Classical Electromagnetism - NTUA

1 ClassicalElectromagnetism:Anintermediate levelcourseRichardFitzpatrickProfessor ofPhysicsTheUniversity .. 'slaw.. 'law.. 'sequation.. ere'sexperiments.. ere'slaw.. ere'scircuitallaw.. 'stheorem..994 Time-dependentMaxwell' .. 'slaw.. 'sfunctions.. elds.. 'slaw.. eld.. 'sequation.. 'sequation.. dielectricmedium.. dielectricmedium.. gaseousmedium.. plasma.. conductor.. collisionalplasma.. ectionata dielectricboundary.. canceoftensors.. eldtensor.. eldtensor.. elds.. movingcharge.. movingcharge.. movingcharge.. haveconsultedmostfrequentlywhilstdevelop ingcoursematerialare:Classicalelectricit y , ,2ndedition(Addison-Wesley, ReadingMA,1962). Feynman, , , (Addison-Wesley, ReadingMA,1964).Specialrelativity:W. Rindler(Oliver& Boyd,Edinburgh& LondonUK,1966).Electromagnetic eldsandwaves:P.

2 Lorrain, ,3rdedition( Co.,SanFranciscoCA,1970). , (JohnWiley& Sons,ChichesterUK,1975).Foundationsof , , Christy,3rdedition(Addison-Wesley, ReadingMA,1980).Theclassicaltheory of , ,4thedition[Butterworth-Heinemann,Oxford UK,1980].Introductionto ths,2ndedition(PrenticeHall,Engle-woodCl iffsNJ,1989). , ,3rdedition(Saun-dersCollegePublishing,F ortWorthTX,1995).Classicalelectrodynamic s:W. Greiner(Springer-Verlag,NewYorkNY, 1998).Inaddition,thesectiononvectorsis largelybasedonmyundergraduatelecturenote stakenfroma coursegivenbyDr. 'sequations. Thesearea setofeight bemoreexact,Maxwell'sequationsconstitute a completedescriptionofthebehaviourofelect ricandmagnetic electricandmagnetic ,fewcananswerthefollowingimportantques-t ion:doelectricandmagnetic eldshavea realphysicalexistence,oraretheymerelythe oreticalconstructswhichweusetocalculatet heelectricandmagneticforcesexertedbychar gedparticlesononeanother?

3 Asweshallsee,theprocessofformulatinganan swertothisquestionenablesustocometoa betterunder-standingofthenatureofelectri candmagnetic elds,andthereasonswhyit anygivenpointin space,anelectricormagnetic eldpossessestwoproper-ties,amagnitudeand adirection. Ingeneral,thesepropertiesvary(continuous ly) is conventionaltorepresentsucha eldintermsofitscomponentsmeasuredwithres pecttosomeconvenientlychosensetofCartesi anaxes( , theconventionalx-,y-, andz-axes).Ofcourse,theorientationofthes eaxesisarbitrary. Inotherwords,differentobserversmaywellch oosedifferentcoordinateaxestodescribethe same , electricandmagnetic canseethatanydescriptionofelectricandmag netic eldsis , thenatureofthe eldsthemselves,and,secondly,ourarbitrary choiceofthecoordinateaxeswithrespecttowh ichwemeasurethese ,Maxwell'sequations theequationswhichdescribethebehaviourof electricandmagnetic elds ,thefundamentallawsof physicswhichgovernthebehaviourof electricandmag-netic elds,and,secondly, wouldbehelpfulif wecouldeasilydistinguishthoseelementsof Maxwell' ,wecanachievethisbyusingwhatmathematicia nscallvector eldtheory.

