A hemispherical section of the surface of a solid glass sphere ( = 1.5 ) of radius r is silvered to render the inner side reflecting. An object is positioned on the axis of the hemisphere at a distance 3r from the centre of the sphere. Light from the object is refracted at the unsilvered part, subsequently reflected from the silvered part, and refracted again at the unsilvered part. Locate the final image formed.

Step-by-step solution

Let the pole of the unsilvered surface be P₁ and the pole of the silvered surface be P₂ . The distance between P₁ and P₂ is the diameter of the sphere, 2r . The object O is at a distance 3r from the centre C , which means it is at a distance 3r - r = 2r from P₁ . For the first refraction at the unsilvered surface P₁ : Object distance, u = -2r Radius of curvature, R = +r Refractive indices, ₁ = 1 (air) and ₂ = 1.5 (glass) Using the spherical refraction formula: ₂ v₁ - ₁ u = ₂ - ₁ R 1.5 v₁ - 1 -2r = 1.5 - 1 r 1.5 v₁