Two narrow slits separated by (0.300 ± 0.005) mm are illuminated by monochromatic light. On a screen (2.00 ± 0.01) m away, the distance across 10 fringe spacings is (42.0 ± 0.5) mm. Determine fringe spacing and wavelength with percentage uncertainty. Explain why measuring many spacings helps, predict what happens when one slit is widened substantially, and distinguish loss of visibility from a change in fringe spacing.

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Fringe spacing = 4.20 mm; wavelength = 630 nm. Maximum fractional uncertainty ≈ 1.19% + 1.67% + 0.50% = 3.36%, so λ ≈ (630 ± 21) nm. Ten spacings occupy 42.0 mm, so fringe spacing s = 4.20 mm. Measuring across many fringes makes the same endpoint-reading uncertainty a smaller fraction of one spacing. For small angles, s = λD/d. Hence λ = sd/D = (4.20 × 10⁻³)(0.300 × 10⁻³)/2.00 = 6.30 × 10⁻⁷ m. Maximum fractional uncertainty is Δs/s + Δd/d + ΔD/D = 0.5/42.0 + 0.005/0.300 + 0.01/2.00 = 0.0336. Thus percentage uncertainty is 3.36% and absolute uncertainty is about 21 nm, giving λ = (630 ± 21) nm at matching precision. Widening one slit changes relative amplitudes and may add a diffraction envelope, reducing visibility because minima no longer cancel perfectly. The geometric fringe spacing λD/d remains essentially unchanged while coherent path separation and wavelength are unchanged.

Resolve wavelength and coherence from fringes

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