A transparent container contains layers of three immiscible transparent liquids \(A\), \(B\) and \(C\) of refractive indices \(n\), \(\dfrac{3n}{4}\) and \(\dfrac{2n}{3}\), respectively. A laser beam is incident at the interface between \(A\) and \(B\) at an angle \(\theta\) as shown in the figure. Prove that the beam does not enter region \(C\) at all for
\[
\sin\theta>\frac{2}{3}.
\]
Show Hint
For problems involving several layers of liquids:
• Use Snell's law successively at each interface.
• Find the critical angle at the final interface.
• Apply the condition for total internal reflection:
\[
i>C
\qquad\text{or}\qquad
\sin i>\sin C.
\]
In this problem,
\[
\boxed{
\sin r=\frac43\sin\theta
}
\qquad\text{and}\qquad
\boxed{
\sin C=\frac89
}
\]
which finally gives
\[
\boxed{
\sin\theta>\frac23.
}
\]
