A solid glass sphere with radius R and an index of refraction `1.5` is silvered over one hemisphere. A small object is located on the axis of the sphere at a distance 2R to the left of the vertex of the unsilvered hemisphere. Find the position of final image after all refractions and reflection have taken place.

Explanation

The ray of light first gets refracted then reflected and then again refracted. For first refraction and then reflection the ray of light travels from left to right while for the last refraction it travels from right to left. Hence, the sign convention will change
First: refraction:Using `mu_(2)/v - mu_(1)/u = (mu_(2)-mu_(1))/R` with proper sign conventions,
we have ` 1.5/v - 1.0/(-2R) = (1.5-1.0)/(+R) v_(1)=oo`
Second: reflection: Using `1/v + 1/u = 1/f =2/R` with proper sign conventions ,
We have , `1/v_(2) + 1/(oo) = - 2/R :. v_(2)= - R/2`
Third: refraction: Again using `mu_(2)/v - mu_(1)/u = (mu_(2)-mu_(1))/R` with reversed sign convention, we have, `1.0/v_(3) - 1.5/(-1.5R) = (1.0-1.5)/(-R)`
or `v_(3)` = -2R i.e., final image is formed on the vertex of the silvered face