Class 12 Physics Wave Optics Constructive Overlap

Constructive Overlap

  • Case 1:-
    • Consider two coherent sources S1 and S2emitting light waves of same frequency and constant phase.
    • The wave fronts of both the sources will overlap with each other.
    • Consider a point P as in the figure to calculate the intensity of disturbance.
    • The distance of point P from S1 and S2 is same. Therefore S1P=S2
    • Let the light wave emitted by wave at S1 y1=a cosωt
      • Where a=amplitude of the wave,y1 =displacement of the wave and cosωt =phase.
    • Light wave emitted by at S2 y2=acosωt
    • Intensity of both the waves =I0∝a2 (equation (1))
    • Resultant displacement of the wave formed by the superposition of the waves
      • y=y1+y2 =2acosωt
    • Intensity I ∝ (Amplitude)2
    • I ∝ (2a)2 => I ∝4a2 where Amplitude=2a.
    • I=4 I0 using equation(1)
    • This means the intensity at point P will be four times the intensity of the individual sources.

 

  • Conclusion: -
    • If a point is equidistant from two sources then the
      • Amplitude as well as the intensity increases.
    • Path difference is defined as the difference in the paths from both the sources to a particular point.
      • This implies S2P - S1P =0.
      • If the path difference is 0 then it will be constructive overlap.
    • Case 2:- Considering a point Q which is not equidistant from the 2 sources and the path difference S1Q - S2Q = 2λ(integral multiple)
      • => As S1Q > S2Q therefore the waves originating from S1 have to travel a greater path than S2.
      • Therefore waves from S2 will reach exactly 2 cycles earlier than waves from S1. Waves reach at S2early by 2λ as compared to S1.
      • One cycle corresponds to λ and two cycles correspond to 2λ.
      • Let the light wave emitted by wave at S1, y1=a cosωt
        • Where a=amplitude of the wave,y1 =displacement of the wave and cosωt =phase.
      • Light wave at S2, y2=acos(ωt -4 π) =a cosωt
        • (Path difference)λ =>2 π (phase difference),therefore 2λ=4π.
        • This shows y1 and y2 are in phase with each other.
        • Resultant y=y1+ y2 =2acosωt.
      • This shows constructive overlap happened when the path difference is 0 or when it is 2λ.
      • The intensity I =4I0.
      • Path difference =n λ; where n=0, 1, 2, 3…

 

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