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

Answer

B

Explanation

`PR = d , PO= d sec theta` , and
`CO = PO cot 2theta = d sec theta cos 2 theta`
Optical path difference between the two rays is
`Delta x = CO+PO=d sec theta+d sec theta cos 2theta + (lambda)/(2)`
Path difference between the two rays is
`Delta phi = 2n pi , n = 0,1,2,3, .....`
Condition for constructive interference should be .....
`Delta x = n lambda , n = 0,1,2,3.....`
`d sec theta [1+cos theta] = (lambda)/(2) ((d)/(cos theta)) (2cos^(2) theta) = (lambda)/(2)`
`cos theta = (lambda)/(4d)`