Waves

Reading cue. θ is measured from the straight-through n = 0 beam, not from the grating face, and 500 lines per mm must first be inverted to give the line spacing d.

1. aCalculate Calculate the distance between the centres of adjacent lines on the grating labelled in Fig. 9.1.

   recall2 marks

   Check answer · 2 marks

   1. d = 1 mm / 500
   2. d = 2.0 × 10⁻⁶ m

2. bDetermine Determine the angle between the two first-order beams drawn either side of the straight-through direction.

   routine3 marks

   Check answer · 3 marks

   1. sin θ = λ/d = 590 × 10⁻⁹ / 2.0 × 10⁻⁶ = 0.295
   2. θ = 17.2°
   3. the two n = 1 beams are symmetrical about n = 0, so the angle between them is 34.3°

3. cDetermine The source is replaced by one giving white light of wavelengths from 400 nm to 700 nm. Determine the angular width of the first-order spectrum.

   demanding3 marks

   Check answer · 3 marks

   1. sin θ = 400 × 10⁻⁹ / 2.0 × 10⁻⁶ = 0.200, giving θ = 11.5°
   2. sin θ = 700 × 10⁻⁹ / 2.0 × 10⁻⁶ = 0.350, giving θ = 20.5°
   3. angular width = 20.5 − 11.5 = 9.0°

4. dDeduce With the white-light source still in place, deduce whether the second-order and third-order spectra overlap.

   top of the paper4 marks

   Check answer · 4 marks

   1. second order at 700 nm: sin θ = 2 × 700 × 10⁻⁹ / 2.0 × 10⁻⁶ = 0.700, θ = 44.4°
   2. third order at 400 nm: sin θ = 3 × 400 × 10⁻⁹ / 2.0 × 10⁻⁶ = 0.600, θ = 36.9°
   3. the third-order violet leaves at a smaller angle than the second-order red, so the two spectra overlap
   4. the third order is incomplete: sin θ would exceed 1 beyond about 670 nm, so its red end is missing