Question

Monochromatic green light, of wavelength 500 nm, illuminates two parallel narrow slits 7.70 mm apart. Calculate the angular deviation ( $\mathit{\theta }$in Fig. 35-10) of the third-order $\left(m=3\right)$bright fringe (a) in radians and (b) in degrees.

Short answer

a. The angular separation in radian is $0.214\text{rad}$.

b. The angular separation in degrees is $12.28°$.

Step-by-step solution

Write the given data from the question

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

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

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

Determine the formulas to calculate the angular separation in the degrees and radian

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.

$d\sin \theta =m\lambda$ …… (1)

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

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

${\mathit{\theta }}_{\mathbf{d}\mathbf{e}\mathbf{g}}\mathbf{=}\frac{\mathbf{180}}{\mathbf{\pi }}\mathbf{\times }\mathit{\theta }$ …… (2)

Calculate the angular separation in radian:

a.

For the small angle, $\sin \theta \approx \theta$.

Calculate the angular separation.

Substitute $7.7\mu m$ for d, 3 for m, 550 nm for $\lambda$, and $\theta$for $\sin \theta$into equation (1).

$7.7\times {10}^{-6}\times \theta =3\times 550\times {10}^{-9}$

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

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

Calculate the angular separation in degrees

Calculate the angular separation.

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

$$\begin{gathered} {\theta }_{\mathrm{deg}}=\frac{180}{\pi }\times 0.214 \\ =\frac{38.52}{\pi } \\ =12.28° \end{gathered}$$

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