Upper Division Lab (Physics 341): BLACKBODY RADIATION
1.
Introduction
In this experiment, the spectrum of an incandescent light bulb is scanned by hand using a prism
spectrometer that measures relative light intensity as a function of angle. The spectrum is recorded with
the computer using the Data Studio software. A broadband light sensor is used with a prism so the
entire spectrum from approximately 400 nm to 2500 nm can be scanned continuously avoiding the
problem of overlapping diffraction orders prone to grating spectrometers. The wavelengths
corresponding to the angles are calculated using the equations for a prism spectrometer. The relative
light intensity can then be plotted as a function of wavelength as the spectrum is scanned, resulting in
the characteristic blackbody curve that you have studied in Optics and your Modern Physics courses.
Decreasing the intensity of the light bulb reduces the temperature, and the scan is repeated to show how
the curves nest with a shift in the peak wavelength (Wien’s Law). The temperature of the filament of
the bulb can be estimated indirectly by determining the resistance of the bulb from the measured
voltage and current. From the temperature, the theoretical peak wavelength can be calculated and
compared to the measured peak wavelength. Moreover, the black body radiation curves can be
compared with Planck’s radiation law and the temperature dependence can be compared to StefanBoltzmann’s Law.
2. Theory
The spectrum of light waves emitted from a hot, glowing object cannot be explained by the classical
theories of radiation and temperature introduced by Rayleigh and Jeans at the end of the 19th century.
Their “prediction” was that the intensity of the light emitted at a given wavelength should vary
inversely as the fourth power of the wavelength, which means that the highest intensity occurs in the
short wavelength or “ultraviolet” region. In fact, the predicted intensity grows without bound as the
wavelength goes to zero so all objects should appear to be blue no matter what their temperature, and
the total intensity of light emitted is infinite. Both these “predictions” are incorrect and this state of
affairs, called the Ultraviolet Catastrophe, persisted until Planck proposed the idea that light waves are
not continuous, but their energy is emitted in small “chunks” called quanta. Each quantum has an
energy E  hf , where h  6.626 1034 J  s is a fundamental constant of nature called Planck's constant.
Introducing this idea, Planck derived his famous distribution function (Planck’s Radiation Law) with
the intensity I of radiation emitted by a black body given by:
I   
2 c 2 h

1
5
e
hc
 kT
,
(1)
1
where c is the speed of light in a vacuum, h is Planck's constant, k is Boltzmann's constant, T is the
absolute temperature of the body, and  is the wavelength of the radiation. This distribution function
has the property that the graph of intensity vs. wavelength peaks at a wavelength that is inversely
proportional to the absolute temperature of the emitter, which is described by Wien Displacement Law:
max 
const 0.002898 m  K

