A glass hemisphere $M$ of  $\mu =\frac{3}{2}$ and radius 10 cm has point object  O  placed at a distance  $20 \mathrm{cm}$behind the flat face which is viewed by an observer from the curved side as shown. Find the location of Final image after two refractions as seen by observer.

Her first refraction at flat surface image is produced at a is distance given by

$\mu \mathrm{h}=\frac{3}{2}\times 20=30 \mathrm{cm}$

For second refraction at spherical surface, for refraction formula we use

u=+40 $\mathrm{cm}; R=+10 \mathrm{cm};{\mu }_1=\frac{3}{2};{\mu }_2=1$

Substituting values in refraction formula, we get

$$\begin{gathered} \frac{{\mu }_2}{v}-\frac{{\mu }_1}{u}=\frac{{\mu }_2-{\mu }_1}{R} \\ \frac{1}{v}-\frac{3}{2\times 40}=\frac{1-\frac{3}{2}}{10} \\ \Rightarrow \frac{1}{v}=\frac{3}{90}-\frac{1}{20}=-\frac{1}{80} \\ \Rightarrow v=-80 \mathrm{cm} \end{gathered}$$

Thus final image is seen by observer at a distance $80 \mathrm{cm}$ from the pole P\ of curved surface and it is a real image.