Chapter 3

At what wavelength does the human body emit the maximum electromagnetic radiation? Use Wien's law from Exercise 14 and assume a skin temperature of ${70}^{\circ }\mathrm{F}$.

Equation $(3-1)$ expresses Planck's spectral energy density as an energy per range $df$ of frequencies. Quite of ten, it is more convenient to express it as an energy per range $d\lambda$ of wavelengths, By differentiating $f=c/\lambda ,$ we find that df $=-c/{\lambda }^{2}d\lambda$. Ignoring the minus sign (we are interested only in relating the magnitudes of the ranges $df$ and $d\lambda$ ). show that, in terms of wavelength. Planck's formula is $\frac{dU}{d\lambda }=\frac{8\pi Vhc}{e^{hc/\lambda k_{B}T}-1}\frac{1}{{\lambda }^5}$

In the photoelectric effect, photoelectrons begin leaving the surface at essentially the instant that light is introduced. If light behaved as a diffuse wave and an electron at the surface of a material could be assumed localized to roughly the area of an atom, it would take far longer. Estimate the time lag. assuming a work function of $4\mathrm{eV}$, an atomic radius of approximately $0.1\mathrm{nm},$ and a reasonable light intensity of $0.01\mathrm{W}/{\mathrm{m}}^2$.

Light of $300\mathrm{nm}$ wavelength strikes a metal plate, and photoclectrons are produced moving as $f$ ast as $0.002c$. (a) What is the work function of the metal? (b) What is the threshold wavelength for this metal?

To expose photographic film. photons of light dissociate silver bromide (AgBr) molecules. which requires an energy of $1.2\mathrm{eV}$. What limit does this impose on the wavelengths that may be recorded by photographic film?

You are an early 20th-century experimental physicist and do not know the value of Planck's constant. By a suitable plot of the following data, and using Einstein's explanation of the photoelectic effect $(\mathrm{K}\mathrm{C})=hf-\varphi$ where $h$ is not known), determine Planck's constant. $$\underline{\begin{array}{cc}\begin{array}{c}\text{ Wavelength of Lighe }\\ \\ (\text{ nm })\end{array}& \begin{array}{c}\text{ Stopping Potential }\\ \\ (\mathrm{V})\end{array}\\ \\ 550& 0.060\\ \\ 500& 0.286\\ \\ 450& 0.563\\ \\ 400& 0.908\\ \end{array}}$$

A typical ionization energy - the energy needed to remove an electron- - for the elements is $10\mathrm{eV}$. Explain why the energy binding the electron to its atom can be ignored in Compton scattering involving an X-ray photon with wavelength about one-tenth of a nanometer.

Determine the wavelength of an X-ray photon that can impart, at most, $80\mathrm{keV}$ of kinetic energy to a free electron.

A photon scatters off of a free electron. (a) What is the maximum possible change in wavelength? (b) Suppose a photon scarters off of a free proton. What is the maximum possible change in wavelength now? (c) Which more clearly demonstrates the particle nature of electromagnetic radiation- - collision with an electron or collision with a proton?

Show that the angles of scatter of the photon and electron in the Compton effect are related by the following form: $$\cot \frac{\theta }{2}=(1+\frac{h}{mc\lambda })\tan \varphi$$