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प्रश्न
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.
उत्तर
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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