Question

A ray of light traveling in a vacuum enters into a medium of the refractive index $\mu$ it is found that the angle of refraction is half of the angle of incidence then the value of angle of incidence is

Answer 1

Step 1: Given

Angle of incidence$=i$

Angle of refraction$=r$

The angle of refraction $\left(r\right)=$$\frac{1}{2}$ The angle of incidence $\left(i\right)$

Step 2: Formula used

$\mu =\frac{\sin i}{\sin r}$

$\sin 2\theta =2\sin \theta \cos \theta$

Step 3: Solution

From snell's law,

$\mu =\frac{\sin i}{\sin r}$

Now, by simplifying the above equation

$\Rightarrow \mu =\frac{2\sin \left({\displaystyle \frac{i}{2}}\right)\cos \left({\displaystyle \frac{i}{2}}\right)}{\sin \left({\displaystyle \frac{i}{2}}\right)}$ $\left[\because \sin 2\theta =2\sin \theta \cos \theta ,r=\frac{i}{2}\right]$

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

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

Hence, the angle of incidence is $2{\cos }^{-1}\left(\frac{\mu }{2}\right)$.