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. `cos theta=3lambda//2d`
B. `cos theta=lambda//4d`
C. `sec theta-cos theta=lambda//d`
D. `sec theta-cos theta=4lambda//d`

Answer

B

Explanation

Path difference between the two rays is given by

`Delta=CO+PO`
`because PR=d`
So, `PO=dsec thetha`
and `CO=PO cos 2theta=dsec thetacos 2theta`
So, `Delta=(d sec theta+d sec theta cos 2 theta)`
Phase difference between two rays is `phi=pi` (as one ray is reflected one and another is direct)
Now, for constructive interference path difference should be even multiple of half wavelength
i.e., `Delta=lambda//2, 3lambda//2`
So, `d sec theta+d sec theta cos 2theta=lambda/2`
or `d sec theta(1+cos 2theta)=lambda/2`
or `d sec theta(2cos^(2)theta)=lambda/2`
`:. cos theta=lambda//4d`