Electrodynamics and Radiave Processes I Lecture 4 Basic theory … · 2019. 8. 19. · Lecture 4...
Transcript of Electrodynamics and Radiave Processes I Lecture 4 Basic theory … · 2019. 8. 19. · Lecture 4...
Lecture4–Basictheoryofradia2onfields&Radia2onfrommovingcharges
Date:19thAugust2019
Bhaswa2BhaCacharyya
ElectrodynamicsandRadia2veProcessesI
August-September2019
IUCAA-NCRAGraduateSchool
Lecture-4Ques2onsraisedintheclass
Einstein’scoefficientsarevalidwhenthermodynamicequilibriumisnotpresent?
WhydonotweseeHIemissioninearth,whatfrac2onspinupwhatfrac2ondown?
Aretherela2onbetweenEinsteincoefficientstrulyindependentoftemperature?
Radia2onpressurederiva2oninRybickiandlightman'stextbook.Consideronlythesolidangleof2πinsteadof4π.Why?
Cantheprotonflipthespininsteadoftheelectron(regarding21cmline)?Reserveforlater
Electricandmagne2cfield
Sta2cchargeàElectricfieldForceonatestcharge
MovingchargeàMagne2cfieldàElectriccurrentMagne2cforceonamovingcharge
Fieldofforce@Maxwell
Spacearoundelectrifiedobject@Faraday
UnitsàNewton/Coulombàvoltsperm
UnitsàNewton/(Coulombm/s)àTeslaàGauss
Electromagne2cflux
Lorentzforce
Rateofworkdone
Currentdensity
Rateofworkdone
Rateofchangeinmechanicalenergyofsystemperunitvolume
Maxwell’sEqua2ons(inGaussianunits)
Maxwell’sEqua2ons(inGaussianunits)
Electromagne2cflux
Usingthefollowingrela2on
Electromagne2cflux
Usingthefollowingrela2on
Poyn2ng’stheorem
RateofchangeofMechanicalenergy
Rateofchangeoffieldenergy
Divergenceoffieldenergyflux
Electromagne2cfluxElectromagne2cfieldenergyperunitvolume
Electromagne2cfluxvectororPoyn2ngvector
Planeelectromagne2cwavesMaxwell’sequa2oninvacuum
Waveequa2onHow?
Waveequa2onwithB?
Planeelectromagne2cwavesMaxwell’sequa2oninvacuum
Waveequa2on
Solu2onsofwaveequa2on
unitvectors
wavevector frequency
complexconstants
Planeelectromagne2cwavesSubs2tu2oninMaxwell’sequa2on
a1anda2areperpendiculartok
a1,a2andkareperpendicular
a1
a2
ka1anda2areperpendicular
Planeelectromagne2cwavesSubs2tu2oninMaxwell’sequa2on
a1,a2andkareperpendicular
a1
a2
k
Planeelectromagne2cwavesSubs2tu2oninMaxwell’sequa2on
a1,a2andkareperpendicular
a1
a2
k
EMwavestravelsatspeedoflight
Planeelectromagne2cwavesEnergyfluxandenergydensity
Timeaveragedpoin2ngvector
Since
Similarly2meaveragedenergydensity
Planeelectromagne2cwavesEnergyfluxandenergydensity
Timeaveragedpoin2ngvector
Since
Similarly2meaveragedenergydensity
Radia2onspectrum
witharadia2onfieldoflengthΔtwecandefinespectrumwithinΔω
TotalEnergyperunitarea
Energyperunitareaperunit2me
Radia2onspectrum
Energyperunitareaperunitfrequency
Radia2onspectrum
a)pulse
PowerspectrumElectricfield
b)sinusoidalpulse
TimeextentofpulseTdetermineswidthoffinestfeatures:Δω~1/TSinusoidal2medependenceinpulseshapecausesspectrumconcentratednearω~ω0
Electromagne2cPoten2alsEandBarereplacedbyΦ(r,t)andA(r,t)
WhyweneedEMpoten2als?1)Onescalarplusonevectorsimplerthantwovectors
2)DeterminingAandΦaresimpler
3)Rela2vis2cEMtheorywillbesimpler
Electromagne2cPoten2alsEandBarereplacedbyΦ(r,t)andA(r,t)
WhyweneedEMpoten2als?
