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A transparent cube of side d , made of a material of refractive index μ 2 , is immersed in a liquid of refractive index μ 1 ( μ 1 < μ 2 ) . A ray is incident on the face A B at an angle θ (shown in the figure). Total internal reflection takes place at the point E on the face B C . Then, θ must satisfy

Options

  1. Aθ < sin - 1 μ 2 2 μ 1 2 - 1
  2. B&#952; &#62; sin - 1 &#956; 1 &#956; 2
  3. C&#952; &#60; sin - 1 &#956; 1 &#956; 2
  4. D&#952; &#62; sin - 1 &#956; 2 2 &#956; 1 2 - 1

Correct answer

A. θ < sin - 1 μ 2 2 μ 1 2 - 1

Step-by-step solution

At surface (1) μ 1 sin θ = μ 2 sin r … 1 At surface (2) μ 1 = μ 2 sin 90 - r μ 1 = μ 2 cos r . . . ( 2 ) From equation ( 1 ) and ( 2 ) ⇒ sin ⁡ θ = sin ⁡ r cos ⁡ r ⇒ sin θ = μ 2 2 - μ 1 2 μ 2 μ 1 μ 2 ⇒ sin θ = μ 2 2 μ 1 2 - 1 θ must satisfy, ⇒ θ < sin - 1 μ 2 2 μ 1 2 - 1

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