Exercises: X-rays and the Photoelectric Effect

1. a) Describe how X-rays are produced in an X-ray tube.
2. b) What is the energy transformation at the target?
3. c) Why is the target cooled?

The work function of a metal is $4.0\ \text{eV}$.

1. a) What is meant by the work function?
2. b) Calculate the threshold frequency. Use $h=6.63\times10^{-34}\ \text{J s}$ and $1\ \text{eV}=1.6\times10^{-19}\ \text{J}$.

Name two uses of X-rays in medicine or industry, and name one use of a photocell (photoelectric cell) such as an automatic door or a light meter.

An X-ray tube in a Dublin hospital operates at $60\ \text{kV}$. Use $h=6.63\times10^{-34}\ \text{J s}$, $e=1.6\times10^{-19}\ \text{C}$ and $c=3.0\times10^{8}\ \text{m s}^{-1}$.

1. a) Calculate the maximum energy of an X-ray photon produced.
2. b) Calculate the corresponding maximum frequency.
3. c) Calculate the minimum wavelength.

Light of wavelength $250\ \text{nm}$ falls on a metal with a work function of $3.2\ \text{eV}$. Use $h=6.63\times10^{-34}\ \text{J s}$, $c=3.0\times10^{8}\ \text{m s}^{-1}$.

1. a) Calculate the photon energy in eV.
2. b) Calculate the maximum kinetic energy of the emitted electrons in eV.
3. c) Calculate the stopping potential.

A photocell is used in a burglar alarm. A light beam falls on the cathode and a photocurrent flows.

1. a) Explain how the alarm is triggered when the beam is interrupted.
2. b) State what happens to the photocurrent if the intensity is doubled at constant frequency.
3. c) State what happens if the frequency is below the threshold frequency, no matter how intense.

Compare the X-ray tube and the photoelectric effect as inverse processes.

1. a) Explain in what sense they are "inverse" in terms of energy conversion between photons and electrons.
2. b) In an X-ray tube, electrons accelerated through $40\ \text{kV}$ strike a target. Calculate the shortest wavelength produced.
3. c) Explain why the X-ray spectrum has a sharp cut-off wavelength and discrete peaks superimposed on a continuous background.
4. d) Explain why the photoelectric effect cannot be explained by the wave model of light.

In a photoelectric experiment with a caesium cathode, the stopping potentials are measured: $0.45\ \text{V}$ for $f=5.5\times10^{14}\ \text{Hz}$ and $1.28\ \text{V}$ for $f=7.5\times10^{14}\ \text{Hz}$. Use $e=1.6\times10^{-19}\ \text{C}$.

1. a) Use $eV_s = hf-\phi$ to calculate $h$ from the two readings.
2. b) Calculate the work function in eV.
3. c) Calculate the threshold frequency.
4. d) Explain how a graph of $V_s$ against $f$ would show these quantities.

An X-ray tube operates at $80\ \text{kV}$ with a beam current of $10\ \text{mA}$. Only $1.0\%$ of the electron beam power is converted to X-rays.

1. a) Calculate the power of the electron beam.
2. b) Calculate the power converted to heat.
3. c) Calculate the minimum wavelength of the X-rays.
4. d) Explain, with reference to the heat, the design of the rotating anode in modern X-ray machines.