Total lines on a grating

A diffraction grating obeys the grating equation (a + b) sin θ = nλ, where (a + b) is the grating element (distance between two adjacent slits) and n is the order. The total number of lines is the width divided by the grating element.

Given: width = 3 cm, n = 2, θ = 33°, λ = 600 nm = 6 × 10⁻⁷ m.

Step 1: grating element

(a + b) = nλ / sin θ
        = 2 × 6×10⁻⁷ / sin 33°
        = 1.2×10⁻⁶ / 0.5446
        = 2.203×10⁻⁶ m = 2.203×10⁻⁴ cm

Step 2: lines per cm

N per cm = 1 / (a + b) = 1 / 2.203×10⁻⁴ ≈ 4539 lines/cm

Step 3: total lines on 3 cm

N = 4539 × 3 ≈ 13 616

Answer: the grating has about 1.36 × 10⁴ lines in total (about 4539 lines per cm).