A capacitor of capacitance `C` is initially charged to a potential difference of `V` volt. Now it is connected to a battery of `2V` with oppoiste polarity. The ratio of heat generated to the final enegry stored in the capacitor will be

Options

A. `1.75`
B. `2.25`
C. `2.5`
D. `1//2`

Explanation

To solve the problem, we need to find the ratio of heat generated to the final energy stored in the capacitor after it is connected to a battery of opposite polarity. Let's break down the solution step by step:

### Step 1: Initial Energy Stored in the Capacitor
The initial energy (U_initial) stored in the capacitor when charged to a potential difference \( V \) is given by the formula:
\[ U_{\text{initial}} = \frac{1}{2} C V^2 \]

### Step 2: Final Charge on the Capacitor
When the capacitor is connected to a battery of \( 2V \) with opposite polarity, the final charge on the capacitor can be calculated. The charge on the capacitor before connecting the battery is \( Q_{\text{initial}} = CV \). After connecting the battery, the total charge on the capacitor becomes:
\[ Q_{\text{final}} = -CV + 2CV = CV \]
This means the final charge on the capacitor is \( -CV \) (since the battery's polarity is opposite).

### Step 3: Final Energy Stored in the Capacitor
The final energy (U_final) stored in the capacitor after connecting to the battery is given by:
\[ U_{\text{final}} = \frac{1}{2} C (2V)^2 = \frac{1}{2} C (4V^2) = 2CV^2 \]

### Step 4: Work Done by the Battery
The work done by the battery (W_battery) while charging the capacitor can be calculated using the formula:
\[ W_{\text{battery}} = Q_{\text{final}} \times V_{\text{battery}} = CV \times 2V = 2CV^2 \]

### Step 5: Heat Generated
The heat generated (Q) during this process can be calculated as the work done by the battery minus the change in energy stored in the capacitor:
\[ Q = W_{\text{battery}} - (U_{\text{final}} - U_{\text{initial}}) \]
Substituting the values:
\[ Q = 2CV^2 - \left(2CV^2 - \frac{1}{2} CV^2\right) \]
\[ Q = 2CV^2 - (2CV^2 - 0.5CV^2) = 2CV^2 - 1.5CV^2 = 0.5CV^2 \]

### Step 6: Ratio of Heat Generated to Final Energy
Now, we need to find the ratio of heat generated to the final energy stored in the capacitor:
\[ \text{Ratio} = \frac{Q}{U_{\text{final}}} = \frac{0.5CV^2}{2CV^2} = \frac{0.5}{2} = \frac{1}{4} \]

### Final Answer
The ratio of heat generated to the final energy stored in the capacitor is \( \frac{1}{4} \).

---