1)Onescalarplusonevectorsimplerthantwovectors
2)DeterminingAandΦaresimpler
3)Rela2vis2cEMtheorywillbesimpler
Vectorpoten2alA(r,t)definedas
Thus
Scalarpoten2alΦ(r,t)definedas
Maxwell’sequa2on
Electromagne2cPoten2alsEandBarereplacedbyΦ(r,t)andA(r,t)
Remember
Electromagne2cPoten2alsThusfromMaxwell’sequa2ons,
Electromagne2cPoten2als
ThusfromMaxwell’sequa2ons,
Electromagne2cPoten2als
ThusfromMaxwell’sequa2ons,
Electromagne2cPoten2alsThusfromMaxwell’sequa2ons,
ThusfromMaxwell’sequa2ons,
Electromagne2cPoten2alsScalarandvectorpoten2alarenotuniquelydeterminedbythecondi2ons
Forexample,theaddi2onofgradientψtoAwillnotchangeB
Electricfieldwillnotchangeifφischangedinfollowingmanner
Suchaltera2onsofAandϕarecalledGaugetransforma2on
LorentzGauge
RetardedPoten2als
Informa2onfrompointr’propagatesatspeedoflight.Thepoten2alatrcanonlybeaffectedbycondi2onsatr’ataretarded2met-|r-r’|/c
r
r’
RetardedPoten2als
ü Retarded2mereferstocondi2onsatthepointr’thatexistedata2meearlierthantby2merequiredforlighttotravelbetweenrandr’
ü Informa2onfrompointr’propagatesatspeedoflight,sopoten2alatrcanbeaffectedbycondi2onsofr’atthisretarded2me
ü Solu2onswithadvanced2mearenotpermiCedphysically
Forgivenchargeandcurrentdensityfirstfindtheretardedpoten2alsandthendetermineEandB
Retardedpoten2alofsinglemovingcharges:Lienard-Wiechartpoten2als
Par2cleofchargeqmovingalongatrajectoryr=r0(t),velocityu(t)=r0(t)
Chargeandcurrentdensity
TotalCharge TotalCurrent
Reterdedpoten2alofsinglemovingcharges:Lienard-Wiechartpoten2als
Chargeandcurrentdensity
ScalarPoten2al
Par2cleofchargeqmovingalongatrajectoryr=r0(t),velocityu(t)=r0(t)
Retardedpoten2alofsinglemovingcharges:Lienard-Wiechartpoten2als
Chargeandcurrentdensity
ScalarPoten2al
Par2cleofchargeqmovingalongatrajectoryr=r0(t),velocityu(t)=r0(t)
Retardedpoten2alofsinglemovingcharges:Lienard-Wiechartpoten2als
δfunc2onAvalueoft’=tretgivenby,
Retardedpoten2alofsinglemovingcharges:Lienard-Wiechartpoten2als
δfunc2onAvalueoft’=tretgivenby,
Fig:Radia2onfieldatRfromposi2onoftheradia2ngpar2cleattheretarded2me
Retardedpoten2alofsinglemovingcharges:Lienard-Wiechartpoten2alsChangethevariablefromt’àt”
Retardedpoten2alofsinglemovingcharges:Lienard-Wiechartpoten2alsChangethevariablefromt’àt”
Retardedpoten2alofsinglemovingcharges:Lienard-Wiechartpoten2als
Changethevariablefromt’àt”
Lienard-WiechartPoten2al
Retardedpoten2alofsinglemovingcharges:Lienard-Wiechartpoten2als
Differfromsta2celectromagne2ctheoryintwoways
1) Extrafactorκ:Importantforveloci2esclosetolight.Tendstoconcentrate/beampoten2alintoanarrowconeaboutpar2clevelocity.Beamingeffect(willbedetailedincominglectures)
2) Quan22esareevaluatedatretarded2me.
Differen2atethepoten2alstogetelectricfield(E)andmagne2cfield(B)(JacksonSec2on14)
VelocityandRadia2onfield
Fig:Radia2onfieldatRfromposi2onoftheradia2ngpar2cleattheretarded2me
Velocityfield Accelera2on/Radia2onfield
Radia2onfield
Velocityfield§ 1/R2dependence§ Onlycontribu2ngtermforpar2clewithconstantvelocity§ Generaliza2onoftheCoulomb’slawtomovingpar2cles,approachestocoulomb’slawwhenu<<c
Accelera2onfield/Radia2onfield1/RdependencePropor2onaltopar2cle’saccelera2onperpendicularton
§ Electricfiledalwayspointtowardscurrentposi2onofthepar2cle
Radia2onfield
Radia2onfield
Erad,Brad,n:mutuallyperpendicular|Erad|=|Brad|
Radia2onfieldsConsiderapar2cleoriginallymovingatconstantvelocityalongxaxisisstoppedatx=0andt=0Att=1thefieldoutsideofaradiuscisradialandpointstotheposi2onwherepar2clewouldhavebeeniftherewasnodecelera2on(sincenoinforma2onIsyetpropagatedtothatdistance)Butfieldinsidetheradiuscisinformed.
Graphicaldemonstra2onof1/Raccelera2onfield
Fromanempiricist’spointofviewthereare4observablesforradia2on•EnergyFlux•Direc2on•Frequency•Polariza0on
Observables
Polarimetry:studyofpolariza2onofincomingradia2on
Polariza2onofelectromagne2cradia2on
ü Polariza2onisproducedinvariousways,includingdirectlyfromsomeradia2onprocesses(e.g.cyclotronandsynchrotronemission),fromdifferen2alabsorp2onofradia2onpassingthroughtheinterstellar
medium,andperhapsmostcommonlyfromthescaCeringofradia2on.
ü PropertyofawavetohaveitsElectricFieldoscilla2nginasingleplane(planepolarizedwave)orinarota2ngplane(ellip2callyorevencircularpolarizedwave).
