Double-slit fringe spacing from grating-derived wavelength Steps: - Grating equation for first order (m=1): d sin 30° = λ, so λ = d × (1/2) = 0.5d, where d is grating line spacing. - Double-slit separation a = 500 × d (given as 500 times greater than line spacing). - Fringe separation formula: β = λ L / a, with screen distance L = 1.0 m. - Substitute: β = (0.5d × 1.0) / (500d) = 0.5 / 500 = 0.001 m = 1.0 × 10^{-3} m. Why C is correct: - Matches β = (sin θ L) / 500 from combining grating equation d sin θ = λ and double-slit formula β = λ L / a with a = 500d. Why the others are wrong: - A: Errors like using θ in radians without sin (≈0.52/500 ≈ 10^{-3} m) or halving incorrectly yield smaller values. - B: Forgets to multiply by L=1.0 m, giving angular spacing λ/a = 10^{-3} rad interpreted wrongly as 10^{-6} m. - D: Same as B; duplicate error in omitting screen distance factor. Final …