Thermal Radiation
Monty Mole
1234567
School of Physics and Astronomy
The University of Manchester
First Year Laboratory Report
Front page follows House
Style for layout and
presentation
April 2019
This experiment was performed in collaboration with Jock Wilson 7654321
Abstract
The Stephan-Boltzmann Law, the inverse square law of a radiating body, and
the absorption of radiation from such body were verified through a series of
experiments measuring values which could be tied to the relevant formulas.
Firstly, the Stephan-Boltzmann Law was confirmed for objects radiating at low
temperature, 130°C, by taking measurements of a Leslie cube. Then the same
law was confirmed for radiating bodies at high temperature, 1000-2500°C, by
taking measurements of the tungsten filament of a light bulb. Successively, the
inverse square law for radiation absorbed at increasing distance from a
radiating object was confirmed, by taking measurements of radiation absorbed
at various distances from the source. Finally, the absorption of radiation formula
with respect to how deep it is felt through an object was confirmed. All these
claims are supported by the graphs featured in this paper as they displayed a
visibly linear fit through the manipulation of the associated equations.
Good clear abstract, but you could make it
slightly shorter with some careful editing to
remove unnecessary words or information.
1. Introduction
In the study of radiation of a blackbody, the Stefan-Boltzmann Law is the
defining formula for all forms of such radiation. By studying the properties of
blackbody, or close to blackbody, radiation in action scientists have been able
to confirm theoretical properties of light emitting bodies. These properties have
long included the Stefan-Boltzmann Law, the inverse square law, and the
formula for absorption of radiation. Each of these can be analysed under similar
circumstances to confirm that their application to everyday objects. Indeed,
these laws apply to radiating sources we might interact with in our everyday life,
such as ovens, lightbulbs, and televisions. However, they also apply to objects
scientists would make measurements of to gain a better understanding of our
universe such as, radiation from the Sun, the absorption of radiation by the
Earth, and the flux at a distance from a star due to that star. Specific
applications include infrared thermography, in which the Stefan-Boltzaman Law
plays the crucial role fo identifying the relationship between small temperatures
changes and the emissive power at those points. [1]
You've tried to give some context in the introduction, but it's mostly at a very
low level for a scientific report. Also, try to link your introduction to the rest of
the report - add a sentence or two to say what you aim to do/discover.
2. Theory
This paragraph
would be better in
the introduction it's clear and
explains well the
context of your
experiment.
The scope of this experiment was to confirm four theoretical predictions of the
behaviour of a radiating body. These were the Stefan-Boltzmann Law at low
temperatures, that at high temperatures, the inverse square law, and the
absorption of radiation formula. The Stefan Boltzmann Law was the product of
the work of Josef Stefan who arrived at the equation by analysing data, and
Ludwig Boltzmann, who derived it theoretically [2]. The equation is as follows,
𝐻 = 𝐴𝑒𝜎𝑇 4 ,
(1)
where 𝐻 is the radiated power, 𝑒 is emissivity, 𝜎 is the Stefan-Boltzmann
constant, and 𝑇 is the absolute temperature. From this the net radiated power
is
𝐻 = 𝐴𝑒𝜎(𝑇 4 − 𝑇04 ) .
(2)
Therefore, by graphing the heat radiated against the difference in the
temperature to the fourth powers one expects to see a linear trend which would
confirm the relationship. The inverse square law was derived from the
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geometric understanding of flux from a source emitting equally to all directions
at a distance, 𝑑, from said source. It states that,
𝐼∝
1
𝑑2
,
(3)
where 𝐼 is the intensity at a distance, 𝑑. Therefore, as one moves the object
absorbing the radiation away from the source, one expects the intensity to drop
off by that distance squared. Finally, the Beer-Lambert Law [3], the formula
describing the absorption of radiation through a medium is,
𝐼 = 𝐼0 𝑒 −𝛼𝑑 .
(4)
Where 𝐼 is the intensity transmitted, 𝐼0 is the intensity of the source, 𝛼 is the
absorption constant, and 𝑑 is the thickness of a material. This could be
confirmed by graphing the log of the ratio of intensity against the number of
layers of PTFE tape each of equal thickness added onto the probe.
3. Experimental Procedure
With the scope of confirming the Stefan-Boltzamnn law at low temperatures,
measurements of resistance, which could be converted to temperature via the
thermistor calibration data table provided, were taken by plugging the Leslie
cube into a multi-meter; the temperature rose to about 130°C. As the Leslie
cube heated due to a thermistor, the resistance could be seen changing and
the heating could be temporarily stopped throughout the process to take
measurements of the two parameters. The second being, measurements of
voltage in the thermocouple probe, which could be converted to heat detected
via a conversion factor, were taken at various intervals throughout the heating
of the cube for all its faces. When it was deemed fit to take a set of
measurements of the faces of the Leslie cube, the heating was stopped
measurements were taken within a short time and resistance frame while
monitoring the resistance of the cube, which again corresponded to its
temperature. The set up for the Leslie cube was as seen in figure 1. The data
collected was then plotted and the line of best fit displayed to confirm its
linearity. The plot for the black matte face of the cube is displayed in figure 2.
Consider changing the order of your material - start with a
description of the equipment (figure 1) you used, then describe the
measurements you took.
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You need to
introduce the
reader to each
piece of equipment
before discussing
how it was used.
