In the figure below, P and Q are two equally intense coherent sources emitting radiation of wavelength 20 m. The separation between P and Q is 5 m and the phase of P is ahead of that of Q by 90ᵒ. A, B and C are three distinct points of observation, each equidistant from the midpoint of PQ. The intensities of radiation at A, B, C will be in the ratio :

Solution

Physics sir · 2 steps

At each point, add P's 90° head start to the phase from the extra path, then find the intensity.

1. Phase from path. A path difference Δ x adds a phase of 2π/λΔ x. With λ = 20 m, the full separation of 5 m adds 2π/20 × 5 = π/2. The wave that travels farther falls behind by this amount.

   I = 4I₀cos² φ/2

   Point	Path effect	Source effect (P ahead)	Net φ and I
   A (beyond Q)	P travels 5 m more, so P lags by π/2	P leads by π/2	φ = 0, I = 4I₀
   B (on bisector)	equal paths, no lag	P leads by π/2	φ = π/2, I = 4I₀cos² 45° = 2I₀
   C (beyond P)	Q travels 5 m more, so Q lags by π/2	Q lags by another π/2	φ = π, I = 0

2. So. I_A : I_B : I_C = 4I₀ : 2I₀ : 0 = 2 : 1 : 0, which is option (b).

The idea to keep. Total phase difference = phase difference of the sources + 2π/λ times the path difference. Always check whether the two parts add up or cancel at each point.

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