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In the adjacent diagram, CP represents a...

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.

A

`costheta=(3lambda)/(2d)`

B

`costheta=(lambda)/(4d)`

C

`sectheta-costheta=(lambda)/(d)`

D

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

Text Solution

Verified by Experts

The correct Answer is:
B

Path difference `=CO+(lambda)/(2)+OP`
for constructive interference at `P`
`CO+OP+(lambda)/(2)=nlambda`
`CO+OP=(2n-1)(lambda)/(2)…(1)`
from `DeltaONP,OP=dsectheta`&from `DeltaCOP`
`(CO)/( d sec theta)=cos2theta`
`CO=d sec theta cos 2theta`
Putting the value of `OP`and `CO`in equation(1)
`d sec theta(1+cos 2theta)=(2n-1)(lambda)/(2)`
`d sectheta 2cos^(2)theta=(2n-1)(lambda)/(2)`
`costheta=(2n-1)(lambda)/(4d) rArr costheta=(lambda)/(4d)`
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