AS 91523 · 2023 paper · diffraction glasses

Question 3(d) Merit

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

Looking at a spot of white light through the glasses, the children see the white spot with spectra on either side (“rainbows”). Give an in-depth explanation: describe where the spectra occur, explain the position of the colours in them and why they are there, and include a labelled sketch.

The 2023 paper, page 7 ↗ Marking schedule, page 6 ↗

Work it out

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

1 Where is the light white, and why?
Straight through, at the centre: at θ = 0 every colour has zero path difference, so they are all bright together and mix into white.
2 For the first order, what path difference does each colour need?
One of its own wavelengths: violet the least (about 400 nm), red the most (about 700 nm).
3 So at what angle is each colour bright?
sin θ = nλ / d: the shorter the wavelength, the smaller the angle. Violet is bright closest to the centre, red furthest out, and the other colours in between.
4 Where do the spectra appear, and what goes in the sketch?
On both sides of the white centre, one complete spectrum for each order (n = 1, 2), each with violet on the inside and red on the outside. Sketch the white central spot labelled n = 0, then on each side the n = 1 and n = 2 spectra, each running violet to red outwards. With these glasses (d = 1.66 × 10⁻⁶ m) the first-order spectrum runs from about 14° (violet) to 25° (red).

Answer A white spot at the centre (n = 0), where every colour has zero path difference, then a complete spectrum on each side for each order. In each spectrum violet is nearest the centre and red furthest out: each colour is bright where the path difference is a whole number of its own wavelengths (d sin θ = nλ), and violet’s short wavelength gets there at a smaller angle than red’s long one.

What the marker wanted

Achieved One idea: a complete spectrum is seen, violet inside and red outside; a whole-wavelength path difference puts the waves in phase; a whole-wavelength path difference gives a maximum; or waves in phase give a maximum.
Merit The right reasoning for violet inside and red outside: red’s longer wavelength needs a larger path difference, so a larger angle in nλ = d sin θ.

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

Common mistake

Putting red on the inside. NZQA’s 2023 report lists spectra drawn the wrong way round among answers that did not reach Achieved. It is the reverse of a prism: with a grating, the shortest wavelength is nearest the centre.

How did it go?

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