A capacitor is charged to store an energy `U`. The charging battery is disconnected. An Identical capacitor is now connected to the first capacitor in parallel. The energy in each capacitor is now.

Options

A. `3U//2`
B. `U`
C. `U//4`
D. `U//2`

Explanation

To solve the problem, let's break it down step by step.

### Step 1: Understanding the Initial Condition
A capacitor (let's call it Capacitor 1) is charged to store energy \( U \). The energy stored in a capacitor is given by the formula:
\[ U = \frac{1}{2} C V^2 \]
where \( C \) is the capacitance and \( V \) is the voltage across the capacitor.

### Step 2: Disconnecting the Battery
Once the capacitor is charged, the battery is disconnected. This means that the charge \( Q \) on Capacitor 1 remains constant. The charge \( Q \) on the capacitor can be expressed as:
\[ Q = C V \]
Since the battery is disconnected, the voltage \( V \) remains constant until we connect another capacitor.

### Step 3: Connecting an Identical Capacitor in Parallel
Now, we connect an identical capacitor (Capacitor 2) in parallel to Capacitor 1. When capacitors are connected in parallel, the total capacitance \( C_{total} \) becomes:
\[ C_{total} = C + C = 2C \]
However, since the charge \( Q \) remains constant, the new voltage \( V' \) across the combined capacitors can be calculated using the formula:
\[ Q = C_{total} V' = 2C V' \]
From the previous equation, we know \( Q = C V \). Therefore, we can equate:
\[ C V = 2C V' \]

### Step 4: Solving for the New Voltage
Dividing both sides by \( C \) (assuming \( C \neq 0 \)):
\[ V = 2 V' \]
Thus, we can express the new voltage \( V' \) as:
\[ V' = \frac{V}{2} \]

### Step 5: Calculating the New Energy Stored
Now, we can calculate the energy stored in each capacitor after connecting them in parallel. The energy stored in each capacitor is given by:
\[ U' = \frac{1}{2} C (V')^2 \]
Substituting \( V' = \frac{V}{2} \):
\[ U' = \frac{1}{2} C \left(\frac{V}{2}\right)^2 = \frac{1}{2} C \cdot \frac{V^2}{4} = \frac{1}{8} C V^2 \]

### Step 6: Finding the Energy in Each Capacitor
Since both capacitors are identical and have the same voltage \( V' \), the energy stored in each capacitor is:
\[ U' = \frac{1}{8} C V^2 \]
Now, we can relate this back to the original energy \( U \):
\[ U = \frac{1}{2} C V^2 \Rightarrow U' = \frac{1}{4} U \]
Thus, the energy in each capacitor after connecting the second identical capacitor in parallel is:
\[ U' = \frac{U}{4} \]

### Final Answer
The energy in each capacitor after connecting the identical capacitor in parallel is \( \frac{U}{4} \).

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