PHY143LAB3:BLACKBODYRADIATION
Introduction
Ablackbodyisdefinedasanobjectthatperfectlyabsorbsall(andthusreflectsnone)ofthe
radiationincidentonitssurface.Whenablackbodyisinthermalequilibriumwithitssurroundings,
itmustalsobeaperfectemittersothatthetemperatureoftheblackbodystaysthesame.Butthis
emittedlightisnotatthesamefrequencyasthelightthatwasinitiallyabsorbed;ratheritis
distributedbetweendifferentfrequenciesinacharacteristicpatterncalledtheblackbodyspectrum.
Themeasurementoftheblackbodyspectrumwasthecenterofacrisisinphysicsduringthe
early20thcenturyknownastheultravioletcatastrophe.Differentclassicalmodelscouldexplain
theblackbodyspectrumoversomefrequencyranges,butbrokedown(inonecasepredicting
infiniteradiationatsomefrequencies).MaxPlanckeventuallyresolvedthecrisisbyintroducingthe
quantizationofenergy,givingbirthtothequantumrevolutionintheprocess.
Inthislabyouwilluseanincandescentlightbulbandaprismspectrometertomeasurethe
blackbodyspectrum.Althoughalightbulbisnotablackbody(itemitsmuchmoreradiationthanit
absorb!)itisagoodapproximationofagreybody:anobjectthatemitsafractionoftheblackbody
spectrumwiththesamefrequencydistribution.
Duetothisapproximationandthesimplicityoftheapparatus,yourintensitydatawillnot
quantitativelymatchthatofablackbody,buttheshapeoftheintensitycurveshouldbe
qualitativelythesame.
THEORY
Planck’slawisderivedintheclasslectures(seepinknotesversion).Herewewilllookat
thecorrespondencebetweenPlanck’sblackbodyfunctionandtheWienandRayleigh-Jeans
functions,whichwerederivedindependently.Theyaregoodapproximations(forshortandlong
wavelengthsrespectively)ofPlanck’slawforemittedpowerperunitareaperunitsolidangleper
unitwavelength,whichis
𝐼 𝜆, 𝑇 =
2ℎ𝑐 !
𝜆!
1
!!
𝑒 !"#
.
−1
Wecanapproximatethisfunctionforsmall(short)wavelengths.Whenλissmall,
andthus𝑒
!!
!"#
!!
!"#
islarge(≫ 1)
≫ 1.Thuswecanapproximate𝐼(𝜆, 𝑡)as
𝐼 𝜆, 𝑇 ≅ 𝐼!"#$ 𝜆, 𝑇 =
2ℎ𝑐 ! ! !!
𝑒 !"# 𝜆!
whichistheWienformula,validonlyforshortwavelengths.
Whatvaluesofλcanweconsidertobesufficientlysmall,e.g.for𝑇 = 5000𝐾?
Nowweapproximateforlongwavelengths.Whenλislarge,
!!
Applyingalinearapproximationto𝑒 !"# for
!!
!"#
!!
!!
!"#
!!
issmallandthus𝑒 !"# iscloseto1.
about0,weget
𝑒 !"# ≅ 1 +
ℎ𝑐
𝜆𝑘𝑇
Puttingthisin𝐼(𝜆, 𝑡)yields
𝐼 𝜆, 𝑇 ≅
2ℎ𝑐 ! 𝜆𝑘𝑇
2𝑐𝑘𝑇
= ! !
𝜆
ℎ𝑐
𝜆
whichistheRayleigh-Jeansformula,validonlyforlongwavelengths.
Whatvaluesofλarelargeenoughforthisapproximation,e.g.for𝑇 = 5000𝐾?
WecanusethePlanckfunctiontocalculatethewavelengthofmaximumintensityforagiven
temperature.Wemaximizethefunctionbysettingitsderivativewithrespecttoλequaltozero,
usingtheproductandchainrules:
!!
!!
𝜕I 𝜆, 𝑇
1 ℎ𝑐
= 2ℎ𝑐 ! ! !
𝑒 !"# 𝑒 !"# − 1
𝜕𝜆
𝜆 𝜆 𝑘𝑇
Thisgivesus
!!
−
!!
5
𝑒 !"# − 1
!
𝜆
!!
= 0
!!
ℎ𝑐 !!
𝑒 !"# = 5 𝑒 !"# − 1 𝜆𝑘𝑇
Or,solvingnumerically,
𝜆𝑇 ≈ 0.2897768 𝑐𝑚 𝐾
Thisrelationshipbetweenthetemperatureandwavelengthofmaximumintensityisknownas
Wien’sdisplacementlaw.
ApparatusSetup
1)PlacetheSpectrophotometer(Rotarymotionsensor+bench+disk)ontheopticstrack.
2)AttachtheBroadSpectrumLightSensorandtheapertureplatetothearmofthe
spectrophotometerusingtheblackrod(imagebelow).PlugtheBroadSpectrumLightSensorinto
AnalogChannelAontheScienceWorkshopinterface.
3)Placethefocusinglensonthespectrophotometerarminbetweenthelightsensorandtheprism,
insideofthewhiteangledmarkings.
4)PluginthepowercableforthepoweramplifierandconnectitscabletoAnalogChannelConthe
ScienceWorkshopinterface.
