Write down the condition of constructive interference.

Explanation

Constructive interference :

According to figure (a) suppose at `S_(1)Q=7lamda` and `S_(2)Q=9lamda` distance Q is the point.
`:.S_(2)Q-S_(1)Q=9lamda-7lamda=2lamda`
Hence, the waves emanating from `S_(1)` will arrive exactly two cycles earlier than the waves from `S_(1)` and will again be in phase.
Thus, if the displacement produced by `S_(1)` is given by
`y_(1)=acosomegat` then the displacement produced by `S_(2)` will be given by
`y_(2)=acos(omegat-4pi)`
[ `:.` Path difference `2lamda` equivalent phase difference `=2xx2pi=4pi`]
`[:.lamda=2pi]`
`:.y_(2)=acosomegat`
`[:.cos(theta-4pi)=costheta]` Hence, both waves will be in phase together.
Here distance between `S_(1)` and `S_(2)` is d and the distance `S_(1)Q` and `S_(2)Q` are much greater than `theta`. So that although `S_(1)Q` and `S_(2)Q` are not equal the amplitude of the displacement produced by each wave very nearly the same.
Hence intensity due to constructive interference at point P will be `4I_(0)`, where `I_(0)` is the intensity of each wave.
Here path difference, `S_(2)Q-S_(1)Q=nlamda" "` [where n=0, 1, 2, 3, .......,]
Conditions of constructive interference : That is if the path difference between superimposed waves at a point is `nlamda`, where n=0, 1, 2, 3, ......... hen the intensity at this point is maximum `(4I_(0))` and constructive interference will be formed. Now we know that `lamda` path difference = `2pi` rad.
`:.` If the phase difference between superimposing waves is 2nt, where n = 0, 1, 2, 3, ... then the intensity will be maximum and constructive interference will be formed.
Now,
According to figure `S_(1)R=9.75lamdaandS_(2)R=7.25lamda`
`:.S_(1)R-S_(2)R=9.75lamda-7.25lamda`
`=2.5lamda`
Thus, the waves emanating from `S_(1)` will arrive exactly 2.5 cycles later than the waves from `S_(y)`
Thus if the displacement produced by `S_(1)` is given by
`y_(1)=acosomegat` then the displacement produced by `S_(2)` will be given by
`y_(2)=acos(omegat+5pi)" "[:.2.5lamda=5pirad]`
`:.y_(2)=-acosomegat`
[where `cos{(2n+pi)+theta}=-costheta`]
he displacements of this both waves is in the opposite phase of each other hence resultant displacement will be zero and intensity will get zero. This is called destructive interference.
Thus, the path difference between waves emanating from `S_(1)` and `S_(2)` at point R is,
`S_(1)R-S_(2)R=(n+(1)/(2))lamda` where n=0, 1, 2, 3,........,
Conditions of destructive interference : If the path difference between superposed waves at a point is `(2n+1)(lamda)/(2)` where n=0, 1, 2, 3, ......, then the intensity of light at that point will be zero. This is called destructive interference.
If the phase difference between superpose waves at a point is `(n+(1)/(2))pior{(2n+1)pi}` where n = 0, 1, 2, 3, ..., then the intensity of light at that point will be zero. This is called destructive interference.