Measuring wavelength with a diffraction grating

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Recall questions on Measuring wavelength with a diffraction grating.

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Questions from this subtopic

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1. 1

   How is the grating spacing d obtained from the number of lines per millimetre?

   Model answer

   The spacing is the reciprocal of the number of lines per unit length. A grating with 300 lines per millimetre has d = 1 ÷ 300 mm = 3.33 × 10⁻⁶ m.

2. 2

   State the diffraction grating equation and define each symbol with its unit.

   Model answer

   The equation is d sin θ = nλ, where d is the grating spacing in metres, θ is the angle of the maximum from the straight-through direction, n is the order as a whole number and λ is the wavelength in metres.

3. 3

   Light of wavelength 500 nm falls normally on a grating with 600 lines per mm. Calculate the angle of the second order maximum.

   Model answer

   The spacing is d = 1 ÷ 600 000 = 1.667 × 10⁻⁶ m, so sin θ = 2λ/d = (2 × 5.00 × 10⁻⁷) ÷ (1.667 × 10⁻⁶) = 0.600 and θ = 36.9°. Substituting back, d sin θ = 1.667 × 10⁻⁶ × 0.600 = 1.00 × 10⁻⁶ m, and twice the wavelength is also 1.00 × 10⁻⁶ m, as required for the second order.

4. 4

   Light of wavelength 500 nm falls normally on a grating with 600 lines per mm. Calculate the highest order that can be observed.

   Model answer

   The spacing is 1.667 × 10⁻⁶ m and the highest order is limited by d/λ = (1.667 × 10⁻⁶) ÷ (5.00 × 10⁻⁷) = 3.33, so the highest order is the third. Substituting back, n = 3 gives sin θ = 0.900 and an angle of 64.2°, while n = 4 would require sin θ = 1.20 and no such angle exists.

5. 5

   Why can the order number n never exceed d/λ?

   Model answer

   The equation gives sin θ = nλ/d, and the sine of an angle cannot be greater than 1. That limits nλ to no more than d.

6. 6

   A grating is illuminated normally with light of wavelength 6.9 × 10⁻⁷ m. Calculate the grating spacing if the first order maximum lies at 12.0°.

   Model answer

   Rearranging d sin θ = nλ with n = 1 gives d = λ / sin θ = (6.9 × 10⁻⁷) ÷ sin 12.0° = (6.9 × 10⁻⁷) ÷ 0.2079 = 3.3 × 10⁻⁶ m, a grating of about 300 lines per mm.

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