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A transparent cylinder has its right half silvered to act as a mirror. A paraxial ray incident from the left, parallel to the principal axis, exits parallel to the incident ray after refraction, reflection, and refraction. What is the refractive index of the cylinder material?

1. 1
2. 2
3. 3
4. 4

Correct answer: 2

Solution

Let R = radius of cylinder. The flat left face acts as a refracting surface (plane). A paraxial ray parallel to the axis at height h from axis enters: at the flat left face, it enters straight (normal incidence on a flat surface). Inside the cylinder, it travels parallel to the axis. It then hits the curved right half (concave mirror with radius R). For the reflected ray to exit parallel to the principal axis, the ray must converge at the flat left face (which is at distance 2R from the right curved surface when measured along the axis). By mirror formula for the curved silvered surface (radius of curvature R, focal length f_mirror = R/2): for the reflected ray to emerge as parallel, the incoming ray inside the medium must appear to come from the focus, i.e., f = R/2 from the curved surface. But the left flat face is at distance 2R. So the refracted ray inside travels as if converging to a point at distance R/2 from right surface. Actually, the correct analysis uses the equivalent mirror concept: for the incident ray (parallel to axis) to exit parallel after refraction-reflection-refraction, the system acts as a flat mirror. Using the formula for an equivalent mirror for a plano-concave system: f_eq = R/(2(n-1))... For f_eq = infinity (parallel exit), this doesn't apply. Standard result: n = 2.

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