianprofessorVITALIYL.GINZBURGforhismanyfas-cinatingandveryelucidatinglectures,commentsandhistoricalfootnotesonelectromagnetic radiation while cruising on the Volga river during our jointRussian-Swedish summer schools.Finally, I would like to thank all students and Internet users who havedownloaded and commented on the book during its life on the World-WideWebI dedicate this book to my son MATTIAS, my daughter KAROLINA, myhigh-school physics teacher, STAFFAN ROsBY, and to my fellow members oftheCAPELLAPEDAGOGICAUPSALIENSIS.BO THIDEUppsala, SwedenNovember,2000/CED/BOODraftreleased13thNovember2000at22:01.0由
“main” 2000/11/13 page xii xii PREFACE the authoring of this book, and to ensure its quality and scope to make it useful in higher university education anywhere in the world, it was produced within a World-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, I have not only received comments on it from fellow Internet physicists around the world, but know, from WWW ‘hit’ statistics that at the time of writing this, the book serves as a frequently used Internet resource. This way it is my hope that it will be particularly useful for students and researchers working under financial or other circumstances that make it difficult to procure a printed copy of the book. I am grateful not only to Per-Olof Fröman and Bengt Lundborg for providing the inspiration for my writing this book, but also to CHRISTER WAHLBERG at Uppsala University for interesting discussions on electrodynamicsin general and on this book in particular, and to my former graduate students MATTIAS WALDENVIK and TOBIA CAROZZI as well as ANDERS ERIKSSON, all at the Swedish Institute of Space Physics, Uppsala Division, and who have participated in the teaching and commented on the material covered in the course and in this book. Thanks are also due to my long-term space physics colleague HELMUT KOPKA of the Max-Planck-Institut für Aeronomie, Lindau, Germany, who not only taught me about the practical aspects of the of highpower radio wave transmitters and transmission lines, but also about the more delicate aspects of typesetting a book in TEX and LATEX. I am particularly indebted to Academician professor VITALIY L. GINZBURG for his many fascinating and very elucidating lectures, comments and historical footnotes on electromagnetic radiation while cruising on the Volga river during our joint Russian-Swedish summer schools. 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 MATTIAS, my daughter KAROLINA, my high-school physics teacher, STAFFAN RÖSBY, and to my fellow members of the CAPELLA PEDAGOGICA UPSALIENSIS. Uppsala, Sweden BO THIDÉ November, 2000 Downloaded from http://www.plasma.uu.se/CED/Book Draft version released 13th November 2000 at 22:01
D1ClassicalElectrodynamicsClassical electrodynamicsdealswith electric andmagnetic fields and inter-actionscausedbymacroscopicdistributionsof electricchargesandcurrentsThis means that the concepts of localised charges and currents assume thevalidity of certain mathematical limiting processes in which it is consideredpossible for the charge and current distributions to be localised in infinitesim-ally small volumes of space. Clearly, this is in contradiction to electromag-netism on a truly microscopic scale, where charges and currents are known tobe spatially extended objects. However, the limiting processes used will yieldresults which are correct on small as well as large macroscopic scales.In this Chapter we start with the force interactions in classical electrostat-ics and classical magnetostatics and introduce the static electric and magneticfieldsandfindtwouncoupledsystemsofequationsforthem,Thenweseehowtheconservation of electric chargeand itsrelationtoelectriccurrent leadstothedynamic connection between electricity and magnetism and howthetwocan be unified in one theory, classical electrodynamics, described by one sys-tem of coupled dynamic field equations.1.1ElectrostaticsThe theory that describes physical phenomena related to the interaction betweenstationaryelectriccharges orchargedistributions inspaceiscalledelectrostat-ics.1.1.1Coulomb'slawIt has been found experimentally that in classical electrostatics the interactionbetweentwostationaryelectricallychargedbodiescanbedescribedintermsofa mechanical force.Let us considerthe simple casedescribed byFigure1.1.1.10由
