Light enters at an angle of incidence in a transparent rod of refractive index n. For what value of the refractive index of the material of the rod the light once entered into it will not leave it through its lateral face whatsoever be the value of angle of incidence.

Explanation

To solve the problem, we need to determine the condition under which light, once it enters a transparent rod with a refractive index \( n \), will not exit through its lateral surface regardless of the angle of incidence. This condition is satisfied when total internal reflection occurs.

### Step-by-Step Solution:

1. **Understanding Total Internal Reflection (TIR)**:
- Total internal reflection occurs when light travels from a medium with a higher refractive index to a medium with a lower refractive index, and the angle of incidence exceeds the critical angle.

2. **Identify the Relevant Angles**:
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