Black Body Radiation
Equipment
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2m Optical bench
large optical bench clamps (3)
Moll’s thermopile
sample thermopile housing
oven
oven insert
cooling jacket
Lab stand
• iris
• Protek TM1300 digital thermometer
• Variac
• Philips PM2535 multimeter
• calipers
• set of connecting leads
Figure 20.1: Equipment setup
• right angle clamp
Preperation
The student should review this lab write-up and the chapter in
the textbook on the historical development of Quantum Mechanics,
specifically those parts relating to Blackbody Radiation. In addition,
the student should ensure that they understand how a thermocouple
and thus a thermopile operates.
Goals of the Experiment
To measure the radiation incident on a detector observing a blackbody source as a function of the distance from the source and the
temperature of the source. To determine the total radiant energy from
the source and to examine the relationship between the temperature
of a black body and the energy of the radiation emitted.
Theory
Frequently, when an object is heated, it changes color and the color
of the object is a good indicator of how hot it is. The red hot element
on an electric range is cooler than the white hot element in a light bulb
but it is hotter than an automobile radiator that does not appear to
be glowing at all. By the 1800’s physicists were able to measure the
R=
Pt
.
A
Forward Hemisphere
radiation given off by a body and they found that it varies in a very
predictable manner that depends on the temperature of the body. It
was found that the intensity of radiation emitted from an ideal nonreflecting surface varies as the fourth power of its temperature. This relationship is now known as the Stefan-Boltzmann law. The idealized
nonreflecting surface that radiates only according to its temperature
is now known as a black body. A black body is an object that emits
electromagnetic radiation which depends only on the temperature of
the body.
The form of the law involving the fourth power greatly interested
physicists. Considerable work was done to understand why the StefanBoltzmann law had this particular form. No one was able to correctly
derive the behaviour of a black body until in 1900 Max Planck realized
that an extra fundamental assumption was required. This was the idea
that radiation is quantized and can only be emitted in discrete energy
packets. This insight was the beginning of quantum theory. So in
a sense, this experiment, which examines the radiation emitted by a
black body, is the originator of quantum theory.
It turns out that a hot furnace with a hole in it is a fairly good
approximation of a black body. The hole in the furnace acts like a radiator of electromagnetic energy. The size of the furnace determines
the amount of power coming out of the hole. The area of hole determines the size of the black body. The temperature of the furnace
determines the spectrum of the emitted radiation. Note that all three
of these properties of the furnace can be varied independently. A blast
furnace at a steel mill emits considerably more power than a pottery
kiln even though they both could be operating at the same temperature. In either furnace, a hole in the side could have different sizes no
matter what the temperature is.
Suppose the black body has area A and emits a total power Pt .
Then the radiance of the black body, R, is defined as the total power
radiated per unit area, so that
Detector (Area a)
r
(20.1)
The radiance of a black body follows the Stefan-Boltzmann law.
Therefore,
R = σT 4 ,
Source (Area A)
(20.2)
where T is the temperature of the body in Kelvins and σ is the StefanBoltzmann constant.
Suppose that the radiant energy from the hole in the furnace is uniformly distributed over the entire forward hemisphere. Also, assume
that the source area is small enough that the radiant energy being emit2
Figure 20.2: Detector Area
ted from it can be approximated as a point source. Then to measure
the black body radiation, a detector has to be placed somewhere inside this forward hemisphere. As seen in Figure Figure 20.2, clearly,
a detector of area, a, at some distance, r, from the oven will intercept
only a small fraction of the emitted power. The amount depends on
the size of the detector aperture as well as its distance from the oven.
Let the amount of power incident on the detector be Pi . Then if the
fact that the detector is flat instead of hemispherical is also ignored,
the amount of incident power is
a
Pt .
(20.3)
2πr2
Taking Equations 20.1 through 20.3 together yields the formula
Pi =
2πr2
P.
(20.4)
aA i
The apparatus is set up as shown in Figure 20.3. A furnace is
arranged at one end of an optical bench and a detector is arranged at
the other end.
R=
Cooling
Jacket
Oven
Figure 20.3: Experimental Setup
Iris
Thermopile
Optical
Bench
Variac
Digital
Thermometer
Troom
Multimeter
The detector outputs a voltage that is proportional to the incident
detected power. A multimeter is used to measure this voltage. The
digital multimeter is a sensitive instrument, and susceptible to thermal
voltages in its circuitry. Therefore, it is best to leave it on for about half
an hour before making observations. This allows the instrument to
reach a uniform operating temperature throughout.
The detector is called a Moll’s thermopile. It is a collection of
constantan-manganin thermocouples. Each thermocouple is a metal
junction in which a heat gradient induces a potential difference. The
conversion factor that relates detector voltage to incident power is
given on the apparatus. This factor takes the form of a number like
mV
. The entrance to the detector is shown in Figure 20.4. It is a
22 mW
polished metal funnel which has a 22◦ angle of aperture. A hole at the
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pointed end is the actual entrance to the detector and it has a diameter
of 0.01 m.
Thermopile
Figure 20.4: Dimensions of the
Moll’s Thermopile
θ
D
d=1cm
11◦
h
x
L
The thermopile is very sensitive to infrared radiation. Unfortunately, every physical body radiates thermal energy that can be detected by the thermopile, which affects the results of the experiment.
Once data taking commences, experimenters should avoid being anywhere where radiation from a body or clothing may be detected by the
thermopile. The apparatus should be located as far as possible from
any reflecting surfaces, both in front and to the side. For protection,
there is a glass cover on the detector. This must be removed before
taking measurements.
The radiation to be examined is that which emanates from the
cylindrical cavity of the oven insert, which is assumed to emit like
a black body. However, surfaces of the oven also radiate energy as
the oven heats up, which would contaminate the black body results.
