In the below diagram, CP represents a wavefront and AO and BP, the corresponding two rays. Find the condition of `theta` for constructive interference at P between the rays BP and reflected ray AOP

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 // 2d`

Answer

b

Explanation

In `Delta OPM`,
`(OM)/(OP) = cos theta`
`implies OP = (d)/(cos theta)`
In `Delta COP`
`cos 2 theta = (OC)/(OP)`
`implies OC = (d cos 2 theta)/(cos theta)`
Path difference the two rays reaching P is
` CO + OP + (lambda)/(2) = (d cos 2 theta)/(cos theta) + (d)/(cos theta) + (lambda)/(2)`
`= (d)/(cos theta) (cos 2 theta + 1) + (lambda)/(2) 2d cos theta + (lambda)/(2)`
For constructive inteference:
Path difference between the two rays reaching P is `n lambda`.
`:. 2 d cos theta + (lambda)/(2) = n lambda implies 2 d cos theta = (n - (1)/(2)) lambda`
`implies 2 d cos theta = ((2n - 1))/(2) lambda implies cos theta = ((2n - 1))/(2) (lambda)/(2d)`
For `n = 1, cos theta = (lambda)/(4d)`
.