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.

Answer 1

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)"`