So 'a Leslie cube'
the first time, then
'the Leslie cube'.
Good clear figure
of your set-up.
Figure 1. [4] Set up of the Leslie cube, showing all its labelled faces and the thermopile
detector as used during the lab. The detector was moved closer to the source during data
collection and measurements of all four sides were taken quickly during the lab.
Good clear figure axes values can be
read clearly and
error bars on data
points are visible.
Figure 2. Plot of H vs. (𝑇 4 − 𝑇04 ) displays the relationship underlying the Stefan-Boltzmann
Law at low temperatures. It should be noted that the values of the x-axis are of the order 1010 .
You need to explain what the figure actually shows - some data points
and a blue line. You need to state that the line is the best-fit straight
line to the data, and then give the best fit parameters.
For the Stefan-Boltzmann Law at high temperatures, a similar procedure was
followed. The source was a tungsten filament lightbulb of which the temperature
could range from 1000-2500°C. Once again measurements of resistance
corresponding to temperature were taken by plugging the lightbulb into a multimeter. However, this time the resistance and its error had to be converted to
instantaneous resistivity of the filament. This was achieved by utilizing the
formula,
𝐿
𝑅 = 𝜌 𝐴,
(5)
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Think about how you order your material
here - the first issue is to measure
current and voltage in order to get
resistance. Then you convert the
resistance into a resistivity by using a
calibration scale.
Which data did you fit with polynomial
curves, what were your conclusions?
where 𝑅 is the resistance, 𝜌is the resistivity, 𝐿 is the length of the filament, and
𝐴 is the cross-sectional area of the current-carrying filament; to find 𝐿/𝐴. We
used the given value for resistivity at 400°K and the resistance of the filament
at that temperature. Resistance could not be measured directly or the filament;
therefore, measurements of current and voltage were taken and divided in
accordance to Ohm’s Law. The data was then fitted with several polynomial
curves to determine which had the best fit, and the result for heat vs.
temperature to the fourth power is shown in figure 3.
Figure 3. Plot of H vs. 𝑇 4 displays the relationship underlying the Stefan-Boltzmann Law at
high temperatures. It should be noted that the values of the x-axis are of the order 1012 .
To confirm the inverse square law, the probe was placed on a stand and a
meter stick was placed on the table surface to take measurements of heat
received at various distances. Expecting a linear relationship, the heat received
was platted against the inverse of the distance squared, as shown in figure 4.
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Figure 4. Plot of 𝐼 vs
1
𝑑2
showing the relationship between the intensity of the radiation and
the distance from the source.
Finally, the absorption of radiation formula was confirmed by adding successive
layers of PTFE tape to the probe opening and holding it to the lightbulb each
time to take a measurement. As it was noticed that the spot at which the
measurement was taken on the lightbulb varied, the same spot was used for all
measurements. In agreement with equation 4, which can be manipulated to
obtain,
ln 𝐼
𝑑
= −𝛼 ln 𝐼0 ,
(6)
the log of the intensity was plotted against the number of layers of tape on the
probe. This plot is shown in figure 5.
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Figure 5. Plot of ln 𝐼 vs 𝑑 showing the relationship between the intensity of radiation and the
distance through a medium.
4. Data Analysis
As the data consisted of the fittings provided by each graph, all the lines of best
fit were agreeably linear. The reduced 𝒳 2 values were all acceptable, being
between 0.5 and 2, except for the absorption of radiation plot for which more
data points are required to have the reduced 𝒳 2 value fall within the threshold.
It should however be mentioned that while the linearity of the plots remains
satisfactory, the number of degrees of freedom were at a satisfactory minimum
due to the number of data sets taken. The reduced 𝒳 2 values for the low
temperature Stefan-Boltzamnn curve is 0.80; that for the high temperature
curve is 1.90; the reduced 𝒳 2 value for the inverse square law linear fitting is
1.41; and that for the absorption of radiation formula fitting is 6.20.
Uncertainty values were very much acceptable with most of the error coming
from all hand measurements, such as the placing of the stand against the meter
stick; and the averages of values for probe measurements where applicable.
Errors were calculated through the partial derivatives of the equations.
5. Conclusion
By analysing the final data, the linearity of the curves, one can confirm that the
associated equation and law is confirmed. Having more data would be an
improvement on the experiment as it would allow for more degrees of freedom
in the plots leading to a better fit; however, the above shown graphs were found
to be satisfactory. Therefore, in agreement with the data, the Stefan-Boltzmann
Law for objects radiating at low temperatures and high temperatures, the
inverse square law, and the formula for absorption of radiation were confirmed.
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References
[1] Application of Infrared Thermography in Sports Science. Place of Publication
Not Identified: SPRINGER, 2019. p.36
[2] Application of Infrared Thermography in Sports Science. Place of Publication
Not Identified: SPRINGER, 2019.p.34-37
[3] Robinson, James W. Atomic Spectroscopy. New York: Marcel Dekker, 1996.
p. 27
[4] rswebsites.co.uk. AQA GCSE Physics Required Practicals. p. 7
[5] Blevin, W. R., and W. J. Brown. "A Precise Measurement of the StefanBoltzmann Constant." Metrologia 7, no. 1 (1971): 15-29. doi:10.1088/00261394/7/1/003.
The total number of words in this document is 1666.
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