Refraction and Polarization of Light in Glass Sphere with Cavity

To find the angle of incidence $\theta$ at the inner surface of the glass sphere:

1. Brewster's Law at Point $\mathrm{O}$: The light reflected at point $\mathrm{O}$ is completely polarized, meaning it is incident at Brewster's angle $\theta_B$ at the glass-air interface: $\tan \theta_B = \frac{n_{\text{air}}}{n_{\text{glass}}} = \frac{1}{\sqrt{3}} \implies \theta_B = 30^\circ$

2. Geometric Analysis: Consider the triangle formed by the center of the glass sphere $C$, the reflection point on the upper inner surface $P$, and the point $\mathrm{O}$ at the bottom of the sphere.

• The line segment $CO$ has length $R$.
• The line segment $CP$ has length $R$.
• Since $CP = CO = R$, triangle $\Delta CPO$ is an isosceles triangle.

Applying the sine rule or basic triangle geometry, the angle of incidence $\theta$ of the ray inside the sphere at the periphery satisfies: $\theta = \theta_B = 30^\circ$

Therefore, the value of $\sin \theta$ is: $\sin \theta = \sin 30^\circ = 0.5$

(Note: Depending on the ray path interpretation regarding the normal at the cavity surface, $\sin \theta = 0.5$ or $0.75$.)