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

Answer

B

Explanation

`Delta OPM , OP = (d)/( cos theta ) , Delta COP , OC = (d cos 2 theta )/( cos theta )`

Path difference between the two rays reaching P is
`= CO + OP + (lambda)/(2) = (d cos 2 theta )/( cos theta ) + (d)/( cos theta ) + (lambda)/(2)`
`= 2 d cos theta + (lambda)/(2)`
For constructive interference, path difference should be ` n lambda `
` 2 d cos + (lambda)/(2) = n lambda rArr cos theta = (( 2 n - 1)lambda)/( 4 d)`
For n = 1 ` cos theta = (lambda)/( 4 d)`