Final answer:
The correct answer is C, indicating that the refractive index of the sphere is 2, which allows a parallel beam of light to emerge when a point source of light lies at its surface.
Step-by-step explanation:
The question pertains to the phenomenon of refraction at a spherical surface. Refraction occurs when a ray of light travels from one medium to another medium with a different refractive index, bending towards or away from the normal. When a parallel beam of light emerges from the opposite surface of a sphere as a point source of light lies at the surface of the sphere, it implies that light is entering a medium with a higher refractive index and then leaving for a medium with a lower refractive index.
Since the light emerges parallel, it must have been focused at a point on the surface of the sphere, which is the definition of the focal point of a lens or spherical surface. This situation resembles the point light source being placed at the focal point of a concave lens, where parallel rays emerge from the other side. Therefore, the refractive index is such that the focal length equals the radius of the sphere. The correct refractive index that satisfies this condition is 2 which is option C. Therefore, when the refractive index of the sphere is 2, a parallel beam of light will emerge from the opposite surface when a point source of light lies at the surface of the sphere.