Challenge a ray-shadow prediction with diffraction width

Establish the model and evidence

Diffraction is wave spreading around obstacles and through openings. A narrow aperture cannot always be represented by a simple ray shadow. Contributions from different parts of one slit interfere, so even a single opening produces a broad central feature and weaker side features. Spreading becomes especially important when aperture size is comparable with wavelength, although it is not identically absent for larger apertures.

For a monochromatic far-field single slit of width a, dark directions satisfy a sinθ=mλ for nonzero integer m. The first minima bracket the central feature. At a screen distance L, the corresponding transverse coordinate is L tanθ, so exact central width is 2L tan(arcsin(λ/a)) when a first minimum exists. A requested order with |mλ/a| greater than one has no real angle; it must not be assigned a fake coordinate.

The small-angle approximation gives width approximately 2Lλ/a. It is useful only with its angle restriction and a suitable precision goal. The current framework uses small-angle reasoning below 10°. Comparing the approximation with the exact geometric expression reveals its direction and size of error rather than treating every formula as equally exact at any angle.

The displayed normalized sinc-squared curve is supplied visualization enrichment, not a required derivation or measured intensity record. A slit minimum relation is not the circular-aperture Rayleigh formula. Finite aperture nevertheless explains qualitatively why a geometrically focused image can still fail to resolve nearby features. This is not an eye test, a telescope specification or an invitation to use lasers or focus sunlight.

Work a reference case

For λ=600 nm and a=6 µm, λ/a=0.10 and the first minimum is at 5.739°. At L=1 m the exact central width is about 201.01 mm; the small-angle estimate is 200 mm. Narrowing the slit increases width, opposite a pure shadow-width expectation.

Predict before investigating

At fixed wavelength and screen distance, will halving the slit width narrow or broaden the central diffraction feature?

Compare widths 2, 6 and 10 µm at 600 nm, then wavelengths 400 and 700 nm at one width. Retain exact angle, exact width and approximate width; flag comparisons outside the stated small-angle range.

Transfer to changed conditions

Use λ=500 nm, a=10 µm and L=2 m. Find the small-angle central width and say whether the exact geometric width is slightly larger or smaller.

Compare your reasoning

The approximate width is 2×2×0.05=0.20 m. The exact width is about 0.20025 m, slightly larger because tan(arcsin(0.05)) exceeds 0.05.

Reflect and qualify the evidence

Why does a zero geometric defocus estimate from Unit 13 not guarantee unlimited wave-optical resolution?

Change the aperture shape as a distinct intervention

Predict and sketch: rotate a long narrow slit by 90 degrees while holding its width, wavelength and screen distance fixed. The direction of strongest diffraction spreading rotates too: spreading is largest perpendicular to the narrow dimension. Replace the slit with a circular opening and the ideal pattern has circular symmetry rather than parallel bands. The slit-zero equation cannot be reused by merely naming the circle’s diameter as a slit width. Keep these shape changes separate from the existing width-ratio calculation. (Current clause 14.7.A.6.)

Sources and boundaries

Original text, calculations and diagrams; checked 30 October 2026 against the current AP Physics 2 course framework and primary textbook verification. Controls generate virtual evidence, not actual measurements or independent practical competence. No lasers, sunlight focusing, loud audio, traffic observations or chemical-film preparation are requested. Public feedback is practice; retain your derivation and critique in the course evidence workspace.

Clause-level source/action reconciliation checked 31 October 2026 against the linked current framework and Fall 2026 clarification. Added packets are original virtual comparisons; no new practical or learner-completion claim.