Two coherent light beams of wavelength λ are superimposed at a point. If the resultant intensity at that point is zero, then the possible path difference between the two waves at that point is:

CONCEPT:

Coherent addition of waves:

• The phenomenon in which two or more waves superpose to form a resultant wave of greater, lower, or the same amplitude is called interference.
• The interference is based on the superposition principle according to which at a particular point in the medium, the resultant displacement produced by a number of waves is the vector sum of the displacements produced by each of the waves.
• Let two sources of wave S1 and S2 are coherent.
• Let the intensity of the waves are Io.

For constructive interference:

• If at any point the waves emerging from the coherent sources S1 and S2 are in phase then we will have constructive interference and the resultant intensity will be 4Io.

⇒ Path difference = nλ and n = 0,1,2,3,...

For destructive interference:

• If at any point the waves emerging from the coherent sources S1 and S2 are in opposite phase then we will have destructive interference and the resultant intensity will be zero.

⇒ Path difference = \(\left ( n+\frac{1}{2} \right )\)λ and n = 0,1,2,3,...

Where λ = wavelength

CALCULATION:

For destructive interference:

• If at any point the waves emerging from the coherent sources S1 and S2 are in opposite phase then we will have destructive interference and the resultant intensity will be zero.

⇒ Path difference = \(\left ( n+\frac{1}{2} \right )\)λ and n = 0,1,2,3,...

Where λ = wavelength

• In the given case the resultant intensity is zero, so there will be destructive interference.
• For destructive interference, the possible path differences between the waves are given as,

For n = 0,

⇒ Path difference = 0.5λ

For n = 1,

⇒ Path difference = 1.5λ

For n = 2,

⇒ Path difference = 2.5λ

• So it is clear that when the path difference is λ and \(\frac{λ}{3}\), there will be no destructive interference.
• The destructive interference will occur when the path difference between the waves is 2.5λ.
• Hence, option 3 is correct.