Infer wavelength while testing geometry and calibration assumptions

Establish the model and evidence

A fringe pattern can support an inference only after its coordinate references and geometry are stated. The synthetic packet uses signed orders −3 through +3, slit separation 0.25 mm and screen distance 1.20 m. Exact generated positions use y=L tan(arcsin(mλ/d)), then add a common coordinate offset and a small declared reading error. The source wavelength is available for testing the model, not because a real instrument has been calibrated.

In the paraxial approximation, a graph of position versus signed order has slope s=Lλ/d and an intercept representing the screen-coordinate zero. A free-intercept fit can separate a common offset from spacing. Differences between symmetric orders also cancel a common zero offset. Neither method removes independent reading errors or a wrong slit separation or screen distance. A small residual is therefore not a certificate for every assumption.

The fit and the generator deliberately use different geometric precision: the generator is exact for the stipulated far-field direction and screen geometry, while the fit is paraxial. Angles in this packet are small, so the difference is slight, but a held-out order can still reveal combined approximation and reading effects. Compare the predicted fourth order with a newly supplied coordinate rather than reporting only the same points used to fit.

For a separate bounded inference, let measured spacing be 2.640±0.020 mm, d=0.250±0.002 mm and L=1.20±0.01 m. With positive quantities, λ=sd/L has conservative endpoints using smallest s and d with largest L, and vice versa: approximately 537.0 to 563.3 nm. This interval is not a probability or confidence level. It excludes unmodeled misidentified orders and coherence failure, which require separate checks.

Work a reference case

For true 550 nm with a +1 mm common offset, the original noisy packet gives fitted slope about 2.63647 mm/order and intercept 1.00000 mm, hence paraxial λ≈549.27 nm. Its fourth-order prediction is 11.54589 mm, while the unperturbed held-out exact coordinate is 11.56041 mm. The difference is small but not silently zero.

Predict before investigating

Will adding the same coordinate offset to every order change a free-intercept spacing slope?

Compare offsets −2, 0 and +2 mm at one wavelength; retain slope, intercept and the symmetric estimate. Then change wavelength and freeze a fourth-order prediction before reading its ledger value. Explain which calibration errors this comparison can and cannot remove.

Transfer to changed conditions

A separate small-angle packet has adjacent-order spacing 3.0 mm, screen distance 1.5 m and slit separation 0.30 mm. Infer wavelength and say what a common +2 mm coordinate offset would do to that spacing estimate.

Compare your reasoning

λ=sd/L=(0.003)(0.00030)/1.5=600 nm. A common offset cancels from adjacent differences, but uncertainties in spacing, screen distance and slit separation still limit the inferred value.

Reflect and qualify the evidence

Why is agreement of two offset-canceling methods not independent verification of the slit-separation value they both use?

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.