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Consider a glass prism immersed in a liquid as shown below. The refractive index of glass and liquid is 1.5 and 1.2, respectively. A ray of light enters the prism perpendicular to the face $A B$. The largest value of angle $\theta$ is, if the ray is totally reflected at the face $A C$, then
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The correct answer is:
$\cos ^{-1}(0.8)$
The given situation is shown in the following figure.
For the ray incident at face $A C$ to be total internally reflected, if
Angle of incidence $=$ Critical angle
$\begin{aligned}
& \Rightarrow \quad i=i_c \Rightarrow \sin (i)=\sin \left(i_c\right) \\
& \Rightarrow \quad \sin (90-\theta)=\frac{\mu_{\text {rarer }}}{\mu_{\text {denser }}} \\
& \Rightarrow \quad \cos (\theta)=\frac{\mu_l}{\mu_g} \Rightarrow \cos (\theta)=\frac{1.2}{1.5} \\
& \Rightarrow \quad \cos \theta=\frac{4}{5} \Rightarrow \cos \theta=0.8 \Rightarrow \theta=\cos ^{-1}(0.8)
\end{aligned}$
For the ray incident at face $A C$ to be total internally reflected, if
Angle of incidence $=$ Critical angle
$\begin{aligned}
& \Rightarrow \quad i=i_c \Rightarrow \sin (i)=\sin \left(i_c\right) \\
& \Rightarrow \quad \sin (90-\theta)=\frac{\mu_{\text {rarer }}}{\mu_{\text {denser }}} \\
& \Rightarrow \quad \cos (\theta)=\frac{\mu_l}{\mu_g} \Rightarrow \cos (\theta)=\frac{1.2}{1.5} \\
& \Rightarrow \quad \cos \theta=\frac{4}{5} \Rightarrow \cos \theta=0.8 \Rightarrow \theta=\cos ^{-1}(0.8)
\end{aligned}$
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