The Grating Measurement That Used the Small-Angle Shortcut
A student uses a diffraction grating labelled 600 lines/mm and a red laser to measure the laser's wavelength. The grating is clamped to a lab bench, and the screen is a sheet of paper taped to a wall 1.20 m away. The student measures the distance from the central maximum to the first-order bright spot as 0.43 m and to the second-order spot as 1.02 m. Using sin θ ≈ tan θ = x/L, the student computes λ = d sin θ = 597 nm for the first order and 602 nm for the second. The laser is labelled 632.8 nm. The student is surprised by the 5–6% discrepancy and asks whether the laser is mislabelled. The teacher notices that the screen is not exactly perpendicular to the beam, the grating plane is tilted slightly relative to the beam, the room lights are on, and the spot positions were measured to the nearest 5 mm. The student must decide whether the small-angle approximation is acceptable, calculate the exact result, and evaluate the evidence.
1. Evaluate the small-angle approximation for the first-order measurement. Calculate the exact angle and wavelength, and compare with the student's 597 nm.
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With d = 1/(600 lines/mm) = 1.667 × 10−6 m, L = 1.20 m, and x = 0.43 m: tan θ = x/L = 0.43/1.20 = 0.3583, so θ = arctan(0.3583) = 19.7°. The exact first-order wavelength is λ = d sin θ = (1.667 × 10−6 m)(sin 19.7°) = (1.667 × 10−6)(0.337) = 5.62 × 10−7 m = 562 nm.
The student's small-angle step used sin θ ≈ tan θ = 0.3583, giving λ = 597 nm. At 19.7° the approximation is not accurate: sin θ is about 6% smaller than tan θ. So the approximation shifts the result upward by about 6%, from 562 nm to 597 nm. This is a real systematic error in the analysis. Interestingly, it moves the answer closer to 632.8 nm, so the approximation partly masks the larger experimental discrepancy.
2. Identify at least three systematic errors in the setup and state how each could shift the measured wavelength. Which is likely the largest?
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- Screen not perpendicular to the beam: the measured x is a distance along the paper, but the diffraction angle is defined relative to the beam direction. If the screen is tilted, x is foreshortened or lengthened depending on the tilt, so θ is wrong. A tilt of a few degrees can change x by a percent or more.
- Grating plane not normal to the beam: the zero-order spot may still be visible, but the grating equation assumes angles measured from the grating normal. A tilt makes the left and right orders asymmetric and changes the effective angle; using the central maximum as the origin does not fully correct for a tilt of the grating plane.
- Grating spacing label: 600 lines/mm is a nominal value; the actual spacing may differ by a percent or so. If d is smaller than assumed, the calculated λ is smaller.
- Measuring to the nearest 5 mm on a 0.43 m distance is a 1.2% random uncertainty, not systematic.
- Room lights add background and make the spot edges fuzzy, which is mainly random but can bias the estimated center.
The largest systematic effect is probably the geometry: screen tilt and grating tilt. A 5–10% error in x or in the effective angle is plausible, and that is the size of the discrepancy.
3. What should the student conclude about the laser wavelength, and what follow-up measurement would be most informative?
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The data do not support the labelled 632.8 nm. The exact first-order analysis gives about 562 nm, and the second-order result is even less consistent, so the experiment has a significant systematic error rather than a small random discrepancy. The student should not conclude that the laser is mislabelled; a classroom grating measurement with uncertain geometry is much more likely to be wrong than a commercial laser label.
The most informative follow-up is to measure the diffraction angle directly rather than infer it from x and L. Mount the grating on a rotation stage or use a spectrometer, and measure the angular positions of the +1 and −1 orders on both sides of the central maximum. This cancels several alignment errors and gives θ directly. Also measure L along the beam direction, ensure the screen is perpendicular, and repeat with a second known grating or a known laser. If the angle measurement still gives 562 nm, then the grating spacing or the laser label becomes the suspect; until then, the geometry is the first thing to fix.