Question

Derive an expression for the potential energy of a charged capacitor.

Hint:

The potential energy stored in a capacitor is given by \( U = \frac{1}{2} C V^2 \), where \( C \) is the capacitance and \( V \) is the potential difference.

Answer 1

Step 1: Capacitor and Potential Energy.
The potential energy stored in a capacitor is given by the formula: \[ U = \frac{1}{2} C V^2 \] where \( U \) is the potential energy, \( C \) is the capacitance of the capacitor, and \( V \) is the potential difference across the plates of the capacitor.
Step 2: Derivation.
The energy stored in the capacitor can also be calculated as the work done to move a small charge \( dq \) from one plate to the other against the potential difference \( V \). The total work done in charging the capacitor from 0 to \( V \) is the potential energy stored in the capacitor: \[ W = \int_0^Q V dq \] where \( Q \) is the total charge and \( V = \frac{Q}{C} \). Substituting this into the integral: \[ W = \int_0^Q \frac{Q}{C} dq = \frac{1}{C} \int_0^Q Q dq = \frac{1}{C} \cdot \frac{Q^2}{2} \] Since \( Q = CV \), we get: \[ U = \frac{1}{2} C V^2 \] Step 3: Conclusion.
Thus, the expression for the potential energy stored in a charged capacitor is: \[ U = \frac{1}{2} C V^2 \]