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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