An uncharged capacitor is fully charged with a battery . The ratio of energy stored in the capacitor to the work done by the battery in this process is

Options

A. `1 : 2 `
B. `2 : 1 `
C. `1: 1 `
D. `2 : 3 `

Answer

To solve the problem of finding the ratio of energy stored in a capacitor to the work done by the battery when charging the capacitor, we can follow these steps:

### Step 1: Understand the basic concepts
- An uncharged capacitor is charged by a battery.
- The capacitance of the capacitor is denoted as \( C \).
- The voltage of the battery is denoted as \( V \).

### Step 2: Calculate the charge on the capacitor
The charge \( Q \) stored in the capacitor when fully charged is given by the formula:
\[ Q = C \times V \]

### Step 3: Calculate the energy stored in the capacitor
The energy \( U \) stored in the capacitor can be calculated using the formula:
\[ U = \frac{Q^2}{2C} \]
Substituting \( Q = C \times V \) into the energy formula:
\[ U = \frac{(C \times V)^2}{2C} = \frac{C^2 V^2}{2C} = \frac{1}{2} C V^2 \]

### Step 4: Calculate the work done by the battery
The work done \( W \) by the battery while charging the capacitor can be calculated as:
\[ W = Q \times V \]
Substituting \( Q = C \times V \):
\[ W = (C \times V) \times V = C V^2 \]

### Step 5: Find the ratio of energy stored to work done
Now, we need to find the ratio of the energy stored in the capacitor to the work done by the battery:
\[ \text{Ratio} = \frac{U}{W} = \frac{\frac{1}{2} C V^2}{C V^2} \]
This simplifies to:
\[ \text{Ratio} = \frac{1}{2} \]

### Conclusion
Thus, the ratio of the energy stored in the capacitor to the work done by the battery is:
\[ \text{Ratio} = \frac{1}{2} \]