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Question
Light waves each of amplitude "a" and frequency "ω", emanating from two coherent light sources superpose at a point. If the displacements due to these waves are given by y1 = a cos ωt and y2 = a cos(ωt + ϕ) where ϕ is the phase difference between the two, obtain the expression for the resultant intensity at the point.
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Solution
Let the displacement of the waves from the sources S1 and S2 at point P on the screen at any time t be given by:
y1 = a cos ωt
and
y2 = a cos (ωt + Φ)
where, Φ is the constant phase difference between the two waves
By the superposition principle, the resultant displacement at point P is given by:
y = y1 + y2
y = a cos ωt + a cos (ωt + Φ)
`=2a[cos((omegat+omegat+phi)/2)cos((omegat-omegat-phi)/2)]`
`y=2acos(omegat+phi/2)cos(phi/2)" ...(i)"`
Let 2 `acos(phi/2)=A ...(ii)"`
Then, equation (i) becomes:
`y=Acos(omegat+phi/2)`
Now, we have:
`A^2=4a^2cos^2(phi/2)" ..(iii)"`
The intensity of light is directly proportional to the square of the amplitude of the wave. The intensity of light at point P on the screen is given by:
`I=4a^2cos^2(phi/2)" ...(iv)"`
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