The energy stored in a capacitor of capacity C and potential V is given by

Options

A. `(CV)/(2)`
B. `(C^(2)V^(2))/(2)`
C. `(C^(2)V)/(2)`
D. `(CV^(2))/(2)`

Answer

To find the energy stored in a capacitor of capacitance \( C \) and potential \( V \), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Capacitor**:
A capacitor stores electrical energy in the form of electrostatic potential energy. The capacitance \( C \) is defined as the ratio of the charge \( Q \) stored on one plate to the potential difference \( V \) across the plates:
\[ V = \frac{Q}{C} \]

2. **Work Done to Charge the Capacitor**:
When a small charge \( dQ \) is brought to the capacitor, the work done \( dW \) in moving this charge against the potential \( V \) is given by:
\[ dW = V \cdot dQ \]

3. **Substituting for Potential**:
Since \( V = \frac{Q}{C} \), we can substitute this into the work done equation:
\[ dW = \left(\frac{Q}{C}\right) dQ \]

4. **Total Work Done (Energy Stored)**:
To find the total work done (or energy stored) in the capacitor as we charge it from 0 to \( Q \), we integrate \( dW \):
\[ W = \int_0^Q dW = \int_0^Q \frac{Q}{C} dQ \]

5. **Performing the Integration**:
The integral can be computed as follows:
\[ W = \frac{1}{C} \int_0^Q Q \, dQ = \frac{1}{C} \left[ \frac{Q^2}{2} \right]_0^Q = \frac{1}{C} \cdot \frac{Q^2}{2} \]

6. **Final Expression for Energy**:
Thus, the energy \( U \) stored in the capacitor becomes:
\[ U = \frac{Q^2}{2C} \]

7. **Substituting for Charge**:
Since \( Q = CV \), we can substitute this into the energy equation:
\[ U = \frac{(CV)^2}{2C} = \frac{C V^2}{2} \]

8. **Conclusion**:
Therefore, the energy stored in a capacitor of capacitance \( C \) and potential \( V \) is given by:
\[ U = \frac{1}{2} C V^2 \]