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

### Step-by-Step Solution:

1. **Understanding the Prism**:
- Consider a prism with an angle \( A \) and two faces, \( AB \) and \( AC \). Light is incident on face \( AB \).

2. **Incident and Reflected Rays**:
- Let the light ray incident on face \( AB \) be represented as \( PQ \). The point of incidence on face \( AB \) is point \( A \).
- The angle of incidence at point \( A \) is denoted as \( I \).

3. **Refraction and Reflection**:
- Upon hitting face \( AB \), the light ray refracts into the prism and travels towards face \( AC \). The angle of refraction at face \( AB \) is denoted as \( R_1 \).
- The light ray then reaches face \( AC \) and the angle of incidence at this face is \( R_2 \).

4. **Condition for No Emergence**:
- For the light ray to not emerge from face \( AC \), it must be totally internally reflected. This occurs when the angle of incidence \( R_2 \) is greater than the critical angle \( C \) for the prism material.
- Therefore, we have the condition: \( R_2 > C \).

5. **Relating Angles**:
- The angle of the prism \( A \) is related to the angles \( R_1 \) and \( R_2 \) by the equation:
\[ A = R_1 + R_2 \]
- Since \( R_2 > C \), we can express this as:
\[ R_1 + R_2 > 2C \]
- This implies:
\[ A > 2C \]

6. **Using Sine Relations**:
- From the relationship of angles and the sine function, we can derive:
\[ \sin\left(\frac{A}{2}\right) > \sin(C) \]
- Using the refractive index \( n \), we know that:
\[ \sin(C) = \frac{1}{n} \]
- Therefore, we can conclude:
\[ n > \frac{1}{\sin\left(\frac{A}{2}\right)} \]

### Conclusion:
The required condition for light incident on one face of a prism to not emerge from the other face is:
\[ n > \frac{1}{\sin\left(\frac{A}{2}\right)} \]