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A pinhole camera has length L and a circular hole of radius a. When illuminated by a parallel beam of light of wavelength λ, the size of the spot formed on the opposite wall is taken as the sum of the geometrical spread and the diffraction spread. The spot has its minimum size b_min when:

1. a = sqrt(λ L) and b_min = sqrt(4 λ L)
2. a = λ²/L and b_min = (2 λ²/L)
3. a = sqrt(λ L) and b_min = (2 λ²/L)
4. a = λ²/L and b_min = sqrt(4 λ L)

Correct answer: a = sqrt(λ L) and b_min = sqrt(4 λ L)

Solution

Total spread b ≈ 2a + λL/a (geometric plus diffraction). Minimizing: db/da = 2 - λL/a² = 0 -> a² = λL/2, so a ~ sqrt(λL). At the optimum b_min is of order sqrt(λL), and among the given forms this matches a = sqrt(λL) with b_min = sqrt(4 λ L) (i.e. b_min ∝ sqrt(λL)).

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