Question

If a ray of light in a denser medium enters into a rarer medium at an angle of incidence $i$, the angle of reflection and refraction are respectively $r$ and $r^{\prime }$. If the reflected and refracted rays are at right angles to each other, the critical angle for the given pair of media is:

• A. ${\sin }^{-1}\left(\tan r^{\prime }\right)$
• B. ${\sin }^{-1}\left(\tan r\right)$
• C. ${\tan }^{-1}\left(\sin i\right)$
• D. $\cot \left(\tan i\right)$

Answer 1

The correct option is B ${\sin }^{-1}\left(\tan r\right)$

From laws of reflection, $i=r$

Sum of supplementary angles is ${180}^o$ and it is given the angle between reflected and refracted rays is ${90}^o$.

Hence, $r+r^{\prime }+{90}^o={180}^o$

$⟹r^{\prime }={90}^o-i$

From Snell's Law, $\frac{\sin i}{\sin r^{\prime }}=\mu$

$⟹\mu =\tan i$

Critical angle is given by, ${\theta }_c={\sin }^{-1}\mu$

$⟹{\theta }_c={\sin }^{-1}\left(\tan i\right)$

$⟹{\theta }_c={\sin }^{-1}\left(\tan r\right)$