- Energy Stored in a Capacitor Definition: A capacitor stores energy by holding an electric charge on its plates.
- Charging Process: When connected to a battery, charges move to the capacitor plates, increasing its voltage and stored energy.
- Work Done to Store Charges: Subsequent charges need work to overcome repulsion from existing charges on the plates.
- Voltage Changes During Charging: The capacitor’s voltage changes until it equals the battery’s voltage.
- Energy Loss During Charging: Half of the energy from the battery is stored in the capacitor, and the other half is lost.
When an uncharged capacitor is connected to a battery, charge builds on its two plates with equal magnitude and opposite sign.
At the start, the plate charge is q = 0 C and the capacitor voltage is V = 0 V. The stored energy is also zero.
As charging begins, the capacitor’s voltage rises from zero. Moving the first small amount of charge onto an uncharged capacitor takes almost no work against its own electric field. Each later increment moves against a larger potential difference. Charge continues to move while the battery voltage exceeds the capacitor voltage.
For an intermediate charge in the capacitor, q, the voltage is q/C. Adding a small charge dq therefore requires the small amount of work dW = (q/C)dq. Integrating this changing work from zero to the final charge gives the total stored energy.
The capacitor voltage is not constant during charging. It rises from zero towards the source voltage.
As charge accumulates, both the voltage and stored energy increase.
The energy of the capacitor is therefore not simply Vq with the final voltage treated as constant; the integral gives U = VQ/2.
The electric field in the dielectric also increases as charge builds, and its direction is from the positive plate to the negative plate.
Here, dx is a small distance along the path between the capacitor plates.
Charge flows until the capacitor reaches the same voltage as the battery.
The energy calculation must cover the full charging process because the work per unit charge changes throughout it.
Suppose the positive plate of the capacitor holds charge q at voltage V = q/C. Moving an additional charge dq requires work dW = Vdq.
Integrating over the full charge range gives
This result is the energy stored in the electric field.
For an initially uncharged capacitor connected to a fixed-voltage source through resistance, the source supplies W = VQ. The capacitor voltage is not fixed during charging, so its stored energy is VQ/2.
The final charge on the capacitor is
The charge delivered by the battery is
Under this resistive-charging assumption, half the supplied energy is stored in the capacitor and half is dissipated in the circuit resistance.
