# Integrals

Exercise 5.12: The Stefan–Boltzmann constant

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

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

Here $\hbar$ is Planck’s constant over 2π, c is the speed of light, and kB is Boltzmann’s constant.

1. a)  Show that the total energy per unit area radiated by a black body is
2. $W = {k_B^4 T^4\over4\pi^2c^2\hbar^3}\int_0^\infty {x^3\over e^x-1} dx.$

3. b)  Write a program to evaluate the integral in this expression. Explain what method you used, and how accurate you think your answer is.
4. c)  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 Stefan’s law: W = σT4, where σ is the Stefan–Boltzmann constant. Use your value for the integral above to compute a value for the Stefan–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.