Question

Sodium light is incident normally on a grating of width 4×10−3m. The total number
of lines on the grating is 2000. Determine the angular separation between the sodium
D-lines in the first order spectrum.

Answer 1

Answer on Question #75230, Physics / Optics

Sodium light is incident normally on a grating of width $4 \times 10^{-3} \, \text{m}$. The total number of lines on the grating is 2000. Determine the angular separation between the sodium D-lines in the first order spectrum.

Solution:

This light contains two closely spaced lines (the well-known sodium doublet) of wavelengths $589 \, \text{nm}$ and $589.59 \, \text{nm}$.

The grating spacing $d$ is given by

$$d = \frac{l}{N} = \frac{4 \times 10^{-3} \, \text{m}}{2000} = 2 \times 10^{-6} \, \text{m}$$

For first line

$$d \sin \theta_1 = m \lambda_1$$

So,

$$\theta_1 = \sin^{-1} \left(\frac{m \lambda_1}{d}\right) = \sin^{-1} \left(\frac{1 \times 589 \times 10^{-9}}{2 \times 10^{-6}}\right) = 0.298932 \, \text{rad} = 17.13{}^\circ = 17{}^\circ 7' 39''$$

For second line

$$\theta_2 = \sin^{-1} \left(\frac{m \lambda_2}{d}\right) = \sin^{-1} \left(\frac{1 \times 589.59 \times 10^{-9}}{2 \times 10^{-6}}\right) = 0.299241 \, \text{rad} = 17.15{}^\circ = 17{}^\circ 8' 43''$$

The angular separation is

$$\Delta \theta = \theta_2 - \theta_1 = (0.299241 - 0.298932) \, \text{rad} = 0.000309 \, \text{rad} = 1' 4''$$

Answer: $0.000309 \, \text{rad} = 1' 4''$

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