,
T
T
(2)
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where T is the absolute temperature of the body and the constant is called Wien’s constant.
The total intensity of the light emitted per m2 of surface area, integrated over all wavelengths, is
proportional to the fourth power of the absolute temperature. This relationship is known as Stefan
Boltzmann’s Law:
I  T 4 ,
(3)
where   5.67 108
W
.
m ×K 4
2
2.1. Determining the Temperature of the Bulb
The temperature T of the blackbody light tungsten filament can be calculated (CRC Handbook, 45th
edition, page E-110) using the resistance of the filament at room temperature R0 . The resistance R of
the bulb is given by
R  R0 1   0 T  T0   .
(4)
where  0 is the thermal coefficient at room temperature and T0 is room temperature. The bulb filament
is made of tungsten, which has a coefficient of  0  4.5 103 / K at room temperature. Solving
equation (4) for the hot temperature T gives
T  T0 
R
1
R0
0
,
(5)
The bulb has an approximate resistance of R0  0.84  at room temperature. For a more exact value,
measure it yourself. You cannot measure the resistance of the bulb while it is still in the holder (why
not?). Solder wire leads to one bulb to ensure a good contact, and use this bulb to determine the
resistance with a multimeter. To find the resistance of the hot bulb, measure the voltage drop across
the bulb and the current passing through it. Use Ohm’s law, where V  IR . You can read off the
current supplied to the bulb directly from the front panel of the Agilent power supply. To measure the
voltage you can use a multimeter and measure across the black body source. The final equation for the
temperature (in Kelvin) of the bulb becomes
V
I 1
0.84
T  300 K 
4.5 103 / K
(6)
2.2 Determining the Wavelength from the Angle
The 60-degree prism is mounted so that the back face is perpendicular to the incoming light, as shown
in Figure 1. A schematic of the prism and the path of light are shown in Figure 2.
2
Figure 1: Experimental Setup of the Pasco Prism spectrometer.
Figure 2: Path of Light through the Prism.
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To measure the spectrum, we take advantage of the dispersion of the prism. In other words different
wavelengths passing through the prism get dispersed at different angles. We need to find which angle
corresponds to which wavelength. In the following, we will go through the analysis to find this
relationship. From Figure 2, since the prism angle is 60 degrees, it can be shown that
 2  3  60o
Using Snell’s law at each interface,
sin  60o   n sin  2  (*) and
sin    n sin 3   n sin  60o   2  ,
which becomes using double angle trig identities:


sin    n sin  60o  cos  2   sin  2  cos  60o  .
This expression can be manipulated to change from cos to sin using trig identities to read
1
1
sin    n sin  60o  1  sin 2 2  2  n sin 2  cos  60o   sin  60o  n2  n2 sin 2 2  2  n sin 2  cos  60o 
Using (*) we can write this as
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sin    sin  60o   n 2  sin 2  60o   2  sin  60o  cos  60o 
Solving for n yields:
2
 sin 

o
n  
 cos 60   sin 2 60o  , which can be simplified to
o

 sin 60

2
 2
1  3
n  
sin     
2  4 
 3
(7)
The dispersion relation (refractive index as a function of wavelength) for glass can be empirically
described by the Cauchy equation as
n   
A
2
B,
(8)
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where A and B depend on the type of glass. Experimental data for the glas prism used is given in
Table 1. A least squares fit to the data given in Table 1 (verify for extra points) to the Cauchy relation
(8) gives the values of A  13900 and B  1.689 . Solving (8) for the wavelength gives

