question_answer

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

A) \[\cos \theta =3\lambda /2d\]

B) \[\cos \theta =\lambda /4d\]

C) \[\sec \theta -\cos \theta =\lambda /d\]

D) \[\sec \theta -\cos \theta =4\lambda /d\]

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  \[\frac{d}{\cos \theta }(2co{{s}^{2}}\theta )=\frac{\lambda }{2}\Rightarrow \cos \theta =\frac{\lambda }{4d}\]