(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Starting with the Planck distribution R([tex]\lambda[/tex],T) for blackbody radiation.

(a) Derive the blackbody Stefan-Boltzmann law (ie total flux is proportional to T^{4}) by integrating the above expression over all wavelengths. Thus show that

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

(b) Show that the maximum value of R([tex]\lambda[/tex],T) occurs for [tex]\lambda[/tex]_{m}T = 2.898 * 10^{-3}mK (this is called Wien's displacement law).

(c) Use Wien's displacement law to determine for the cosmic background radiation with T = 2.7 K

(i) the value of [tex]\lambda[/tex]_{m}for peak intensity

(ii) the energy in eV of photons at this peak intensity, and

(iii) the region of the electromagnetic spectrum corresponding to the peak intesnsity.

2. Relevant equations

Planck distribution for blackbody radiation:

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

3. The attempt at a solution

That was a mouthful. Help please.

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# Derive Stefan-Boltzmann Law from Planck Distribution for blackbody radiation

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