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Section 53.7 Quantum Nature of Light Bootcamp

Subsection 53.7.1 Photon and Blackbody Radiation

Subsection 53.7.2 Photoelectric effect

Subsection 53.7.3 Compton effect

Subsection 53.7.4 Miscellaneous

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.

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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*}
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(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?

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(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.

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Solution

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 \(x = hc/\lambda k_B T\text{.}\)

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