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 surface of the sphere.

Step-by-step solution

For refraction at a single spherical surface, the formula is ₂ v - ₁ u = ₂ - ₁ R . Here, the incident light is a parallel pencil of rays, so u = - . The light enters from air ( ₁ = 1 ) into the transparent sphere ( ₂ = ). The radius of curvature of the first surface is R = +r . Substituting these values into the formula: v - 1 - = - 1 r v = - 1 r v = r - 1 For the rays to be focused at the opposite surface of the sphere, the image distance v from the first surface must be equal to the diameter of the sphere, which