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=(4lamda)/(d)`

Answer

B

Explanation

In `DeltaOPR,(PR)/(OP)=costhetaimpliesOP=(d)/(costheta)`
in `DeltaCOPcos2theta=(OC)/(OP)` ltbr. `impliesOC=OPcos2theta=(dcos2theta)/(costheta)`
So path difference `=CO+OP+(lamda)/(2)`
`=(dcos2theta)/(costheta)+(d)/(costheta)+(lamda)/(2)`
`=(d(2cos^(2)theta-1))/(costheta)+(d)/(costheta)+(lamda)/(2)=2dcostheta+(lamda)/(2)`
Now for constructive interference at P between B P and OP, path difference `=nlamda`
`implies2dcostheta+(lamda)/(2)=nlamdaimplies2dcostheta=(n-(1)/(2))lamda`
`impliescostheta=((2n-1)/(4d))lamda`:
For `n=1,costheta=(lamda)/(4d)`