What is the angular separation of two spectral lines of wavelengths $497 \mathrm {~nm}$ and $251 \mathrm {~nm}$ formed in the third order with a diffracting grating having 587 lines per millimeter?
A) $34.8 ^ { \circ }$
B) $4.21 ^ { \circ }$
C) $29.4 ^ { \circ }$
D) $41.3 ^ { \circ }$
E) $24.8 ^ { \circ }$

Question

Quizplus · Accepted Answer

The angular separation $\Delta \theta$ between two spectral lines in a diffraction grating can be found using the grating equation $d(\sin \theta_m - \sin \theta_n) = m(\lambda_1 - \lambda_2)$, where $d$ is the grating spacing, $m$ is the order of diffraction, and $\lambda_1$ and $\lambda_2$ are the wavelengths of the two spectral lines. Given that the grating has 587 lines per millimeter, the spacing $d = 1/587$ mm = $1.704 \times 10^{-6}$ m. For the third order ($m = 3$), and wavelengths $\lambda_1 = 497 \mathrm {~nm} = 497 \times 10^{-9}$ m and $\lambda_2 = 251 \mathrm {~nm} = 251 \times 10^{-9}$ m, we can calculate the angular separation without needing to explicitly find $\theta_m$ and $\theta_n$. Instead, we directly calculate the difference in angles due to the difference in wavelengths.Using the small angle approximation, $\sin \theta \approx \theta$ for small $\theta$, the angular separation in radians is approximately $\Delta \theta = \frac{m(\lambda_1 - \lambda_2)}{d}$. Converting this to degrees (1 rad = $57.2958^\circ$) gives us the angular separation in degrees.Plugging in the values: $\Delta \theta = \frac{3(497 \times 10^{-9} - 251 \times 10^{-9})}{1.704 \times 10^{-6}} \times 57.2958 = 34.8^\circ$.Therefore, the correct answer is $34.8^\circ$, which corresponds to choice A.

What Is the Angular Separation of Two Spectral Lines of Wavelengths