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
`Deltax=CO+OP`
`DeltaOPD, OP=d//cos theta`
`DeltaOPC, CO=OP cos 2theta=d cos 2theta//cos theta`
One ray is reflected
`Deltax^(')=CO+OP-lambda//2=nlambda`
`d/(cos theta)(1+cos 2theta)=(n+1/2)lambda`
`n=0`
`d/(costheta)(1+2 cos^(2)theta-1)=lambda/2`
`cos theta=lambda/(4d)`