Question

Which of the following conditions hold true if the ray (shown in figure) always undergo total internal reflection at the surface $\text{AC}$. The critical angle between prism and air is $i_c$.

• A. $A>i_c$
• B. $A<i_c$
• C. $A>2i_c$
• D. $A<2i_c$

Answer 1

The correct option is C $A>2i_c$


For the ray to never exit face $\text{AC}$ , $\text{TIR}$ should always occur at $\text{AC}$ irrespective of angle of incidence $i$ and angle of refraction $r_2$.

Condition for $\text{TIR}$,
$(r_2{)}_{min}>i_c$

$\Rightarrow$$r_2=A-r_1$ $($From figure$)$

$\Rightarrow$$(A-r_1{)}_{min}>i_c$

$(A-r_1)$ is minimum , when $r_1$ is maximum.

$\therefore A-(r_1{)}_{max}>i_c$

Now,
$r_1$ will be maximum when $i$ is maximum.

Thus,
$i_{max}={90}^{\circ }\Rightarrow r_1=i_c$ $($From reciprocity theorem$)$

So, $(r_1{)}_{max}=i_c$

$\therefore A-i_c>i_c$

$\Rightarrow A>2i_c$

If this condition is satisfied, the ray always show $\text{TIR}$ at $\text{AC}$.

Hence, option (c) is the correct answer.