Dr. Cristian Bahrim
Laboratory#2 – Phys4480/5480 (Optics)
Blackbody radiation
In this experiment we will investigate the radiation emitted by a
glowing object. The experiment uses the phenomenon of dispersion of
light through a transparent prism (which is a dielectric material transparent
to visible light). The blackbody object is simulated by the filament of a light
bulb enclosed in a black cavity. The goal of this experiment is to determine
accurate experimental values of the Wien’s constant (of theoretical value
2.898 × 10 6 n m ⋅ K ) and of the Stefan-Boltzmann constant (of theoretical
(
)
value 5.67 × 10 −8 W m 2 ⋅ K 4 ). The setup is able to provide accurate values
of the surface temperature and of the flux density of glowing objects using
the wavelength at maximum radiancy. This technique enlarges the domain
of applicability of a light sensor.
Any object having a body temperature emits radiation. When the radiation
emitted by objects is analyzed, the reflected light by their surface should be
eliminated. In order to study the radiation which originates from the inside of an
object, a theoretical model called “blackbody” ( or “black cavity” ) at thermal
equilibrium can be used. Typically, a black cavity is considered as being a hole
in the walls of an empty metal box. We note that the blackbody is the hole itself
and not the box! Any radiation entering through the hole in the cavity will have a
negligible chance to exit. The blackbody radiation that one observes is formed
inside the black cavity and is due solely to the temperature of the object.
Figure 1 shows the radiancy of a blackbody object as a function of
wavelength at four different temperatures. The radiancy of an object can be
defined as the energy emitted per unit area, per wavelength, and per second by
a glowing object. Planck’s theory of radiation based on the photon concept
successfully reproduces these curves. The Planck’s formula for radiancy is
 c  8π
R(λ ) =   4
 4  λ

1
  hc 
   hc λkT

− 1
  λ  e
(1)
where c is the speed of light in free space, λ is the wavelength, h is Planck’s
constant, k is the Boltzmann constant, and T is absolute temperature [in Kelvin].
Figure 1 shows that the peak of the radiancy shifts toward smaller wavelengths
as the temperature increases. Wien generated an experimental relationship
between the wavelength at the maximum radiancy and the temperature of the
glowing object know as the Wien’s displacement law
λmaxT = 2.898 × 10 6 nm.K .
(2)
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This law can be found by taking the derivative of the Planck’s formula (1) with
respect to the wavelength.
In figure 1, the area below the curve of radiancy represents the flux
density of the light emitted by a glowing object at certain temperature, and is
related to the temperature by a formula known as the Stefan-Boltzmann law.
∞
∫
F = R(λ ) dλ = σ T 4
(3)
0
where σ is the Stefan-Boltzmann constant ( 5.6705 × 10 −8 W / m 2 K 4 ). The flux
density is an average measurement of the total amount of energy emitted
from each square meter of a light source per second. If the flux density is
multiplied by the area of the light source then we have the luminosity of a glowing
object, which is a useful quantity for astronomers, and also, it provides one of the
seven fundamental units of the international system of units, called candela.
2.5
Theorectical Blackbody Curves
2
2800 K
Radiancy(relative units)
2600 K
1.5
1
2300 K
1900 K
0.5
Wavelength (nm)
0
0
500
1000
1500
2000
2500
3000
3500
4000
4500
Figure 1
The theoretical radiancy emitted by a blackbody
versus the wavelength at several temperatures.
Experimental setup
The setup is presented in figure 2. An infrared (or broad spectrum) light sensor
attached to the light sensor arm is used to analyze the light emitted from a
blackbody-like light source. The light first passes through a set of collimating slits
and a lens before it reaches the prism. The light is then dispersed by the prism,
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focused by a lens and passes through an aperture slit before it reaches the light
sensor. A prism is mounted in the center of a degree plate (rotary table). A light
sensor arm is attached to the plate and it can be rotated so that the sensor
collects the entire spectrum of radiation dispersed by the prism. The angular
position of the arm is recorded by a rotary motion sensor which has a small
pinion in contact with the rotary table. The rotary motion sensor and the light
sensor transfer the data to a laptop computer via a PASCO Science Workshop
Interface (Model 750). Finally, the data is analyzed by the DataStudio™ software
(IM-BB 1999) to create accurate plots of the radiancy.
Figure 2 The PASCO setup for the study of the radiation
emitted by a blackbody. The path of the light is shown by
two thick arrows. In inset we show the position of the prism
with respect to the incoming and outgoing light.
