Question

A ray of light passes from vacuum into a medium of refractive index $\mu$. If the angle of incidence is twice the angle of refraction, then the angle of incidence is,

• A. ${\cos }^{-1}\left(\frac{\mu }{2}\right)$
• B. ${\sin }^{-1}\left(\frac{\mu }{2}\right)$
• C. $2{\cos }^{-1}\left(\frac{\mu }{2}\right)$
• D. $2{\sin }^{-1}\left(\frac{\mu }{2}\right)$

Answer 1

The correct option is C $2{\cos }^{-1}\left(\frac{\mu }{2}\right)$

Applying Snell's law.
$\mu =\frac{\sin i}{\sin r}$
$\Rightarrow \mu \sin r=\sin i$

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

$\Rightarrow \mu \sin \left(\frac{i}{2}\right)=2\sin \left(\frac{i}{2}\right)\cos \left(\frac{i}{2}\right)$

$\Rightarrow \frac{\mu }{2}=\cos \left(\frac{i}{2}\right)$

$\Rightarrow i=2{\cos }^{-1}\left(\frac{\mu }{2}\right)$

Hence, $\left(C\right)$ is the correct answer.