1.

On fig. CP represents a wavefront and AO and BP, the corresponding two rays. Find the condition on theta for constructive interference at P between the ray BP and reflected ray OP :

Answer»

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

Solution :The path difference at POINT P between the RAY BP and reflected ray OP is `Deltax` = O + OC...(1)
In `DeltaOPS , OP = (PS)/(cos theta) = (d)/(cos theta)`" `(because PS = d)`
In `DeltaCOP , (OC)/(OP) = cos 2 theta`
`therefore` From equation (1) , we GET
`Deltax = OP+ OP cos 2 theta = OP(1 + cos 2 theta)`
` = (d)/(cos theta) (1 + 2 cos^(2) theta - 1) = 2 d cos theta`

The condition of constructive interference at P is :
`Deltax = (lambda)/(2) .(3lambda)/(2)`,...etc
or `2d cos theta = (lambda)/(2),(3 lambda)/(2)`...etc
or `cos theta = (lambda)/(4d),(3 lambda)/(4d)`...etc.


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