A point source of light is placed at a distance of $u=-15\text{ cm}$ from a glass sphere of radius $R=10\text{ cm}$ and refractive index $\mu =1.5$. Find the position of the final image formed by the refraction at both surfaces of the sphere.

Solution

1. For the first surface, use $\frac{{\mu }_2}{v_1}-\frac{{\mu }_1}{u_1}=\frac{{\mu }_2-{\mu }_1}{R_1}$ with $$\begin{array}{rl}{\mu }_1& =1,{\mu }_2\\ & =1.5,u_1\\ & =-15,R_1\\ & =10\end{array}$$.
2. Solving gives $\frac{1.5}{v_1}-\frac{1}{-15}=\frac{0.5}{10}$, which yields $v_1=-60\text{ cm}$.
3. For the second surface, the object distance is $$\begin{array}{rl}{u}_2& =v_1-2R\\ & =-60-20\\ & =-80\text{ cm}\end{array}$$ with $$\begin{array}{rl}{\mu }_1& =1.5,{\mu }_2\\ & =1,R_2\\ & =-10\end{array}$$.
4. Using $\frac{1}{v_2}-\frac{1.5}{-80}=\frac{1-1.5}{-10}$ gives $$\begin{array}{rl}\frac{1}{v_2}& =0.05-0.01875\\ & =0.03125\end{array}$$.
5. Thus $v_2=32\text{ cm}$ from the second surface, which is $10\text{ cm}$ from the center, hence the answer is (C).