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Planck’s Law

for

Blackbody Radiation;

a Mathematica Project

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Physics—Mathematics ILAP

Interdisciplinary Lively Applications Project

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.

[pic] 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 for the 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 Society entitled, “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 that had 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 [pic]. (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 perfect radiator—it absorbs and then re-radiates all incident electromagnetic radiation, reflecting none. Interestingly, the spectral radiancy of a blackbody depends only 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

[pic]

where

[pic] = spectral radiancy at temperature T in watts per meter squared per meter (W/(m2 m))

h = Planck’s constant; 6.6260693 x 10-34 in 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 10-23 Joule per Kelvin (J/K)

λ = 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 [pic] for the above temperatures. Make sure to label your axes correctly, including units.

Tip: the wavelength range of interest is [pic].

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

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,

[pic],

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 and infinitesimally 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

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