“main” 2000/11/13 page 1 1 Classical Electrodynamics Classical electrodynamics deals with electric and magnetic fields and interactions caused by macroscopic distributions of electric charges and currents. This means that the concepts of localised charges and currents assume the validity of certain mathematical limiting processes in which it is considered possible for the charge and current distributions to be localised in infinitesimally small volumes of space. Clearly, this is in contradiction to electromagnetism on a truly microscopic scale, where charges and currents are known to be spatially extended objects. However, the limiting processes used will yield results which are correct on small as well as large macroscopic scales. In this Chapter we start with the force interactions in classical electrostatics and classical magnetostatics and introduce the static electric and magnetic fields and 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 dynamic connection between electricity and magnetism and how the two can be unified in one theory, classical electrodynamics, described by one system of coupled dynamic field equations. 1.1 Electrostatics The theory that describes physical phenomena related to the interaction between stationary electric charges or charge distributions in space is called electrostatics. 1.1.1 Coulomb’s law It has been found experimentally that in classical electrostatics the interaction between two stationary electrically charged bodies can be described in terms of a mechanical force. Let us consider the simple case described by Figure 1.1.1. 1
E2CLASSICALELECTRODYNAMICSMOqx0FIGURE1.1:Coulomb's law describes how a static electric chargeq,locatedatapointxrelativetotheoriginO,experiencesanelectrostaticforcefrom a static electric chargeglocated atx'Let F denote theforce acting on a charged particlewith chargeq located at x,due to the presence of a charge q' located at x'. According to Coulomb's lawthisforce is, in vacuum,given bythe expressionqqx-xqq1F(x) =(1.1)4元00-x34元20wherewehave used resultsfromExampleM.6onpage172.InSIunits,whichwe shall use throughout, the force F is measured in Newton (N), the chargesq and q' in Coulomb (C) [= Ampere-seconds (As)], and the length |x -x'l inmetres (m). The constant 80 = 107 /(4c2) ~ 8.8542 × 10-12 Farad per metre(F/m) is the vacuum permittivity and c2.9979×108m/s is the speed of lightinvacuum.InCGSunitso=1/(4)andtheforceismeasuredindyne,thechargeinstatcoulomb,and lengthincentimetres(cm)1.1.2 The electrostatic fieldInstead of describing the electrostatic interaction in terms of a“force actionat a distance,it turns out that it is often more convenient to introduce theconcept of a field and to describe the electrostatic interaction in terms of astatic vectorial electricfield Estat definedby the limiting processFEstat def,lim(1.2)-q-0qwhereF is the electrostatic force, as defined in Equation (1.1), from a netcharge q'on the test particle with a small electric net charge q.Since theDaae/CED/BOokDraft version released 13th November 2000 at 22:01.tp://①由