Details about the theoretical model
You will use a DataStudio file already configured for measurements,
and also, detectors calibrated for proper analysis. You will need to plot the
radiancy versus wavelength (as in figure 1) for several temperatures (T)
and to find the wavelength λmax at which the radiancy reaches a maximum
value. This wavelength is a function of temperature as predicted by the Wien’s
displacement law (2). The temperature of the blackbody is varied by adjusting the
voltage supplied across it. A voltmeter and an ammeter are used to find the
temperature as a function of the supplied voltage and the current going through
the blackbody, as described below.
The resistance of the filament depends linearly on temperature as
R = R0 [1 + α (T − T0 )]
(4)
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where R is the resistance at temperature T and α is the temperature coefficient of
resistivity. Our light bulb, which is used as a blackbody, has α = 4.5 x 10 −3 K −1 .
The subscript “0” in (4) labels the parameters at room temperature (300 K). The
resistance at room temperature is R0 = 0.84 Ω . Solving equation (4) for T leads to
T = T0 +
R
−1
R0
(5)
α0
The resistance R can be found using Ohm’s law ( V = I R ):
R=
V
I
(6)
Both the voltage and current are measured using meters. Finally, the expression
for temperature is:
T = 300 K +
V
−1
0.84 I
(7)
4.5 × 10 −3
During this experiment, the temperature given by equation (7) is compared
with the value predicted by Wien’s law. The agreement is typically within 15%.
The blackbody radiation is incident on a 60 degree triangular prism (see figure 3).
Figure 3 The path of the
light through the prism.
By using Snell’s law at each face of the prism, one can show that the index of
refraction is given by the expression
 2
1
n = 
sin θ + 
2
 3
2
+
3
4
(8)
The Cauchy equation gives a relationship between the index of refraction and λ:
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n(λ ) =
A
λ2
+B
(9)
The coefficients A and B are specific to the dispersive medium. The coefficients
A and B will be derived in a later lab. For our acrylic prism these coefficients are
A=13,900 and B=1.689.
Solving equation (9) for wavelength gives
λ2 =
A
.
n−B
(10)
Substituting the index of refraction n from equation (8) into equation (10) leads to
the following equation which is necessary for getting the wavelength from the
angular position
λ=
13900
 2
1

sin θ + 
2
 3
(11)
2
+
3
− 1.689
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Remark: The entire domain of λ , from 1000 nm to1500 nm, of interest in our
study (see figure 4.a) corresponds to a change in θ (which is the table-angle) of
only 0.02 radians or 1.146 degrees (!!!) (see figures 4 below). Therefore, our
measurement demands a very precise calibration and alignment of the setup.
Figure 4.b An enlarged view of
the range for the wavelengths
used in this experiment.
Figure 4.a Experimental values of
wavelength versus the angle θ. The
region between the dashed lines
indicates the range of interest for
the present study: (1000-1500) nm.
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Details about the experimental procedure
Some special considerations need to be made related to formula (11). The
deviation angle θ from figure 3 is not the same with the angle measured by the
rotary motion sensor. There are two measurements which need to be done:
(1) The angular ratio which will be simply called “Ratio” represents the
ratio between the angle of rotation of a small pinion attached to the rotary motion
sensor and the angle of the degree plate:
Ratio =
pin angle
plate angle
(12)
Light
Sensor
θ
α
Stop
INIT
Initial
position
Prism
Figure 5 Diagram of the angles
used during the experiment. Note
that all angles are measured with
respect to the stop.
Incident
Light
(2) Figure 3 shows that the angle θ needed in equation (11) is measured
from the optical axis or the normal to the back side of the prism (where the light
exits). However, the angular position measured by the rotary motion sensor
(which is rotated clockwise during measurements) is the angle α as shown in
figure 5. We name INIT (see figure 5), the angle between the Stop (or the
initial position of the rotary arm) and the normal to the back face of the
prism. From figure 5 we see that θ = (INIT – α ). In figure 5 all the angles are
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measured with respect to the degree plate. However, the Data Studio software
measures the angles using the pinion of the rotary motion sensor. So, the actual
angle displayed by Data Studio is the pin-angle. Therefore, all angles should be
divided by Ratio (which is given in equation (12)):
θ=
INIT − α
Ratio
(13)
The value of INIT is critical for this experiment. Our study proved that the
success of the entire experiment depends on the accuracy of the INIT angle. In
order to have a good measurement, we need an angular precision that should
reach a hundredth of a radian. This difficulty will be overcome by using a
calibration technique for a particular voltage (7 volts) where the setup is the most
sensitive.
Experimental Steps
using DataStudio (DS) and an Excel spreadsheet.
Phase 1. Measuring the RATIO.
1. Turn on the PASCO interface, the Power Amplifier and the two meters. The
Power amplifier delivers dc currents. Set the voltmeter on scale DC V 20 and the
ammeter on DC A 20A.
