Question

In the given diagram, $CP$ represents a wavefront and $AO$ and $BP$, the corresponding two rays. The condition of $\theta$ for constructive interference at $P$ between the rays $BP$ and reflected ray $AOP$ is found to be $\cos \theta =Q\frac{\lambda }{d}$. The value of $Q$ is______.

Answer 1

Answer: 0.25

Path difference between the two rays is given by
$\Delta =CO+PO+\frac{\lambda }{2}$

so, $PO=d\sec \theta$
and $CO=PO\cos 2\theta =d\sec \theta \cos 2\theta$
so, $\Delta =(d\sec \theta +d\sec \theta \cos 2\theta )+\frac{\lambda }{2}$
Phase difference between two rays is $\varphi =\pi$ (as one ray is reflected one and another is direct).
Now, for constructive interference, path difference should be even multiple of half wavelength
i.e., $\lambda ,2\lambda ,3\lambda$
$\therefore d\sec \theta +d\sec \theta \cos 2\theta +\frac{\lambda }{2}=\lambda$
$\therefore d\sec \theta (1+\cos 2\theta )=\frac{\lambda }{2}$
$\therefore d\sec \theta \left(2{\cos }^2\theta \right)=\frac{\lambda }{2}$
$\therefore \cos \theta =\frac{\lambda }{4d}$
Comparing with $Q\frac{\lambda }{d}$
$\Rightarrow Q=\frac{1}{4}=0.25$