STATEMENT-1 `:` If the distance between plates of a charged isolated capacitor increases, then the potential energy in the electric field of capacitor increases.
and
STATEMENT-2 `:` The energy stored in a capacitor is always directly proportional to separation between the plates.

Options

A. Statement-1 is True, Statement-2 is True, Statement-2 is a correct explanation for Statement-1
B. Statement-1 is True, Statement-2 is True, Statement-2 is NOT a correct explanation for Statement-1
C. Statement-1 is True, Statement-2 is False
D. Statement-1 is False , Statement-2 is True

Answer

To analyze the statements provided in the question, we need to understand the relationship between the capacitance of a capacitor, the potential energy stored in it, and the distance between its plates.

### Step-by-Step Solution:

1. **Understanding Capacitance**:
The capacitance \( C \) of a parallel plate capacitor is given by the formula:
\[ C = \frac{\varepsilon_0 A}{d} \]
where \( \varepsilon_0 \) is the permittivity of free space, \( A \) is the area of one of the plates, and \( d \) is the distance between the plates.

2. **Effect of Increasing Distance**:
If the distance \( d \) between the plates increases, the capacitance \( C \) decreases because they are inversely related.

3. **Potential Energy in a Capacitor**:
The potential energy \( U \) stored in a capacitor can be expressed as:
\[ U = \frac{1}{2} C V^2 \]
or alternatively,
\[ U = \frac{Q^2}{2C} \]
where \( Q \) is the charge on the capacitor and \( V \) is the voltage across the capacitor.

4. **Isolated Capacitor**:
For an isolated capacitor, the charge \( Q \) remains constant when the distance \( d \) is increased. As \( C \) decreases (due to the increase in \( d \)), the potential energy \( U \) must increase because \( U \) is inversely proportional to \( C \) when \( Q \) is constant.

5. **Analyzing the Statements**:
- **Statement 1**: "If the distance between plates of a charged isolated capacitor increases, then the potential energy in the electric field of capacitor increases."
- This statement is **True** because as \( C \) decreases, \( U \) increases.
- **Statement 2**: "The energy stored in a capacitor is always directly proportional to separation between the plates."
- This statement is **False** because the energy stored in a capacitor is inversely proportional to the capacitance, which decreases as the separation increases.

### Conclusion:
- **Statement 1 is True**.
- **Statement 2 is False**.