In the adjacent diagram, CP represents a wavefront and A O & B P,… - Vidyadip

MCQ

In the adjacent diagram, $CP$ represents a wavefront and $A O \& B P$, the corresponding two rays. Find the condition on $\theta$ for constructive interference at $P$ between the ray $B P$ and reflected ray $O P$

A
$\cos \theta=3 \lambda / 2 d$
✓
$\cos \theta=\lambda / 4 d$
C
$\sec \theta-\cos \theta=\lambda / d$
D
$\sec \theta-\cos \theta=4 \lambda / d$

✓

Answer

Correct option: B.

$\cos \theta=\lambda / 4 d$

$\because P R=d \Rightarrow P O=d \sec \theta \text { and } C O=P O \cos 2 \theta$
$=d \sec \theta \cos 2 \theta \text { is }$

Path difference between the two rays
$\Delta=C O+P O=(d \sec \theta+d \sec \theta \cos 2 \theta)$
Phase difference between the two rays is $\phi=\pi$ (One is reflected, while another is direct)
Therefore condition for constructive interference should be
$\Delta=\frac{\lambda}{2}, \frac{3 \lambda}{2} \ldots \ldots$
$\text { or } d \sec \theta(1+\cos 2 \theta)=\frac{\lambda}{2} $
$\text { or } \frac{d}{\cos \theta}\left(2 \cos ^2 \theta\right)=\frac{\lambda}{2} \Rightarrow \cos \theta=\frac{\lambda}{4 d}$

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