The electrostatic force between the metal plate of an isolated parallel plate capacitor `C` having charge `Q` and area `A`, is

Options

A. Proportional to the square root of the distance between the plates
B. Linearly proportional to the distance between the plates
C. Independent of the distance between the plates
D. Inversely proportional to the distance between the plates.

Explanation

To find the electrostatic force between the metal plates of an isolated parallel plate capacitor with capacitance \( C \), charge \( Q \), and area \( A \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Concept of Electric Field**:
The electric field \( E \) between the plates of a parallel plate capacitor can be expressed in terms of the surface charge density \( \sigma \). The surface charge density is given by:
\[ \sigma = \frac{Q}{A} \]
where \( Q \) is the charge on one plate and \( A \) is the area of the plates.

2. **Calculate the Electric Field**:
The electric field \( E \) between the plates of a parallel plate capacitor is given by:
\[ E = \frac{\sigma}{\epsilon_0} \]
where \( \epsilon_0 \) is the permittivity of free space. Substituting \( \sigma \):
\[ E = \frac{Q}{A \epsilon_0} \]

3. **Determine the Force on a Charge**:
The electrostatic force \( F \) acting on a charge \( Q \) in an electric field \( E \) is given by:
\[ F = Q \cdot E \]
Substituting the expression for \( E \):
\[ F = Q \cdot \left(\frac{Q}{A \epsilon_0}\right) \]
Simplifying this gives:
\[ F = \frac{Q^2}{A \epsilon_0} \]

4. **Analyze the Dependence on Distance**:
From the derived formula for force \( F \), we observe that there is no term involving the distance \( d \) between the plates. This indicates that the force is independent of the distance between the plates.

5. **Conclusion**:
Therefore, the electrostatic force between the metal plates of an isolated parallel plate capacitor is independent of the distance between the plates.

### Final Answer:
The electrostatic force between the metal plates of an isolated parallel plate capacitor is independent of the distance between the plates.