Explain how the principle of superposition accounts for the formation of a dark fringe in a single-slit diffraction pattern at an angle where the slit width equals twice the wavelength of the incident light.

IB DP Physics Higher Level (2023 syllabus) — C.3 Wave phenomena · Explain · 4 marks · View as Markdown

Written & reviewed by James Millett — Biology (Imperial College London), PGCE Science (University of Cambridge).

Monochromatic light of wavelength λ is incident on a single slit of width b = 2λ. A dark fringe is observed on a distant screen at the first minimum of the diffraction pattern.

Model answer (4 marks)

The slit can be divided into two equal halves, each of width λ, which act as sets of secondary Huygens sources.
For the first minimum the path difference between a wavelet from the top of the slit and the wavelet from the midpoint of the slit is λ/2.
A path difference of λ/2 gives a phase difference of π (180°), so each wavelet from the upper half is exactly out of phase with the corresponding wavelet from the lower half.
By the principle of superposition the amplitudes of all paired wavelets cancel everywhere in the slit, giving a resultant amplitude of zero and therefore a dark fringe at that angle.

Examiner tips

  • Use the phrase ‘principle of superposition’ explicitly. Show the path‑difference calculation λ/2. State the phase difference π and the cancellation. Mention the slit is split into two halves of width λ.

Common mistakes

  • Failing to split the slit into two equal halves. Using the wrong path‑difference (e.g. λ instead of λ/2). Not linking the π phase shift to cancellation of amplitudes.

Mark scheme (4 marks)

  1. The slit can be divided into two equal halves, each of width λ, acting as a set of secondary sources (Huygens' sources).
  2. For the first minimum, the path difference between the wavelet from the top of the slit and the wavelet from the midpoint of the slit is λ/2 (half a wavelength).
  3. A path difference of λ/2 corresponds to a phase difference of π (180°), so each wavelet from the upper half is exactly out of phase with its corresponding wavelet from the lower half.
  4. By the principle of superposition, the amplitudes of all paired wavelets cancel throughout the entire slit, giving zero resultant amplitude and hence a dark fringe at that angle.

Key terms in this question

principle of superposition · single-slit diffraction

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