Question

Consider a transparent hemisphere $(n=2)$ in front of which a small object is placed in air $(n=1)$ as shown.
For which value of $x$, of the following, will the final image of the object at $O$ be virtual?

• A. $2R$
• B. $3R$
• C. $R/2$
• D. $1.5R$

Answer 1

Answer: (C)

$R/2$

(i) $\frac{{\mu }_2}{v}-\frac{{\mu }_1}{u}=\frac{{\mu }_2-{\mu }_1}{R}$
Taking refraction first at curved surface,
$\frac{2}{v_1}+\frac{1}{x}=\frac{1}{R}\Rightarrow v_1=\frac{2Rx}{x-R}$
For plane surface,
$v^{\prime \prime }=v_1-R\Rightarrow v^{\prime }=\frac{xR+R^2}{x-R}$
$\Rightarrow \frac{1}{v}-\frac{2(x-R)}{R(x+R)}=0$
$\frac{1}{v}=\frac{2(x-R)}{R(x+R)}$
For virtual image,
$\frac{1}{v}<0\Rightarrow \frac{2(x-R)}{R(x+R)}<0$
$x<R$
So, only one option is possible that is $\frac{R}{2}$