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.

Options

A. (a) `costheta=3lambda//2d`
B. (b) `costheta=lambda//4d`
C. (c) `sectheta-costheta=lambda//d`
D. (d) `sectheta-costheta=4lambda//d`

Answer

B

Explanation

`:'PR=dimpliesPO=dsectheta`
and `CO=POcos2theta=dsecthetacos 2theta` is

Path difference between the two rays
`D=CO+PO=(dsectheta+dsec theta cos 2theta)`
Phase difference between the two rays is
`phi=pi` (One is reflected, while another is direct)
Therefore condition for constructive interference should be `Delta=lambda/2, (3lambda)/(2)........`
or `d sec theta(1+cos 2 theta)=lambda/2`
or `(d)/(cos theta)(2 cos^2theta)=lambda/2impliescos theta=(lambda)/(4d)`