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In the adjacent diagram,$ CP$ represents a wavefront and $AO$ & $BP$, the corresponding two rays. Find the condition on $\theta$ for constructive interference at $P$ between the ray $BP$ and reflected ray $OP$
$cos \theta = 3 \lambda /2d$
$cos \theta = \lambda /4d$
$sec \theta -cos \theta = \lambda /d$
$sec \theta -cos \theta = 4 \lambda /d$
Solution
(b)$PR = d ==> PO = d sec \theta$ and $CO = PO\, cos\, 2 \theta = d \,sec \theta \cos 2 \theta $ is
Path difference between the two rays
$\Delta = CO + PO = (d \,sec \theta + d\, sec\, \theta \,cos\, 2 \theta )$
Phase difference between the two rays is
$\phi = \pi$ (One is reflected, while another is direct)
Therefore condition for constructive interference should be $\Delta = \frac{\lambda }{2},\frac{{3\lambda }}{2}……$
or $d\sec \theta (1 + \cos 2\theta ) = \frac{\lambda }{2}$
or $\frac{d}{{\cos \theta }}(2{\cos ^2}\theta ) = \frac{\lambda }{2}$ ==> $\cos \theta = \frac{\lambda }{{4d}}$
