A-Level Physics May/June 2024 Q8(c)(iii): the minimum wavelength of X-rays produced at the target.

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  The minimum wavelength of the X-ray photon corresponds to the maximum possible photon energy. This occurs when an electron loses all of its kinetic energy in a single collision, with this energy being converted into a single photon.

  The energy of a photon is given by $E = hf$. The relationship between frequency and wavelength is $c = f\lambda$.

  Combining these two equations to make wavelength the subject: $f = \frac{c}{\lambda} \implies E = \frac{hc}{\lambda}$

  Rearranging for the minimum wavelength, $\lambda_{min}$: $\lambda_{min} = \frac{hc}{E_{max}}$

  The maximum energy of the photon is the kinetic energy of the electron, which from the previous part of the question is $E = 0.032 \times 1.60 \times 10^{-13}$ J.

  Substituting the values: $\lambda_{min} = \frac{(6.63 \times 10^{-34} \text{ J s}) \times (3.00 \times 10^8 \text{ m s}^{-1})}{0.032 \times 1.60 \times 10^{-13} \text{ J}}$

  $\lambda_{min} = \frac{1.989 \times 10^{-25}}{5.12 \times 10^{-15}}$

  $\lambda_{min} = 3.884... \times 10^{-11}$ m

  $\lambda_{min} = 3.9 \times 10^{-11}$ m (to 2 s.f.)

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  How the marks are awarded

  • C1 — Stating the two fundamental relationships for photon energy ($E=hf$) and wave speed ($c=f\lambda$).
  • C1 — Correctly combining the two initial equations to derive the working formula for wavelength, $\lambda = hc/E$.
  • A1 — Correctly substituting all values into the formula and calculating the final answer of $3.9 \times 10^{-11}$ m.

  Common mistakes

  • Forgetting to convert the electron's kinetic energy from MeV (or keV) into the SI unit of Joules by multiplying by the elementary charge and the relevant power of 10.
  • Incorrectly rearranging the equations, for example calculating $\lambda = E/(hc)$ instead of $\lambda = hc/E$.
  • Using an incorrect value for the energy, perhaps from a miscalculation in a previous part of the question (though error-carried-forward may apply).
  • Stating the final answer to an inappropriate number of significant figures, such as one, or writing out the full calculator display.

  Examiner tip: Practice combining fundamental equations from different topics to derive a single working formula for the unknown quantity.

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