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Calculating average energy and black body energy spectrum using Boltzman probability

  1. Sep 2, 2008 #1
    The question is as follows:

    The black body energy spectrum is [tex]\rho[/tex](T,v)dv=[tex]\frac{8\piv2<E(v)>}{c3}[/tex]dv where v is the frequency of the EM wave and <E(v)> is the average energy at v. Assuming the energy of a EM wave of v can only take multiples of hv, the from the Boltzman probability P(E)=exp(-E/kbT/[tex]\sumexp(-E/kbT[/tex]), calculate the average energy <E(v)>=[tex]\sumEP(E)[/tex] and [tex]\rho[/tex](T,v). Discuss the result in the limits of hv<<kbT and hv>>kbT, and compare the results with the Rayleigh-Jean Law and Wien's result.

    I am fairly sure that answering this question intails basically deriving [tex]\rho[/tex](T,v) which involves putting the probabilty series within the average energy series. Then taking the simplified expression for average energy and inserting it into the black body spectrum equation. I know from online research that [tex]\rho[/tex](T,v)=[tex]\frac{2hv3}{c2}[/tex][tex]\frac{1}{e\frac{hv}{kbT}-1}[/tex].

    Basically I get to the point where <E>=-[tex]\partiallnZ[/tex]/[tex]\partial[/tex][tex]\beta[/tex] where z=[tex]\sum[/tex]exp(-nhv[tex]\beta[/tex]) and [tex]\beta[/tex]=1/kbT.

    I can see that if I am able to continue the derivation then the exp term can allow me to answer the questions about limit behavior. My problem is with finishing the derivation. Can anyone help?
     
  2. jcsd
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