Question

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

Answer 1

Given that,

Wave front = CP

Reflect ray = OP

AO and BP the corresponding two rays

Now, according to diagram

$PR=d$

$PO=d\sec \theta$

$CO=PO\cos 2\theta$

Now, put the value of PO

$CO=d\sec \theta \cos 2\theta$

Now, path difference between the two rays

$\Delta =CO+PO$

$\Delta =d\sec \theta +d\sec \theta \cos 2\theta$

Now, phase difference between the two rays

$\varphi =\pi$

Now, for constructive interference should be

$\Delta =\frac{\lambda }{2},\frac{3\lambda }{2}....$

$dsec\theta +dsec\theta cos2\theta =\frac{\lambda }{2}$

$d\sec \theta (1+\cos 2\theta )=\frac{\lambda }{2}$

$\frac{d}{\cos \theta }\left(2{\cos }^2\theta \right)=\frac{\lambda }{2}$

$\cos \theta =\frac{\lambda }{4d}$

$\theta ={\cos }^{-1}\left(\frac{\lambda }{4d}\right)$

Hence, the condition on $\theta$ is ${\cos }^{-1}\left(\frac{\lambda }{4d}\right)$