Infer slit separation from a fringe pattern

Change one quantity and predict the result

Set wavelength to 500 nm and separation to 0.20 mm. The first positive bright fringe is 5.000016 mm from the center. Double separation to 0.40 mm: that position becomes 2.500002 mm, nearly half as far out. Larger separation creates more path difference at a given screen angle, so one full cycle of difference is reached sooner.

Return separation to 0.20 mm and increase wavelength to 700 nm. Now the first bright fringe is 7.000043 mm from the center. A longer wavelength needs a larger path difference to complete one cycle. Compare captures while naming the unchanged quantities; changing both controls at once cannot isolate either effect.

Derive the useful small-angle rule

At a bright fringe, d sin θ = mλ. Near the center, sin θ and tan θ are approximately equal, while x = L tan θ. This gives xₘ ≈ mλL/d and near-axis fringe spacing Δx ≈ λL/d. Here x and Δx are screen lengths, λ is wavelength, L screen distance, d slit-center separation and m an integer order.

For λ = 500 nm = 5 × 10⁻⁷ m, L = 2 m and d = 0.20 mm = 2 × 10⁻⁴ m, spacing is approximately (5 × 10⁻⁷ × 2)/(2 × 10⁻⁴) = 0.005 m = 5 mm. Nanometres, millimetres and metres cannot be mixed as bare numbers.

With the retained screen geometry, let q = mλ/d. Then xₘ = Lq/√(1 − q²) for |q| below one. The browser’s positive feature table uses this expression. Consequently adjacent fringes are only approximately equally spaced on the flat screen. The table’s six decimals describe the ideal calculation, not physical measurement precision.

At the default setting the near-axis estimate of 5 mm differs from the first-feature position by approximately 0.000313%. That percentage quantifies a mathematical approximation relative to this model. It is not an uncertainty estimate for a ruler, camera or fabricated slit.

Infer an unknown separation

Suppose an explicitly synthetic small-angle exercise gives bright positions −15, −10, −5, 0, 5, 10 and 15 mm for orders −3 through +3. These rounded practice values are not physical measurements or the exact higher-order coordinates of the browser. The span from order −3 to +3 is 30 mm across six intervals, so estimated spacing is 5 mm.

Rearrange the rule to d ≈ λL/Δx. With wavelength 500 nm and screen distance 2 m, this gives d ≈ 0.20 mm. Counting six intervals matters: there are seven bright positions, but only six gaps between the outer ones.

Using a multi-fringe span can reduce the relative effect of endpoint-reading error compared with reading one short gap. A shared origin offset cancels in a difference, but a wrong ruler scale or screen distance does not. Repeating a calculated pattern is not an independent measurement. For real data, retain calibrated positions, units, wavelength and screen-distance uncertainty before making a precision claim.

Diagnose a misleading pattern comparison

A wider-looking pattern might mean a longer wavelength, a larger screen distance or a smaller slit separation. One observed spacing cannot identify all three unknowns. State which quantities are known before inverting the equation. Also check that finite slit widths, limited coherence and camera saturation do not invalidate the ideal feature interpretation.

The browser keeps L fixed at 2 m. Predictions for other screen distances are transfer calculations, not additional controls secretly simulated by the plot. Its 0.2 mm sampling grid need not land exactly on a bright peak; the analytic table supplies the feature coordinate.

Transfer an inference

A synthetic near-axis pattern has spacing 4.0 mm at wavelength 600 nm on a screen 1.5 m away. Estimate separation. If the same apparatus uses wavelength 450 nm, predict the new spacing.

Check the fringe-spacing transfer

d ≈ (600 × 10⁻⁹ × 1.5)/0.004 = 0.000225 m = 0.225 mm. At fixed d and L, spacing scales with wavelength: 4.0 × 450/600 = 3.0 mm. These are small-angle estimates, not exact finite-angle coordinates.

OpenStax: Young’s double-slit interference gives the path-difference conditions. Next, single-slit diffraction distinguishes slit width from separation between two openings.