2. Open the “Blackbody_Data_Studio.ds” file located on the Desktop.
3. Open the “Blackbody_spreadsheet.xls” file located on the Desktop.
4. Set the light sensor at 60 degrees away from the 0-180 degrees axis, placing it
opposite to your position with respect to the central axis (as indicated in figure 2).
5. Press Start in DS and look to the window “Angular position”. This window
displays the pin-angle in radians.
6. Record the pin-angle on the first Excel spreadsheet called “Ratio”. Go in steps of
5 degrees on the table-angle as indicated in the first column of the spreadsheet.
7. After reading all the angles from 0 down to 60 degrees, and calculating the pinangle in degrees open from “Tools”, the Data Analysis and Regression and
perform a linear regression of the Pin Angle (y) vs. Table Angle (x) and set the
intercept to zero (for that click “Set Constant is zero”).
Record the Ratio with uncertainty: ________±_________ degrees
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Phase 2. Measuring the INIT angle.
8. Rotate the arm of the light sensor counterclockwise until it stops. A STOP is
fixed under the rotary arm. Delete all the data runs in the Data Studio windows.
9. Access the window called ”Radiancy vs. Alpha” in DS. Enlarge the window.
For this step you will need to rotate the light sensor arm in the clockwise
direction from STOP until you align it with the prism and the light source.
How to do the measurement?
a. Turn ON the blackbody source: click ON in the “Signal Generator” window of
DS which is initially set up at 7 volts.
b. Check if you see a rainbow on the aperture bracket in front of the light sensor.
c. Press the START button on DS and before you start rotating the rotary arm,
TARE the light sensor. This will zero the plot for “Radiancy vs. Alphas”. The
Tare button is on the light sensor. This action eliminates the background
radiation from your measurements.
d. Rotate the light-sensor arm in the clockwise direction SMOOTHLY and
STEADLY until you get two peaks on the ”Radiancy vs. Alpha” window at
about 14 and 72 degrees.
e. After recording all the data press STOP and turn off the “Signal Generator”.
10. Using a cross-hair read the angles αmax = _______________ degrees
and
INIT = _______________ degrees
Remark: The peak at smaller angle represents αmax (see figure 5 and equation
(13) ) and the peak at larger angles represents INIT (equation (13)). This value of
INIT is just an approximated value. You will find a more precise value for INIT
later on.
11. Go on EXCEL and open the second spreadsheet called “Initial” (2nd page of the
Excel file). Introduce the voltage V and the current I measured by the two meters:
V = ___________ volts
I = ___________ amps
On the line called “run#1” you have to evaluate the temperature according to
equation (7) and calculate λpredicted according to the Wien’s law (2):
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Report:
λpredicted = _______________ degrees
12. On the “Initial” spreadsheet, type in the value for Ratio (from item 7) and αmax
(from item 10) in the first two columns. Next, in the 3rd column introduce a range
of angles with a step of 0.1 degrees near the estimated value for INIT from item
10. The first value should be the closest lower integer to the estimated INIT
(example: type in 73 for INIT = 73.6). Next, calculate λ in column 4 (according to
formula (11)). Adjust the value of the INIT angle until λ associated matches
the value of λpredicted (from item 11) within a precision of 0.1 nm. Typically, the
angle should have three decimals for the best agreement with λpredicted. This is the
most precise value of INIT that you can measure. This measurement for INIT
takes care of the imprecision in the alignment of the prism.
Record the INIT : ____________ degrees
13. Go on DS look under DATA (upper left window) for the formula “theta=…”.
Open a “Calculator” window (clicking on the formula for “theta =…”) and insert
the Variables INIT and RATIO. Next, press “Accept” on the “Calculator”. Look
to the changes in the plot “Radiancy vs. wavelength” (on DS) when these
variables are introduced. Close the “Calculator”.
14. In the window “Radiancy vs. wavelength” check with a cross-hair if the value of
λmax for the maximum radiancy at 7 volts agrees with the value of λ after the best
adjustment of INIT using the spreadsheet (item 12) within 1 %. If not, you have
first to check if the two meters still display the same numbers as when you have
recorded them (there is a chance that the readings has changed because the circuit
needed some time to reach thermal stability), and next, please check the formulas
for “theta” (given in equation (13)) and “wavelength” (given in equation (11))
which are typed in “Calculator”. If the problem still persists, you will need to restart collecting new data again.
Now you are ready to start collecting the data for measuring the Wien’s
constant and the Stephan-Boltzmann’s constant.
Phase 3. Measuring the Wien’s constant.
