Question

A ray of light enters a medium of refractive index ${\mu }_m$. It is observed that the angle of refraction is half the angle of incidence. Then, the angle of incidence will be :

• A. ${\cos }^{-1}\mu$
• B. $2{\cos }^{-1}\frac{\mu }{2}$
• C. $2{\cos }^{-1}\mu$
• D. ${\cos }^{-1}\frac{\mu }{3}$

Answer 1

The correct option is B $2{\cos }^{-1}\frac{\mu }{2}$

From Snell's Law, $\mu =\frac{\sin i}{\sin r}$

As $r=\frac{i}{2}$

So, $\mu =\frac{\sin i}{\sin \frac{i}{2}}=\frac{2\sin \frac{i}{2}\cos \frac{i}{2}}{\sin \frac{i}{2}}$

$=2\cos \frac{i}{2}$

$\Rightarrow \cos \frac{i}{2}=\frac{\mu }{2}$

$\Rightarrow \frac{i}{2}={\cos }^{-1}\frac{\mu }{2}$

$\Rightarrow i=2{\cos }^{-1}\frac{\mu }{2}$.