Correct Answer - Option 3 :
\(\frac{\pi}{3}\)
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, and Resultant intensity I = 3Io
The resultant intensity at that point is given as,
\(⇒ I=4I_ocos^2\left ( \frac{ϕ}{2} \right )\)
\(⇒ 3I_o=4I_ocos^2\left ( \frac{ϕ}{2} \right )\)
\(⇒ cos\left ( \frac{ϕ}{2} \right )=\frac{\sqrt3}{2}\)
\(⇒ \frac{ϕ}{2}=\frac{\pi}{6}\)
\(⇒ ϕ=\frac{\pi}{3}\)
- Hence, option 3 is correct.
