Studyoftheblackbodyradiation
1. Purpose:
Analyze the blackbody radiation coming from different surfaces and verify the StefanBoltzmannlaw[1].Theemittedradiationismeasuredwitharadiometer.
2. Apparatus:
RadiometerEG-45Daedalon
Leslie’scubeandthermometerEH-10Daedalon
Stefan-BoltzmannsourceEH-15Daedalonwithpowersupplyequippedwithacurrentlimiter.
TwomultimeterstomeasurethecurrentandvoltageoftheEH-15.
3. Descriptionofexperiment:
Measurement of radiant energy is vital, among other, to the understanding of the energy
balance of the earth and its atmosphere. All objects emit electromagnetic radiation with
intensity that depends on the absolute temperature of their surface. At the same time, all
objects absorb energy from their surrounding such that at equilibrium temperature, the
absorbed and emitted energies are equal. A radiating hot body is said to emit “black body
radiation”. However, a black body is an ideal, theoretical object able to absorb the entire
energyfallingonitssurface.Inreality,allobjectspartiallyreflecttheincomingradiation.The
radiated power (integrated over all frequencies) is increasing with the increase of body’s
temperature.SuchdependenceisgivenbytheStefan-Boltzmannlaw[1]:
P(T)=εσT4(1)
whereP(T)istheirradiance, σ=0.56686·10-7W/(m2K4)isknownasStefan’sconstantand εis
theemissivityparameter(ε=1foranidealblackbodyand ε<1forrealobjects).Theirradiance
representsthepowerperunitareaandisthereforemeasuredinW/m2.
Inthisexperiment,theirradianceemittedbyvariousobjectsismeasuredwithradiometerEG45capabletointegrateawiderangeoffrequencies,fromthewarmthofahandprintleftona
table to direct sunlight. The radiometer is an evaporated thin film resistor array known as a
“thermopile”.It’smadeoftungsten(W)whichresistancevarieswithtemperature.Tungsten’s
specific heat is very small (~30 times smaller than water specific heat) so the instrument
respondsveryquicklytoatemperaturechange.Thedetectorisenclosedinasmallcasefilled
withArhavingawindowmadeofpotassiumbromide(KBr)whichallowsradiationtransmission
even in the far infrared region. The ensemble is mounted in a massive aluminum housing to
maintain thermal equilibrium. The detector measures the radiation balance between the
source and its own temperature, so it is important to keep the detector at constant
temperature during the entire experiment. The entrance aperture is conical with a 60°
acceptanceangleandisclosedwithshutter(mountedandthermalizedonthesamealuminum
enclosure).Theshutteristobeopenedonlyduringmeasurementsoftheirradianceincoming
throughthecone.TheRANGEswitchindicatesthefull-scalevalue,soareadingof10onRange
100means100W/m2(read[3],[4]and[5]!).
1
WewillverifytwoaspectsofStefan-Boltzmannlaw,namelythetemperaturedependenceand
thevariationofεfromobjecttoobject[2].First,oneusestheStefan-Boltzmann(S-B)source
EH-15toverifytheT4lawandnext,theLeslie’scubetostudytheemissivityofvarioussurfaces.
PART1. The radiation from a hot W filament is measured with the EG-45 radiometer as a
function of filament temperature T (see Fig. 1 from [4]). To measure the temperature, one
measuresthefilamentresistanceandthenusesitstemperaturedependence:
R(T)=R0[1+α(T-T0)](2)
where α=0.0045 K-1 while R0 and T0 are the room temperature resistance and temperature,
respectively(solveforT!).TheresistanceRisfoundusinga4wireconfigurationtoreducethe
effect of contact resistances. In this part, one studies the temperature and distance
dependenceoftheradiatedpower.
Fig.1.ExperimentalsetuptomeasureStefan-Boltzmannlaw.
PART2. The Leslie’s cube provides a way to measure the emission from 4 different faces of a
cubeandcomparetheiremissivity(seeFig.2from[5]).Thecubeisfilledwithwarmwaterto
ensureidenticaltemperaturesonallfaces.
Fig.2.Experimentalsetuptomeasuretheemissivityof4differentfacesofLeslie’scube.
Inthebackgroundsectionofyourreport,givetheerrorpropagationanalysis.
4. Measurementprocedure:
2
PART1.
4.1 TurnONtheradiometerandsetitonastablesupportfacingtheS-Bsource.Measure
the distance from the filament to detector (located in the same plane as the shutter control
rod)andadjustitto10cm.
4.2 IMPORTANT:makesurethepowersupplyhastheoutputturnedtozero!Otherwisethe
lampwillburnoutimmediately;NEVERexceed1.7Ampsinthefilamentcircuit!TurntheVandI
limiterknobsfullytozero(anticw),shortcutthe+and–outputswithabananacable,turnon
thepowersupply,turntheVknobjustfewdegreescwtoincreaseabitthevoltagelimit.Then
increasethecurrentlimittillyouread1.7Aonthepowersupplydisplay.YouDON’Tchangethe
currentlimiterfromhereon!!ConnecttheS-Bsourcepowersupplyandmultimeters(Fig.1).
