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Tricky Derivation of Blackbody Equations.

  1. Apr 28, 2010 #1
    1. The problem statement, all variables and given/known data

    Starting with the Planck distribution:

    [tex] R(\lambda,T) = \frac{c}{4} \frac{8 \pi}{\lambda^4} (\frac{hc}{\lambda})(\frac{1}{e^{hc/(\lambda kT)}-1})[/tex]

    Derive the blackbody Stefan-Boltzmann law (ie total flux is proportional to T4) by integrating the above expression over all wavelengths. Thus show that

    R(T) = [tex] \frac{2 \pi^5 k^4}{15h^3 c^2} T^4 [/tex]

    and [tex]\int[/tex] [tex] \frac{x^3}{e^x -1} dx = \frac{\pi^4}{15}[/tex]

    3. The attempt at a solution

    I know I need to substitute [tex]x = \frac{hc}{kT} \frac{1}{\lambda}[/tex]. And somehow I think I can use the form KR([tex]\lambda[/tex],T) = A([tex]\lambda[/tex])B([tex]\lambda[/tex])
  2. jcsd
  3. Apr 28, 2010 #2
    Just use that substitution and try to get it into the form of the integral provided.
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