Question

A ray of light falls on a transparent sphere with its centre at $\text{C}$, as shown in the figure. The ray emerges from the sphere parallel to the line AB. Then, the refractive index of the sphere is

• A. $\sqrt{2}$
• B. $\sqrt{3}$
• C. $\frac{3}{2}$
• D. $\frac{1}{2}$

Answer 1

The correct option is B $\sqrt{3}$


Let the refractive index of the sphere be $\mu$.

Applying Snell's law at point $\text{P}$,
$1\times \sin {60}^{\circ }=\mu \sin r.....\left(1\right)$

From the figure, we can see that $\Delta PQC$ in an isosceles triangle.

Here, angle of deviation by sphere $={60}^{\circ }=2(i-r)$
$\Rightarrow i-r={30}^{\circ }$
$\Rightarrow {60}^{\circ }-r={30}^{\circ }\Rightarrow r={30}^{\circ }......\left(2\right)$

Using eq.(2) in (1),

$1\times \sin {60}^{\circ }=\mu \sin {30}^{\circ }$
$\Rightarrow \frac{\sqrt{3}}{2}=\mu \times \frac{1}{2}$
$\Rightarrow \mu =\sqrt{3}$