## Section53.7Quantum Nature of Light Bootcamp

### Subsection53.7.4Miscellaneous

An oven is heated to a high temperature and the electromagnetic radiation coming out of the oven through a tiny hole in the oven is analyzed for radiance $R_T(\lambda)\text{,}$ which is the power content per unit wavelength range per unit cross-section area of the hole. The data obtained at five wavelengths are:

\begin{align*} \amp (0.3\ \mu\textrm{m}, 1.1\times 10^{13}\:\textrm{W/m}^3),\\ \amp (0.4\ \mu\textrm{m}, 2.7\times 10^{13}\:\textrm{W/m}^3), \\ \amp (0.5\ \mu\textrm{m}, 3.8\times 10^{13}\:\textrm{W/m}^3), \\ \amp (0.6\ \mu\textrm{m}, 4.0\times 10^{13}\:\textrm{W/m}^3), \\ \amp (0.7\ \mu\textrm{m}, 3.7\times 10^{13}\:\textrm{W/m}^3), \\ \amp (0.8\ \mu\textrm{m}, 3.2\times 10^{13}\:\textrm{W/m}^3). \end{align*}

(a) Plot $R_T$ versus $\lambda\text{.}$ (b) From the data find the temperature of the oven. (c) Find the total power radiated per unit area of cross-section of the hole.

A photon of energy $hf$ collides head-on with a nearly free electron at rest. Let $E_0$ be the rest energy of an electron. Show that the kinetic energy of the recoiled electron is given by

\begin{equation*} K = \dfrac{2h^2f^2}{2hf + E_0}. \end{equation*}

(a) Prove that in the Compton scattering, a photon cannot transfer all of its energy to an electron. (b) Is there a maximum percentage of energy that a photon can transfer to an electron at rest? If so, what is it? If not, why not?

(a) Treating $R_T(\lambda)$ as a function of $\lambda\text{,}$ prove that the maximum of the radiance occurs at a wavelength $\lambda_{\textrm{max}}$ whose product with temperature is a constant.

\begin{equation*} \lambda_{\textrm{max}} = \textrm{constant}. \end{equation*}

(b) Find the value of the constant.

Integrate $R_T(\lambda)$ over all values of $0\le \lambda \le \infty$ to deduce the Stefan-Boltzmann law.

\begin{equation*} I = \int_0^\infty R_T(\lambda) d\lambda = \sigma T^4. \end{equation*}

Hint: Let $x = hc/\lambda k_B T\text{.}$