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Grating lines per cm, superposed lines

Short answer

Equate n×540=(n+1)×405 gives n=3. Grating spacing d=3×540e−9/sin30°=3.24e−6 m. Lines per cm N=1/d=3086 lines/cm.

A diffraction grating used at normal incidence gives a line (540 nm) in a certain order superposed on the violet line (405 nm) of the next higher order. How many lines per cm are there in the grating if the angle of diffraction is 30°?

A line of wavelength λ₁ in order n is superposed on a line of λ₂ in order (n+1) when the same angle θ satisfies the grating equation for both: d sin θ = nλ₁ = (n+1)λ₂.

Given: λ₁ = 540 nm, λ₂ = 405 nm, θ = 30°.

  1. Equate: n × 540 = (n + 1) × 405
  2. 540n − 405n = 405, so 135n = 405, n = 3. The 540 nm line is third order; the 405 nm line is fourth order.
  3. Grating element: d sin θ = nλ₁
d = nλ₁ / sin θ = 3 × 540×10⁻⁹ / sin 30°
  = 1.62×10⁻⁶ / 0.5
  = 3.24×10⁻⁶ m = 3.24×10⁻⁴ cm
  1. Number of lines per cm:
N = 1/d = 1 / (3.24×10⁻⁴ cm) = 3086 lines/cm

Answer: about 3.09 × 10³ lines per cm (≈ 3086 lines/cm).

✓ Answer checked by Claude Opus 5.5 (high effort)

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