Consider Planck's radiation law given by (1.2).
(a) Derive an expression for the spectral energy density $\rho(\lambda)$ as a function of wavelength.
(b) Locate the maxima $\nu_{\text {max }}$ and $\lambda_{\text {max }}$ of the spectral energy densities $\rho(\nu)$ and $\rho(\lambda)$, respectively. Explain why $\nu_{\max } \neq c / \lambda_{\text {max }}$.
(c) How does the total energy density $\rho$, that is, $\rho(v)$ integrated over the frequency $\nu$, depend on the temperature $T$ ?
Hints: the algebraic equation $\mathrm{e}^z(3-z)=3$ has roots $z=0$ and $z \approx 2.821$. Likewise, the equation $\mathrm{e}^z(5-z)=5$ has roots $z=0$ and $z \approx 4.965$. The following identity holds: $\int_0^{\infty} z^3[\exp (z)-1]^{-1} \mathrm{~d} z=\pi^4 / 15$.