What is the required condition, if the light incident on one face of a prism, does not emerge from the other face?

Options

A. `n lt "cosec "((A)/(2))`
B. `n lt sec ((A)/(2))`
C. `n gt sec A`
D. `n gt "cosec "((A)/(2))`

Answer

To solve the problem of determining the required condition for light incident on one face of a prism not to emerge from the other face, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Prism Setup**:
- Consider a prism with two faces: AB and AC.
- A light ray PQ is incident on face AB at an angle of incidence \( I \).

2. **Refraction at Face AB**:
- When the light ray hits face AB, it refracts according to Snell's Law. Let \( R_1 \) be the angle of refraction at face AB.
- The relationship can be expressed as:
\[ n_1 \sin I = n_2 \sin R_1 \]
- Assuming the light is coming from air, \( n_1 = 1 \) and \( n_2 = n \) (the refractive index of the prism).

3. **Angle of Prism**:
- The angle of the prism is denoted as \( A \).
- According to the geometry of the prism, we have:
\[ A = R_1 + R_2 \]
- Here, \( R_2 \) is the angle of incidence at face AC.

4. **Condition for Total Internal Reflection**:
- For the light ray not to emerge from face AC, the angle \( R_2 \) must be greater than the critical angle \( C \).
- The critical angle \( C \) is defined as:
\[ C = \sin^{-1}\left(\frac{1}{n}\right) \]
- Therefore, we need:
\[ R_2 > C \]

5. **Combining the Conditions**:
- Since \( A = R_1 + R_2 \) and \( R_2 > C \), we can express this condition as:
\[ R_1 + R_2 > 2C \]
- This implies:
\[ A > 2C \]

6. **Using the Critical Angle**:
- Substituting the expression for the critical angle, we have:
\[ A > 2 \sin^{-1}\left(\frac{1}{n}\right) \]

7. **Final Condition**:
- Taking the sine of both sides, we find:
\[ \sin\left(\frac{A}{2}\right) > \frac{1}{n} \]
- Rearranging gives:
\[ n > \csc\left(\frac{A}{2}\right) \]

### Conclusion:
The required condition for the light incident on one face of a prism not to emerge from the other face is:
\[ n > \csc\left(\frac{A}{2}\right) \]