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

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

so, $P O=d \sec \theta$
and $C O=P O \cos 2 \theta=d \sec \theta \cos 2 \theta$
so, $\Delta=(d \sec \theta +d \sec \theta \cos 2 \theta)+\frac{\lambda}{2}$
Phase difference between two rays is $\phi=\pi$ (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., $\lambda, 2 \lambda, 3 \lambda$
$\therefore d \sec \theta+ d \sec \theta \cos 2 \theta+\frac{\lambda}{2}=\lambda$
$\therefore d \sec \theta(1+\cos 2 \theta)=\frac{\lambda}{2}$
$\therefore d \sec \theta\left(2 \cos ^{2} \theta\right)=\frac{\lambda}{2}$
$\therefore \cos \theta=\frac{\lambda}{4 d}$
Comparing with $Q \frac{\lambda}{d}$
$\Rightarrow Q=\frac{1}{4}=0.25$