Question

As shown in figure, light is incident normally on one face of the prism. A liquid of refractive index $\mu$ is placed on the horizontal face $\text{AC}$. The refractive index of prism is $3/2$. If total internal reflection taken place on face $\text{AC}$, $\mu$ should be less than $\frac{l\sqrt{3}}{4},$ where $l$ is an integer. Find the value of $l$.

Answer 1

Critical angle, ${\theta }_c$ between glass and liquid,

$\sin {\theta }_c=\frac{\mu }{\left(3/2\right)}=\frac{2\mu }{3}....\left(1\right)$

Light ray will pass the surface $\text{AB}$ without bending since it is incident normally.

As, it strikes the surface $\text{AC}$ at an angle of incidence $i={60}^{\circ }$.

For Total internal reflection $\text{(TIR)}$ at the face $\text{AC}$, angle of incidence must be greater than the crtical angle . i.e. $i>{\theta }_c$

Taking $\sin$ both sides,

$\Rightarrow \sin {60}^{\circ }>\sin {\theta }_c$

Substituting the value of $\sin {\theta }_c$ from equation $\left(1\right)$, we get

$\frac{\sqrt{3}}{2}$ $>\frac{2}{3}\mu \Rightarrow$ $\mu <\frac{3\sqrt{3}}{4}$

From the data given in the question,

$\mu <\frac{l\sqrt{3}}{4}$

Comparing both we get , $l=3$.

Why this Question : This question will enhance the ability of solving critical problems on $\text{TIR}$ and this problem provides conceptual understanding to solve similar problems for JEE.