Question

Two identical coherent waves of intensity I_o are superimposed at two different points A and B. At point A the waves show constructive interference and at point B the phase difference between the waves is \(\frac{\pi}{3}\). Find the ratio of the intensity of the resultant wave at point A to point B.
1. \(\frac{4}{3}\)
2. \(\frac{3}{4}\)
3. \(\frac{2}{\sqrt3}\)
4. \(\frac{\sqrt3}{2}\)

Answer 1

Correct Answer - Option 1 : \(\frac{4}{3}\)

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 two 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 two 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

• Let at any point the phase difference between the displacements of the two waves is ϕ.
• So the displacement of the two waves is given as,

⇒ y1 = a.cos(ωt)

⇒ y2 = a.cos(ωt + ϕ)

• When the waves are superimposed at this point the resultant displacement is given as,

\(⇒ y=2acos\left ( \frac{ϕ}{2} \right )cos\left ( \omega t+ \frac{ϕ}{2} \right )\)

• The amplitude of the resultant displacement is given as,

\(⇒ A=2acos\left ( \frac{ϕ}{2} \right )\)

• The intensity at that point is given as,

\(⇒ I=4I_ocos^2\left ( \frac{ϕ}{2} \right )\)

EXPLANATION:

Given Intensity = Io

At point A: (Constructive interference)

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

⇒ I_A = 4I_o     -----(1)

At point B: (ϕ = \(\frac{\pi}{3}\))

• The intensity at that point is given as,

\(⇒ I_B=4I_ocos^2\left ( \frac{ϕ}{2} \right )\)

\(⇒ I_B=4I_ocos^2\left ( \frac{\pi}{6} \right )\)

⇒ I_B = 3I_o     -----(2)

By equation 1 and equation 2,

\(⇒ \frac{I_A}{I_B}= \frac{4I_o}{3I_o}\)

\(⇒ \frac{I_A}{I_B}= \frac{4}{3}\)

• Hence, option 1 is correct.