Classical Electrodynamics Theoretical PhysicsII Manuscript EnglishEdition Franz Wegner Institutfu¨r Theoretische Physik Ruprecht-Karls-Universita¨t Heidelberg 2003 2 c2003FranzWegnerUniversita¨tHeidelberg (cid:13) Copyingforprivatepurposeswithreferencetotheauthorallowed.Commercialuseforbidden. Iappreciatebeinginformedofmisprints. I am grateful to Jo¨rg Raufeisen, Andreas Haier, Stephan Frank, and Bastian Engeser for informing me of a numberofmisprintsinthefirstGermanedition. SimilarlyIthankBjo¨rnFeuerbacher,SebastianDiehl,Karsten Frese,MarkusGabrysch,andJanTomczakforinformingmeofmisprintsinthesecondedition. IamindebtedtoCorneliaMerkel,MelanieSteiert, andSonjaBartschforcarefullyreadingandcorrectingthe textofthebilingualedition. Books: Becker,Sauter: TheoriederElektrizita¨tI Jackson,ClassicalElectrodynamics Landau,Lifschitz: LehrbuchderTheoretischenPhysikII:KlassischeFeldtheorie Panofsky,Phillips,ClassicalElectricityandMagnetism Sommerfeld: Vorlesungenu¨berTheoretischePhysikIII:Elektrodynamik Stratton,ElectromagneticTheory Stumpf,Schuler: Elektrodynamik A Basic Equations c2003FranzWegnerUniversita¨tHeidelberg (cid:13) IntroductoryRemarks I assume that the student is already somewhat familiar with classical electrodynamics from an introductory course. Therefore I start with the complete set of equations and from this set I spezialize to variouscases of interest. InthismanuscriptIwilluseGaussianunitsinsteadoftheSI-units. Theconnectionbetweenbothsystemsand themotivationforusingGaussianunitswillbegiveninthenextsectionandinappendixA. FormulaeforvectoralgebraandvectoranalysisaregiveninappendixB.Awarningtothereader: Sometimes (B.11, B.15, B.34-B.50 and exercise after B.71) he/she should insert the result by him/herself. He/She is re- questedtoperformthecalculationsbyhim/herselforshouldatleastinserttheresultsgiveninthisscript. 1 Basic Equations of Electrodynamics Electrodynamicsdescribeselectricandmagneticfields,theirgenerationbychargesandelectriccurrents,their propagation(electromagneticwaves),andtheirreactiononmatter(forces). 1.a Chargesand Currents 1.a.(cid:11) ChargeDensity Thechargedensity(cid:26)isdefinedasthecharge(cid:1)qpervolumeelement(cid:1)V (cid:1)q dq (cid:26)(r)= lim = : (1.1) (cid:1)V 0(cid:1)V dV ! ThereforethechargeqinthevolumeV isgivenby q= d3r(cid:26)(r): (1.2) ZV Ifthechargedistributionconsistsofpointchargesq atpointsr,thenthechargedensityisgivenbythesum i i (cid:26)(r)= q(cid:14)3(r r); (1.3) i i (cid:0) i X whereDirac’sdelta-function(correctlydelta-distribution)hastheproperty f(r ) ifr V ZVd3rf(r)(cid:14)3(r(cid:0)r0)=( 00 ifr00 2<V : (1.4) Similarlyonedefinesthechargedensityperarea(cid:27)(r)atboundariesandsurfacesaschargeperarea dq (cid:27)(r)= ; (1.5) df similarlythechargedensityonaline. 3 4 ABasicEquations 1.a.