BLACKBODY RADIATION
PHYSICS 359E
INTRODUCTION
In this laboratory, you will make measurements intended to illustrate the Stefan-Boltzmann Law
for the total radiated power per unit area Itot (in W m−2 ) emitted by a blackbody at temperature
T to its surroundings
(1)
Itot = σSB T 4 ,
where σSB is the Stefan-Boltzmann constant. The Planck blackbody radiation Law for the dependence of monochromatic emission I(λ) on wavelength and temperature (energy radiated per second
per unit area per unit wavelength interval, radiated into one steradian of solid angle; W m−2 m−1
ster−1 ) is
2c2 h
.
(2)
I(λ) = 5
λ [exp(hc/kT λ) − 1]
From the data you will obtain during this experiment, you should be able to estimate both the
Stefan-Boltzmann constant σSB , and the constant hc/k. You will also learn something about the
detection of infrared radiation and the use of lock-in amplifiers.
PRE-LABORATORY PREPARATION
1. Familiarize yourself with the Stefan-Boltzmann Law, the Wien Displacement Law, and the
Blackbody Radiation Law (see Refs. 1 through 4; be sure to read at least one of these before class).
2. Look up the variation of the resistivity and emissivity of tungsten with temperature (see Ref.
5; the resistivity data are also available as a table and a graph from the course Web page).
3. Look over the manual for the lock-in amplifier to become familiar with the theory of its operation
and how to use it.
APPARATUS
To carry out this experiment you will need a blackbody, a detection system, and some device to
place between source and detector for wavelength selection.
Blackbody
Our approximation to a blackbody is a tungsten filament lamp powered by a stable current source.
Temperatures as high as 2500 K can be readily reached with such a lamp. (In what sense does
such a lamp differ from a true blackbody?). To obtain the temperature of the lamp, we measure
both the current Il flowing through it and the voltage Vl appearing across it. The resistance which
can be calculated from these values can be used to obtain the filament temperature, provided that
the resistivity of tungsten as a function of temperature is known, and provided that the value of
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the resistance of the lamp at some independently known temperature (usually room temperature)
is known.
Measurement of Vl and Il also allow one to calculate the electrical power being supplied to the
filament. Since the electrical input power will be equal to the net power radiated per second from
the filament (why?), we can use Vl and Il to determine the radiated power Itot emitted by the lamp.
Detection system
If we make measurements in the spectral region from 500 nm to 2500 nm as the temperature of
the lamp is increased to 2500 K, we will explore much of the blackbody curve. (You should verify
this statement using Wien’s displacement law). To do this may require two detectors: a silicon
photodiode which is sensitive in the visible and the near infrared (to about 1060 nm) spectral
regions, and a lead sulphide photoconductor which is sensitive to about 3000 nm. Use of the silicon
photodiode is straightforward: we let the light fall on it and measure the current that flows through
the diode.
The lead sulphide detector is a bit more complicated. The conductivity γ of the detector varies
with the amount of radiation I falling on it as
γ = γ1 + γ2 I.
(3)
For low light intensity, the second term in this equation is small compared to the first, and the
resistance of the lead sulphide detector will decrease linearly with I (convince yourself of this).
Consequently, the simplest way to use the lead sulphide detector is shown below, where the load
resistor R is chosen to be about equal to the dark resistance of the detector. The change in the
voltage V measured across the load, relative to the value for I = 0, is directly proportional to the
radiation intensity I (again, convince yourself).
However, if one measures the noise (i.e. fluctuations in the measured voltage V ) from such
a system as a function of frequency f , one finds that the noise grows dramatically towards low
frequencies (as 1/f ) as shown in the log–log plot below:
The presence of this characteristic ’1/f ’ noise at low frequency means that we could measure
smaller changes in V if the signal were modulated at a frequency high enough to avoid the noise.
We achieve this by putting a ’chopper’ in the beam to interrupt the light at a suitable frequency
f . (The best place to put the chopper is as close to the lamp as possible. Why?) The voltage
V across the load resistor will now be a tiny square wave at frequency f (superimposed on a DC
offset). We want to measure the amplitude of this small square wave.
For this we use a ’lock-in amplifier’. This is a very useful instrument whose main purpose is to
measure the amplitude of the fundamental component of a signal at frequency f within a bandwidth
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df while rejecting any input (i.e. noise) outside this bandwidth. To do this it needs a reference
signal at the required frequency, and the chopper just happens to provide such a signal. A valuable
characteristic of the lock-in is that it will remain locked on to the reference frequency even if it drifts
(slowly enough!) and will still measure the required amplitude. (See the manual for the lock-in for
a more detailed description of how it works. It can also be used for phase measurements.)
A noise source is provided to allow you to observe how the lock-in amplifier can significantly improve the signal-to-noise ratio (S/N) in the measurement of the amplitude of a periodic signal. You
can choose between ’white’ noise, line frequency noise (predominantly 60 Hz), and noise generated
by light sources in the lab such as fluorescent lights.
