3. The energy density of photons in the frequency range (v, v+ dv), is given by the blackbody or Planck function: \varepsilon(\nu) d \nu=\frac{8 \pi h}{c^{3}} \frac{v^{3} d \nu}{\exp (h v / k T)-1} a.Derive that the peak of this function occurs at an energy hv=2.82 kT. What isthis relation or law usually called in the literature? b. Derive that, integrated over all frequencies, the energy density is equal to: \begin{array}{l}

\varepsilon_{\gamma}=\alpha T^{4} \quad \text { where } \\

\alpha=\frac{\pi^{2}}{15} \frac{k^{4}}{\hbar^{3} c^{3}}=7.56 \times 10^{-16} \mathrm{~J} \mathrm{~m}^{-3} \mathrm{~K}^{-4}

\end{array} What is this relation generally known as? c. Hence, calculate the total energy density of the Cosmic MicrowaveBackground (assume T = 2.725 K) and its photon density.