4 ThistheoryenablesustowriteMaxwell'sequat ionsin a ,Maxwell'sequationslooka lotsimplerwhenwrittenina coordinate-freemanner. Infact,insteadofeight rst-orderpartialdifferentialequations,we onlyrequirefoursuchequationswithinthecon textofvector briefreviewofthoseelementsofvector eldtheorywhicharereleventtoMaxwell' , wederivethetime-independentversionofMaxw ell' , Maxwell'sequationsto ,weincorporatedielectricandmagneticmedia intoMaxwell' ' , weexaminehowMaxwell' , weemployMaxwell' conclude, , witha discussionoftherelativisticformulationof Maxwell' Wang-JungYoon[ChonnamNationalUniversity, RepublicofKorea(South)] ,weshallgivea briefoutlineof thoseaspectsof vectoralgebra,vec-torcalculus,andvector eldtheorywhichareneededtoderiveandunders tandMaxwell' largelybasedonmyundergraduatelecturenote sfroma coursegivenbyDr.

5 :Inappliedmathematics,physicalquantities are(predominately) ,denotedscalars, arerepre-sentedbyrealnumbers. Others,denotedvectors, ,!PQ( ).Notethatlineelements(and,there-fore,ve ctors)aremovable, ,vectorsjustpossessa magnitudeanda direction,whereasscalarspossessa ,vectorquantitiesaredenotedbybold-facedc haracters( ,a) intypesetdocuments,andbyunderlinedcharac -ters( ,a) , butthesameunitsmustbeused, parallelogram:!PR=!PQ+!QR( ).Supposethata !PQ !SR, :b !QR !PS, andc !PR. It is ,a+b=b+a. It ,a+ (b+c) = (a+b) + basedonlineelementsin de nedbyCartesiancoordinates, ,wegenerallyadoptthesecondapproach,becau seit is ,a vectoris denotedastherowmatrixofitscom-ponentsalo ngeachoftheCartesianaxes(thex-,y-, andz-axes,say):a (ax; ay; az):( )Here,axis thex-coordinateofthe head ofthevectorminusthex-coordinateofits tail.

6 Ifa (ax;ay;az)andb (bx;by;bz)thenvectoradditionis de neda+b (ax+bx; ay+by; az+bz):( )Ifais a vectorandnis a scalarthentheproductof a scalaranda vectoris de nedna (nax; nay; naz):( )It is ,n(a+b) =na+ yy qFigure3:Unitvectorscanbede nedinthex-,y-, andz-directionsasex (1;0;0),ey (0;1;0), andez (0;0;1). Anyvectorcanbewrittenintermsoftheseunitv ectors:a=axex+ayey+azez:( )Inmathematicalterminology, thethreevectorsaremutuallyperpendiculart hentheyaretermedorthogonalbasisvectors. However, ,r= (x; y; z);( )andvelocities,v=drdt=lim t!0r(t+ t) -r(t) t:( )Supposethatwetransformtoa neworthogonalbasis,thex0-,y0-, andz0-axes,whicharerelatedtothex-,y-, andz-axesviaa rotationthroughanangle aroundthez-axis( ).Inthenewbasis,thecoordinatesofthegener aldisplacementrfromtheoriginare(x0;y0;z0 ). Thesecoordinatesarerelatedtothepreviousc oordinatesviathetransformation:x0=xcos +ysin ;( )y0=-xsin +ycos ;( )z0=z:( ) is stilldenotedr.

7 Thereasonforthisis , theymustdependin a veryspeci cmanner[ , Eqs.( ) ( )] displacementfromanorigin(thisisjusta specialcaseofa directedlineelement),it followsthatthecomponentsofa generalvectoramusttransforminananalogous mannertoEqs.( ) ( ).Thus,ax0=axcos +aysin ;( )ay0=-axsin +aycos ;( )az0=az;( )withsimilartransformationrulesforrotati onaboutthey- ,Eqs.( ) ( ) constitutethede nitionofa vector. Thethreequantities(ax,ay,az) arethecomponentsofa vectorprovidedthattheytransformunderrota tionlikeEqs.( ) ( ). Conversely, (ax,ay,az)cannotbethecomponentsofa vectorif theydonottransformlikeEqs.( ) ( ). ,theindividualcom-ponentsofa vector(ax, say)arerealnumbers, ,andallvectorsderivedfromdisplacements,a utomaticallysatisfyEqs.( ) ( ).Thereare,however, otherphysicalquantitieswhichhavebothmagn itudeanddirection, We cande nea vectorareaSwhosemagnitudeisS, andwhosedirectionis perpendiculartotheplane,inthesensedeterm inedbytheright-handgripruleontherim( ).