PROCEDURE
1. You will start by learning about the lock-in amplifier using artificially generated signals. Signals are generated by a Wavetek function generator, which can make a wide range of calibrated
waveforms for you at frequencies between mHz and MHz and with amplitudes down to mV. This
signal source is supplemented by a noise source: as the signal from the Wavetek is passed through
the noise source, noise is added to the signal. Use the oscilloscope to familiarize yourself with the
outputs obtainable from these two devices.
Now to explore the ability of the lock-in amplifier to detect weak signals buried in noise, we must
provide the lock-in with a reference frequency. The reference input of the lock-in requires a periodic
signal of amplitude between 0.2 and 5 V. The SYNC output on the back of the Wavetek function
generator is suitable (check this on the oscilloscope!). A light on the front panel of the lock-in will
go off, indicating that it has accepted the reference. Connect a sine wave output from the Wavetek
at the same frequency as the reference to the noise source and observe the SUM output directly
on the oscilloscope. Choose amplitudes such that the sum output will not exceed the maximum
signal input for the lock-in. Connect the sum output to the signal input of the lock-in. Adjust the
gain and phase as necessary and record the S/N of the output for various time constant settings.
Repeat this exercise for different input S/N values. You can also experiment with different noise
sources, such as the fluorescent lights, using the photodetector in the noise box. If you do this, try
tuning the frequency generated by the Wavetek slowly around a noise frequency and its harmonics
present in the source (e.g. 60Hz, 120 Hz,..) and observe the lock-in output. Tabulate the results
of your experiments and explain them.
2. Measure the room temperature resistance of the filament of the tungsten lamp. This is an important measurement since any error will lead to a systematic error in the inferred lamp temperature.
Since a digital multimeter puts very little current through the resistance, and thus heats it very
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little above room temperature, it is safe to use one to measure the room temperature resistance.
3. Measure the intensity of the lamp for at least 3 – 4 different wavelengths, using the filters
provided, at 8 or more different temperatures obtained by increasing the voltage and current to the
lamp. Record your measurements of both V and I (why?). Do a series of temperatures with a single
filter, then move on to the next filter. Be very careful not to move anything (the lamp, any lenses,
the filter, the detector) in the optical path while you are doing one temperature series (why?). In
setting up the apparatus you will have to decide how far to put the detectors from the lamp and
whether to use a lens to image the lamp onto the detector. The basis for your decision should be
whether you have enough signal at the lowest temperature without saturating the detector at the
highest temperature.
Do not exceed a voltage of 125 V on the bulb!
4. Estimate the area of the filament. You need this to get σSB . Note that the filament is not a
straight wire, but a tiny, tightly wound spiral coil.
ANALYSIS
1. Find a value for hc/k from your data. (With care, your value could be within 5% of the accepted
value – aim for this.) Try plotting ln[λ5 I(λ)] vs. 1/λT . (Should you use a series of data at one
value of λ, or at one value of T ? Why?) There should be a straight line whose slope will give you
hc/k. Why is there a straight line region? Will the data for all filters lie on the same line?
2. Verify the Stefan-Boltzmann Law by plotting Itot vs T 4 and obtain an approximate value for
σSB . Note that since the tungsten filament is not a perfect blackbody, the Stefan-Boltzmann Law
should be used in the modified form
Itot = eσSB (T 4 − T04 )
(4)
where e is the emissivity (e depends on temperature; its variation can be found in Ref. 5) and T0
is the room temperature (the T0 term includes in the net power radiated [Itot ] the power absorbed
by the lamp from the room).
REPORT
Include in your report:
• a summary of your experiments using the lock-in to detect weak periodic signals in the
presence of noise.
• an example of one of your data sets of I(λ) vs T , displayed graphically with suitable scales.
• your analysis leading to a value of hc/k (including a relevant graph as an example), with
an estimate of uncertainty; include an explanation of why the graph you use enables you to
determine hc/k.
• your analysis confirming the Stefan-Boltzmann Law (include a graph of Itot vs T 4 ), and a
discussion of your value for σSB , again with an error analysis.
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REFERENCES
1. D. Halliday, R. Resnick and K. S. Krane, “Physics”, extended version (John Wiley & Sons,
New York, 1992), Secs 49-1 and 49-2.
2. R. Eisberg and R. Resnick, “Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles” (John Wiley & Sons, New York, 1985), Ch. 1.
3. P.A. Tipler, “Modern Physics”, (Worth, New York, 1978), S3-4.
4. F.K. Richtmyer, E.H. Kennard, and J.N. Cooper, “Introduction to Modern Physics”, 6 ed.
(McGraw-Hill, 1969), Ch.5.
5. “Handbook of Chemistry and Physics” (CRC Press).
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