To avoid this contamination, a cooling jacket is placed in front of the
oven, with its aperture centered on the oven opening. Cold tap water
circulating through the cold jacket keeps its’ temperature at about 10
◦ C, which should reduce surface radiation to insignificant levels.
The iris between the oven and the detector serves to limit the field
of view of the detector, ideally just to the aperture inside the cold
jacket. In reality this is difficult to achieve, but a good approximation would be to set the iris opening to be slightly smaller than that
of the detector funnel. To minimize losses in the detector cone due to
multiple reflections (each reflection causes a slight reduction in intensity), the detector should be far enough from the source that each ray
is reflected at most once. There is a minimum distance beyond which
the detector should be placed so that multiple reflections cannot occur.
This minimum distance can be determined from the geometry of the
funnel seen in Figure 20.4 and the diameters of the source and detector
apertures.
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Experimental Procedure
1. Check to see if the multimeter and the oven have been warmed up
prior to the lab period. The oven should be warmed up to at least
600 ◦ C. If this has not been done, turn on the multimeter and the
Variac, increase the Variac power to 85%, and go have a coffee! Also,
make sure that cold water is circulating through the cold jacket. It
will take about 50-60 minutes for the equipment to be ready, if it is
starting cold.
2. Find the calibration label on the thermopile that you will be using
during this experiment and record the calibration factor in your lab
notebook.
3. Using the vernier calipers, measure the openings in the cold jacket
and the funnel entrance. Set the detector at 40 cm from the source,
with the iris midway between them, and adjust the iris opening
to an appropriate size. Make sure all components are properly
aligned. Do NOT poke anything into the inner aperture of the
funnel. The thermopile detector is fragile and expensive and cannot
be disturbed.
4. To test the dependence of radiative intensity on distance from the
source, keep the oven temperature constant and take readings of
power (voltage) for varying detector positions, from 40 cm to 150
cm. Keep the iris at a constant distance from the detector, about 20
cm away. At least ten different distances should be measured.
5. Determine a suitable separation between the source and the detector to keep reflections in the funnel at one or less. To test the relationship of radiance and temperature, turn the oven off and take
measurements every 30 ◦ C from 600 ◦ C down to 300 ◦ C. Again,
keep the iris about 20 cm from the detector.
Error Analysis
Most of the measurement errors will be standard reading errors,
tempered with the judgment of the experimenter. The thermopile voltmeter combination introduces some significant uncertainties into
the calculations. Due to fluctuations in voltmeter readings, an error of
0.01 mV is suggested for the voltmeter. The student is free to investigate this and come up with alternatives, within reason. If the PASCO
thermopile is used, the absorbing surface occupies only 1/6 of the area
at the back of the funnel. This suggests that about 1/6 of the radiation incident on the funnel opening will be recorded by the voltmeter,
but warming of adjacent surfaces may increase this amount. Relative
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to these uncertainties, measurement errors in aperture sizes and oven
temperatures should be insignificant.
To be handed in to the laboratory instructor
Prelab
1. Find a literature value for the Stefan-Boltzmann constant present in
Equation 20.2.
2. What is the surface area of a hemisphere? Provide a derivation of
Equation 20.4, noting the assumptions that are needed.
3. Suppose that two quantities, x and y, are related by the formula
y=kxb , where k and b are constants. A formula of this form is
known as a power law. In some experiment, the parameter x is
varied and for each value of x a corresponding y value is measured.
Explain how to plot a graph of this power law data so that the points
appear on a straight line and the slope and intercept of this line can
be used to obtain values for the constants k and b.
4. From Figure 20.4, derive a formula for the angle θ knowing that the
cone angle is 22◦ and using any of the parameters D, d, h, x, and L.
Data Requirements
5. The calibration factor for your thermopile.
6. Measurements of aperture sizes and calculated areas, with uncertainties. A calculation of the minimum distance from source to detector to satisfy the single reflection requirement.
7. A table of values for voltmeter readings as a function of sourcedetector separation, and an appropriately linearized graph. Note
the oven temperature. Find the equation of the regression line and
comment. Include uncertainties in all parameters.
8. For decreasing temperature, a table of oven temperature, voltage,
incident power on the thermopile (using the given conversion factor), and a graph of Pi versus T4 , with error bars. Note the distance
of the detector from the source. Find the equation of the graph,
with uncertainties.
9. Let the intercept of the graph above be P0 . Draw a graph of some
function of Pi and P0 versus some function of T such that the power
law data in the previous graph appears as a straight line and find
the slope. What does this graph imply? What does the slope of the
graph imply?
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10. From Equation 20.4 and your graphs, determine a value of the
Stefan-Boltzmann constant.
Discussion
11. How does your value for the Stefan-Boltzmann constant compare
with the accepted value that you found in Prelab exercise 1?
12. Discuss the accuracy and significance of your results, making ample
reference to sources of error in the lab.
13. Explain how your source separation graph reflects on the assumption that the black body is acting like a point source radiator. Explain whether your results support or refute the assumption that
the black body acts like a point source.
14. Discuss any one of the following items.
(a) What applications in Physics or Astrophysics can you suggest
that utilize the general result of this laboratory, specifically Equation 20.2.
(b) The black body temperature of the cloudless night sky background is about 250 K. Explain what this statement means.
(c) The black body temperature of the background radiation of space
is about 3 K. Explain what is meant by this statement.
(d) An angle is some portion of an arc of a circle of radius 1, so that
a full circle has an angle of 2π radians. Similarly, a solid angle
is some portion of the surface of a sphere of radius 1, so that a
full sphere has a solid angle of 4π steradians. Compute the solid
angle subtended by the detector in this experiment.
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