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. `costheta=(3lamda)/(2d)`
B. `costheta=(lamda)/(4d)`
C. `sectheta-costheta=(lamda)/(d)`
D. `sectheta-costheta=(lamda)/(d)`

Answer

B

Explanation

In `triangleOPM,OP=(d)/(costheta)`
In `triangleCOP`,`OC=(dcos2theta)/(costheta)`
Path difference between the two rays reaching P is `=CO+OP+(limda)/(2)=(dcos2theta)/(costheta)+(d)/(costheta)+(limda)/(2)`
`=(d)/(costheta)(cos2theta+1)+(limda)/(2)``=2d costheta+(limda)/(2)`
For constructive interference, path difference should be n`limda`
:' `2d costheta+(limda)/(2)` `=n`limda` `impliescostheta=((2n-1))/(4) (lambda)/(d)`
For `n=1`, `costheta=(lambda)/(4d)`