“main” 2000/11/13 page 2 2 CLASSICAL ELECTRODYNAMICS O x 0 x q x−x 0 q 0 FIGURE 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 0 located at x 0 . Let F denote the force acting on a charged particle with charge q located at x, due to the presence of a charge q 0 located at x 0 . According to Coulomb’s law this force is, in vacuum, given by the expression F(x) = qq0 4πε0 x−x 0 |x−x 0 | 3 = − qq0 4πε0 ∇ ✁ 1 |x−x 0 | ✂ (1.1) where we have used results from Example M.6 on page 172. In SI units, which we shall use throughout, the force F is measured in Newton (N), the charges q and q 0 in Coulomb (C) [= Ampère-seconds (As)], and the length |x−x 0 | in metres (m). The constant ε0 = 107 /(4πc 2 ) ≈ 8.8542 × 10−12 Farad per metre (F/m) is the 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, the charge in statcoulomb, and length in centimetres (cm). 1.1.2 The electrostatic field Instead of describing the electrostatic interaction in terms of a “force action at a distance,” it turns out that it is often more convenient to introduce the concept of a field and to describe the electrostatic interaction in terms of a static vectorial electric field E stat defined by the limiting process E stat def ≡ lim q→0 F q (1.2) where F is the electrostatic force, as defined in Equation (1.1), from a net charge q 0 on the test particle with a small electric net charge q. Since the Downloaded from http://www.plasma.uu.se/CED/Book Draft version released 13th November 2000 at 22:01
④31.1ELECTROSTATICSpurpose of the limiting process is to assure that the test charge q does notinfluence the field, the expression for Estat does not depend explicitly on g butonly on the charge q' and the relative radius vector x-x'.This means that wecan say that any net electric charge produces an electric field in the space thatsurroundsit, regardless ofthe existence of a second chargeanywherein thisspace.!Using formulae (1.1) and (1.2), we find that the electrostatic field Estat atthefield pointx (alsoknown as theobservation point),due to afield-producingcharge q'at the sourcepoint x', isgivenbyqx-x0Estat(x) =(1.3)4元0--4元0Inthe presence ofseveralfield producingdiscrete chargesq',atx,=1,2,3,.respectively,the assumption of linearity of vacuumallows us to superimposetheirindividual Efields into atotal EfieldEsta(x)=Z, X-x(1.4)24元80-Ifthe discrete charges are small and numerous enough, we introduce the chargedensityplocatedatx'andwritethetotal fieldas1X-XEstat(x) =d'x=Px (1.5)4元0J4元0x-x'Twhere, in the last step, we used formula Equation (M.68) on page 172. Weemphasise that Equation (1.5) above is valid for an arbitrary distribution ofcharges, including discrete charges, in which case p can be expressed in termsofoneormoreDiracdeltafunctions'In the preface to thefirst edition of the first volume ofhis book A Treatise on Electricityand Magnetism, first published in 1873,James Clerk Maxwell describes this in thefollowing,almost poetic, manner [6]"For instance,Faraday, in his mind's eye, saw lines offorce traversing all spacewhere themathematicians saw centres of forceattracting at adistance:Faradaysawa medium wheretheysawnothingbut distance:Faraday sought the seat ofthephenomena inreal actionsgoing oninthemedium,theyweresatisfied thatthey had found it in a power of action at a distance impressed on the electricfluids."2infact,vacuum exhibits aquantummechanicalnonlinearityduetovacuumpolarisationeffects manifesting themselves in themomentary creationand annihilation of electron-positronpairs, but classically this nonlinearity is negligibleDraft versied13thNoveDer2000at 22:01.Downloadedfromhttp://w/CED/BOOK由
“main” 2000/11/13 page 3 1.1 ELECTROSTATICS 3 purpose of the limiting process is to assure that the test charge q does not influence the field, the expression for E stat does not depend explicitly on q but only on the charge q 0 and the relative radius vector x−x 0 . This means that we can say that any net electric charge produces an electric field in the space that surrounds it, regardless of the existence of a second charge anywhere in this space.1 Using formulae (1.1) and (1.2), we find that the electrostatic field E stat at the field point x (also known as the observation point), due to a field-producing charge q 0 at the source point x 0 , is given by E stat(x) = q 0 4πε0 x−x 0 |x−x 0 | 3 = − q 0 4πε0 ∇ ✁ 1 |x−x 0 | ✂ (1.3) In the presence ofseveral field producing discrete charges q 0 i , at x 0 i , i = 1,2,3,. , respectively, the assumption of linearity of vacuum2 allows us to superimpose their individual E fields into a total E field E stat(x) = ∑ i q 0 i 4πε0 x−x 0 i ✄ ✄ x−x 0 i ✄ ✄ 3 (1.4) If the discrete charges are small and numerous enough, we introduce the charge density ρ located at x 0 and write the total field as E stat(x) = 1 4πε0 ☎V ρ(x 0 ) x−x 0 |x−x 0 | 3 d 3 x 0 = − 1 4πε0 ☎V ρ(x 0 )∇ ✁ 1 |x−x 0 | ✂ d 3 x 0 (1.5) where, in the last step, we used formula Equation (M.68) on page 172. We emphasise that Equation (1.5) above is valid for an arbitrary distribution of charges, including discrete charges, in which case ρ can be expressed in terms of one or more Dirac delta functions. 1 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: [6] “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 actions 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.” 2 In fact, vacuum exhibits a quantum mechanical nonlinearity due to vacuum polarisation effects manifesting themselves in the momentary creation and annihilation of electron-positron pairs, but classically this nonlinearity is negligible. Draft version released 13th November 2000 at 22:01. Downloaded from http://www.plasma.uu.se/CED/Book
E4CLASSICALELECTRODYNAMICSSince, according to formula Equation (M.78) on page 175, V×[Vα(x)] = 0for any 3D R3 scalar field a(x), we immediately find that in electrostatics17Vd3xV×Estat(x)=o(x)4元801(1.6)0(x)Vxd'x4元20=0Ie., Estat is an irrotational field.Taking the divergence ofthe general Estat expressionfor an arbitrary chargedistribution,Equation (1.5)on the preceding page, and using the representationof the Dirac delta function, Equation (M.73) on page 174, we find thatX-Xd3xV·Estat(x)=- V42Jx-xT1p(xv.(Ix-x'l4元80J1:p(x)v2d'x(1.7)x4元0元p(x)s(x-x)dx= P(x)80whichisGauss'slawindifferential form1.2MagnetostaticsWhile electrostatics deals with static charges, magnetostatics deals with sta-tionary currents, i.e., charges moving with constant speeds, and the interactionbetweenthese currents.1.2.1 Ampere's lawExperimentsontheinteractionbetweentwosmallcurrentloopshaveshownthat they interact via a mechanical force, much the same way that chargesinteract.LetFdenote sucha forceactingon a small loop Ccarryinga currentJ located atx,dueto thepresenceof a small loopC'carryinga currentJ/CED/BoOkDraft veion released 13thNovember2000 at 22:01.n-f1①由
“main” 2000/11/13 page 4 4 CLASSICAL ELECTRODYNAMICS Since, according to formula Equation (M.78) on page 175, ∇×[∇α(x)] ≡ 0 for any 3D ✆3 scalar field α(x), we immediately find that in electrostatics ∇×E stat(x) = − 1 4πε0 ∇× ☎V ρ(x 0 ) ✝∇ ✁ 1 |x−x 0 | ✂✟✞ d 3 x 0 = − 1 4πε0 ☎V ρ(x 0 )∇× ✝∇ ✁ 1 |x−x 0 | ✂✟✞ d 3 x 0 = 0 (1.6) I.e., E stat is an irrotational field. Taking the divergence of the general E stat expression for an arbitrary charge distribution, Equation (1.5) on the preceding page, and using the representation of the Dirac delta function, Equation (M.73) on page 174, we find that ∇·E stat(x) = ∇· 1 4πε0 ☎V ρ(x 0 ) x−x 0 |x−x 0 | 3 d 3 x 0 = − 1 4πε0 ☎V ρ(x 0 )∇·∇ ✁ 1 |x−x 0 | ✂ d 3 x 0 = − 1 4πε0 ☎V ρ(x 0 )∇ 2 ✁ 1 |x−x 0 | ✂ d 3 x 0 = 1 ε0 ☎V ρ(x 0 )δ(x−x 0 )d 3 x 0 = ρ(x) ε0 (1.7) which is Gauss’s law in differential form. 1.2 Magnetostatics While electrostatics deals with static charges, magnetostatics deals with stationary currents, i.e., charges moving with constant speeds, and the interaction between these currents. 1.2.1 Ampère’s law Experiments on the interaction between two small current loops have shown that they interact via a mechanical force, much the same way that charges interact. Let F denote such a force acting on a small loop C carrying a current J located at x, due to the presence of a small loop C 0 carrying a current J 0 Downloaded from http://www.plasma.uu.se/CED/Book Draft version released 13th November 2000 at 22:01