Module 5 : MODERN PHYSICS
Lecture 23 : Black Body Radiation
  Stefan's Law
  The power radiated by the black body per unit area is
$\displaystyle U$ $\displaystyle =$ $\displaystyle \int_0^\infty I(\lambda) d\lambda$
$\displaystyle =$ $\displaystyle 2\pi hc^2\int_0^\infty \frac{1}{\lambda^5}\frac{1} {\exp(hc\beta/\lambda )-1}d\lambda$
  To evaluate the integral, substitute $ x =hc\beta/\lambda$, so that $ dx = -hc\beta/\lambda^2 d\lambda$. We get
 
$\displaystyle U = 2\pi hc^2 \frac{1}{(hc\beta)^4}\int_0^\infty \frac{x^3}{e^x-1}dx$
  The value of the integral $ \int_0^\infty x^3/(e^x-1)$ is known to be $ \pi^4/15$, so that
 
$\displaystyle U = \frac{2\pi (kT)^4}{h^3c^2}\frac{\pi^4}{15} = \sigma T^4$
  where
 
$\displaystyle \sigma = \frac{2\pi^5 k^4}{15h^3 c^2} = 5.67\times 10^{-8} \ {\rm J. K^4/m^2.s}$
   
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