question_answer

In the adjacent diagram, CP represents a wavefront and AO & BP, the corresponding two rays. Find the condition on q for constructive interference at P between the ray BP and reflected ray OP                                [IIT-JEE (Screening) 2003]

A)            cosq = 3l/2d

B)            cosq = l/4d

C)            secq ? cosq = l/d

D)            secq ? cosq = 4l/d

Correct Answer: B

Solution :

             \[\because \]PR = d  Þ PO = d secq  and CO = PO cos 2q \[=d\sec \theta \cos 2\theta \] is Path difference between the two rays D = CO + PO = (d secq + d secq cos 2q) Phase difference between the two rays is f = p (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 \[\frac{d}{\cos \theta }(2{{\cos }^{2}}\theta )=\frac{\lambda }{2}\] Þ \[\cos \theta =\frac{\lambda }{4d}\]