Question

A transparent cylinder has its right half polished so as to act as a mirror. A paraxial light ray incident from left, that is parallel to the pricipal axis, exits parallel to the incident ray as shown. The refractive index $n$ of the material of the cylinder is

• A. $1.2$
• B. $1.5$
• C. $1.8$
• D. $2.0$

Answer 1

The correct option is D $2.0$

For the refraction of incident ray at the spherical surface.

From above figure,
$u=-\infty ;v=+2R$
[$\because$ Light ray is intersecting at $2R$ from pole of spherical surface]

Using the formula of refraction through curved surface,
$\frac{{\mu }_2}{v}-\frac{{\mu }_1}{u}=\frac{{\mu }_2-{\mu }_1}{R}$
$\Rightarrow \frac{n}{+2R}-\frac{1}{(-\infty )}=\frac{n-1}{R}$
$(\because {\mu }_2=n,{\mu }_1={\mu }_{air}=1)$
$\Rightarrow \frac{n}{2R}=\frac{n-1}{R}$
$\Rightarrow 2n-2=n$
$\therefore n=2$
Hence, option $\left(d\right)$ is the correct answer.

Why this question? Tip: After first refraction a real image is formed at distance $2R$ from spherical surface when $u\to -\infty$ .