Interference pattern is observed at 'P' due to superimposition of… - Vidyadip

MCQ

Interference pattern is observed at $'P'$ due to superimposition of two rays coming out from a source $'S'$ as shown in the figure. The value of $'I'$ for which maxima is obtained at $'P'$ is: ( $R$ is perfect reflecting surface)

A
$I\, = \,\frac{{2n\lambda }}{{\sqrt 3 - 1}}$
B
$I\, = \,\frac{{(2n - 1)\lambda }}{{2(\sqrt 3 - 1)}}$
✓
$I\, = \,\frac{{(2n - 1\,\lambda )\sqrt 3 }}{{4(2 - \sqrt 3 )}}$
D
$I\, = \,\frac{{(2n - 1)\lambda }}{{\sqrt 3 - 1}}$

✓

Answer

Correct option: C.

$I\, = \,\frac{{(2n - 1\,\lambda )\sqrt 3 }}{{4(2 - \sqrt 3 )}}$

c
From the figure straight path $\mathrm{SP}=2 l$

Reflected path $\mathrm{SP}=2 l$ sec $30^{\circ}$

So path difference is $2 l\left(\sec 30^{\circ}-1\right)$

Also the ray, when reflected by the mirror, suffers a phase change of $\pi$

So the total difference in phase is $2 l\left(\sec 30^{\circ}-1\right) \times \frac{2 \pi}{\lambda}+\pi$

For constructive interference

$2 l\left(\sec 30^{\circ}-1\right) \times \frac{2 \pi}{\lambda}+\pi=2 n \pi$

Solving this, we get $l=\frac{(2 n-1) \lambda \sqrt{3}}{4(2-\sqrt{3})}$

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