The diagram shows five isosceles right angled prisms. A light ray incident at 90o at the first face emerges at same angle with the normal from the last face. Which of the following relations will hold regarding the refractive indices?

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$μ_{1}^{2} + μ_{3}^{2} + μ_{5}^{2} = μ_{2}^{2} + μ_{4}^{2}$

none of these

$μ_{1}^{2} + μ_{3}^{2} + μ_{5}^{2} = 1 + μ_{2}^{2} + μ_{4}^{2}$

$μ_{1}^{2} + μ_{3}^{2} + μ_{5}^{2} = 2 + μ_{2}^{2} + μ_{4}^{2}$

answer is B.

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Detailed Solution

$\begin{array}{l} 1^{st} surface: 1 \sin 90^{\circ} = μ_{1} \sin r_{1} \\ 2^{nd} surface: μ_{1} \cos r_{1} = μ_{2} \sin r_{2} \\ 3^{rd} surface: μ_{2} \cos r_{2} = μ_{3} \sin r_{3} \\ 4^{th} surface: μ_{4} \sin r_{4} = μ_{3} \cos r_{3} \\ 5^{th} surface: μ_{4} \cos r_{4} = μ_{5} \sin r_{5} \\ 1 . \sin 90^{\circ} = μ_{5} \cos r_{5} \end{array}$