The hydrogen spectrum has a red line at 656 nm, and a blue line at 434 nm. What is the first order angular separation between the two spectral lines obtained with a diffraction grating with 5000 rulings/cm?
A) 7.7°
B) 16.6°
C) 6.6°
D) 3.2°
E) 19.2°

Question

Quizplus · Accepted Answer

Using the grating equation, 
$$ n\lambda = d\left(\sin	heta_m+\sin	heta_iight)$$ where $n$ is the order of the spectrum, $\lambda$ is the wavelength of light, $d$ is the grating spacing (5000 rulings/cm = 50000 rulings/m = 5 $	imes$ 10$^{-5}$ m/ruling), $	heta_m$ is the angle between the m-th diffracted beam and the grating normal, and $	heta_i$ is the angle between the incident light and the grating normal.

To find the angular separation between the red and blue lines, we want to find the difference between the angles for which the red and blue lines are observed at first order spectrum ($n=1$). Thus,
\begin{align*}
d\sin	heta_{1,r} &= \lambda_r \
d\sin	heta_{1,b} &= \lambda_b \
\Rightarrow d\left(\sin	heta_{1,r} - \sin	heta_{1,b}ight) &= \lambda_r - \lambda_b \
\Rightarrow \sin\frac{\Delta	heta_1}{2} &= \frac{\lambda_r - \lambda_b}{2d} \
\Rightarrow \Delta	heta_1 &= 2\sin^{-1}\left(\frac{\lambda_r - \lambda_b}{2d}ight)
\end{align*}

Substituting the values, we get $\Delta	heta_1 \approx 6.6^{\circ}$. Therefore, the answer is C).

The Hydrogen Spectrum Has a Red Line at 656 Nm