Energy stored in a capacitor equals:

The energy stored in a capacitor is equal to the work done by a battery to move charge onto its plates. 1. **Relationship between Charge and Voltage**: The charge $Q$ on a capacitor is proportional to the potential difference $V$ across it, defined by $Q = CV$, where $C$ is the capacitance. 2. **Deriving Work (Energy)**: As we move a small increment of charge $dq$, the small amount of work done is $dW = V dq$. Since $V = q/C$ at any instant: $$W = \int_{0}^{Q} \frac{q}{C} dq = \left[ \frac{q^2}{2C} \right]_{0}^{Q} = \frac{Q^2}{2C}$$ 3. **Expressing in terms of C and V**: Substituting $Q = CV$ into the energy equation: $$U = \frac{(CV)^2}{2C} = \frac{1}{2} CV^2$$ Therefore, the energy stored in a capacitor is $\frac{1}{2} CV^2$.