A narrow pencil of parallel light falls normally on a solid transparent sphere of radius r . Determine the required refractive index if the pencil is to be focused at the centre of the sphere.

Step-by-step solution

For refraction at the first spherical surface, the formula is ₂ v - ₁ u = ₂ - ₁ R . Here, the light is incident from air ( ₁ = 1 ) to the transparent sphere ( ₂ = ). Since the incident light is a parallel pencil, the object distance is u = - . The radius of curvature of the surface is R = +r . The light is to be focused at the centre of the sphere, which is at a distance r from the pole of the first surface. So, the image distance is v = +r . Substituting these values into the refraction formula: r - 1 - = - 1 r r