In a Young's double slit experiment, each of the two slits A and B,...

In a Young's double slit experiment, each of the two slits A and B, as shown in the figure, are oscillating about their fixed center and with a mean separation of

0.8 \mathrm{~mm}

. The distance between the slits at time

t

is given by

d=(0.8+0.04 \sin \omega t) \mathrm{mm}

, where

\omega=0.08 \mathrm{rad} \mathrm{s}^{-1}

. The distance of the screen from the slits is

1 \mathrm{~m}

and the wavelength of the light used to illuminate the slits is

6000 Ã…

. The interference pattern on the screen changes with time, while the central bright fringe (zeroth fringe) remains fixed at point

O

In a Young's double slit experiment, each of the two slits A and B, as shown in the figure, are oscillating about their

The maximum speed in

\mu \mathrm{m} / \mathrm{s}

at which the

8^{\text{th }}

bright fringe will move is ________ .

Correct answer

Step-by-step solution

\begin{aligned} & \mathrm{y}=\mathrm{n} \cdot \frac{\lambda \mathrm{D}}{\mathrm{d}} \\ & \mathrm{v}=\frac{\mathrm{dy}}{\mathrm{dt}}=-\mathrm{n} \cdot \frac{\lambda \cdot \mathrm{d}}{\mathrm{d}^2} \cdot \frac{d(d)}{d t} \\ & d=0.8+0.04 \sin \omega t \\ & \frac{d(d)}{d t}=0.04 \omega \cos \omega t \end{aligned}

\text{\text{ for } } \mathrm{v} \rightarrow \max \Rightarrow \frac{\mathrm{d}(\mathrm{d})}{\mathrm{dt}} \rightarrow \max \text{. }

For

\frac{\mathrm{d}(\mathrm{d})}{\mathrm{dt}} \rightarrow \max

.

\begin{aligned} & \cos \omega \mathrm{t}=1 \Rightarrow \sin \omega \mathrm{t}=0 \\ & \Rightarrow\left(\frac{\mathrm{d}(\mathrm{d})}{\mathrm{dt}}\right)_{\max }=0.04 \\ & \Rightarrow \mathrm{d}=0.8 \mathrm{~mm} \\ & \mathrm{v}_{\max }=\frac{8 \times 6000 \times 10^{-10} \times 1 \times 0.04 \times 0.08}{0.8 \times 0.8 \times 10^{-6} \times 10^{-3}}=24 \mu \mathrm{m} / \mathrm{s} . \end{aligned}

$ (range is from 23.5 to 24.5)

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