Integrals

Exercise 5.12: The Stefanā€“Boltzmann constant

The Planck theory of thermal radiation tells us that in the (angular) frequency interval Ļ‰ to Ļ‰ + dĻ‰, a black body of unit area radiates electromagnetically an amount of thermal energy per second equal to I(Ļ‰) dĻ‰, where

I(\omega) = {\hbar\over4\pi^2c^2} \, {\omega^3\over(e^{\hbar\omega/k_BT}-1)}.

HereĀ \hbar is Planckā€™s constant over 2Ļ€, c is the speed of light, and kB is Boltzmannā€™s constant.

  1. a) Ā Show that the total energy per unit area radiated by a black body is

    2. W = {k_B^4 T^4\over4\pi^2c^2\hbar^3}\int_0^\infty {x^3\over e^x-1} dx.

  2. b) Ā Write a program to evaluate the integral in this expression. Explain what method you used, and how accurate you think your answer is.
  3. c) Ā Even before Planck gave his theory of thermal radiation around the turn of the 20th century, it was known that the total energy W given off by a black body per unit area per second followed Stefanā€™s law: W = ĻƒT4, where Ļƒ is the Stefanā€“Boltzmann constant. Use your value for the integral above to compute a value for the Stefanā€“Boltzmann constant (in SI units) to three significant figures. Check your result against the known value, which you can find in books or on-line. You should get good agreement.