Diffraction Grating: Choosing the Right Measurement
A student uses a diffraction grating labelled '300 lines per mm' with a red laser of wavelength 632.8 nm. A screen is placed 1.20 m from the grating. The student measures the distance from the central maximum to the first-order bright spot on the screen as 0.28 m, and to the second-order spot as 0.62 m. Another student says the second-order measurement must be wrong because 'the spacing should double'. The room lights are on during the first measurements. The grating label is printed, not verified.
1. Use the first-order measurement to calculate the grating spacing and the implied number of lines per mm, and compare with the label.
Read the worked answer
Slit spacing d = 1 / (300 × 10³ m⁻¹) = 3.33 × 10⁻⁶ m from the label.
For the first order, tan θ = 0.28 / 1.20 = 0.2333, so θ = 13.1°. Then sin θ = 0.2267.
Using d sin θ = λ: d = λ / sin θ = 632.8 × 10⁻⁹ / 0.2267 = 2.79 × 10⁻⁶ m.
Lines per metre = 1 / d = 3.58 × 10⁵, i.e. about 358 lines per mm, not 300.
So either the label is inaccurate or the distance measurement is off. The discrepancy is about 19%, which is too large to ignore. A good answer notes that the label should be treated as a claim to be tested, not a given fact, and that the measured geometry gives a more direct value of d.
2. Check whether the second-order measurement is consistent with the first-order result, and explain why the spacing does not simply double.
Read the worked answer
Using d = 2.79 × 10⁻⁶ m from the first order, predict the second-order position.
sin θ₂ = 2λ / d = 2 × 632.8 × 10⁻⁹ / 2.79 × 10⁻⁶ = 0.4537, so θ₂ = 27.0°.
Then tan θ₂ = 0.5095, and the screen distance y = 1.20 × 0.5095 = 0.611 m.
The measured 0.62 m is close to 0.61 m, so the second-order measurement is consistent. The claim that the spacing should double is wrong because the condition is d sin θ = nλ, and sin θ is not proportional to tan θ. For small angles the two are nearly equal and doubling is approximately true, but at 27° the difference is significant. The student's error is applying a small-angle approximation outside its range.
3. Suggest two improvements to the experiment and justify each in terms of reducing uncertainty.
Read the worked answer
Turn off the room lights and use a dark room. Stray light makes faint higher-order spots hard to locate and can shift the apparent centre of a bright fringe, increasing reading uncertainty. Removing it sharpens the pattern so the position can be measured more precisely.
Measure the distance between the two symmetric first-order spots (2y) rather than from the centre to one spot, then halve it. This removes any error in locating the exact centre of the central maximum and cancels a systematic offset in the ruler placement.
Other acceptable improvements: use a travelling microscope or a ruler clamped to the screen to avoid parallax; increase the screen distance to spread the pattern so a millimetre reading is a smaller fraction of the total; take several orders and plot sin θ against n, using the gradient to find d, which averages random errors. Each must be justified by the specific uncertainty it reduces.