A ray of light is incident on a parallel slab of thickness t and refractive index n. If the angle of incidence `theta` is small, than the lateral displacement in the incident and emergent ray will be

Options

A. `(t theta(n-1))/(n)`
B. `(t theta)/(n)`
C. `(t theta n)/(n-1)`
D. `(t theta(n+1))/(n)`

Answer

To find the lateral displacement of a ray of light passing through a parallel slab of thickness \( t \) and refractive index \( n \) when the angle of incidence \( \theta \) is small, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Geometry**:
- Consider a parallel slab of thickness \( t \) and refractive index \( n \).
- A ray of light strikes the slab at an angle \( \theta \) (the angle of incidence).

2. **Refraction at the Slab**:
- When the ray enters the slab, it bends towards the normal due to the higher refractive index of the slab compared to air.
- Let \( R \) be the angle of refraction in the slab.

3. **Using Snell's Law**:
- According to Snell's Law:
\[ n_1 \sin \theta = n_2 \sin R \]
- Here, \( n_1 = 1 \) (refractive index of air) and \( n_2 = n \) (refractive index of the slab).
- Thus, we have:
\[ \sin \theta = n \sin R \]
- For small angles, we can approximate \( \sin \theta \approx \theta \) and \( \sin R \approx R \).

4. **Finding \( R \)**:
- From the equation \( \sin \theta = n \sin R \), we can express \( \sin R \):
\[ \sin R = \frac{\sin \theta}{n} \approx \frac{\theta}{n} \]

5. **Calculating Lateral Displacement**:
- The lateral displacement \( D \) can be calculated using the formula:
\[ D = \frac{t}{\cos R} \left( \sin \theta - \sin R \right) \]
- For small angles, \( \cos R \approx 1 \) and \( \sin R \approx \frac{\theta}{n} \):
\[ D \approx t \left( \theta - \frac{\theta}{n} \right) \]
- Simplifying this gives:
\[ D \approx t \theta \left( 1 - \frac{1}{n} \right) \]

6. **Final Expression for Lateral Displacement**:
- Therefore, the lateral displacement \( D \) is:
\[ D = \frac{t \theta (n - 1)}{n} \]

### Conclusion:
The lateral displacement of the incident and emergent ray is given by:
\[ D = \frac{t \theta (n - 1)}{n} \]