91Ó°ÊÓ

The wavelength, $\mathrm{λ}=550\text{nm}$.

The order of the bright fringe, $m=3$.

The distance between the slits, $d=7.7\mathrm{μ}\text{m}$.

Young's double-slit experiment. When monochromatic light passing through two narrow slits illuminates a distant screen, a characteristic pattern of bright and dark stripes is observed. This interference pattern is caused by the superposition of overlapping light waves originating from the two slits.

The condition for the maxima in Young’s experiment is given as follows.

$d\sin \mathrm{θ}=m\mathrm{λ}$ …… (1)

Here, d is the distance between the slits, $\mathit{λ}$ is the wavelength, m is the order and $\mathit{θ}$is the angular separation.

The expression to calculate the angular separation in degrees is given as follows.

${\mathit{θ}}_{\mathbf{d}\mathbf{e}\mathbf{g}}\mathbf{=}\frac{\mathbf{180}}{\mathbf{π}}\mathbf{×}\mathit{θ}$ …… (2)

a.

For the small angle, $\sin \mathrm{θ}≈\mathrm{θ}$.

Calculate the angular separation.

Substitute $7.7\mathrm{μ}m$ for d, 3 for m, 550 nm for $\mathrm{λ}$, and $\mathrm{θ}$for $\sin \mathrm{θ}$into equation (1).

$7.7×{10}^{-6}×\mathrm{θ}=3×550×{10}^{-9}$

$$\begin{gathered} \mathrm{θ}=\frac{3×550×{10}^{-9}}{7.7×{10}^{-6}} \\ =\frac{1650×{10}^{-3}}{7.7} \\ =0.214\text{rad} \end{gathered}$$

Hence, the angular separation in radian is $0.214\text{rad}$.

Calculate the angular separation.

Substitute $0.214\text{rad}$for $\mathrm{θ}$into equation (2).

$$\begin{gathered} {\mathrm{θ}}_{\mathrm{deg}}=\frac{180}{\mathrm{π}}×0.214 \\ =\frac{38.52}{\mathrm{π}} \\ =12.28° \end{gathered}$$

Hence,the angular separation in degrees is $12.28°$.