AS 91523 · 2023 paper · diffraction glasses

Question 3(c) Merit to Excellence

At a fair, children buy “Rainbow Glasses” made of diffraction gratings in a cardboard frame. Steve shines a laser pointer of wavelength 643 nm (6.43 × 10⁻⁷ m) through one of the gratings onto a wall. The pattern is a bright red spot at the centre, with slightly dimmer red spots either side. The wall is 1.43 m from the grating, and the distance from the central bright spot to the second dimmer spot is 1.75 m.

λ	643 nm (6.43 × 10⁻⁷ m)
grating to wall	1.43 m
centre to 2nd spot	1.75 m

Calculate the slit separation in the grating.

The 2023 paper, page 6 ↗ Marking schedule, page 5 ↗

Work it out

Try it on paper first. Stuck? Take one hint at a time.

1 Which order is the spot 1.75 m from the centre?
The second dimmer spot out: n = 2.
2 Is the angle small enough for nλ = dx / L?
No. A spot 1.75 m out on a wall only 1.43 m away is at a large angle: tan θ = 1.75 / 1.43 = 1.2238, so θ = 50.75°.
3 Now use the grating equation.
d = nλ / sin θ = 2 × 6.43 × 10⁻⁷ / sin 50.75° = 1.286 × 10⁻⁶ / 0.7744 = 1.661 × 10⁻⁶ m
4 Does that fit the pattern on the wall?
n_max = d / λ = 1.661 × 10⁻⁶ / 6.43 × 10⁻⁷ = 2.58, so orders 0, 1 and 2 exist: five spots, as the paper shows. (About 600 lines per millimetre.)

Answer 1.66 × 10⁻⁶ m.

What the marker wanted

Achieved The small-angle formula with the values put in correctly, but the wrong answer (x / 2 = 0.875 m with n = 1 is allowed).
Merit The calculation with one error, or the small-angle formula carried through, giving 1.05 × 10⁻⁶ m.
Excellence The right calculation, 1.66 × 10⁻⁶ m.

In our words, from NZQA’s 2023 marking schedule.

Common mistake

Using nλ = dx / L. At 50.75° the small-angle form is far off and gives 1.05 × 10⁻⁶ m, which the schedule credits at Merit only. NZQA’s 2023 report names seeing that it fails at large angles as an Excellence skill.

How did it go?

Got it moves it further out. Review brings it back tomorrow.