The top plate of a charged plane capacitor is fixed, whereas the bottom one is kept in equilibrium by the gravitational and electrostatic forces.

Options

A. When the capacitor is isolated, the plate is in stable equilibrium
B. When the capacitor is connected to a battery, the plate is In unstable equilibrium
C. When the capacitor is isolated, the plate is in neutral equilibrium
D. When a capacitor is connected to a battery, the plate is in neutral equilibrium.

Answer

To analyze the situation described in the question, we can break it down step by step:

### Step 1: Understanding the Capacitor Setup
We have a plane capacitor with two plates. The top plate is fixed in position, while the bottom plate is held in equilibrium by the balance of gravitational and electrostatic forces.

**Hint:** Consider the forces acting on the bottom plate and how they relate to the capacitor's functionality.

### Step 2: Forces Acting on the Bottom Plate
The bottom plate experiences two main forces:
- **Gravitational Force (Weight)**: This force acts downward and is equal to \( mg \), where \( m \) is the mass of the plate and \( g \) is the acceleration due to gravity.
- **Electrostatic Force**: This force acts upward due to the electric field created by the charge on the top plate. The electrostatic force can be expressed as \( F_e = E \cdot q \), where \( E \) is the electric field strength and \( q \) is the charge on the plate.

**Hint:** Remember that the electrostatic force is dependent on the charge and the electric field between the plates.

### Step 3: Equilibrium Condition
For the bottom plate to be in equilibrium, the gravitational force must equal the electrostatic force:
\[ mg = E \cdot q \]

**Hint:** This equation shows the balance of forces and is crucial for understanding the stability of the system.

### Step 4: Effect of Connecting to a Battery
When the capacitor is connected to a battery, it will charge up to a certain voltage. This increases the charge \( q \) on the plates, which in turn increases the electrostatic force acting on the bottom plate. The increased electrostatic force can disturb the equilibrium condition.

**Hint:** Think about how increasing the charge affects the forces and the stability of the system.

### Step 5: Stability Analysis
- **Isolated Capacitor**: When the capacitor is isolated (disconnected from the battery), it retains its charge. The equilibrium can be stable if the forces balance out correctly.
- **Connected Capacitor**: When connected to a battery, the capacitor is in unstable equilibrium because any small displacement of the bottom plate will lead to a change in the electrostatic force, potentially causing it to move further away from equilibrium.

**Hint:** Consider how the nature of equilibrium changes with the connection to the battery.

### Step 6: Conclusion
From the analysis:
1. When isolated, the capacitor can be in stable equilibrium.
2. When connected to a battery, the capacitor is in unstable equilibrium.

Thus, the correct statements are:
- The capacitor is in unstable equilibrium when connected to the battery.
- The capacitor is in neutral equilibrium when isolated.

**Final Answer:** The correct options are 2 and 3.