An air capacitor is completely charged upto the energy U and removed from battery. Now distance between plates is increased slowly by an external agent. If work done by external agent is 3U then ratio of final separation between the plates to the initial separation :

Options

A. 5
B. 4
C. 3
D. `1.5`

Explanation

To solve the problem, we will analyze the situation step by step.

### Step 1: Understand the Initial Conditions
- An air capacitor is charged to an energy \( U \) and then disconnected from the battery. This means that the charge \( Q \) on the capacitor remains constant.

### Step 2: Work Done by the External Agent
- The work done by the external agent while increasing the separation between the plates is given as \( 3U \).

### Step 3: Relate Work Done to Change in Potential Energy
- The work done by the external agent is equal to the change in potential energy of the capacitor. We can express this as:
\[ W = U_f - U_i \]
where \( U_f \) is the final potential energy and \( U_i \) is the initial potential energy.

### Step 4: Substitute Known Values
- We know that the initial potential energy \( U_i = U \) and the work done \( W = 3U \). Therefore, we can write:
\[ 3U = U_f - U \]
Rearranging gives:
\[ U_f = 3U + U = 4U \]

### Step 5: Use the Formula for Potential Energy of a Capacitor
- The potential energy \( U \) of a capacitor is given by the formula:
\[ U = \frac{Q^2}{2C} \]
where \( C \) is the capacitance. For a parallel plate capacitor, the capacitance \( C \) is given by:
\[ C = \frac{\varepsilon_0 A}{d} \]
where \( A \) is the area of the plates and \( d \) is the separation between the plates.

### Step 6: Analyze the Change in Capacitance
- Since the charge \( Q \) remains constant and the separation \( d \) is increased, the new capacitance \( C_f \) can be expressed as:
\[ C_f = \frac{\varepsilon_0 A}{d_f} \]
where \( d_f \) is the final separation.

### Step 7: Relate Initial and Final Potential Energies
- The initial potential energy can be expressed as:
\[ U = \frac{Q^2}{2C_i} = \frac{Q^2 d_i}{2 \varepsilon_0 A} \]
and the final potential energy as:
\[ U_f = \frac{Q^2}{2C_f} = \frac{Q^2 d_f}{2 \varepsilon_0 A} \]

### Step 8: Set Up the Ratio of Energies
- We know \( U_f = 4U \), thus:
\[ \frac{Q^2 d_f}{2 \varepsilon_0 A} = 4 \cdot \frac{Q^2 d_i}{2 \varepsilon_0 A} \]
This simplifies to:
\[ d_f = 4d_i \]

### Step 9: Find the Ratio of Final to Initial Separation
- The ratio of the final separation \( d_f \) to the initial separation \( d_i \) is:
\[ \frac{d_f}{d_i} = 4 \]

### Final Answer
The ratio of the final separation between the plates to the initial separation is \( 4 \).

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