Question

In interference, two individual amplitudes are A_(0)each and the intensity is I_(0)each. Find resultant amplitude and intensity at a point, where (a) phase difference between two waves is 60^(@). (b) path difference between two waves is lambda/3.

Answer 1

Given :

• Amplitude of waves = A₀
• Intensity = I₀

To find :

• Resultant amplitude and intensity at a point where,

1. phase difference between two waves is 60°
2. path difference between two waves is λ/3

Solution :

(a) Phase difference(Ф) between two waves is 60° :

• We know that, resultant amplitude is given by :

        A = √[(A₁²) + (A₂²) + 2 A₁.A₂.cosФ]

• A = √[(A₀)² + (A₀)² + 2 A₀.A₀.cosФ]

           = A₀√[2(1+cosФ)]

           = A₀ √[2(2 cos²(Ф/2)]

           = A₀.2cos(Ф/2)        ....(1)

           = A₀ . 2.cos(60/2)

        A = A₀ . $\sqrt{3}$

• We know that, I = A² = [2A₀cos(Ф/2)]²
• Hence, I = 4A₀²cos²(Ф/2)    .....(2)

                       = 4A₀²[cos(60/2)]²

                       = 4A₀²(3/4)

                     I = 3 I₀    ....( A₀² = I₀)

(b) Path difference(Δx) between two waves is λ/3 :

• We know that, Phase difference   Ф = (2π/λ) . Δx
• So, Ф = (2π/λ) . (λ/3)

              Φ = (2π/3)

• From equation (1),

        A = A₀.2cos(π/3)

        A = A₀  

• From equation (2),

        I = 4A₀²cos²(Ф/2)

          = 4A₀²cos²(π/3)

        I = A₀²

        I = I₀

Answer : (a) A = $\sqrt{3}$ A₀ , I = 3 I₀

                (b) A = A₀ , I = I₀

Answer 2

The resultant amplitude is A = (√3) A0 and I = 3I0  when phase difference is 60∘  and A = A0 and I = I0  when path difference is λ / 3

Explanation:

(a) Substituting ϕ = 60∘ in the equations

A = 2A0 (cos) ϕ / 2    -------(1)

and I = 4I0 cos^2 (ϕ / 2)   ------(2)

We get, A = (√3)A0 and I = 3I0

(b) Given data:

Δx = λ3

∴ϕ or Δϕ = (2π / λ).Δx

Δϕ = (2π / λ) (λ / 3) = 2π / 3 or 120∘

Now, substituting ϕ = 120∘ in the above two equations we get

A = A0 and I = I0

Hence the resultant amplitude is A = (√3) A0 and I = 3I0  when phase difference is 60∘  and A = A0 and I = I0  when path difference is λ / 3