Question

A point object is placed at $6\text{cm}$ to the left of point $P$. The final image is formed at $\frac{32}{3}\text{cm}$ to right of point $C$. The hemisphere is made of glass $(\mu =1.5)$. Find its radius (in $\text{cm}$).

Answer 1

For curved surface:
$\frac{1.5}{v_1}-\frac{1}{-6}=\frac{1.5-1}{R}$
$\frac{1.5}{v_1}=\frac{1}{2R}-\frac{1}{6}=\frac{3-R}{6R}$
$v_1=\frac{9R}{3-R}$
Object distance for refraction at curved surface is
$u_2=\frac{9R}{3-R}-R=\frac{6R+R^2}{3-R}$
Image distance is given as
$v_2=32/3$
For plane refraction
$\frac{1.5}{u_2}=\frac{1}{v_2}$
Putting the value of $v_2$, we get
$u_2=16$
$6R+R^2=48-16R$
$R^2+22R-48=0$
$R=\frac{-22\pm \sqrt{484+192}}{2}=2\text{cm}$