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In the adjacent diagram, CP represents a wavefront and AO   &   BP , the corresponding two rays. Find the condition of θ for constructive interference at P between the ray BP and reflected ray OP .

Options

  1. Acos θ = 3 λ 2 d
  2. Bcos θ = λ 4 d
  3. Csec θ - cos θ = λ d
  4. Dsec θ - cos θ = 4 λ d

Correct answer

B. cos θ = λ 4 d

Step-by-step solution

P R = d   ∴  P O = d sec θ and C O = P O cos 2 θ = d sec θ cos 2 θ   Path difference between the two rays is, Δ x = P O + O C  = d sec θ + d sec θ cos 2 θ phase difference between the two rays is Δ ϕ = π   oneisreflected , whileanotheris direct Therefore, condition for constructive interference should be Δ x = λ 2 , 3 λ 2 ...... or, d sec θ 1 + cos 2 θ = λ 2 or, d cos θ 2 cos 2 θ = λ 2

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