(cid:12) CurrentandCurrentDensity ThecurrentI isthechargedqthatflowsthroughacertainareaF pertimedt, dq I = : (1.6) dt Bev(r;t)theaveragevelocityofthechargecarriersandntheunitvectornor- maltotheareaelement. Thenvdtisthedistancevectortraversedduringtime n dt. Multipliedbynoneobtainsthethicknessofthelayerv ndtofthecarriers (cid:1) v whichpassedthesurfaceduringtimedt.Multipliedbythesurfaceelementdf oneobtainsthevolumeofthecharge,whichflowsthroughthearea.Additional multiplicationby(cid:26)yieldsthechargedqwhichpassesduringtimedtthesurface df dq = vdt ndf(cid:26) (1.7) (cid:1) ZF I =dq=dt = v(r;t)(cid:26)(r;t) n(r)df = j(r;t) df (1.8) (cid:1) (cid:1) ZF ZF withthecurrentdensityj=(cid:26)vandtheorientedareaelementdf =ndf. 1.a.(cid:13) ConservationofChargeandEquationofContinuity ThechargeqinafixedvolumeV q(t)= d3r(cid:26)(r;t) (1.9) ZV changesasafunctionoftimeby dq(t) @(cid:26)(r;t) = d3r : (1.10) dt @t ZV This charge can only change, if some charge flows through the surface @V of the volume, since charge is conserved.WedenotethecurrentwhichflowsoutwardbyI. Then dq(t) = I(t)= j(r;t) df = d3rdivj(r;t); (1.11) dt (cid:0) (cid:0) (cid:1) (cid:0) Z@V ZV wherewemakeuseofthedivergencetheorem(B.59).Since(1.10)and(1.11)holdforanyvolumeandvolume element,theintegrandsinthevolumeintegralshavetobeequal @(cid:26)(r;t) + divj(r;t)=0: (1.12) @t Thisequationiscalledtheequationofcontinuity.Itexpressesindi(cid:11)erentialformtheconservationofcharge. 1.b Maxwell’sEquations The electric charges and currents generate the electric field E(r;t) and the magnetic induction B(r;t). This relationisdescribedbythefourMaxwellEquations @E(r;t) 4(cid:25) curlB(r;t) = j(r;t) (1.13) (cid:0) c@t c divE(r;t) = 4(cid:25)(cid:26)(r;t) (1.14) @B(r;t) curlE(r;t)+ = 0 (1.15) c@t divB(r;t) = 0: (1.16) TheseequationsnamedafterMaxwellareoftencalledMaxwell’sEquationsinthevacuum.However,theyare alsovalidinmatter. Thechargedensityandthecurrentdensitycontain allcontributions,the densitiesoffree chargesandpolarizationcharges,andoffreecurrentsandpolarization-andmagnetizationcurrents. Oftenonerequiresasaboundaryconditionthattheelectricandthemagneticfieldsvanishatinfinity. 1BasicEquationsofElectrodynamics 5 1.c CoulombandLorentzForce The electric field E and the magnetic induction B exert a force K on a charge q located at r, moving with a velocityv v K=qE(r)+q B(r): (1.17) c (cid:2) HereEandBarethecontributionswhichdonotcomefromqitself. Thefieldsgeneratedbyqitselfexertthe reactionforcewhichwewillnotconsiderfurther. Thefirst contributionin (1.17)is theCoulombforce, thesecond onethe Lorentzforce. Onehas c=299 792 458m/s. Laterwewillseethatthisisthevelocityoflightinvacuum. (Ithasbeendefinedwiththevaluegiven aboveinordertointroduceafactorbetweentimeandlength.) Theforceactingonasmallvolume(cid:1)V canbe writtenas (cid:1)K = k(r)(cid:1)V (1.18) 1 k(r) = (cid:26)(r)E(r)+ j(r) B(r): (1.19) c (cid:2) kiscalledthedensityofforce. TheelectromagneticforceactingonthevolumeV isgivenby K= d3rk(r): (1.20) ZV 6 ABasicEquations 2 Dimensions and Units 2.a GaussianUnits InthiscourseweuseGaussianunits. Weconsiderthedimensionsofthevariousquantities. Fromtheequation ofcontinuity(1.12)andMaxwell’sequations(1.13to1.16)oneobtains [(cid:26)]=[t] = [j]=[x] (2.1) [B]=[x] = [E]=([c][t])=[j]=[c] (2.2) [E]=[x] = [B]=([c][t])=[(cid:26)]: (2.3) Fromthisoneobtains [j] = [(cid:26)][x]=[t] (2.4) [E] = [(cid:26)][x] (2.5) [B] = [(cid:26)][c][t]=[(cid:26)][x]2=([c][t]); (2.6) and [c] = [x]=[t] (2.7) [B] = [(cid:26)][x]: (2.8) From(2.7)oneseesthatc reallyhasthedimensionofavelocity. Inordertodeterminethedimensionsofthe otherquantitieswestillhavetouseexpression(1.19)fortheforcedensityk [k]=[(cid:26)][E]=[(cid:26)]2[x]: (2.9) Fromthisoneobtains [(cid:26)]2 = [k]=[x]=dyncm 4 (2.10) (cid:0) [(cid:26)] = dyn1=2cm 2 (2.11) (cid:0) [E]=[B] = dyn1=2cm 1 (2.12) (cid:0) [j] = dyn1=2cm 1s 1 (2.13) (cid:0) (cid:0) [q] = [(cid:26)][x]3 =dyn1=2cm (2.14) [I] = [j][x]2 =dyn1=2cms 1: (2.15) (cid:0) 2.b Other SystemsofUnits Theunitforeachquantitycanbedefinedindependently.Fortunately,thisisnotusedextensively. BesidestheGaussiansystemofunitsanumberofothercgs-systemsisusedaswellastheSI-system(interna- tionalsystemofunits,Giorgi-system). Thelastoneisthelegalsysteminmanycountries(e.g. intheUSsince 1894,inGermanysince1898)andisusedfortechnicalpurposes. WhereasallelectromagneticquantitiesintheGaussiansystemareexpressedincm,gunds,theGiorgi-system usesbesidesthemechanicalunitsm,kgandstwootherunits,A(ampere)undV(volt). Theyarenotindepen- dent,butrelatedbytheunitofenergy 1kgm2s 2 =1J=1Ws=1AVs: (2.16) (cid:0) The conversion of the conventional systems of units can be described by three conversion factors (cid:15) , (cid:22) and 0 0 . The factors (cid:15) and (cid:22) (known as the dielectric constant and permeability constant of the vacuum in the 0 0 SI-system)andtheinterlinkingfactor (cid:13)=cp(cid:15) (cid:22) (2.17) 0 0 cancarrydimensionswhereas isadimensionlessnumber.Onedistinguishesbetweenrationalsystems =4(cid:25)) andnon-rationalsystems( =1)ofunits. Theconversionfactorsofsomeconventionalsystemsofunitsare 2DimensionsandUnits 7 SystemofUnits (cid:15) (cid:22) (cid:13) 0 0 Gaussian 1 1 c 1 electrostatic(esu) 1 c 2 1 1 (cid:0) electromagnetic(emu) c 2 1 1 1 (cid:0) Heaviside-Lorentz 1 1 c 4(cid:25) Giorgi(SI) (c2(cid:22) ) 1 4(cid:25) Vs 1 4(cid:25) 0 (cid:0) 107Am ThequantitiesintroduceduntilnowareexpressedinGaussianunitsbythoseofothersystemsofunits(indicated byanasterisk)inthefollowingway E= (cid:15) E 1dyn1=2cm 1=ˆ3 104V/m (2.18) 0 (cid:3) (cid:0) (cid:1) B= p =(cid:22)0B(cid:3) 1dyn1=2cm(cid:0)1=ˆ10(cid:0)4Vs/m2 (2.19) 1 q=p q 1dyn1=2cm=ˆ10 9=3As,similarly(cid:26);(cid:27);I; j: (2.20) (cid:3) (cid:0) p (cid:15) 0 Anexampleofconversion:TheCoulomb-Lorentz-forcecanbewritten 1 q p 1 1 K=q(E+ v B)= (cid:3) ( (cid:15) E + v B )=q (E + v B )=q (E + v B ): (2.21) c (cid:2) p (cid:15)0 0 (cid:3) cp(cid:22)0 (cid:2) (cid:3) (cid:3) (cid:3) cp(cid:15)0(cid:22)0 (cid:2) (cid:3) (cid:3) (cid:3) (cid:13) (cid:2) (cid:3) p The elementary chargee is 4:803 10 10 dyn1=2 cm in Gaussian units and 1:602 10 19 As in SI-units. The 0 (cid:0) (cid:0) (cid:1) (cid:1) electroncarriescharge e , theprotone , a nucleuswith Z protonsthe chargeZe , quarksthecharges e =3 0 0 0 0 (cid:0) (cid:6) and 2e =3. 0 (cid:6) Theconversionofotherquantitiesisgivenwheretheyareintroduced. AsummaryisgiveninAppendixA. 2.c MotivationforGaussianUnits IntheSI-systemtheelectricalfieldEandthedielectricdisplacementDaswellasthemagneticinductionBand themagnetic fieldH carry di(cid:11)erentdimensions. Thisleads easily tothe misleadingimpressionthat these are independentfields. Onamicroscopiclevelonedealsonlywithtwofields,EandB,(1.13-1.16)(Lorentz1892). However, the second set of fields is introduced only in order to extract the polarization and magnetization contributionsofchargesandcurrentsinmatterfromthetotalchargesandcurrents,andtoaddthemtothefields. (Section6and11). Thiscloserelationisbetterexpressedincgs-units,whereEandDhavethesamedimension,aswellasBand H. Unfortunately,theGaussian systembelongsto theirrationalones, whereasthe SI-systemisarational one, so thatinconversionsfactors4(cid:25)appear.IwouldhavepreferredtousearationalsystemlikethatofHeavisideand Lorentz. However,intheusualtextbooksonlytheSI-systemandtheGaussianoneareused. Idonotwishto o(cid:11)ertheelectrodynamicsinasystemwhichinpracticeisnotusedinothertextbooks. 8 ABasicEquations B Electrostatics c2003FranzWegnerUniversita¨tHeidelberg (cid:13) 3 Electric Field, Potential, Energy of the Field 3.a Statics First we consider the time-independent problem: Statics. This means, the quantities depend only on their location, (cid:26) = (cid:26)(r), j = j(r), E = E(r), B = B(r). Then the equation of continuity (1.12) and Maxwell’s equations(1.13-1.16)separateintotwogroups divj(r)=0 curlB(r)= 4(cid:25)j(r) divE(r)=4(cid:25)(cid:26)(r) c divB(r)=0 curlE(r)=0 (3.1) magnetostatics electrostatics k = 1j(r) B(r) k =(cid:26)(r)E(r) ma c (cid:2) el The first group of equations contains only the magnetic induction B and the current density j. It describes magnetostatics. ThesecondgroupofequationscontainsonlytheelectricfieldEandthechargedensity(cid:26). Itis thebasisofelectrostatics. Theexpressionsforthecorrespondingpartsoftheforcedensitykisgiveninthelast line. 3.b ElectricFieldand Potential 3.b.