A capacitor is charged by a battery and the energy stored is U. The battery is now removed and the separation distance between the plates is doubled. The energy stored now is

Options

A. `U//2`
B. U
C. 2U
D. 4U

Explanation

To solve the problem step by step, we will analyze the situation involving a capacitor, the energy stored in it, and the effects of changing the distance between its plates.

### Step 1: Understand the Initial Conditions
Initially, a capacitor is charged by a battery, and the energy stored in the capacitor is given as \( U \). The energy stored in a capacitor can be expressed using the formula:
\[ U = \frac{Q^2}{2C} \]
where \( Q \) is the charge on the capacitor and \( C \) is the capacitance.

### Step 2: Recall the Capacitance Formula
For a parallel plate capacitor, the capacitance \( C \) is given by:
\[ C = \frac{\epsilon_0 A}{D} \]
where \( \epsilon_0 \) is the permittivity of free space, \( A \) is the area of the plates, and \( D \) is the separation between the plates.

### Step 3: Analyze the Changes After Removing the Battery
Once the battery is removed, the charge \( Q \) on the capacitor remains constant because there is no external circuit to allow charge to flow.

### Step 4: Change the Distance Between the Plates
Now, the distance between the plates is doubled, so the new distance \( D' \) is:
\[ D' = 2D \]

### Step 5: Calculate the New Capacitance
With the new distance, the new capacitance \( C' \) becomes:
\[ C' = \frac{\epsilon_0 A}{D'} = \frac{\epsilon_0 A}{2D} = \frac{C}{2} \]

### Step 6: Calculate the New Energy Stored
Now, we can find the new energy stored \( U' \) in the capacitor using the formula for energy:
\[ U' = \frac{Q^2}{2C'} \]
Substituting \( C' \) into the equation gives:
\[ U' = \frac{Q^2}{2 \left(\frac{C}{2}\right)} = \frac{Q^2}{C} = 2 \left(\frac{Q^2}{2C}\right) = 2U \]

### Conclusion
Thus, the new energy stored in the capacitor after the distance between the plates is doubled is:
\[ U' = 2U \]