Blackbody Radiation;
Planck’s Law
for
Blackbody Radiation;
a Mathematica Project
Physics—Mathematics ILAP
Interdisciplinary Lively Applications Project
Planck’s Law
Title:Planck’s Law for Blackbody Radiation; a Mathematica Project
Authors: Jerry McCoy, Dept of Physics and Engineering Physics
Donna Farrior, Dept of Math and Computer Sciences
The University of Tulsa
Mathematics
Classifications:Calculus II—MATH 2024
Disciplinary
Classification:Introductory Modern Physics—PHYS 2073
Prerequisite
Skills:Optimization, integration, curve fitting, graphing and Mathematica
Physical Concepts
Examined:Blackbody Radiation, Wein’s Displacement Law, Stefan’s Law, Cosmic Microwave Background Radiation
Materials Available:1.Problem statement and discussion (student)
2.Sample solution (instructor)
Computing Requirements:Mathematica
Class Requirements:Half class period for introduction
Students will work in teams of three
References:Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles, by Eisberg and Resnick, John Wiley & Sons publishers, 1985, p. 2 – 20.
“The Cosmic Microwave Background Spectrum from the Full COBE FIRAS Data Set,” by D. J. Fixsen, et. al., Astrophysical Journal, 1996, p. 473, 576.
Partial support for this work was provided by the National Science Foundation's Course, Curriculum, and Laboratory Improvement program under grant #0410653
Problem Statement and Discussion:
Quantum mechanics is the science of the very small. It is easily one of the most successful physical theories, accurately accounting forthe behavior of microscopic particles and integral to the design of much modern technology.
The birthday of quantum mechanics is dated December 14, 1900 when Max Planck read a paper before the German Physical Societyentitled, “On the Theory of the Energy Distribution Law of the Normal Spectrum.” In his paper, Planck correctly derived the spectral radiancy of blackbody radiation, a feat thathad eluded his contemporaries up to that point. This project concerns his result, now referred to as Planck’s Law.
Any object with temperature above absolute zero radiates energy as electromagnetic waves. (Electromagnetic waves include radio waves, TV waves, infrared radiation, visible light, ultraviolet light, X-rays, and gamma rays.) If the radiator is a solid, the energy is spread smoothly over a continuous spectrum of wavelengths. The distribution of the intensity of these waves versus their wavelength is called the body’s spectral radiancy and is denoted . (Intensity is the energy per second [power] transmitted through a unit area perpendicular to the wave. The eye perceives intensity in visible light as brightness and wavelength as color.)
A blackbody is a perfectradiator—it absorbs and then re-radiates all incident electromagnetic radiation, reflecting none. Interestingly, the spectral radiancy of a blackbody dependsonly on the body’s temperature and not on the material out of which it is made. It is the spectral radiancy of a blackbody that Planck was the first to correctly derive. Planck’s law is given by
where
= spectral radiancy at temperature T in watts per meter squared per meter (W/(m2 m))
h= Planck’s constant; 6.6260693 x 10-34in Joule seconds (J s)
c= speed of light; 2.99792458 x 108 in meters per second (m/s)
k= Boltzmann’s constant; 1.3806505 x 10-23Joule per Kelvin (J/K)
= wavelength in meters (m)
T= absolute temperature in Kelvins (K)
As an example, the graph on the front page is a plot of Planck’s law for a blackbody radiating at a temperature of 6000 K.
In this project, you will plot and describe Planck’s law for several temperatures. You will then derive two important laws from Planck’s law, Wein’s Displacement law and the Stefan-Boltzmann law. Finally, you will use your results to analyze cosmic microwave background radiation data taken by the Cosmic Background Explorer (COBE) satellite.
Do all your work within Mathematica. Make sure to use the Help Browser to aid in resolving difficulties. When you have completed the various steps, please clean up your Mathematica work for readability; that is, add comments, edit out unnecessary material, and put things in their proper order.
A.Spectral Radiancy of a Blackbody Radiator: understanding the characteristics of a blackbody spectrum
1.Overlay plots of the spectral radiancy of a blackbody radiator for absolute temperatures ranging from 3500 K to 6000 K in increments of 500 K (there should be six plots). That is, plot for the above temperatures. Make sure to label your axes correctly, including units.
Tip:the wavelength range of interest is .
