Consider two wave equation Y1=acosωt & Y2=acos(ω+ϕ) the displacement equation producing interference pattern. Show that intensity is given by l=4a2cos2(ϕ/2)
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Solution
Let the displacement of the wave from the sources S1 and S2 at point P on the screen at any time t be given by:
y1=acosωt
y2=acosωt+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=acosωt+acos(ωt+ϕ)
y=2a[cos(ωt+ωt+ϕ2)cos(ωt−ωt−ϕ2)
y=2acos(ωt+ϕ2)cos(ϕ2)....(i)
Let 2acos(ϕ2)=A....(ii)
Then, equation (i) becomes:
y=Acos(ωt+ϕ2)
Now, we have
A2=4a2cos2(ϕ2)
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: