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 $\varphi =\pi$(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 $\theta \left(1+cos2\theta \right)=\frac{\lambda }{2}$
or $\frac{d}{cos\theta }\left(2co{s}^2\theta \right)=\frac{\lambda }{2}$
$\Rightarrow$ cos $\theta =\frac{\lambda }{4d}$