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A prism has apex angle A. One of its refracting surfaces is silvered (acts as a mirror). A ray of light incident on the first surface at an angle of incidence 2A retraces its path after reflecting off the silvered surface. What is the refractive index mu of the prism?

1. 2 sin A
2. 2 cos A
3. (1/2) cos A
4. tan A

Correct answer: 2 cos A

Solution

For a ray to retrace its path after reflecting from the silvered surface, it must be incident normally (at 0 degrees to the normal) on that surface. In a prism with apex angle A, if the ray refracts at the first surface with refraction angle r, then it hits the second (silvered) surface at angle (A - r) from the normal of that surface. For normal incidence: A - r = 0 => r = A. At the first surface, by Snell's law (assuming air outside, mu₁ = 1): sin(2A) = mu * sin(A). Using double angle formula: sin(2A) = 2 sin A cos A. Therefore mu = 2 sin A cos A / sin A = 2 cos A.

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