phy143 lab 3: blackbody...
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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?
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