Module 5 : MODERN PHYSICS
Lecture 23 : Black Body Radiation
where the constant of proportionality $ h$ is called Planck's constant . Its value in SI units is $ 6.626\times 10^{-34}$ J-s. Thus the possible energy of a mode with frequency $ \nu$ is $ nh\nu$ where $ n=0,1,2\ldots$. According to Boltzmann distribution, the probability of a mode having an energy $ E$ at a temperature $ T$ is given by $ \exp(-\beta E)$, where $ \beta = 1/kT$. Here, $ k$ is the Boltzmann constant and $ T$ is the absolute temperature. Thus the average energy of a mode is
 
$\displaystyle \bar\varepsilon$ $\displaystyle =$ $\displaystyle \frac{\sum_{n=0}^\infty nh\nu\exp(-nh\beta\nu)} { \sum_{n=0}^\infty \exp(-nh\beta\nu)}$
$\displaystyle =$ $\displaystyle \frac{h\nu}{\exp(h\nu\beta)-1}$ (3)
  Exercise 2
 

Exercise 3

  Exercise 4
 

Example-1
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