Question

A plane mirror is placed horizontally inside liquid of refractive index $7/5$ . As ray is made incident normally on the mirror it is rotated through angle $\theta .$ The value of $\theta$ for which ray does not come out of the liquid surface is $\theta >\frac{1}{2}{\left(sin\right)}^{-1}(\frac{x}{7})$ where ${}^{\prime }{x}^{\prime }$ has the value____.

Answer 1

Answer: 5

As plane mirror is rotated through angle $\theta ,$ the reflected ray turns through angle $2\theta .$ For total internal reflection to take place at liquid-air boundary, $sin2\theta >sin{i}_c$
$sin2\theta >\frac{1}{{\mu }_d}$ i.e.,
$sin2\theta >\frac{1}{7/5}$
$2\theta >{\sin }^{-1}(5/7)$
$\theta >\frac{1}{2}{\sin }^{-1}(5/7)$
$\Rightarrow x=5$