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Problem from Eisberg: Blackbody Radiation

  1. Oct 15, 2008 #1
    1. The problem statement, all variables and given/known data
    In case of Cavity Radiation(Blackbody radiation) let [tex]\rho_{T}(\nu)d\nu[/tex] denote the energy density of radiation having frequency in the interval [tex]\nu[/tex] and [tex]\nu + d\nu[/tex].
    Then we need to show that [tex]\frac{\int_{0}^{\lambda_{max}} \rho_{T}(\nu)d\nu}{\int_{0}^{\infty} \rho_{T}(\nu)d\nu} \approx \frac{1}{4} [/tex]
    where [tex]\lambda_{max}[/tex] is the wavelength at which the radiation is most intense.

    2. Relevant equations
    The obvious rela\evant equation is
    [tex]\rho_{T}(\nu)d\nu = \frac{8\pi h\nu^{3}}{c^{3}}\cdot \frac{d\nu}{e^{\frac{h\nu}{kT}}-1}[/tex] . But the problem is the integration part. Can some one please help me with that?

    3. The attempt at a solution
  2. jcsd
  3. Oct 15, 2008 #2
    OK, so:

    [tex]\int \nu ^3 \cdot (e^{\frac{h\nu}{k_b T}}-1)^{-1} \cdot \delta \nu[/tex]

    Is the equation you need to solve, so you can use:

    [tex]\int A \delta B = AB - \int B \delta A[/tex]

    (from the differentiation product rule). There's a start. Let me know if it helped.

    Sam :smile:
  4. Oct 15, 2008 #3
    Oops, and you'll probably need this standard integral:

    [tex]\int_0^{\infty} \frac{x^3}{e^x -1}\delta x = \frac{\pi^4}{15}[/tex]

    Sam :smile:
  5. Oct 15, 2008 #4
    Yes i could evaluate the denominator using the standard integral u have mentioned, however the Numerator is creating trouble. I have also tried a parts argument, doesnt seem to be taking me any where :cry:
    Some1 please help asap
  6. Oct 16, 2008 #5
    I can't do this without some assumptions, have you stated the WHOLE question from the start?

  7. Oct 16, 2008 #6
    Yes I have indeed given all the necessary details the problem demands. Essentially my doubt boils down to evaluating the integral
    [tex]\int_{0}^{a} \frac{t^{3}}{e^{t}-1}dt[/tex] ; [tex]a[/tex] being some constant.
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