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 TIR takes place at this surface. For any angle of incidence, this condition will be satisfied, provided we have 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 i$=i_{max}={90}^{\circ }$
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

$A>2C$