A-Level Physics October/November 2024 Q8(b)(ii): For ultraviolet radiation with a frequency of 11 × 10¹⁴Hz, calculate the maximum speed…

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  According to Einstein's photoelectric equation: $hf = \Phi + E_{k,max}$ where the maximum kinetic energy is $E_{k,max} = \frac{1}{2}mv_{max}^2$.

  Therefore, $hf = \Phi + \frac{1}{2}mv_{max}^2$.

  Substituting the given values: $(6.63 \times 10^{-34}) \times (11 \times 10^{14}) = (5.8 \times 10^{-19}) + \frac{1}{2}(9.11 \times 10^{-31})v_{max}^2$

  First, calculate the energy of the incident photon, $E = hf$: $E = 7.293 \times 10^{-19}$ J

  Now, rearrange the equation to find the maximum kinetic energy: $E_{k,max} = hf - \Phi$ $E_{k,max} = (7.293 \times 10^{-19}) - (5.8 \times 10^{-19}) = 1.493 \times 10^{-19}$ J

  Finally, solve for $v_{max}$: $\frac{1}{2}mv_{max}^2 = 1.493 \times 10^{-19}$ $v_{max}^2 = \frac{2 \times 1.493 \times 10^{-19}}{9.11 \times 10^{-31}} = 3.2777 \times 10^{11}$ $v_{max} = \sqrt{3.2777 \times 10^{11}} = 572513$ m s⁻¹

  Rounding to 2 significant figures (as per the data provided): $v_{max} = 5.7 \times 10^5$ m s⁻¹

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  How the marks are awarded

  • C1 — Awarded for correctly stating Einstein's photoelectric equation, $hf = \Phi + \frac{1}{2}mv_{max}^2$.
  • C1 — Awarded for the correct substitution of all values for Planck's constant, frequency, work function, and electron mass into the photoelectric equation.
  • A1 — Awarded for the final correct answer of $5.7 \times 10^5$ m s⁻¹, which is correctly rounded to 2 significant figures and includes the correct unit.

  Common mistakes

  • Incorrectly rearranging the photoelectric equation, for example, adding the work function to the photon energy instead of subtracting it.
  • Forgetting to take the square root at the final step to find $v_{max}$ after calculating $v_{max}^2$.
  • Stating the final answer to an inappropriate number of significant figures, such as copying the full calculator display, instead of matching the precision of the data in the question.
  • Forgetting the factor of $\frac{1}{2}$ in the kinetic energy formula, i.e., using $E_k = mv^2$.

  Examiner tip: Always start a calculation question by writing down the relevant fundamental equation before substituting any values, as this often secures the first mark.

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