Question

For a Prism of refractive index $\mu$ , if $C<A\le 2C$, where $C$ is critical angle and $A$ is angle of prism. Find the value of angle of incidence, $i_0$ for which if,
$0\le i<i_0\to$$\text{TIR}$ occurs, ray does not come out of $\text{BC}$.
And if $i_0<i\le {90}^{\circ }\to$ Ray emerges from face $\text{BC}$.

[Assume, $i$ be the angle of incidence of incident ray as shown in the figure below]

• A. $i_0={\sin }^{-1}\left[\mu \sin \right(A-{\sin }^{-1}\mu \left)\right]$
• B. $i_0={\sin }^{-1}\left(\mu \sin A\right)$
• C. $i_0={\sin }^{-1}\left[\mu \sin (A-{\sin }^{-1}\frac{1}{\mu })\right]$
• D. $i_0={\sin }^{-1}\left[\mu \sin (A+{\sin }^{-1}\frac{1}{\mu })\right]$

Answer 1

The correct option is C $i_0={\sin }^{-1}\left[\mu \sin (A-{\sin }^{-1}\frac{1}{\mu })\right]$


Given that,

If angle of incidence $i$ is less than $i_0$ , $\text{TIR}$ will take place and if it is greater than $i_o$ , ray emerges out of face $\text{BC}$.

Condition for grazing emergence is $A=2C$.

If the ray just emerges from face $\text{BC}$,

$e={90}^{\circ }$ and $r_2=C......\left(1\right)$

Applying Snell's law at face $\text{AB}$

${\mu }_a\sin i_0={\mu }_p\sin (A-C)$

where ,
${\mu }_a\to$ Refractive index of surroundings.
${\mu }_p\to$ Refractive index of prism.

$\Rightarrow 1\times \sin i_0=\mu \sin (A-C)$

$\Rightarrow i_0={\sin }^{-1}\left[\mu \sin \right(A-C\left)\right]$

$\because$ Critical angle , $C={\sin }^{-1}\frac{1}{\mu }$

we get,

$i_0={\sin }^{-1}\left[\mu \sin (A-{\sin }^{-1}\frac{1}{\mu })\right]$

Thus, option (c) is the correct answer.