Proof of Stefan Boltzmann Law
According to Planck's radiation law, the radiation density is\[I= \frac{2 h}{c^2} \int^{\infty}_0 \frac{f^3}{e^{\frac{hf}{kT}} -1} df=\frac{2 h}{c^2} \int^{\infty}_0 \frac{f^3}{1-e^{- \frac{hf}{kT}}} e^{-\frac{hf}{kT}} df\]
.Substitute
\[\frac{hf}{kT}
=x \rightarrow f=\frac{kT}{h} x \rightarrow df= \frac{kT}{h} dx\]
then the integral becomes\[\begin{equation} \begin{aligned} I &= \frac{2 h}{c^2} \frac{k^4T^4}{h^4} \int^{\infty}_0 \frac{x^3}{e^x-1}dx \\ &= \frac{2 h}{c^2} \frac{k^4T^4}{h^4} frac{2 \pi^4}{15} = \frac{2 \pi^5k^4}{15c^2h^3}T^4 \end{aligned} \end{equation}\]
where
\[\frac{2 \pi^5k^4}{15c^2h^3}=5.67 \times 10^{-8} W/m^2/K^4\]
is the Stefan Boltzmann constant.This is the Stefan Boltzmann Law (radiation radiated per unit area from a body is proprtional to the fourth power of temperature).