Refraction at Spherical Boundary of Coated Glass Cylinder

To determine the focal lengths $f_1$ and $f_2$, we consider the refraction of light through a spherical boundary between two media coated with a thin film.

Refraction at a Thin Coated Spherical Boundary

Since the coating is a thin film of uniform and negligible thickness ($t \to 0$), the total refraction across the two interfaces (air-film and film-glass) can be analyzed by combining the refraction equations at each boundary:

1. Refraction at the first surface (medium $a$ to medium $b$ of radius $R$): $\frac{n_b}{v_1} - \frac{n_a}{u} = \frac{n_b - n_a}{R}$

2. Refraction at the second surface (medium $b$ to medium $c$ of radius $R$): $\frac{n_c}{v} - \frac{n_b}{v_1} = \frac{n_c - n_b}{R}$

Adding the two equations yields the net refraction formula, showing that the intermediate refractive index $n_b$ cancels out: $\frac{n_c}{v} - \frac{n_a}{u} = \frac{n_c - n_a}{R}$

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1. Rays traveling from Air to Glass ($f_1$)

• Incident medium: Air ($n_a = 1$)
• Final medium: Glass ($n_c = 1.5$)
• Object distance: Parallel rays imply $u = -\infty$
• Radius of curvature: The surface is convex towards air, so its center of curvature lies inside the glass cylinder in the direction of light propagation ($R = +R$).

Using the refraction formula: $\frac{1.5}{f_1} - 0 = \frac{1.5 - 1.0}{+R}$ $\frac{1.5}{f_1} = \frac{0.5}{R}$ $f_1 = 3R \implies |f_1| = 3R$

This matches Option (A).

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2. Rays traveling from Glass to Air ($f_2$)

• Incident medium: Glass ($n_a = 1.5$)
• Final medium: Air ($n_c = 1$)
• Object distance: Parallel rays inside the glass imply $u = -\infty$
• Radius of curvature: As light travels from glass to air, it encounters a surface that is concave relative to the incident direction. Therefore, the center of curvature lies opposite to the direction of light propagation ($R = -R$).

Using the refraction formula: $\frac{1.0}{f_2} - 0 = \frac{1.0 - 1.5}{-R}$ $\frac{1.0}{f_2} = \frac{-0.5}{-R} = \frac{0.5}{R}$ $f_2 = 2R \implies |f_2| = 2R$

This matches Option (C).

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Conclusion

The correct options are (A) and (C).