Tip:you might consider using the Mathematica PlotLegend command to label the overlaid curves; to do so will require you to first load the graphics package with command
< Graphics`Legend`
2.Inspect the overlaid plots from #1; describe the effect of increasing temperature on the spectral radiancy curves.
Tip:there are two major trends that you should be able toidentify.
3.Based on your answer to #2,describe thevisual effect of increasing temperature on blackbody radiation. That is, describe the light you would see emitted by the blackbody as its temperature is increased. Can you think of an actual example of this effect? (The body doesn’t have to be a perfect blackbody.)
Tip:the visible range of the spectrum is ; you can’t see electromagnetic waves outside this range.
Tip:a wavelength of 4x10-7 m corresponds to violet light, while 7x10-7 m corresponds to red light, with all the colors of the rainbow in between in order: violet, indigo, blue, green, yellow, orange, red.
B.Wein’s Displacement Law: exploring the effect of temperature on the wavelength of maximum radiancy (m)
1.Note that each of the overlaid plots in part A has a fairly obvious peak. This peak is called the maximum radiancy for that plot; the wavelength at which this peak occurs is called the wavelength of maximum radiancy and is designated m.
Plot the wavelength of maximum radiancy versus temperature for each of the six temperatures you investigated in part A. That is, plot m(T) for the six temperatures; your plot should have six separate points on it. Make sure to label your axes correctly, including units.
Tip:to find m for, say, 3500 K, first define the spectral radiancy for 3500 K by substituting 3500 K in for T in . Then use Mathematica command“FindMaximum” to locate the peak:
FindMaximum[r3500[], {,0, min, max}]
If you supply the wavelength 0 at which to begin the search and the endpoints of the interval min and max that you want to search (get all these from the graph), then this function will numerically find the maximum radiancy and the wavelength at which it occurs. Repeat this for each of the six temperatures. Input the resulting ordered pairs (T, m) into Mathematica plotting function “ListPlot.”
2.Suppose you wish to derive a formula for the points you just plotted. That is, suppose you would like to derive a formula for , the wavelength of maximum radiancy as a function of temperature. (This equation is called “Wein’s Displacement Law.”) Describe the calculus procedure by which you could do this.
3.Now apply your answer to #2 to derive Wein’s Displacement Law, . Overlay a plot of the result on the graph in the previous step.
Tip:you’ll have to be very careful in carrying out your procedure from #2 or Mathematica will choke on the math.
First, you can reduce what Mathematica has to handle by defining a simplified version of Planck’s law in which the exponent is replaced by x:
Next, take the derivative of rx() with respect to and simplify the result using the Mathematica command “Simplify.” Even after simplification, the expression is still a rather nasty-looking fraction.
At this point, your response to #2 might lead you to set the nasty fraction equal to zero and then let Mathematica solve for m, but it is still too complicated. You can again reduce what Mathematica has to handle by recognizing that if a fraction is to equal zero, it can only be because the numerator equals zero; there is no value the denominator can have except that would cause the fraction to equal zero, and is not an allowed value. That is, if , then N = 0.
Therefore, use the Mathematica command “Numerator” to strip off the numerator from the nasty fraction, set this numerator equal to zero, then apply “NSolve” to numerically solve the resulting expression for x. Though Mathematica will spit out a warning message or two, it will ultimately yield a result for x which can then be set equal to and solved for m. After all this, you will have your equation for Wein’s Displacement Law that you can overlay on the previous graph.
Tip:you’ll know that you got it right if your plot of exactly overlays the points you plotted in #1.
4.State Wein’s Displacement Law in words. Describe how you could use this law to determine the temperature of a blackbody if you were given data for its spectral radiancy.
5.The spectral radiancy curve for a star shows that it is very nearly a blackbody radiator. The wavelength of maximum radiancy for our Sun is 5.10x10-7 m, which is squarely in the middle of visible spectrum. What is our Sun’s temperature?
6.If the human body radiated as a blackbody, what would beits wavelength of maximum radiancy? What portion of the electromagnetic spectrum is this in? (visible? ultraviolet? etc.)
Tip:most introductory physics textbooks will show the wavelengths associated with various parts of the electromagnetic spectrum; you can also find the information online.
