Question

A small diameter parallel light beam is directed towards the center of a large solid sphere made of transparent plastic. The beam is brought to a focus on the opposite side of the sphere. Find the refractive index of the plastic.

Answer 1

2

Answer 2

For a parallel light beam incident on a transparent sphere, if the beam is focused on the opposite side of the sphere (i.e., at the second pole), we can use the spherical refraction formula. Assuming the final image is formed at a distance of $2R$ from the first surface due to the first refraction itself (a common simplification for this type of problem):

1. First Refraction (Air to Plastic):
   • Object distance, $u = -\infty$ (parallel beam).
   • Image distance, $v = +2R$ (final image at the second pole, measured from the first pole).
   • Radius of curvature, $R_1 = +R$.
   • Refractive indices: $\mu_1 = 1$ (air), $\mu_2 = \mu$ (plastic).
2. Apply Refraction Formula: $$\frac{\mu_2}{v} - \frac{\mu_1}{u} = \frac{\mu_2 - \mu_1}{R_1}$$ $$\frac{\mu}{2R} - \frac{1}{-\infty} = \frac{\mu - 1}{R}$$ $$\frac{\mu}{2R} = \frac{\mu - 1}{R}$$
3. Solve for $\mu$: $$\mu = 2(\mu - 1)$$ $$\mu = 2\mu - 2$$ $$\mu = 2$$