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
$r^{\prime }=\frac{1}{2}r$
As $\mu =\frac{\sin i}{\sin r^{\prime }}$ 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 $r=2{\cos }^{-1}\left(\frac{\mu }{2}\right)$
$\because i=r$
$\therefore i=2{\cos }^{-1}\left(\frac{\mu }{2}\right)$