Two waves of intensity ratio 1: 9 cross each other at a point. The resultant intensities at that point, when (a) Waves are incoherent is I₁ (b) Waves are coherent is I₂ and differ in phase by 60°. If I₁/I₂=10/x then x= _________.

Solution

Physics sir · 4 steps

Incoherent waves add intensities; coherent waves add amplitudes, which brings in an extra interference term.

1. Set the intensities. Take the two intensities as I₀ and 9I₀, which matches the ratio 1:9.
2. Incoherent case. With no fixed phase relation, the interference term averages to zero and the intensities add directly: I₁ = I₀ + 9I₀ = 10I₀.

   I₂ = Iₐ + I_b + 2√(Iₐ I_b)cos φ

3. Coherent case. Here √(I₀ × 9I₀) = 3I₀ and cos 60° = 1/2. So I₂ = 10I₀ + 2 × 3I₀ × 1/2 = 13I₀.
4. So. I₁/I₂ = 10I₀/13I₀ = 10/13, so x = 13.

The common slip: 16. Leaving out cos 60° makes the extra term 6I₀ and gives 16, not 13.

The idea to keep. Incoherent sources: add the intensities. Coherent sources: add 2√(Iₐ I_b)cos φ to that sum, and this term carries all the phase information.

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