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(B) From the geometry,the distance from $O$ to the line containing $P$ is $d$. Thus,$PO = d \sec 	heta$.Since $CP$ is a wavefront,the optical path le

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$\cos 	heta = \lambda / 4d$

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(B) From the geometry,the distance from $O$ to the line containing $P$ is $d$. Thus,$PO = d \sec 	heta$.Since $CP$ is a wavefront,the optical path length from $C$ to $P$ is equal to the path length from $O$ to $P$ along the ray $AO$. The path difference between the ray $BP$ and the reflected ray $OP$ is $\Delta = CO + OP$.In $	riangle COP$,$CO = PO \cos 2	heta = d \sec 	heta \cos 2	heta$.Thus,$\Delta = d \sec 	heta + d \sec 	heta \cos 2	heta = d \sec 	heta (1 + \cos 2	heta) = d \sec 	heta (2 \cos^2 	heta) = 2d \cos 	heta$.Since the ray $OP$ undergoes reflection at the surface $QR$,there is an additional phase shift of $\pi$,which corresponds to a path difference of $\lambda / 2$.For constructive interference,the total path difference must be an odd multiple of $\lambda / 2$ (due to the $\pi$ phase shift): $\Delta = \lambda / 2$.$2d \cos 	heta = \lambda / 2 \implies \cos 	heta = \lambda / 4d$.

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

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