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 reaction and then reflection the ray of light travels from left to right while for the last reflection it travels from right to left. Hence, the sign convention will change accordingly.
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) therefore v_1= infty `
Second reflection
Using `1/v+1/u=1/f=2/R` with proper sign conventions
we have `1/v_2+1/infty=2/R therefore 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.