Assertion : A charged capacitor is disconnected from a battery. Now, if its plate are separated further, the potential energy will fall.
Reason : Energy stored in a capacitor is equal to the work done in charging it.

Options

A. If both Assertion and Reason are true and Reason is the correct explanation of Assertion.
B. If both Assertion and Reason are true but Reason is not correct explanation of Assertion.
C. If Assertion is true but Reason is false.
D. If Assertion is false but Reason is true.

Answer

To solve the problem, we need to analyze both the assertion and the reason given in the question.

### Step 1: Understand the Assertion
The assertion states that when a charged capacitor is disconnected from a battery and its plates are separated further, the potential energy will fall.

- When a capacitor is charged and then disconnected from the battery, the charge \( Q \) on the capacitor remains constant.
- The capacitance \( C \) of a parallel plate capacitor is given by the formula:
\[ C = \frac{A \epsilon_0}{d} \]
where \( A \) is the area of the plates, \( \epsilon_0 \) is the permittivity of free space, and \( d \) is the separation between the plates.

### Step 2: Analyze the Effect of Increasing Plate Separation
When the separation \( d \) is increased while the charge \( Q \) remains constant, the capacitance \( C \) decreases because \( C \) is inversely proportional to \( d \).

- The potential energy \( U \) stored in a capacitor is given by:
\[ U = \frac{Q^2}{2C} \]
- As \( C \) decreases (due to increased \( d \)), the potential energy \( U \) actually increases because \( U \) is inversely related to \( C \).

### Conclusion for Assertion
Thus, the assertion that the potential energy will fall is **false**.

### Step 3: Understand the Reason
The reason states that the energy stored in a capacitor is equal to the work done in charging it.

- The work done \( W \) in charging a capacitor can be expressed as:
\[ W = \int_0^Q V \, dQ \]
where \( V \) is the voltage across the capacitor.
- Since \( V = \frac{Q}{C} \), we can substitute this into the integral to find that the work done in charging the capacitor is indeed equal to the energy stored in it.

### Conclusion for Reason
The reason is **true** because the energy stored in a capacitor is equal to the work done in charging it.

### Final Conclusion
Since the assertion is false and the reason is true, the correct answer is that the assertion is false while the reason is true.

### Summary of Steps
1. Analyze the assertion about potential energy when the capacitor is disconnected and plates are separated.
2. Understand the relationship between capacitance, charge, and potential energy.
3. Evaluate the reason regarding the work done in charging the capacitor.
4. Conclude that the assertion is false and the reason is true.