Two beams of light having intensities I and 4I interfere to produce a fringe pattern on a screen. The phase difference between the two beams are π / 2 and π / 3 at points A and B respectively. The difference between the resultant intensities at the two points is x I. The value of x will be ________.

Solution

Physics sir · 4 steps

Only the cross term 2√(I₁ I₂)cos φ changes between A and B, so compare it at each point.

1. Resultant intensity. When two coherent beams of intensities I₁ and I₂ meet with phase difference φ, the resultant intensity is given below.

   I_R = I₁ + I₂ + 2√(I₁ I₂)cos φ

2. Put in intensities. With I₁ = I and I₂ = 4I, the cross-term coefficient is 2√(I × 4I) = 4I. So I_R = 5I + 4I cos φ.
3. At A and B. At A, φ = π/2 and cos φ = 0, so I_A = 5I. At B, φ = π/3 and cos φ = 1/2, so I_B = 5I + 2I = 7I.
4. So. I_B - I_A = 7I - 5I = 2I, which gives x = 2.

The common slip: 1. Dropping the factor 2 in front of √(I₁ I₂)cos φ makes the cross term 2I cos φ, so the difference comes out as 1, not 2.

The idea to keep. Intensities never add directly in interference. Add 2√(I₁ I₂)cos φ to I₁ + I₂, and take the square root of the product before using it.

Answer from the NTA final answer key. Solution by PrepWiser.