Question

A light ray is moving from denser (refractive index $=\mu$) to air. If the angle of incidence is half the angle of refraction, find out the angle of refraction.

Answer 1

Given: A light ray is moving from denser(refractive index=μ) to air. If the angle of incidence is half the angle of refraction,

To find out the angle of refraction.

Solution:

As per the given criteria,

Angle of incidence, $i=\frac{r}{2}$, where $r$ is the angle of refraction

We know as per Snell's law

$\frac{\sin i}{\sin r}=\frac{{\mu }_2}{{\mu }_1}⟹\frac{\sin \frac{r}{2}}{\sin r}=\frac{1}{\mu }$

But we know, $\sin 2A=2\sin A\cos A⟹\sin r=2\sin \frac{r}{2}.\cos \frac{r}{2}$

$⟹\frac{1}{\mu }=\frac{\sin \frac{r}{2}}{2\sin \frac{r}{2}.\cos \frac{r}{2}}⟹\frac{1}{\mu }=\frac{1}{2\cos \frac{r}{2}}⟹\frac{\mu }{2}=\cos \frac{r}{2}⟹r=2{\cos }^{-1}\frac{\mu }{2}$

is the angle of refraction.