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Question

Derive the expression for the intensity at a point where interference of light occurs. Arrive at the conditions for the maximum and zero intensity.

Solution
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Let y1 and y2 be displacement of two waves having same amplitude a and phase difference ϕ between them.
y1=asinωt
y2=asin(ωt+ϕ)

Resultant displacement is: y=y1+y2
y=asinωt+asin(ωt+ϕ)=asinωt(1+cosϕ)+cosωt(asinϕ)
Rcosθ=a(1+cosϕ)
Rsinθ=asinϕ
y=Rsin(ωt+θ)
Where, R is resultant amplitude at P, I is intensity, squaring the equations we get,
I=R2=a2(1+cosϕ)2+a2(sinϕ)2=2a2(1+cosϕ)=4a2cos2ϕ2

Maximum intensity:
cos2ϕ2=1
ϕ=2nπ where n=0,1,2,3,....
Therefore, Imax=4a2
Minmum intensity:
cos2ϕ2=0
ϕ=(2n+1)π where n=0,1,2,3,....
Therefore, Imax=0

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