C.Stefan’s Law: exploring the effect of temperature on total radiancy (RT)
1.As you (hopefully) noted in part A, the area under each of the overlaid plots of grows strongly with increasing temperature. The total area under each curve is the total radiancy RT of the blackbody for temperature T; it represents the total intensity radiated by the blackbody over all wavelengths at temperature T.
Plot the total radiancy RTversus temperature for each of the six temperatures you investigated in part A. That is, plot for the six temperatures; your plot should have six separate points on it. Make sure to label your axes correctly, including units.
Tip:the total radiancy may be found analytically (that is, you don’t need to do a numerical integration) with an improper integral over wavelengths from zero to infinity ().
2.Suppose you wish to derive a formula for the points you just plotted. That is, suppose you would like to derive a formula for , the total radiancy as a function of temperature. (This formula is called “Stefan’s Law.”) Describe the calculus procedure by which you could do this.
3.Apply your answer to #2 to derive Stefan’s Law, . Overlay a plot of the result on the graph in the previous step.
Tip:if you reduce the output of your integration using the “Simplify” command with the assumption that T > 0, you will get Stefan’s Law in simple form.
4.State Stefan’s Law in words.
5.A 40-W light bulb radiates from a tungsten filament operating at 3300 K. Assuming the bulb radiates like a blackbody, what percentage of the bulb’s power is radiated in the visible range? Are such lights an efficient means of producing light?
6.Using the temperature for the Sun that you found in #5 of part B, what is its power output over all wavelengths? (Assume that the Sun radiates uniformly in all directions.)
Tip:note that total radiancy is intensity, and intensity is power (P) divide by the area (A)of the blackbody radiator; that is,
,
where in this case, A is the surface area of the Sun. Data necessary to calculate the Sun's surface area may be found in most introductory physics texts or online.
D.Cosmic Microwave Background Radiation: exploring data from a real blackbody radiator
The Big Bang theory is the accepted scientific explanation of the beginnings of the universe. According to the theory, the entire universe exploded into being from an inconceivably hot and dense andinfinitesimally small collection of matter and radiation. This unimaginably extreme mixture expanded and cooled for as much as 14 billion years to become the present universe of stars and galaxies we see today. Though the Big Bang model may seem implausible, it is founded on solid, experimental evidence. Some of the strongest confirmation of the theory is the cosmic microwave background.
For the first half-million years after the creation instant, the Big Bang theory asserts that electromagnetic radiation was trapped within the very hot plasma comprising the expanding universe. At the end of this era, the universe had cooled sufficiently to release the trapped radiation, allowing it to stream freely throughout the cosmos. The Big Bang model predicts that today this radiation should permeate the universe, an “echo” of the creation event. In the eons since the Big Bang, the radiation should have cooled to a few Kelvins and be found largely as microwaves with a characteristic blackbody spectrum. This ancient light is referred to as the cosmic microwave background radiation.
The Big Bang prediction of cosmic microwave background radiation was verified experimentally in 1965 by two Bell Lab employees, Arno Penzias and Robert Wilson. For this accomplishment the pair was awarded the 1978 Nobel Prize in physics. In the early 1990’s, a NASA satellite called the Cosmic Background Explorer (COBE) was launched to collect data on the cosmic microwave background. As the final part of this project, you will analyze data from that mission to determine the temperature of the cosmic microwave background radiation. The data you are to analyze may be found in Mathematica notebook “cobedata.nb” on the ILAPs website: (The data was originally published in ApJ by Fixsen et al., 1996, supplied courtesy of David Leisawitz of NASA’s Goddard Space Flight Center, Greenbelt, MD.)
1.Plot the spectral radiancy of the COBE data making sure to label your axes correctly, including units. Comment on whether the spectrum looks like that of a blackbody radiator.
2.Use Wein’s law to estimate the temperature T of the cosmic background radiation.
3.Fit Planck’s law to the data. Overlay a plot of the resulting expression on the graph from #1. Use the resulting expression to determine a more precise value for the temperature of the radiation.
Tip:to fit Planck’s law with Mathematica, you must first load the statistics package with command
<Statistics`NonlinearFit`
Tip:use Mathematica command “NonlinearFit” to fit Planck’s law to the data:
NonlinearFit[data,model, , {T, T0}]
where
data is the COBE dataset
model is Planck’s law
T0 is a starting value for the estimation of T (use the value you found in #2)
- What is the total power incident on the Earth from the cosmic background radiation?
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