Question

A ray of light falls on a transparent sphere with centre C as shown in the figure. The ray emerges from sphere parallel to line AB. The refractive index of sphere is:

Answer 1

Given: A ray of light falls on a transparent sphere with centre C as shown in the figure. The ray emerges from sphere parallel to line AB.

To find the refractive index of sphere

Solution:

The ray diagram is as shown in above fig.

$\angle XAY=\angle PAC={60}^{\circ }$ [vertically opposite angles]

In $\Delta ACD$,

Side AC=side CD

Hence $\angle DAC=\angle CDA=r$

Hence, $\angle DCB=2r$ [external angle property]

As AB is parallel to PD,

$\delta ={60}^{\circ }$

i.e., $\delta =2r⟹r={30}^{\circ }$

Now applying snell's law,

$\mu =\frac{\sin i}{\sin r}⟹\mu =\frac{\sin \left({60}^o\right)}{\sin \left({30}^o\right)}=\frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}}⟹\mu =\sqrt{3}$

Hence the refractive index of the sphere is $\sqrt{3}$