15. In the window called “Radiancy vs. wavelength” you have the experimental plot
for R=f(λ). Read λmax @ 7 volts measured with the cross-hairs.
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16. Open the “Data-Wien constant” spreadsheet and type in “run 1” the voltage and
the current indicated by meters, for 7 volts delivered by the “Signal Generator”,
and type in λmax @ 7 volts measured by DS.
17. Now you will start setting up the voltage on the “Signal Generator” at 4, 6, 8,
and 10 volts, taking one voltage at the time and collecting the data. For that
please start with the lowest voltage (4 volts). The rotation of the arm should be
done smoothly and steadily in a clockwise direction. After each run you have to
rotate back (counter-clockwise) the rotary arm until it reaches the STOP position.
BEFORE you start rotating the arm of the light sensor you need to TARE the light
sensor!!! For each voltage you need to collect the current and the voltage
read by meters and to measure λmax at maximum radiancy using the crosshairs tool on “Radiancy vs. wavelength”. Rotate the light sensor until the DS
plot reaches 4,000 nm (on the horizontal axis), and so it completely includes
the peak of the radiancy.
You collect four runs in addition to the measurement used for calibration
(run#1 @ 7 volts). After you set up a new voltage wait about 1 minute before
you will re-start collecting new data. The voltage and current should be
stable when you do the readings.
18. On the “Data-Wien constant” spreadsheet for each run you will evaluate the
temperature measured with meters and indicated as T=f(V,i). Applying the Wien’s
law (2) you will calculate λmax from the measured temperature (in the column “λ
- from T”). This represents the expected value for λmax at each T.
19. Using the data from column F called “λ - from DS” you need to calculate “T =
f(λ - DS)” based on the Wien’s law (2). The precision of this measurement for
getting λ is compared with “λ - from T”. You will see that the error is the largest
at 4 volts (and is expected to be about 20%). For the rest of the voltages, the
precision for λ should be within 10%.
20. Plot λ = f( 1/T ) using the DS values for λ and the 1/T calculated based on the
measurements using meters ( T = f(V,i) ). You should get ideally a straight line.
21. Run a “Linear regression” (using “Tools” - “Data analysis” in Excel) of λ
from DS vs. 1/T from f(V,i). The slope represents the Wien’s constant, which
should be about 5% of the theoretical value (which is 2,898,000 nm.K).
Report the Wien’s constant: (_______±_______) x 106 [nm.K]
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22. Estimate the percentage error between the theoretical and the experimental values
for the Wien’s constant. For that, copy your value of the Wien’s constant in the
cell next to “Report”.
Estimate the precision in measuring the Wien’s constant = ____%.
Phase 4. Measuring the Stephan-Boltzmann’s constant.
23. Go on “Planck Radiancy vs. Wavelength” (enlarge the window) and take each of
the five runs at the time. The window is set up so that you can run an integral of
the radiancy over λ. This represents “the flux density” defined in (3), and is
indicated graphically as the shaded area under the curve of radiancy. For getting
this area, you have to activate “Σ - Show selected statistics” – “Area”.
24. Go on the Excel spreadsheet called “Flux density” and write the temperature
measured with meters ( T = f(V,i) ) and next, calculate T4. The temperatures T
= f(V,i) should be copied from “Data – Wien constant” spreadsheet (Attention:
when you “Paste”, you have to select “Paste Special” and click “Values”).
25. For each run you need to update the value of the temperature (T) in the formula
for “planck radiancy=…” using the “Calculator”, and next, you need to calculate
the area below the curve of radiancy on “Planck Radiancy vs. Wavelength”.
26. Enter in the “Flux density” spreadsheet the area calculated with DS for each
run, and next, convert this area in units that will be cancelled by dλ [in nm]
from equation (3). This corresponds to column F (labeled “area*10^-9”).
27. The error bar can now be estimated by comparing the two areas. It should be
within a few percents.
28. Plot “Flux density” (given by DS) versus T4 (given by meters). For this plot
you need to have the data in order (i.e. the increasing order of temperatures: move
run#1 between runs #3 and #4).
29. Run a linear regression of the “Flux density” (column F) given by DS with
the T4 (column C). The agreement between theory and experiment should be
within 5%.
30. Estimate the deviation error (in percentage) between the experimental and the
theoretical Stephan-Boltzmann’s constant (which is ( 5.67 × 10 −8 W m 2 ⋅ K 4 ).
(
11
)
Report the Stephan-Boltzmann’s constant: (_____±_____) x 10-8 [W/m2.K4]
Estimate the precision in measuring the Stefan-Boltzman constant = _____ %.
You need to print out and show the four Excel spreadsheets
with the results and the plots you have done.
SETUP – BLACKBODY RADIATION
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