Separate interference orders from a finite-slit envelope

Establish the model and evidence

Coherent slit contributions have a stable phase relationship, so their path differences can produce repeatable bright and dark directions. For slit spacing d, nominal constructive orders satisfy d sinθ=mλ. Increasing wavelength moves a given noncentral order farther from the center. White light has a common central order but dispersed noncentral colors, with longer-wavelength red farther out than violet for the same order.

A real-width slit also produces diffraction. The multiple-slit interference structure sits inside a single-slit envelope controlled by width a. Slit spacing and slit width are therefore different parameters with different roles. If a nominal constructive order lies at an envelope minimum, that order can be missing even though the path-difference condition between slit centers is satisfied.

The original model uses d=20 µm and a=5 µm. Because d/a=4, the fourth nominal interference order coincides with the first single-slit minimum. Adding coherent slits narrows principal interference peaks without changing the center-to-center spacing condition. The plotted curve is computed from a supplied coherent sum as optional numerical enrichment; learners can explain orders and missing peaks with algebraic path arguments rather than complex-number arithmetic.

All curves are normalized at their central peak. This makes shape comparisons possible but removes absolute brightness information: equal displayed heights across different slit counts do not establish equal physical power. The dense angular sampling resolves the bounded model’s features, while tables display a labeled sample of coordinates. A missing observed peak could also result from detector limitations or a wrong coherence assumption, so real inference needs more than one absent point.

For two equally illuminated coherent slits, destructive interference between their contributions occurs when the path difference is Δ=(m+1/2)λ, with integer m. This is different from a single-slit minimum a sinθ=mλ and from a missing principal order where the two mechanisms coincide. Predict the first two-slit dark direction for d=20 µm and λ=600 nm: d sinθ=λ/2 gives sinθ=0.015, so θ≈0.8595°. Select two slits and check that this dark direction lies between the central and first bright directions; it is not the first single-slit envelope zero.

Work a reference case

At λ=600 nm, nominal order 1 is arcsin(0.03)=1.719°. Order 4 is arcsin(0.12)=6.892°, but a sinθ=5 µm×0.12=600 nm, the first envelope minimum. It is therefore not a bright fourth peak in this finite-width model.

Predict before investigating

Can satisfying the slit-center constructive condition guarantee a bright peak when each slit has finite width?

Compare 2, 4 and 8 slits at 600 nm, then 400 and 700 nm at four slits. Retain first-order angle and the missing-order explanation. Distinguish peak narrowing from wavelength dispersion and from changes in normalized peak height.

Transfer to changed conditions

A different array has spacing 12 µm and slit width 3 µm. For λ=600 nm, find the first nominal order angle and the first missing interference order. Do not use a small-angle formula for every higher order automatically.

Compare your reasoning

Order 1 is arcsin(0.05)≈2.866°. The first missing interference order is again 4 because d/a=4; its nominal angle is arcsin(0.20)≈11.537°, outside the declared below-10° small-angle range.

Reflect and qualify the evidence

Which additional evidence would distinguish envelope suppression from a detector that simply failed to register weak light?

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.