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Homework Help: Planck's Law

  1. Feb 7, 2005 #1

    My question is in regards to Planck's Law and a blackbody:

    For the single lambda case I can readily find the spectral exitance. Alternately, if I substitute to create an integral in the form of x^3 / (e^x - 1) and integrate over all lambda, I reach Stefan-Boltzmann. No problems there, but I am really interested in finding the (average?) spectral exitance over a small, finite spectral band. Any tips on how to go about this?

    Thanks in advance.
  2. jcsd
  3. Feb 7, 2005 #2
    i am not quite sure what your question is, but it seems to me you are looking for the answer of this integral:
    [tex] \int_{b}^a \frac{x^3}{e^x-1} dx [/tex],
    the above integral has no exact solution... unless a=0, b=infinity, or a=b.... the best you could do is use numerical analysis....
  4. Feb 7, 2005 #3


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    If you don't have a computer software to give you the result,learn that Debye integral ([itex] D_{3} [/itex]) values are tabulated...

  5. Feb 7, 2005 #4
    Thank you both for your quick responses. To state my question more directly: "What is the area under the curve generated by a blackbody at some given temperature for some finite lambda range?" (Say in the visible region only.)

    I can generate an approximate answer by using small increments of area and summing, but I wanted to compare my answer the solution obtained from a definite integral. However, from your answers I believe it is not possible (I do not have any computer software - just pencil and paper!).

    I have taken your advice and will study more about numerical analysis and Debye integrals. Thank you again for your help.
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