Plank’s Law for Blackbody Radiation;
A Mathematica Project
By
Group 1
Monther al-Sulaiman
Vanessa Andrews
Brad Beene
Nathan Buchanan
Submitted to Professor Donna Farrior
Calculus II Math 2073 Section 1
9 March 2005
Executive Summary
This project deals with the radiation of blackbodies, which are objects that absorb and re-radiate all incident electromagnetic radiation, reflecting none. The spectral radiancy of a blackbody depends only on the body’s temperature, which means that the radiance of an object of a given temperature can be given as a function of wavelength, known as Plank’s law.
We begin this project by plotting the radiance at several temperatures. We then use Plank’s law to derive Wein’s Displacement law, which gives the wavelength of maximum radiance as a function of temperature. This law can be used to calculate both the temperature of objects for which we know the wavelength of maximum radiance, such as the sun, and the wavelength of maximum radiance for objects for which we know the spectral radiance. We calculated the temperature of the sun to be approximately 5682 K and found that the maximum frequency of spectral radiance of the human body, were the human body to be a blackbody radiator, would be approximately 9.35 * 10-6 meters, which lies within the thermal infrared portion of the electromagnetic spectrum.
We then derived Stefan’s law, which states a formula for the total radiancy, which is the improper integral from 0 to infinity of spectral radiancy, as a function of temperature. We applied this formula to calculate the percentage of light radiated by a 40-W light bulb that is in the visible range by dividing the amount of light radiated in the visible range by the total amount of radiation and found that less than 12% of the light bulb’s light is radiated in the visible range. We also used Stefan’s law to estimate the power output of the Sun; as total radiancy equals power multiplied by the area, we multiplied the total radiancy by surface area of the sphere whose radius extends from the Sun to the Earth to estimate power output.
Finally, we investigated data collected by the Cosmic Background Explorer regarding cosmic microwave background radiation, one of the major pieces of evidence for the Big Bang. We calculated the temperature of this radiation to be approximately 2.76 Kelvin and showed that it does give off a characteristic blackbody spectrum. We also calculated the total power incident on the Earth from the cosmic background radiation to be approximately 1.68 * 106 Watts.
The methods that we describe in this report can be used for many other problems involving blackbody radiators.
References
[1] The Electromagnetic Spectrum. The University of Tennessee. http://csep10.phys.utk.edu/astr162/lect/light/spectrum.html
[2] The Earth, Sun, and Moon. ThinkQuest Library. http://library.thinkquest.org/29033/begin/earthsunmoon.htm
[3] Human Body Temperature Control. May Wong. The University of Sydney.
http://www3.fhs.usyd.edu.au/bio/homeostasis/Human_BodyTC_Pg01.htm
[4] Earth’s Mean Radius. Whatis.com. http://searchsmb.techtarget.com/sDefinition/0,,sid44_gci816253,00.html
Acknowledgements
The software package Mathematica, v. 5.0, by Wolfram Research, was used to complete this project.
Andrew Aguirre assisted with some portions of the Mathematica work.