PHY143 LAB3 : BLACKBODYRADIATION

Introduction

Ablackbodyisdefinedasanobjectthatperfectlyabsorbsall(andthusreflectsnone)oftheradiationincidentonitssurface.Whenablackbodyisinthermalequilibriumwithitssurroundings,itmustalsobeaperfectemittersothatthetemperatureoftheblackbodystaysthesame.Butthisemittedlightisnotatthesamefrequencyasthelightthatwasinitiallyabsorbed;ratheritisdistributedbetweendifferentfrequenciesinacharacteristicpatterncalledtheblackbodyspectrum.

Themeasurementoftheblackbodyspectrumwasthecenterofacrisisinphysicsduringtheearly20thcenturyknownastheultravioletcatastrophe.Differentclassicalmodelscouldexplaintheblackbodyspectrumoversomefrequencyranges,butbrokedown(inonecasepredictinginfiniteradiationatsomefrequencies).MaxPlanckeventuallyresolvedthecrisisbyintroducingthequantizationofenergy,givingbirthtothequantumrevolutionintheprocess.

Inthislabyouwilluseanincandescentlightbulbandaprismspectrometertomeasuretheblackbodyspectrum.Althoughalightbulbisnotablackbody(itemitsmuchmoreradiationthanitabsorb!)itisagoodapproximationofagreybody:anobjectthatemitsafractionoftheblackbodyspectrumwiththesamefrequencydistribution.

Duetothisapproximationandthesimplicityoftheapparatus,yourintensitydatawillnotquantitativelymatchthatofablackbody,buttheshapeoftheintensitycurveshouldbequalitativelythesame.

THEORY

Planck’slawisderivedintheclasslectures(seepinknotesversion).HerewewilllookatthecorrespondencebetweenPlanck’sblackbodyfunctionandtheWienandRayleigh-Jeansfunctions,whichwerederivedindependently.Theyaregoodapproximations(forshortandlongwavelengthsrespectively)ofPlanck’slawforemittedpowerperunitareaperunitsolidangleperunitwavelength,whichis

𝐼 𝜆,𝑇 =2ℎ𝑐!

𝜆!1

𝑒!!!"# − 1

.

Wecanapproximatethisfunctionforsmall(short)wavelengths.Whenλissmall, !!!"#

islarge(≫ 1)

andthus𝑒!!!"# ≫ 1.Thuswecanapproximate𝐼(𝜆, 𝑡)as

𝐼 𝜆,𝑇 ≅ 𝐼!"#$ 𝜆,𝑇 =2ℎ𝑐!

𝜆!𝑒!

!!!"#

whichistheWienformula,validonlyforshortwavelengths.

Whatvaluesofλcanweconsidertobesufficientlysmall,e.g.for𝑇 = 5000𝐾?

Nowweapproximateforlongwavelengths.Whenλislarge, !!!"#

issmallandthus𝑒!!!"#iscloseto1.

Applyingalinearapproximationto𝑒!!!"#for !!

!"#about0,weget

𝑒!!!"# ≅ 1 +

ℎ𝑐𝜆𝑘𝑇

Puttingthisin𝐼(𝜆, 𝑡)yields

𝐼 𝜆,𝑇 ≅2ℎ𝑐!

𝜆!𝜆𝑘𝑇ℎ𝑐

=2𝑐𝑘𝑇𝜆!

whichistheRayleigh-Jeansformula,validonlyforlongwavelengths.

Whatvaluesofλarelargeenoughforthisapproximation,e.g.for𝑇 = 5000𝐾?

WecanusethePlanckfunctiontocalculatethewavelengthofmaximumintensityforagiventemperature.Wemaximizethefunctionbysettingitsderivativewithrespecttoλequaltozero,usingtheproductandchainrules:

𝜕I 𝜆,𝑇𝜕𝜆

= 2ℎ𝑐!1𝜆!

ℎ𝑐𝜆!𝑘𝑇

𝑒!!!"# 𝑒

!!!"# − 1

!!−5𝜆!

𝑒!!!"# − 1

!!= 0

Thisgivesus

ℎ𝑐𝜆𝑘𝑇

𝑒!!!"# = 5 𝑒

!!!"# − 1

Or,solvingnumerically,

𝜆𝑇 ≈ 0.2897768 𝑐𝑚 𝐾

ThisrelationshipbetweenthetemperatureandwavelengthofmaximumintensityisknownasWien’sdisplacementlaw.

ApparatusSetup

1)PlacetheSpectrophotometer(Rotarymotionsensor+bench+disk)ontheopticstrack.

2)AttachtheBroadSpectrumLightSensorandtheapertureplatetothearmofthespectrophotometerusingtheblackrod(imagebelow).PlugtheBroadSpectrumLightSensorintoAnalogChannelAontheScienceWorkshopinterface.

3)Placethefocusinglensonthespectrophotometerarminbetweenthelightsensorandtheprism,insideofthewhiteangledmarkings.

4)PluginthepowercableforthepoweramplifierandconnectitscabletoAnalogChannelContheScienceWorkshopinterface.

