If a ray of light in denser medium strikes a rarer medium at angle of incidence $i$, the angles of…

If a ray of light in denser medium strikes a rarer medium at angle of incidence $i$, the angles of reflection and refraction are $r$ and $r^{\prime}$ respectively. If the reflected and refracted rays are at right angles to each other, the critical angle for the given pair of media is
  1. $\sin ^{-1}\left(\tan r^{\prime}\right)$
  2. $\tan ^{-1}(\sin \mathrm{i})$
  3. $\sin ^{-1}(\tan r)$
  4. $\cot ^{-1}(\tan \mathrm{i})$

Solution

According to Snell's Law, $\begin{array}{ll} & \frac{\sin i}{\sin r}=\frac{1}{n} \\ & \text { But } i=r, \\ \therefore \quad & \frac{\sin r}{\sin r^{\prime}}=\frac{1}{n} \\ & \text { From figure } r^{\prime}+r+90=180^{\circ} \\ \therefore \quad & r^{\prime}=180^{\circ}-90^{\circ}-r=90^{\circ}-r \\ \therefore \quad & \Rightarrow \frac{\sin r}{\sin \left(90^{\circ}-r\right)}=\frac{1}{n} \\ & \frac{\sin r}{\cos r}=\frac{1}{n} \end{array}$ $\therefore \quad \tan \mathrm{r}=\frac{1}{\mathrm{n}}$ Critical angle is given by $\sin i_c=\frac{1}{n}$ $\begin{array}{ll} \therefore & \tan r=\sin i_c \\ \therefore & i_c=\sin ^{-1}(\tan r) \end{array}$ :

Asked in: MHT CET 2023 (09 May Shift 2)

Practice more Ray Optics questions on Aicharya