Question

a transparent cylinder of radius R and height H made from material fo refractove index n=1.5 is placed on a horizontal tabletop and a point light soruce is placed at a hight h above the centre of top circular face of the cylinder. if unilluminated area pn the tabletop outide the cylinder is?

Answer 1

$A=\pi R^2\Bigl(\frac{h}{\sqrt{n^2-1}}-1\Bigr)^2\quad\Longrightarrow\quad A=\pi R^2\Bigl(\frac{2h}{\sqrt5}-1\Bigr)^2\quad\text{for } n=1.5\,.$

Answer 2

By tracing rays from a point light source (at height $h$ above the centre of the top face) that enter the cylinder and then reach the table after refraction at the lateral wall (with the critical‐angle condition $\sin i_c=1/n$) one finds that the envelope of the emerging rays on the table is the circle $r=\frac{h}{\sqrt{n^2-1}}$. Since the cylinder’s base already covers a circle of radius $R$, the dark (unilluminated) area outside is

$$A=\pi\Bigl(\frac{h}{\sqrt{n^2-1}}-R\Bigr)^2$$

which may be written as

$$A=\pi R^2\Bigl(\frac{h}{R\sqrt{n^2-1}}-1\Bigr)^2.$$

For $n=1.5,$ using $\sqrt{1.5^2-1}=\sqrt{1.25}=\sqrt5/2,$ we get

$$A=\pi R^2\Bigl(\frac{2h}{\sqrt5}-1\Bigr)^2.$$