In the adjacent diagram, $CP$ represents a wavefront and $AO$ and $BP$, the corresponding two rays. Find the condition on $ heta$ for constructive interference at P between the ray $BP$ and reflected ray $OP$ [image:10_image5.emf]

Question

In the adjacent diagram, $CP$ represents a wavefront and $AO$ and $BP$, the corresponding two rays. Find the condition on $	heta$ for constructive interference at P between the ray $BP$ and reflected ray $OP$ [image:10_image5.emf]

Dr. Abubacker · Accepted Answer

(b) $PR = d$
$PO = d\sec	heta$
And $CO = PO\cot{\ 2	heta} = d\sec	heta\cos{2	heta}$
Path difference between the two rays is,
$\mathrm{\Delta}x = CO + PO$
$= (d\sec	heta + \ d\sec	heta\cos{2	heta})$
Phase difference between the two rays is
$\mathrm{\Delta}\phi = \pi$ (one is reflected, while another is direct)
Therefore condition for constructive interference should be
[image:10_image40.emf]
$\Delta x = \frac{\lambda}{2},\frac{3\lambda}{2}\ldots.$
Or $d\sec	heta\left( 1 + \cos{2	heta} ight) = \frac{\lambda}{2}$
Or $\left( \frac{d}{\cos	heta} ight)\left( 2\cos^{2}	heta ight) = \frac{\lambda}{2}$
Or $\cos	heta = \frac{\lambda}{4d}$

Q. 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 $OP$

Explanation

Reviewed by Dr. Abubacker

Want more practice?