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
Here is Planck’s constant over 2π, c is the speed of light, and kB is Boltzmann’s constant.
- a) Show that the total energy per unit area radiated by a black body is
- b) Write a program to evaluate the integral in this expression. Explain what method you used, and how accurate you think your answer is.
- 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.