Diffraction grating equation for angular positions of maxima

Steps:

• Calculate wavelength: λ = c / f = (3.0 × 10^8 m/s) / (8.7 × 10^14 Hz) ≈ 3.45 × 10^{-7} m.
• Calculate line spacing: d = 1 / (4.0 × 10^3 lines/m) = 2.5 × 10^{-4} m.
• Apply grating equation sin θ_m = m λ / d to find θ_2 = arcsin(2 λ / d) ≈ 21° and θ_3 = arcsin(3 λ / d) ≈ 33°.
• Compute separation: θ_3 - θ_2 ≈ 12°.

Why A is correct:

• The grating equation d sin θ_m = m λ defines positions of order m maxima; the difference θ_3 - θ_2 gives the angle between adjacent second- and third-order maxima on the same side.

Why the others are wrong:

• B. 21° matches θ_2, the position of the second-order maximum from center.
• C. 33° matches θ_3, the position of the third-order maximum from center.
• D. 54° matches θ_2 + θ_3, the span between second-order on one side and third-order on the opposite side.

Final answer: A