An object lies in front if a thick parallel glass slab, the bottom of which is published if the distance between m first two images formed by bottom surface is `4 cm` then find the thickness of the slab [Assume `n_(glass) = 3//2` paraxial rays].

Options

A. a.2 cm
B. b.4 cm
C. c.6 cm
D. d.none of these

Explanation

To solve the problem, we need to determine the thickness of the glass slab based on the given information about the images formed by the bottom surface. Let's break down the solution step by step.

### Step-by-Step Solution:

1. **Understanding the Setup**:
- We have a thick parallel glass slab with a polished bottom surface.
- An object is placed in front of the slab, and the refractive index of the glass is given as \( n = \frac{3}{2} \).
- The distance between the first two images formed by the bottom surface of the slab is given as \( 4 \, \text{cm} \).

2. **Apparent Distance Calculation**:
- When light rays pass through the glass slab, they bend due to refraction. The apparent distance of the object from the top surface of the slab can be calculated using the formula:
\[ X' = \frac{X}{n} \]
where \( X \) is the actual distance of the object from the top surface and \( n \) is the refractive index of the glass.

3. **Total Distance to the Bottom Surface**:
- The total distance from the bottom surface of the slab to the object is:
\[ \text{Distance to bottom} = X' + T = \frac{X}{n} + T \]
where \( T \) is the thickness of the slab.

4. **Finding the First Image**:
- The first image formed by the bottom surface of the slab is at the same distance from the bottom surface as the apparent distance calculated above:
\[ \text{First Image Position} = \frac{X}{n} + T \]

5. **Finding the Second Image**:
- The second image will be formed at a distance of \( \frac{X}{n} + 2T \) from the bottom surface, as it reflects back after passing through the slab.

6. **Distance Between the Two Images**:
- The distance between the first and second images is given as \( 4 \, \text{cm} \):
\[ \left( \frac{X}{n} + 2T \right) - \left( \frac{X}{n} + T \right) = 4 \, \text{cm} \]
- Simplifying this gives:
\[ 2T - T = 4 \, \text{cm} \implies T = 4 \, \text{cm} \]

7. **Final Calculation**:
- From the above equation, we find:
\[ T = 2 \, \text{cm} \]

### Conclusion:
The thickness of the slab \( T \) is \( 2 \, \text{cm} \).