The diffraction grating

AS Level Physics · Superposition. A short explanation of the idea, the rules to remember, the mistake to avoid, a worked example and practice questions with answers.

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The idea

A diffraction grating is a slide with thousands of equally spaced slits. Light spreads out from every slit, and the waves add up strongly only in a few directions. In these directions the path difference between light from neighbouring slits is a whole number of wavelengths.

Each bright direction is called an order. The zero order goes straight through. The first order (n = 1) appears on both sides of it, then the second order, and so on. Because there are so many slits, the maxima are sharp, bright and far apart.

To find a wavelength, shine the light at right angles onto a grating of known slit spacing, measure the angle of one order and use the grating equation.

Rules to remember

Common mistake

Students use the number of lines per mm as d. First change it to lines per metre, then take 1 ÷ that number to get the slit spacing d in metres.

Worked example

A grating has 400 lines per mm. Light of wavelength 650 nm is incident normally. At what angle is the second-order maximum? (sin 31.3° = 0.52)

  1. 400 lines per mm = 4.0 × 105 lines per m, so d = 1 ÷ (4.0 × 105) = 2.5 × 10−6 m.
  2. sin θ = nλ/d = (2 × 6.5 × 10−7) ÷ (2.5 × 10−6) = 1.3 × 10−6 ÷ 2.5 × 10−6 = 0.52.
  3. θ = sin−1(0.52) = 31.3°.

Answer: The second-order maximum is at 31.3° to the straight-through direction.

Practice questions

Try each one, then open the answer.

1. A diffraction grating has 300 lines per mm. Light of wavelength 550 nm is incident normally. At what angle to the normal is the second-order maximum?

  1. A
    9.5°
  2. B
    19.3°
  3. C
    29.7°
  4. D
    41.3°
Show answer

Answer: B. d = 1 ÷ 300 mm = 3.33 × 10−6 m. sin θ = 2 × 550 × 10−9 ÷ 3.33 × 10−6 = 0.330, so θ = 19.3°. 9.5° is the first order.

2. Why is a diffraction grating better than a double slit for measuring the wavelength of light?

  1. A
    it gives sharp, bright maxima at large angles, which can be measured precisely
  2. B
    it needs no light source
  3. C
    it produces more closely spaced fringes
  4. D
    it works only with white light
Show answer

Answer: A. Many slits make the maxima much sharper and brighter, and the angles are large, so the percentage uncertainty in θ is small.

3. Using the same grating (d = 2.0 × 10−6 m) and 500 nm light, what is the highest order of maximum that can be observed?

  1. A
    2
  2. B
    3
  3. C
    4
  4. D
    5
Show answer

Answer: C. nmax = d/λ = 2.0 × 10−6 ÷ 5.0 × 10−7 = 4 (sin θ cannot exceed 1).

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