5)Placetheincandescentlampsourceonthetrackandconnecttothepoweramplifieroutputswith
thebananaplugs.
6)AttachtheVoltageSensor(bananaplugsononeendandanalogchannelinputontheother)to
theterminalsofthelampandAnalogChannelB.Youcanplugthebananaplugsintothebackofthe
onescomingfromthepoweramplifier.Thiswillallowthecomputertomeasurethevoltageacross
thelampterminals.
7)Placethecollimatingslitholderandthenthecollimatinglensinfrontoftheincandescencelamp.
Makesurethatthecollimatinglensisabout12cmfromthecollimatingslits.Thelampshouldslide
intothebackofthecollimatingslitholder.Havesomeonewith20/20vision(correctedwith
glassesisok)lookthroughthecollimatinglensattheslits.Adjustthecollimatinglensuntiltheslits
areinsharpfocus.Thecollimatinglensshouldbeabout10cmfromthecollimatingslits.
8)Movethespectrophotometerclosetothecollimatinglens,thefocusinglensshouldnowbeabout
10cmfromthecollimatinglens.
9)OpentheblackbodyCapstonefileonthecomputer.OntheleftsideofthescreenclickHardware
Setup.OntheimageoftheScienceWorkshopinterfaceclickonAnalogChannelC.Scrolldownthe
listandclickonPowerAmplifier.ClickHardwareSetupagaintoclosethemenu.
10)ClickSignalGeneratorontheleftsideofthescreen.TheboxnexttoAmplitudeishowyou
changethevoltage.ClickOntoturnontheincandescentlamp.Turningupthewillincreasethe
brightness.Pleasedonotincreasethevoltageabove7voltsasitdrasticallydecreasesthelifeofthe
bulb.
11)PositiontheApertureBracketsothatyoucanseethe
thinbeamofwhitelight.Movethefocusinglenssothat
yougetthemostinfocusbeamoflightontheBracket
(Thisshouldbetowardstherearoftheangledbox).
•
Howshouldyouchose
whichslittouseduring
yourexperiment?Hint:
boththecollimatingslits
andtheapertureslits
shouldbethesame
number.Whatarethe
advantagesand
disadvantagesofusinga
largercollimatingslit?
COMPUTERSETUP
1) OpentherotarysensorcalibrationCapstonefile.Thepurposeofthisprogramisto
determinetherelationshipbetweentherotationofthespectrometerarmandtherotation
recordedbytherotarysensor.
2) Click“Record”,thenrotatethespectrophotometerarm
• Shouldyousweepthrough
betweentwodegreemarks.Ifthereadinggoesnegative,
asmallorlargeangleto
reversetherotarysensor’sconnectiontothe
maketheproceduremore
ScienceWorkshopinterface.
accurate?
•
Howwillambientlight
3) Writedownthenumberofradianstherotarymotion
affectyour
sensorrotates(shownonthescreen)foryourgiven
measurements?Whatare
rotation.
thesourcesofambient
lightaroundyour
experiment,andhowcan
youminimizethem?
4) Takethenumberofdegreesthatyourotatedthespectrophotometerarmanddivideit
bythenumberofradiansthatyougot.Thenumberyoushouldgetshouldbearound
0.96.
5) OpentheblackbodyCapstonefile.ClickonCalculatorfoundontheleftsideofthe
screen.Online7,replacethenumber.9569withthenumberthatyougotinthe
previousstep.ClickAccept,thenclickCalculatoragain.
6) Movethesensorarmtoitsstartingposition(whereithitsthesideofthemount,sothat
youcanrepeatedlystartfromthesamepoint).
7) HitRecord.Beforemovingthesensorarm,hittheTAREbuttononthesensor.This
mustbedonepriortoeachrun.
8) Slowlymovethedetectorarmarounduntilitpassesthebrightreferenceband.
9) OntheAngleGraphwindow,findtheangle(inradians)ofthereferenceband.
10) ClickonCalculator.Online5replace68.9withtheangleyoufoundfromthestepabove.
ClickonCalculatoragaintoclosethismenu.ThiscalibrationwillallowCapstonetocalculate
anddisplaytheintensityasafunctionofwavelength.
11) DatarunsyounowtakewillhavecorrectlycalibratedIntensityvs.Wavelengthgraphs.
YoumaynowclickontheBlackbodytabtostarttakingdata.
PROCEDURE
Usethespectrometertorecordtheblackbodyspectrumat
fivedifferenttemperatures.Thetemperaturecanbesetby
changingvoltageoverthelightbulbfilament.Trytochoose
temperaturesthatgivenoticeablydifferentblackbody
curves.
FityourdatainIGORprotocalculatetheapproximate
temperatureofthefilamentforeachmeasurement.Findthe
wavelengthofpeakemission.Doesyourmeasurement
agreewithWien'sLaw?
THINGSTOTHINKABOUT
-Howshouldyoudecidewhatslitapertures
andsensorgaintouse?
-Whatarethesourcesoferrorinthe
experimentalapparatus?
-Canyouqualitativelyexplainthecalculation
thatCapstoneisdoingbehindthescenesto
convertanglesintowavelengths?Whatwas
thepurposeoftheinitangleandtherotation
sensorratio?