A light ray falls on a rectangular glass slab as shown in the figure. If total internal reflection occurs at the vertical face of the slab at point \( B \), the refractive index of glass is:
Show Hint
For total internal reflection problems, first determine the critical angle using:
\[
\sin \theta_c = \frac{1}{n}
\]
Then verify the geometry to ensure the incident angle \( \theta \) exceeds \( \theta_c \).
Step 1: Understanding Total Internal Reflection
Total internal reflection (TIR) occurs when light passes from a denser medium to a rarer medium at an angle greater than the critical angle \( \theta_c \), which satisfies:
\[
\sin \theta_c = \frac{1}{n}
\]
where \( n \) is the refractive index of glass.
Step 2: Identifying the Geometry
The incident ray undergoes refraction at the first surface and reaches point \( B \) at the vertical face. At this point, the angle of incidence \( \theta \) at \( B \) must be equal to or greater than the critical angle for TIR to occur.
Using Snell’s law at the first refraction:
\[
n \sin \theta_1 = \sin \theta_2
\]
where \( \theta_2 \) is determined by the geometry and refraction angles.
Step 3: Computing Refractive Index
The given figure suggests a relationship where:
\[
n = \frac{(\sqrt{3} + 1)}{2}
\]
Thus, the correct answer is:
\[
\frac{(\sqrt{3} + 1)}{2}
\]
