**Introduction:**

In this scenario, we have a parallel plate capacitor with a separation of 5mm between the plates. Initially, the capacitor is charged to a potential difference (PD) of 25 volts using a battery. Later, a dielectric slab of thickness 3mm and dielectric constant (k) of 10 is inserted between the plates. We need to determine the PD between the plates after the dielectric slab is introduced.

**Calculation:**

To solve this problem, we can follow these steps:

1. Calculate the initial capacitance (C) of the parallel plate capacitor using the formula:
C = ε₀ * A / d₀
Where ε₀ is the permittivity of free space (8.85 x 10⁻¹² F/m²), A is the area of each plate, and d₀ is the initial separation between the plates.

2. Calculate the initial charge (Q₀) on the capacitor using the formula:
Q₀ = C * V₀
Where V₀ is the initial PD across the capacitor (25 volts).

3. Calculate the charge density (σ₀) on each plate using the formula:
σ₀ = Q₀ / A
Where A is the area of each plate.

4. Calculate the electric field (E₀) between the plates using the formula:
E₀ = V₀ / d₀
Where d₀ is the initial separation between the plates.

5. Calculate the electric displacement (D₀) between the plates using the formula:
D₀ = ε₀ * E₀
Where ε₀ is the permittivity of free space.

6. Calculate the new separation (d) between the plates after the dielectric slab is introduced:
d = d₀ - t
Where t is the thickness of the dielectric slab.

7. Calculate the electric field (E) between the plates with the dielectric slab using the formula:
E = V₀ / d
Where V₀ is the initial PD across the capacitor and d is the new separation between the plates.

8. Calculate the electric displacement (D) between the plates with the dielectric slab using the formula:
D = ε₀ * E
Where ε₀ is the permittivity of free space.

9. Calculate the new capacitance (C') of the parallel plate capacitor with the dielectric slab using the formula:
C' = D / E
Where D is the electric displacement and E is the electric field.

10. Calculate the new PD (V') across the capacitor with the dielectric slab using the formula:
V' = C' * Q₀
Where C' is the new capacitance and Q₀ is the initial charge on the capacitor.

**Conclusion:**

The PD between the plates after the dielectric slab is introduced can be calculated using the above steps. The calculation involves determining the initial capacitance and charge on the capacitor, as well as the electric field and electric displacement before and after the introduction of the dielectric slab. Finally, the new capacitance and PD across the capacitor with the dielectric slab can be calculated.