Polariza2onofelectromagne2cradia2on
ü Polariza2onisproducedinvariousways,includingdirectlyfromsomeradia2onprocesses(e.g.cyclotronandsynchrotronemission),fromdifferen2alabsorp2onofradia2onpassingthroughtheinterstellarmedium,andperhapsmostcommonlyfromthescaCeringofradia2on.
ü Frac2onalpolariza2onsdetectedfromastronomicalobjectscanbeveryhigh(pulsars:almostfullylinearlypolarised)to,verylow(sun:oneofthemostsensi2vepolariza2onmeasurementsevermadewasbyJamesKempin1987,whoshowedthatthefrac2onallinearpolariza2onoflightfromtheSunwas~10-7)
ü Polarimetry,isamethodusedtostudythepolariza2onofincomingradia2onandcanprovidesubstan2alcluestothenatureofthesource.
ü Polarimetryisusedtoextractinforma2onsuchasthestrengthof
magne2cfieldsintheinterstellarmedium(ISM),provideevidenceforinfla2onbyobserva2onsoftheCMBpolariza2on,mo2vateaunifiedmodelforac2vegalac2cnuclei(AGN),probingemissiongeometryforpulsarsetc.
Polariza2onofelectromagne2cradia2on
Polariza2onofelectromagne2cradia2on
ü Studyofpolariza2onofelectromagne2cplanewavesfromastrophysicalsourcesandmodifica2onofthepolariza2oninthe
medium.ü Planewavesaredescribedbyoscilla2ngelectricandmagne2c
fields,whosefieldvectorsareorthogonaltoeachotherandthedirec2onofpropaga2on.
ü Byconven2on,astronomersdescribethepolariza2onoflightonlyintermsoftheelectricfieldvector(becauseEandBareorthogonal).
Maximumobservedorexpecteddegreeofpolariza2onfordifferentastronomicalobjects
TheseareapproximatenumbersMaynotbeupdated
Credit:AgnieszkaSłowikowska
Stokesparameters
StokesparametersweredefinedbyGeorgeGabrielStokesin1852,asamathema2callyconvenientalterna2vetothemorecommondescrip2onofincoherentorpar2allypolarizedradia2onintermsofitstotalintensity(I),(frac2onal)degreeofpolariza2on(p),andtheshapeparametersofthepolariza2onellipse
ü Thepolariza0oncanbedescribedbytheshapethatthe0pofEtracedoutoverthecourseofaperiod,anditcanbelinear,circular,orellip0cal.
Polariza2onofelectromagne2cradia2on
SpecificcaseWediscussedaboutmonochroma2cplanewave
Oscillatesalonga1Superposi2onoftwosuchoscilla2onsinperpendiculardirec2on
β
E1andE2arecomplexamplitudeandcanbewriCenas
Polariza2onofelectromagne2cradia2on
Superposi2onoftwosuchoscilla2onsinperpendiculardirec2on
β
ConsideringrealpartofE,physicalcomponentofelectricfieldsalongxandydirec2on
Theseequa2onsdescribeTipofEinx-yplane
Polariza2onofelectromagne2cradia2on
β
Figuretracedoutby2pofEisanellipse
Equa2onsdescribing2pofEinx-yplane
Equa2onsforageneralellipserela2vetoitsprincipalaxesx’andy’
Polariza2onofelectromagne2cradia2on
Superposi2onoftwosuchoscilla2onsinperpendiculardirec2on
Ellip2callyPolarized
Equa2onof2pofelectricfieldvectordeterminestypeofpolarisa2on
Ellip2callyPolarized
0<β<π/2 ClockwiseellipseRight-handedpolariza2on
-π/2<β<0 An2-ClockwiseellipseLe�-handedpolariza2on
β
Polariza2onofelectromagne2cradia2on
Twospecialcasesofellip2calpolariza2on
β=±π/4 Circular β=0,β=π/2 Linear
Right-handedcircularlypolarizedLe�handedcircularlypolarized
Polariza2onandstokesparameters
Thus
Polariza2onandstokesparameters
Thus
However,
Consider,
Polariza2onandstokesparameters
Stokesparameters
Polariza2onandstokesparametersStokesparameters
ü IisPropor2onaltointensityofwave(+ve)
ü Circularityparametermeasurera2osofaxesoftheellipse+veforRight-handedpolariza2on-veforle�handedpolariza2onV=0forlinearpolariza2on
ü Q / Umeasuresorienta2onofellipserela2vetox-axisQ=U=0forcircularpolariza2on
ValidforMonochroma2cwave
Polariza2onandstokesparametersQuasimonochroma2cwaves,
Degreeofpolariza2on,
FurtherreadingPoincareSphere:agraphicaltooltovisualizedifferenttypesofpolarizedradia2on
FurtherreadingMuellerMatrix:MethodfortransformingStokesparameters
EndofLecture4
Nextlecture:22ndAugust
Reference:RybickiLightmanChapter2,3
TopicofnextLecture:Radia2onfrommovingcharges(con2nued)(Chapter3ofRybicki&Lightman)Prepara2on:Lecture4