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 find the ratio of the energy stored in a capacitor to the work done by the battery while charging it, we can follow these steps:

### Step 1: Understand the Energy Stored in a Capacitor
The energy (U) stored in a capacitor can be expressed using the formula:
\[ U = \frac{1}{2} C V^2 \]
where \(C\) is the capacitance and \(V\) is the voltage across the capacitor.

### Step 2: Calculate the Work Done by the Battery
The work done (W) by the battery to charge the capacitor is given by the formula:
\[ W = C V^2 \]
This is because the battery moves charge \(Q = CV\) through a potential difference \(V\), and the work done is the product of charge and voltage.

### Step 3: Set Up the Ratio
Now, we need to find the ratio of the energy stored in the capacitor to the work done by the battery:
\[ \text{Ratio} = \frac{U}{W} = \frac{\frac{1}{2} C V^2}{C V^2} \]

### Step 4: Simplify the Ratio
In the ratio, we can cancel out \(C\) and \(V^2\):
\[ \text{Ratio} = \frac{\frac{1}{2} C V^2}{C V^2} = \frac{1}{2} \]

### Step 5: Conclusion
Thus, the ratio of the energy stored in the capacitor to the work done by the battery is:
\[ \frac{U}{W} = \frac{1}{2} \]

### Final Answer
The ratio is \( \frac{1}{2} \).

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