8 It a truevector?We knowthatif thenormaltothesurfacemakesanangle :theareaseenlookingalongthex-directionis Scos x. Thisis thex-componentofS. Similarly, if thenormalmakesanangle ywiththey-axisthentheareaseenlookingalon gthey-directionisScos y. Thisis they-componentofS. Ifwelimitourselvestoa surfacewhosenormalis perpendiculartothez-directionthen x= =2- y= . It followsthatS=S(cos ;sin ; 0). If werotatethebasisaboutthez-axisby degrees,whichis equivalenttorotatingthenormaltothesurfac eaboutthez-axisby- degrees,thenSx0=Scos( - ) =Scos cos +Ssin sin =Sxcos +Sysin ;( )whichis thecorrecttransformationruleforthex-comp onentofa vector. vectorareais ,theprojectedareaoftwoplanesurfaces,join edtogetherata line,lookingalongthex-direction(say)is ,formanyjoined-upplaneareas,theprojected areainthex-direction,whichis thesameastheprojectedareaoftheriminthex- direction,is thex-componentoftheresultantofallthevect orareas:S=XiSi:( )If weapproacha limit,bylettingthenumberofplanefacetsinc rease,andtheirareasreduce,thenweobtaina :S=Xi Si:( )It is clearthattheprojectedareaoftherimin thex-directionis justSx.

9 , ,a loop(notallinoneplane)hasa vectorareaSwhichis , a closedsurfacehasS=0, sinceit doesnotpossessa scalarquantityis scalaroutofsomecombinationofthecompo-nen tsofone,ormore,vectors?Supposethatwewere tode nethe ampersand product,a&b=axby+aybz+azbx=scalarnumber; ( )forgeneralvectorsaandb. Isa&binvariantundertransformation,asmust bethecaseif it is a scalarnumber? (1; 0; 0)andb= (0; 1; 0). It is easilyseenthata&b=1. Letusnowrotatethebasisthrough45 ,a= (1=p2;-1=p2; 0)andb= (1=p2; 1=p2; 0), givinga&b=1=2. Clearly,a&bisnotinvariantunderrotational transformation,sotheabovede nitionis a , now, thedotproductorscalarproduct:a b=axbx+ayby+azbz=scalarnumber:( )Letusrotatethebasisthough Eqs.( ) ( ), inthenewbasisa btakestheforma b=(axcos +aysin ) (bxcos +bysin ) +(-axsin +aycos ) (-bxsin +bycos ) +azbz( )=axbx+ayby+azbz:Thus,a caneasilybeshownthatit is alsoinvariantunderrotationaboutthex- ,a bis a truescalar, sotheabovede nitionis a ,a bis theonlysimplecombinationofthecomponentso ftwovectorswhichtransformslikea scalar.

10 Itis easilyshownthatthedotproductis commutativeanddistributive:a b=b a;a (b+c) =a b+a c:( )Theassociativepropertyis meaninglessforthedotproduct,becausewecan nothave(a b) c, sincea bis haveshownthatthedotproducta bis thephysicalsigni canceof this?Considerthespecialcasewherea=b. Clearly,a b=a2x+a2y+a2z=Length(OP)2;( )ifais thepositionvectorofPrelativetotheoriginO . So,theinvarianceofa ais equivalenttotheinvarianceofthelength,orm agnitude, usuallydenotedjaj( themodulusofa ) orsometimesjusta, soa a=jaj2=a2:( )b ( )is(b-a) (b-a) =jaj2+jbj2-2a b:( )However, accordingtothe cosinerule oftrigonometry,(AB)2= (OA)2+ (OB)2-2(OA) (OB)cos ;( )where(AB)denotesthelengthofsideAB. It followsthata b=jaj jbjcos :( )Clearly, theinvarianceofa bundertransformationis equivalentto b=0theneitherjaj=0,jbj=0, orthevectorsaandbareperpendicular.