Question

A ray of light from a denser medium strikes a rarer medium at an angle of incidence $i$ as shown in the figure. The reflected and refracted rays make an angle of ${90}^{\circ }$ with each other. The angles of reflection and refraction are r and r'. The critical angle is:

• A. $si{n}^{-1}\left(tanr\right)$
• B. $si{n}^{-1}\left(tani\right)$
• C. $si{n}^{-1}\left(tan{r}^{\prime }\right)$
• D. $ta{n}^{-1}\left(tani\right)$

Answer 1

The correct option is A $si{n}^{-1}\left(tanr\right)$

From figure, i = r and r' = ${90}^{\circ }-r$
Now, $n=\frac{sin{r}^{\prime }}{sinr}=\frac{sin({90}^{\circ }-r)}{sinr}$
$\Rightarrow n=\frac{cosr}{sinr}=\frac{1}{tanr}$
We know that, $n=\frac{1}{sinC}$
where C = critical angle
$\therefore \frac{1}{sinC}=\frac{1}{tanr}$
$\Rightarrow C=si{n}^{-1}\left(tanr\right)$