Optics Question 462
Question: In the adjacent diagram, CP represents a wave front 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
Options:
A) $ \cos \theta =3\lambda /2d $
B) $ \cos \theta =\lambda /4d $
C) $ \sec \theta -\cos \theta =\lambda /d $
D) $ \sec \theta -\cos \theta =4\lambda /d $
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Answer:
Correct Answer: B
Solution:
[b]
$ \therefore PR=d\Rightarrow 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+cos2\theta )=\frac{\lambda }{2} $
Or $ \frac{d}{\cos \theta }(2cos^{2}\theta )=\frac{\lambda }{2}\Rightarrow \cos \theta =\frac{\lambda }{4d} $
