Question

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

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

Answer 1

Answer: (B)

$\because PR = d \Rightarrow PO =d$ sec $\theta$ and
$CO = PO$ cos $2\theta =d$ sec $\theta$ cos $2\theta$ is Path difference between the two rays

$\Delta = CO +PO =$ (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
$\Delta =\frac{\lambda}{2} \frac{3\lambda}{2}...$
or d sec $\theta \left(1 +cos 2\theta\right) = \frac{\lambda}{2}$
or $\frac{d}{cos \theta} \left(2 cos ^{2} \theta\right) =\frac{\lambda}{2}$
$\Rightarrow $ cos $\theta = \frac{\lambda}{4d}$