Question

A transmission diffraction grating with 600 lines/mm is used to study the line spectrum of the light produced by a hydrogen discharge tube with the setup shown above. The grating is 1.0 m from the source (a at the center of the meter stick). An observer sees the first-order red line at a distance yr = 428 mm from the .

a. Calculate the wavelength of the red line in the hydrogen spectrum.

b. Qualitatively describe how the location of the first-order red line would change if a diffraction grating with 800 lines/mm were used instead of the one with 600 lines/mm.

Answer 1

(a) Diffraction grating: Since the screen distance is 1 m and the first order line is at 0.428 m, the angle is not small and the small angle approximation cannot be used. Instead we find the angle with tan θ and use mλ = d sin θ. First find d. d = 600 lines / mm = 1/600 mm / line = 0.00167 mm / line = 1.67x10–6 m / line.

tan θ = o/a … tan θ = 0.428 / 1 … θ = 23° … Then, mλ = d sin θ … (1) λ = (1.67 x 10–6) sin 23° λ = 6.57 x 10–7 m = 657 nm

(b) Referring to the calculation of d above .. d = 1/800 mm / line which is a smaller d value. Less d means sin θ must increase so the angle is larger and the location of the line would be further out.