Calculating average energy and black body energy spectrum using Boltzman probability

  • #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?
 

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