PHY143 LAB 3: BLACKBODY RADIATION Introduction A blackbody is defined as an object that perfectly absorbs all (and thus reflects none) of the radiation incident on its surface. When a blackbody is in thermal equilibrium with its surroundings, it must also be a perfect emitter so that the temperature of the blackbody stays the same. But this emitted light is not at the same frequency as the light that was initially absorbed; rather it is distributed between different frequencies in a characteristic pattern called the blackbody spectrum. The measurement of the blackbody spectrum was the center of a crisis in physics during the early 20th century known as the ultraviolet catastrophe. Different classical models could explain the blackbody spectrum over some frequency ranges, but broke down (in one case predicting infinite radiation at some frequencies). Max Planck eventually resolved the crisis by introducing the quantization of energy, giving birth to the quantum revolution in the process. In this lab you will use an incandescent light bulb and a prism spectrometer to measure the blackbody spectrum. Although a light bulb is not a blackbody (it emits much more radiation than it absorb!) it is a good approximation of a grey body: an object that emits a fraction of the blackbody spectrum with the same frequency distribution. Due to this approximation and the simplicity of the apparatus, your intensity data will not quantitatively match that of a blackbody, but the shape of the intensity curve should be qualitatively the same.
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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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