 Solution

ID: past-exam-of-the-physics-a-course-of-the-university-of-cambridge/2018/paper-2/section-b/b7/c/solution

Represent equal narrow slits at

x = \pm D /2

by two contributions with equal complex amplitudes. In a direction

θ

the path difference is

D sin θ

, so

ψ (θ) \propto e^{i q D /2} + e^{- i q D /2} = 2 cos (q D /2), q = \frac{2 π}{λ} sin θ .

I (θ) \propto cos^{2} (\frac{π D sin θ}{λ}) .

D sin θ = mλ

, with a central maximum. In the small-angle region,

Δ θ ≃ λ / D = 4.00 \times 1 0^{- 3} rad = 4.00 mrad .

Dark fringes lie halfway between bright ones. Exactly, spacing is uniform in

sin θ

, not in

θ

at large angles. The first panel of the pattern figure below shows the narrow-slit result.

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