5)Placetheincandescentlampsourceonthetrackandconnecttothepoweramplifieroutputswiththebananaplugs.

6)AttachtheVoltageSensor(bananaplugsononeendandanalogchannelinputontheother)totheterminalsofthelampandAnalogChannelB.Youcanplugthebananaplugsintothebackoftheonescomingfromthepoweramplifier.Thiswillallowthecomputertomeasurethevoltageacrossthelampterminals.

7)Placethecollimatingslitholderandthenthecollimatinglensinfrontoftheincandescencelamp.Makesurethatthecollimatinglensisabout12cmfromthecollimatingslits.Thelampshouldslideintothebackofthecollimatingslitholder.Havesomeonewith20/20vision(correctedwithglassesisok)lookthroughthecollimatinglensattheslits.Adjustthecollimatinglensuntiltheslitsareinsharpfocus.Thecollimatinglensshouldbeabout10cmfromthecollimatingslits.

8)Movethespectrophotometerclosetothecollimatinglens,thefocusinglensshouldnowbeabout10cmfromthecollimatinglens.

• Howshouldyouchosewhichslittouseduringyourexperiment?Hint:boththecollimatingslitsandtheapertureslitsshouldbethesamenumber.Whataretheadvantagesanddisadvantagesofusingalargercollimatingslit?

9)OpentheblackbodyCapstonefileonthecomputer.OntheleftsideofthescreenclickHardwareSetup.OntheimageoftheScienceWorkshopinterfaceclickonAnalogChannelC.ScrolldownthelistandclickonPowerAmplifier.ClickHardwareSetupagaintoclosethemenu.

10)ClickSignalGeneratorontheleftsideofthescreen.TheboxnexttoAmplitudeishowyouchangethevoltage.ClickOntoturnontheincandescentlamp.Turningupthewillincreasethebrightness.Pleasedonotincreasethevoltageabove7voltsasitdrasticallydecreasesthelifeofthebulb.

11)PositiontheApertureBracketsothatyoucanseethethinbeamofwhitelight.MovethefocusinglenssothatyougetthemostinfocusbeamoflightontheBracket(Thisshouldbetowardstherearoftheangledbox).

• Shouldyousweepthroughasmallorlargeangletomaketheproceduremoreaccurate?

• Howwillambientlightaffectyourmeasurements?Whatarethesourcesofambientlightaroundyourexperiment,andhowcanyouminimizethem?

COMPUTERSETUP

1) OpentherotarysensorcalibrationCapstonefile.Thepurposeofthisprogramistodeterminetherelationshipbetweentherotationofthespectrometerarmandtherotationrecordedbytherotarysensor.

2) Click“Record”,thenrotatethespectrophotometerarmbetweentwodegreemarks.Ifthereadinggoesnegative,reversetherotarysensor’sconnectiontotheScienceWorkshopinterface.

3) Writedownthenumberofradianstherotarymotionsensorrotates(shownonthescreen)foryourgivenrotation.

4) Takethenumberofdegreesthatyourotatedthespectrophotometerarmanddivideitbythenumberofradiansthatyougot.Thenumberyoushouldgetshouldbearound0.96.

5) OpentheblackbodyCapstonefile.ClickonCalculatorfoundontheleftsideofthescreen.Online7,replacethenumber.9569withthenumberthatyougotinthepreviousstep.ClickAccept,thenclickCalculatoragain.

6) Movethesensorarmtoitsstartingposition(whereithitsthesideofthemount,sothatyoucanrepeatedlystartfromthesamepoint).

7) HitRecord.Beforemovingthesensorarm,hittheTAREbuttononthesensor.Thismustbedonepriortoeachrun.

8) Slowlymovethedetectorarmarounduntilitpassesthebrightreferenceband.

9) OntheAngleGraphwindow,findtheangle(inradians)ofthereferenceband.

10) ClickonCalculator.Online5replace68.9withtheangleyoufoundfromthestepabove.ClickonCalculatoragaintoclosethismenu.ThiscalibrationwillallowCapstonetocalculateanddisplaytheintensityasafunctionofwavelength.

11) DatarunsyounowtakewillhavecorrectlycalibratedIntensityvs.Wavelengthgraphs.YoumaynowclickontheBlackbodytabtostarttakingdata.

PROCEDURE

Usethespectrometertorecordtheblackbodyspectrumatfivedifferenttemperatures.Thetemperaturecanbesetbychangingvoltageoverthelightbulbfilament.Trytochoosetemperaturesthatgivenoticeablydifferentblackbodycurves.

FityourdatainIGORprotocalculatetheapproximatetemperatureofthefilamentforeachmeasurement.Findthewavelengthofpeakemission.DoesyourmeasurementagreewithWien'sLaw?

THINGS TO THINKABOUT

-Howshouldyoudecidewhatslitaperturesandsensorgaintouse?

-Whatarethesourcesoferrorintheexperimentalapparatus?

-CanyouqualitativelyexplainthecalculationthatCapstoneisdoingbehindthescenestoconvertanglesintowavelengths?Whatwasthepurposeoftheinitangleandtherotationsensorratio?