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.

Answer

-2R

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 accordingly.

First refraction Using `mu_2/v=mu_1/u=(mu_2-mu_1)/R` with proper
sign conventions, we have
`1.5/v_1-1.0/-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.