To understand why the dielectric constant of the dielectric slab is 5, let's break down the problem into different sections:

Given Information:
- The plates of a parallel plate capacitor are charged up to 200 volts.
- A dielectric slab of thickness 4 mm is inserted between the plates.
- To maintain the same potential difference between the plates, the distance between the plates is increased by 3.2 mm.

1. Capacitance of the Parallel Plate Capacitor:
The capacitance of a parallel plate capacitor is given by the formula:

C = ε₀ * (A/d)

Where:
- C is the capacitance
- ε₀ is the permittivity of free space (constant)
- A is the area of the plates
- d is the distance between the plates

2. Effect of Dielectric on Capacitance:
When a dielectric material is inserted between the plates of a capacitor, it increases the capacitance by a factor of the dielectric constant (K) of the material. The formula for the new capacitance is:

C' = K * C

3. Initial Capacitance:
Before the dielectric slab is inserted, the initial capacitance (C₁) is given by:

C₁ = ε₀ * (A/d₁)

Where d₁ is the initial distance between the plates.

4. Final Capacitance:
After the dielectric slab is inserted and the distance between the plates is increased to d₂, the final capacitance (C₂) is given by:

C₂ = K * ε₀ * (A/d₂)

5. Maintaining the Same Potential Difference:
To maintain the same potential difference (V) between the plates, the charge on the plates must remain constant. The formula for the charge (Q) on a capacitor is given by:

Q = C * V

Since the charge remains constant, we can equate the initial and final charges:

C₁ * V₁ = C₂ * V₂

Where V₁ is the initial potential difference and V₂ is the final potential difference.

6. Calculation:
Substituting the values into the equation:

ε₀ * (A/d₁) * V₁ = K * ε₀ * (A/d₂) * V₂

Canceling out common terms:

V₁/d₁ = K * V₂/d₂

Rearranging the equation:

V₂/V₁ = d₂/(K * d₁)

We know that the distance between the plates has increased by 3.2 mm:

d₂ = d₁ + 3.2 mm

Substituting this value into the equation:

V₂/V₁ = (d₁ + 3.2 mm)/(K * d₁)

Since the potential difference remains the same, V₂/V₁ = 1. Therefore:

1 = (d₁ + 3.2 mm)/(K * d₁)

Simplifying the equation:

K = (d₁ + 3.2 mm)/d₁

Given that the increase in distance is 3.2 mm and the original distance is 4 mm:

K = (4 mm + 3.2 mm)/4 mm = 7.2 mm/4 mm = 1.8

However, the question asks for the dielectric constant, not the ratio of the distances. The dielectric constant is defined as the ratio of the capacitance with the dielectric to the capacitance