This is an Applied Project from James Stewart's Calculus 6E - Instructor's Edition. It might take a long time, so don't get into it if you're in a hurry. Have fun.
Any object emits radiation when heated. A blackbody is a system that absorbs all the radiation that falls on it. For instance, a matte black surface or a large cavity with a small hole in its wall (like a blastfurnace) is a blackbody and emits blackbody radiation. Even the radiation from the sun is close to being a blackbody radiation.
Proposed in the late 19th century, the Rayleigh-Jeans Law expresses the energy density of blackbody radiation of wavelength as
where is measured in meters, is te temperature in kelvins , and is Boltzmann's constant. The Rayleigh-Jeans Law agrees with experimental measures for long wavelengths but disagrees drastically for short wavelengths. [The law predicts that as but experiments have shown that .] This fact is known as the ultraviolet catastrophe.
In 1900 Max Planck found a better model (known now as Planck's Law) for blackbody radiation:
where is measured in meters, is the temperature (in kelvins), and
speed of light
Boltzmann's constant .
I. Use L'Hôpital's Rule to show that
for Planck's Law. So this law models blackbody radiation better than the Rayleigh-Jeans Law for short wavelengths.
II. Use a Taylor polynomial to show that, for large wavelengths, Planck's Law gives approximately the same values as the Rayleigh-Jeans Law.
III. Graph as given by both laws on the same screen and comment on the similarities and differences. Use (the temperature of the sun (on surface)). (You may want to change from meters to the more convenient unit of micrometers: ).
IV. Use your graph in Problem 3 to estimate the value of for which is a maximum under Plack's Law.
V. Investigate how the graph of changes as varies. (Use Planck's Law.) In particular, graph for the stars Betelgeuse (), Procyon (), and Sirius () as well as the sun. How does the toal radiation emitted (the area under the curve) var with ? Use the graph to comment on why Sirius is known as a blue star and Betelgeuse as a red star.
Problem credit: James Stewart: _Calculus: _6th Edition, Instructor's Edition.