Question

If the critical angle for the medium of a prism is $C$ and the angle of prism is $A$, then there will be no emergent ray when

• A. $A\le 2C$
• B. $A<2C$
• C. $A=2C$
• D. $A>2C$

Answer 1

The correct option is D $A>2C$

The light entering the prism at surface $AB$, will not be able to come out from the surface $AC$, if total internal reflection $\left(TIR\right)$ takes place at this surface.

As $TIR$ takes place at surface $AC$

$(r_2{)}_{min}>C$

Since $r_1+r_2=A$

$\Rightarrow r_2=A-r_1$

$\Rightarrow (r_2{)}_{min}=A-(r_1{)}_{max}\dots \left(1\right)$

But $(r_1{)}_{max}$ is possible when incident angle is maximum $\Rightarrow i_{max}={90}^{\circ }$
i.e., incident ray grazes the interface $AB$.

Now, applying Snell's Law at $AB$,

$1\times sini=\mu sin{r}_1$

$\Rightarrow sin\left({90}^{\circ }\right)=\mu sin{r}_1$

$\Rightarrow r_1=si{n}^{(-1)}\left(\frac{1}{\mu }\right)$

$\Rightarrow r_1=C\dots \left(2\right)$

From equations (1) and (2), we get

$r_1+r_2>2C\dots \left(3\right)$

Since, $r_1+r_2=A$

Therefore, the condition becomes

$A>2C$