Lecture4–Basictheoryofradia2onfields&Radia2onfrommovingcharges

Date:19thAugust2019

Bhaswa2BhaCacharyya

ElectrodynamicsandRadia2veProcessesI

August-September2019

IUCAA-NCRAGraduateSchool

bhaswa2@ncra.2fr.res.in

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