(cid:11) ElectricPotential Nowwe introducethe electric Potential (cid:8)(r). For this purposewe consider the path integraloverE along to di(cid:11)erentpaths(1)and(2)fromr tor 0 r r dr E(r)= dr E(r)+ dr E(r); (3.2) r0 (cid:1) r0 (cid:1) (cid:1) Z(1) Z(2) I wherethelastintegralhastobeperformedalongtheclosedpathfromr along(1) 0 torandfromthereinoppositedirectionalong(2)tor .Thislaterintegralcanbe (2) r 0 transformedbymeansofStokes’theorem(B.56)intotheintegral df curlE(r) overtheopensurfaceboundedby(1)and(2),whichvanishesduetoM(cid:1)axwell’s F R equation curlE(r)=0(3.1). r (1) 0 Thereforetheintegral(3.2)isindependentofthepathandonedefinestheelectricpotential r (cid:8)(r)= dr E(r)+(cid:8)(r ): (3.3) 0 (cid:0) (cid:1) Zr0 Thechoiceofr andof(cid:8)(r )isarbitrary,butfixed. Therefore(cid:8)(r)isdefinedapartfromanarbitraryadditive 0 0 constant. Fromthedefinition(3.3)wehave d(cid:8)(r)= dr E(r); E(r)= grad(cid:8)(r): (3.4) (cid:0) (cid:1) (cid:0) 9 10 BElectrostatics 3.b.(cid:12) ElectricFluxandCharge From divE(r)=4(cid:25)(cid:26)(r),(3.1)oneobtains d3rdivE(r)=4(cid:25) d3r(cid:26)(r) (3.5) ZV ZV andthereforewiththedivergencetheorem(B.59) df E(r)=4(cid:25)q(V); (3.6) (cid:1) Z@V idesttheelectricfluxofthefieldEthroughthesurfaceequals4(cid:25)timesthechargeqinthevolumeV. Thishasasimpleapplicationfortheelectricfieldofarotationalinvariantchargedistribution(cid:26)(r) = (cid:26)(r)with r= r.Forreasonsofsymmetrytheelectricfieldpointsinradialdirection,E=E(r)r=r j j r r 4(cid:25)r2E(r)=4(cid:25) (cid:26)(r )r2dr d(cid:10)=(4(cid:25))2 (cid:26)(r )r2dr ; (3.7) 0 0 0 0 0 0 Z0 Z0 sothatoneobtains 4(cid:25) r E(r)= (cid:26)(r )r2dr (3.8) 0 0 0 r2 Z0 forthefield. Asaspecialcaseweconsiderapointchargeintheorigin. Thenonehas q r 4(cid:25)r2E(r)=4(cid:25)q; E(r)= ; E(r)= q: (3.9) r2 r3 Thepotentialdependsonlyonrforreasonsofsymmetry.Thenoneobtains rd(cid:8)(r) grad(cid:8)(r)= = E(r); (3.10) r dr (cid:0) whichafterintegrationyields q (cid:8)(r)= +const. (3.11) r 3.b.(cid:13) PotentialofaChargeDistribution We start outfrom point chargesq at locationsr. The correspondingpotential andthe field is obtainedfrom i i (3.11)und(3.10)byshiftingrbyr i q (cid:8)(r) = i (3.12) r r i i j (cid:0) j X q(r r) E(r) = grad(cid:8)(r)= i (cid:0) i : (3.13) (cid:0) r r 3 Xi j (cid:0) ij We change now from point charges to the charge density (cid:26)(r). To do this we perform the transition from q f(r)= (cid:1)V(cid:26)(r)f(r)to d3r (cid:26)(r)f(r),whichyields i i i i i i 0 0 0 P P R (cid:26)(r) (cid:8)(r)= d3r 0 (3.14) 0 r r Z j (cid:0) 0j FromE= grad(cid:8)and divE=4(cid:25)(cid:26)oneobtainsPoisson’sequation (cid:0) (cid:8)(r)= 4(cid:25)(cid:26)(r): (3.15) 4 (cid:0) Pleasedistinguish = and(cid:1)=Delta. Wecheckeq. (3.15). Firstwedetermine 4 r(cid:1)r r r a (cid:8)(r)= d3r0(cid:26)(r0) 0(cid:0) = d3a(cid:26)(r+a) (3.16) r r r3 a3 Z j 0(cid:0) j Z