×

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 $$\lambda$$ as

$$f(\lambda)=\frac{8\pi kT}{\lambda^4}$$

where $$\lambda$$ is measured in meters, $$T$$ is te temperature in kelvins $$K$$, and $$k$$ 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 $$f(\lambda)\rightarrow\infty$$ as $$\lambda \rightarrow 0%+$$ but experiments have shown that $$f\lambda \rightarrow 0$$.] 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:

$$f(\lambda)=\frac{8\pi hc\lambda^{-5}}{e^{\frac{hc}{(\lambda KT)}}-1}$$

where $$\lambda$$ is measured in meters, $$T$$ is the temperature (in kelvins), and

$$h=$$ Planck's constant $$=$$$$6.6262\times 10^{-34}J\cdot s$$

$$c=$$ speed of light $$=2.997925\times 10^8m/s$$

$$k=$$ Boltzmann's constant $$=1.3807\times 10^{-23}J/K$$.

I. Use L'Hôpital's Rule to show that

$$\lim _{ \lambda\rightarrow0^+ }{ f(\lambda) }=0$$

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 $$f$$ as given by both laws on the same screen and comment on the similarities and differences. Use $$T=5700K$$ (the temperature of the sun (on surface)). (You may want to change from meters to the more convenient unit of micrometers: $$1\mu=10^{-6}m$$).

IV. Use your graph in Problem 3 to estimate the value of $$\lambda$$ for which $$f(\lambda)$$ is a maximum under Plack's Law.

V. Investigate how the graph of $$f$$ changes as $$T$$ varies. (Use Planck's Law.) In particular, graph $$f$$ for the stars Betelgeuse ($$T=3400K$$), Procyon ($$T=6400K$$), and Sirius ($$T=9200K$$) as well as the sun. How does the toal radiation emitted (the area under the curve) var with $$T$$? 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.