A hemispherical portion of the surface of a solid glass sphere (mu = 1.5) of radius r is silvered to make the inner side reflecting. An object is placed on the axis of the hemisphere at a distance 3r from the centre of the sphere. The light from the object is refracted at the unsilvered part, then reflected from the silvered part and again refracted at the unsilvered part. Locate the final image formed.

Explanation

We need to use refraction formula and then mirror formula and then refraction formula again .
Let us first solve for refraction from air to glass .
`(mu_(2))/(v) - (mu_(1))/(u) = (mu_(2) - mu_(1))/(R)`
`implies " " (1.5)/(v) - (1)/(-2R) = (1.5 - 1)/(+R)`
`implies " " (1.5)/(v) + (1)/(2R) = (1)/(2R)`
`implies v = oo`
Hence light ray will become parallel to principal axis after refraction from air to glass .
Silvered part acts like a concave mirror .
Light rays coming parallel to principal axis are converged to focus of concave mirror which is half of the radius of curvature of the surface . we are solving for para-axial rays . Hence after reflection from the concave mirror (formed on the surface of sphere) light ray will converge at a distance R/2 from point B that is pole of the concave mirror formed on polishing the surface of sphere . In the following figure point I is the image formed after reflection from silvered part .

Now point I will act as an object for the refraction from glass to air . Note that after reflection direction of light ray is reversed and so convention of positive and negative will be reversed . Now for the next refraction from glass to air distance will be measured from point A . Distance of point I from point A to R + R /2 = 3R/2 .
Let us apply refraction formula to locate the final image.
`(mu_(2))/(v) - (mu_(1))/(u) = (mu_(2) - mu_(1))/(R)`
`implies " " (1)/(v) - (1.5)/(-3R//2) = (1 - 1.5)/(-R)`
`implies " " (1)/(v) + (1)/(R) = (1)/(2R)`
`implies " " (1)/(v) = (1)/(2R) - (1)/(R) = - (1)/(2R)`
`implies " " v = - 2 R `
-2R distance from point A means point B which is pole of the concave mirror . Hence , final image is formed at point B and it is virtual .