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

Answer

A

Explanation

PC is wavefront, hence points P and C are in same phase. Paht difference between two interfering rays becomes
`Deltax = OC + OP " " …..(1)`
In triangle OPS we can write the following :
`cos theta = (PS)/(OP) = d/(OP)`
`implies OP = d sec theta " ".....(2)`
In triangle OCP we can write the following
`cos 2 theta = (OC)/(OP)`
`implies OC = OP cos 2 theta`
From equation (2) we can substitute `OP = d//cos theta`
`implies OC = d/(cos theta) cos 2 theta`
`implies OC = d/(cos theta) (2 cos^2 theta - 1)`
`implies OC = 2 d cos theta - d sec theta" ".....(3)`
Substituting values form equations (2) and (3) in equation (1) we get the following:
`Delta x = OC + OP = (2 d cos theta - d sec theta) + (d sec theta)`
`Delta x = 2d cos theta`
For constructive interference path difference between the waves must be integral multiple of wavelength `(lambda)`.
`Delta x = 2 d cos theta = n lambda`
`implies cos theta = (n lambda)/(2d)`
`n = 0,1,2,3,4,.......`.