Question

When a light ray enters a refracting medium, it is found that the magnitude of the angle of refraction is equal to half the angle of reflection. If $\mu$ is the refractive index of the medium, then the angle of incidence is

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

Answer 1

Answer: (B)

According to given problem,
Angle of refraction $=\frac{1}{2}$ (Angle of reflection) or
$ r'=\frac{1}{2} r$
As $\mu=\frac{\sin i}{\sin r'} $ where $i$ is the angle of incidence)
$\mu=\frac{\sin r}{\sin \left(\frac{1}{2} r\right)}$ $(\because$ Angle of incidence, i= Angle of reflection, r)
$\mu=\frac{2 \sin \left(\frac{r}{2}\right) \cos \left(\frac{r}{2}\right)}{\sin \left(\frac{1}{2} r\right)}$ or $\frac{\mu}{2}=\cos \left(\frac{r}{2}\right)$
$\cos ^{-1}\left(\frac{\mu}{2}\right)=\frac{r}{2}$ or $r=2 \cos ^{-1}\left(\frac{\mu}{2}\right) $
$\because i=r$
$ \therefore i=2 \cos ^{-1}\left(\frac{\mu}{2}\right)$