Question

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

• A. $co{s}^{-1}\left(n/2\right)$
• B. $si{n}^{-1}\left(n/2\right)$
• C. $2co{s}^{-1}\left(n/2\right)$
• D. $2si{n}^{-1}\left(n/2\right)$

Answer 1

The correct option is C $2co{s}^{-1}\left(n/2\right)$

Let, the angle of incidence $=i$ and angle of refraction $=r$

Given, $i=2r$ or $r=i/2$

Refractive index of vacuum is 1 and refractive index of the medium is given $n.$

Using Snell's law, $1sini=nsinr$

or $sini=nsin\left(i/2\right)$

or $2sin\left(i/2\right)cos\left(i/2\right)=nsin\left(i/2\right)$

or $sin\left(i/2\right)\left[2cos\right(i/2)-n]=0$

So, $sin\left(i/2\right)=0$ and $2cos\left(i/2\right)-n=0$

or $sin\left(i/2\right)=sin\pi$ and $cos\left(i/2\right)=n/2$

or $i/2=\pi$ and $i/2=co{s}^{-1}\left(n/2\right)$

or $i=2\pi$ and $i=2co{s}^{-1}\left(n/2\right)$

Angle of incidence can not be $2\pi$.

Thus, the angle of incidence should be $2co{s}^{-1}\left(n/2\right)$