Question

In YDSE, the width of one slit is different from other, so that the amplitude of light from one slit is double that from the other. If Im is the maximum intensity, the intensity when they interfere with a phase difference Ф is

Answer 1

I = Im/9 (1 + 8Cos² ∅/2)

Explanation:

A₂ = 2A₁  

Intensity ∝ Amplitude²

I₂/I₁ = (A₂/A₁)² = (2A₁/A₁)² = 4

I₂ = 4I₁

Max intensity Im = (√I₁ + √I₂)²

 = √I₁ + √4I₁)²

 = (3√I₁)²

 = 9I₁

I₁ = Im/9 -------------------(1)

Resultant Intensity I =  I₁ + I₂ + 2√I₁I₂Cos∅

  I = I₁ + 4I₁ + 2√I₁.4I₁.Cos∅

    = 5I₁ + 4I₁.Cos∅

    = I₁ + 4I₁ + 4I₁.Cos∅

  I = I₁ + 4I₁(1 + Cos∅)

We know that 1 + Cos∅ = 2Cos² ∅/2. So:

 I = I₁ + 8I₁.Cos² ∅/2)

 I = I₁ (1 + 8Cos² ∅/2)

Substituting value of I₁ from equation (1), we get:

I = Im/9 (1 + 8Cos² ∅/2)