A
nB
(9)
Thus the final equation for the wavelength (in
nanometers) is given by

13900
2
 2
1  3
sin       1.689

2  4 
 3
(10)
Thus measuring the angle with the rotary motion sensor
in In DataStudio, lets you calibrate measured angle 
values into wavelength . There is a way to do this
calibration in DataStudio using the Calculator Window.
Alternatively, you can calibrate your angle
measurements in Origin.
3. Equipment
PASCO spectrophotometer, IR sensor, Blackbody Light
Source with Light Aperture and Lens assemblies,
Agilent power supply, Digital Multimeter (DMM),
Science Workshop interface with Data Studio software.
The equipment should be already assembled for you and your instructor will show you how to make
the fine adjustments. Please refer to the Spectroscopy laboratory manual for details, for instance how to
calibrate the angular position readout of the sensor into actual angle.
4. Procedure
The PASCO prism spectrometer used in this experiment allows for the measurement of the intensity of
infrared radiation as a function of its wavelength. The source of the radiation is a light bulb that is
heated to a known temperature by adjusting the voltage V across its filament. The temperature can be
calculated from the resistance of the filament at room temperature, the voltage across the bulb and the
current through the bulb. The wavelength of the light is related to the angle at which it emerges from
the prism. However, since the index of refraction of the prism is a function of the wavelength of the
light, the relationship between wavelength and angle is somewhat complicated. In this experiment, the
corresponding equation has been determined for the spectrometer you are using and has been described
in the theory section before.
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4.1. Determination of Intensity as a Function of Wavelength.
The light intensity as a function of angle will be measured using the Data Studio program, for several
different voltages across the bulb in the range from 1-7 V. The computer interface has been set up for
you. Please also look at the instructions in the Spectroscopy manual for details. The procedure to make
a measurement is as follows:
Open the Data Studio program and set up the experiment just like you did for the Spectroscopy
experiment. Make sure the IR sensor and rotary motion sensors are selected.
Carefully rotate the “light sensor” arm on the spectrometer in a counter-clockwise direction until it hits
the stop on the angular indicator of the circular table. You will be finding a good position for placing
this stop after some experimentation. All data runs will start in this position so they can be compared
side by side later on. Set the voltage across the light bulb, using the power supply to a value in the
range from 1-7V, and record this value (double check whether you can trust the reading of the power
supply or whether you need to use the multimeter) as well as the current through the light bulb. One
person will start and stop the data run and view the resulting intensity vs. wavelength plot, while a
second person will “tare” the IR sensor and move the arm of the spectrometer to scan the spectrum.
Click on the green START button of Data Studio next to the timer at the top of the screen. This will
become the red STOP button, at which time the data taking can begin. Tare the IR sensor by pushing
its tare button (located on the top front of the sensor) while holding your hand in front of the lens to
prevent light from the bulb from reaching the sensor. This establishes the “zero intensity” IR level.
Next, uncover the lens and begin to rotate the arm of the spectrometer clockwise, slowly and steadily.
At first nothing may seem to be happening, but eventually the intensity level on the Intensity vs. angle
plot will begin to rise and then fall. Experiment where to start and stop your scan and place the stop on
the angular indicator of the circular table and record the corresponding angle value. You will always
start your experiment from that location. This entire wavelength-scan process should take about one
minute or so. Repeat the above process for at least four different light-bulb currents and voltages
(record the values in your lab notebook and calculate the temperature). After changing the bulb current
give the bulb some time to come to a steady temperature. Each lab partner should make at least one
scan.
If you are happy with your spectra, export the data into Origin for further analysis (look at the
Spectroscopy manual to show you how). At this point, data-taking is complete and the remainder of
the analysis can be done elsewhere.
5. Analysis
In Origin, convert your angular position values into angles, using the calibration procedure described in
the Spectroscopy lab. Then calibrate the x- axis to read actual wavelength in nm. It should cover a
range between 500-2500nm.
The data plots you made will look like the Planck distribution going to zero quickly at short
wavelengths and less quickly at long wavelengths. The location of the peak of the distribution should
be rather well determined, and the integral from 500-2500 nm should be a reasonably good (though not
entirely accurate) measure of the total output of the source.
(1) In Origin, compare your data to theory described by Planck’s radiation law. Since you do not know
the absolute normalization of your measurements, you will need to normalize your theory curve to
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one of your data points. Does the shape of your measured curve match the theoretical curve? Can
the bulb really be considered a black body?
(2) Does the peak shift toward shorter or longer wavelengths as the temperature is lowered? How does
the intensity change as the temperature is increased? Determine the constant in the Wien
Displacement Law by multiplying the wavelength at maximum IR output by the absolute
temperature, for all four runs (use the last graph that you printed, the one with all the curves, to do
this). Compare your values with the standard value of 2.898 10-3 mK . To what degree of precision
do these values agree? How did the color of the bulb change with temperature? How did the color
composition of the spectrum change with temperature? Considering the peak wavelengths, why is a
bulb’s filament red at low temperatures and white at high temperatures? At about what wavelength
is the peak wavelength of our Sun? What color is our Sun? Why?
(3) Check the Stefan-Boltzmann Law by integrating over your spectra from 500-2500 nm and dividing
by the fourth power of the measured temperature for each run (it is convenient to express the
temperature in units of thousands of degrees K to avoid small numbers). To what degree of
accuracy do the data follow the Stefan-Boltzmann Law?
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