Two coherent sources `S_(1)` and `S_(2)` separated by distance `2lamda` emit light of wavelengths `lamda` is phase as shown in the figure. A circular wire of radius `100lamda` is placed in such a way that `S_(1)S_(2)` lies in its plane and the mid-point of `S_(1)S_(2)` is at the centre of wire.

(i). Find the angular positions `theta` on the wire for which constructive interference takes place. Hence or otherwise find the number of maxima.
(ii). Find the angular position `theta` on the wire of which intensity reduces to half of its maximum value.

Answer

A, B, D

Explanation

As shown in the ure

`(S_1P)^2=(S_1x)^2+(Px)^`………1
`(S_2P)^2=(S_2M)^2+(PM)^2`……..2
from 1 and 2
`(S_1P)^2-(S_2P)^2=(Px)^2-(PM)^2`
`=(1.5lamda+Rcostheta)^2-(Rcostheta-1.5lamda)^2`
`=6lamdacostheta`
`rarr (S_1P-S_2P)^2=(6lamdaRcostheta)/(2R)`
`=3costheta`
For constructive interferene
`(S_1P-S-2P)^2=x=3lamdacostheta=nlamda`
`rarr costheta=n/3`
`theta=cos^-1(n/3)`
`where n=0,1,2,..............
`rarr theta=0^@, 48^@, 2^@, 7^@, 5^@, 90^@ and` similar points in other quadrants.