A ray of light travelling in a denser medium strikes the boundary of a rarer medium. The ray gets partially reflected and refracted. The angle of reflection is $r$ and that of refraction is $r^{\text{'}}$. If the reflected and refracted rays make an angle of $90°$ with each other then the critical angle for the pair of media is

• Correct answer
  A

  sin^-1 (tan r)
• B

  tan^-1 (sin r)
• C

  sin^-1 (tan r')
• D

  tan^-1 (sin r')

Solution

${}_{\text{D}}{\text{μ}}_{\text{R}}=\frac{\text{sin r}}{{\text{sin r}}^{\prime }}\text{-----(1)}$

$\text{At critical angle}{}_{\text{D}}{\text{μ}}_{\text{R}}=\frac{\text{sin C}}{{\text{sin 90}}^{\text{o}}}\text{-----(2)}$

from equation (1) & (2)

$\text{sin C}=\frac{\text{sin r}}{\text{sin}\left(\text{90}-\text{r}\right)}\Rightarrow \text{sin r}=\text{tan r}\Rightarrow \text{C}={\text{sin}}^{-1}\text{tan r}$

Hence, the correct answer is option: 'A'.