4.3 Darkentheroom(ifpossible),closetheshutterandzerothemeterwiththeZEROAdjust
knobonRange1.Dothiseverytimeyouchangethescaleandcheckeditfromtimetotime.
4.4
Opentheshutterandnotethereading,iflargerthan0.2findthesourceofradiation.
4.5 Measurethefilamentresistanceatlowcurrent/voltage(tensofmAshouldcorrespond
to~30mV).ThiswillbeR0andyouhavetomeasureitverycarefully.MeasureT0oftheroom.
4.6
Checktheradiometerzero.Opentheshutterandrecordthesignal.
4.7 Take~20datapoints(V,I,PowerP)byincreasingthevoltage.Showspectrafor~1Aand
~1.3 A; discuss their differences (use the ASEQ spectrometer manual). Do NOT overpass 1.7
Amps.Checkthezeroeverytimeyouhavetochangescale.Setcurrentto1.4Aandgradually
increasethedistancedfilament–detector.Take~20datapoints(V,I,P,d).
PART2
4.8 Makesureyourhandsareclean.Takethecubeoutoftheplasticbag.NEVERtouchthe
polishedbrasssurface.Inspectthesurfacesandifyoufindthemdirty,cleanthemcarefully.
4.9 Arrange the cube as shown in Fig. 2. Fill it with hot water (~60-70° C). You have to
performtheexperimentinashortamountoftimesuchthatthetemperatureisconstant.You
willmeasureandnotewatertemperatureatthebeginningandtheendoftheexperiment.
4.9 Position each of the four faces in front of the radiometer and measure the irradiance.
Checkandadjustthezeroifneedisbeforeeachmeasurement.
5. Analysis:
PART1
5.1 For each data point you should present the Voltage, Current, Resistance, Temperature
and Irradiance values (V,I,R,T,P), each with its own uncertainty (so 10 columns). Break the
informationin2-3tables.ForinstancegiveatableforV,I,Rtogetherwiththeiruncertainties.
GiveanothertableforP,itsuncertainty,TanditsuncertaintytogetherwithacolumnforT4.
5.2 PlotPvsT4.Whyisalinearfitadequate?Usingtheslopemanditsuncertainty,estimate
an effective radius Rf of the filament and its uncertainty. Assume ε=1. All the power (that is,
irradiance σT4 times surface) of the ~spherical filament exits through a surface 4πRf2 and
reaches another spherical surface 4πd2 with d the filament-detector distance. Express the
3
irradianceseenbydetectorandthustheslopemasafunctionofσ,Rfandd.
5.3 MakeatableforlogP,logTandtheuncertaintyoflogP(explainhowyoucalculateit!).
PlotlogPvslogT.Discusstheshapeoftheplot,isalinearfitadequate?Extractthevalueofthe
slopeanditsuncertainty.Compareittoyourtheoreticalexpectation.
5.4 Asyoucansee,Pisdecreasingwhenincreasingd.Whatisthetheoreticaldecreasethat
youexpect?Knowingthat,howshouldyouplotthisdecreasesuchthatyouobtainalinearfit
(anddotheplot).Usingtheslopem’anditsuncertainty,estimateaneffectiveradiusRfofthe
filamentanditsuncertainty(remember5.2!).ComparetheresultwithRfcalculatedat5.2.
PART2
5.5 Tabulatethevaluesofirradiancevsthetypeofsurface(blackpaint,whitepaint,rough
brass,polishedbrass).Explainwhyyouseedifferentvalues.
5.6 Assume that the black painted side approximates a blackbody emitter and find the
relativeemissivitiesofthethreesides.Compareyourresultswithexpectedvalues(trytofind
adequateliterature).Forinstance,thepolishedbrassvalueshouldbearound0.03.
6.Questions:
6.1
WhatistheEarth’sgreenhouseeffect?
6.2WhatisthesolarirradianceattheentranceinEarth’satmosphere?(readthePDFs!!)
6.3 Give the formula of the blackbody spectra, also known as Planck’s law, as P(λ). Plot
ln(P(λ))vs λfortwotemperatures(sayT=1000Kand3000Kfor λfrom0.2to9µm);youmay
needtolimitthe λaxisinyourplots.Findthevaluesof λwhereP(λ)ismaximum.Bymeansof
firstderivative,describethedemonstrationofWien’slawandverifyitagainstthetwomaxima
thatyoufound.
6.4 Explain which of the two plots (T4 and log-log) is more robust against an incorrect
irradiancezerooffset?(hint:supposeanoffsetpisaddedtotherighthandsideofEq.(1)).
6.5Whyisimportanttohaveuniformtemperatureinsidethecube?Whywaterisbetterthan
justroomtemperatureair?
7.References
[1]ModernPhysicsforScientistsandEngineers,byS.T.ThorntonandA.Rex,4thEd.,Cengage
Learning(2012),Chapter3.5.
[2]TheArtofExperimentalPhysics,byDarylW.PrestonandEricR.Dietz,JohnWileyandSons
(1991),Chapter8.
[3]EG-45Radiometerinstructionmanual,PDFavailableonline.
[4]EH-15Stefan-BoltzmannSourceinstructionmanual,PDFavailableonline.
[5]EH-10Leslie’scubeinstructionmanual,PDFavailableonline.
4