Question

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

Answer 1

Let $i$ be the incident angle and $r$ be the refracted angle

given, $1=2r\Rightarrow r=\frac{i}{2}$

the refractive index of vacuum is 1 and that of another medium is n

from snell's law,

$1\sin i=n\sin r$

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

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

$\Rightarrow \sin \left(\frac{i}{2}\right)[2\cos \left(\frac{i}{2}\right)-n]=0$

$\sin \left(\frac{i}{2}\right)=0=\sin \pi \Rightarrow i=2\pi$

$[2\cos \left(\frac{i}{2}\right)-n]=0\Rightarrow i=2{\cos }^{-1}\left(\frac{n}{2}\right)$

therefore the value of incident angle $=2{\cos }^{-1}\left(\frac{n}{2}\right)$