A laser of wavelength 632.8 nm illuminates a single slit. The screen is 2.00 m away. Measured first-minimum distances from the centre are y = 4.20, 6.31, 8.39 and 10.52 mm for screen distances L = 1.00, 1.50, 2.00 and 2.50 m. Treat y/L as small. Determine the best estimate of slit width, explain the expected graph of y against L, calculate the full central-maximum width at L = 2.00 m, and evaluate one random and one systematic uncertainty.

Question

Accepted Answer

The gradient y/L ≈ 4.21 × 10⁻³, giving slit width a ≈ 1.50 × 10⁻⁴ m. The full central maximum at 2.00 m is about 16.8 mm. The first single-slit minimum satisfies a sinθ = λ. For small angles, sinθ ≈ tanθ = y/L, so y = (λ/a)L. A y–L graph should be linear through the origin with gradient λ/a. The ratios cluster near 4.21 mm m⁻¹ = 4.21 × 10⁻³. Thus a = λ/gradient = (632.8 × 10⁻⁹)/(4.21 × 10⁻³) = 1.50 × 10⁻⁴ m. At L = 2.00 m the first minima lie about 8.42 mm on either side, so the full central-maximum width is 2y ≈ 16.8 mm. Repeat measurements of both left and right minima to reduce random judgement error in locating a dark fringe. A mismeasured screen distance or non-normal screen introduces a systematic scale error; include uncertainties on the fitted gradient rather than trusting one ratio.

Determine slit width from diffraction data

What a complete response should do

First-minimum displacement against screen distance

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First-minimum displacement against screen distance — supplied values
Screen distance L / m	First-minimum displacement y / mm
1	4.2
1.5	6.31
2	8.39
2.5	10.52