Light is incident normally on a grating 0.25 cm wide with 1250 lines.

Data : Number of lines / cm (N)= 5000

m=1

λ1=5895 Å

λ2=5901 Å

Formula :

(a+b)sinθ=Nλ

mN=λ/dλ

Calculations :

—1

—-2

—3

Number of lines / cm required for grating is 983

Number of lines / cm available for grating is 5000

angular separation is 0.0180

This lines will be resolved well by grating

Angular separation ≈ 0.018° (about 1.08′); yes, the lines are resolved, because the grating has 1250 lines while only about 983 are needed.

Grating element:

\[d = \frac{0.25\times10^{-2}\ \text{m}}{1250} = 2\times10^{-6}\ \text{m}\]

First order (n = 1): \(d\sin\theta = n\lambda\)

\[\sin\theta_1 = \frac{5895\times10^{-10}}{2\times10^{-6}} = 0.29475\]

\[\theta_1 = 17.143^\circ\]

\[\sin\theta_2 = \frac{5901\times10^{-10}}{2\times10^{-6}} = 0.29505\]

\[\theta_2 = 17.161^\circ\]

\[\Delta\theta = 0.018^\circ \approx 1.08'\]

Resolution: resolving power needed is

\[\frac{\lambda}{d\lambda} = \frac{5895}{6} \approx 983\]

Resolving power of the grating in first order is \(nN = 1 \times 1250 = 1250\), where \(N\) is the total number of lines.

Since 1250 > 983, the two sodium lines can be seen as separate lines in the first order.