Capacitor Energy⚠ unverified
Electrical / Power Electronics · Compute the energy stored in a capacitor
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| C | C | F | 1.0 | Capacitance |
| V | V | V | 1.0 | Voltage across the capacitor |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | E | J | Stored energy, in joules (J) |
The science & history
Understanding the Parameters
- Capacitance $C$ — the charge held per volt, $Q = CV$; set by plate area, spacing, and dielectric.
-
Voltage $V$ — the potential difference across the plates. Energy grows with the square of voltage, so doubling $V$ quadruples the stored energy (and the dielectric stress).
-
$E$ — the field energy, in joules; the same energy is available on discharge (the basis of camera flashes, defibrillators, and DC‑link buffering).
Derivation (Approaching a Proof)
Charging is not free: pushing an extra charge $dq$ onto a capacitor already at voltage $v = q/C$ costs work $dW = v\,dq = \dfrac{q}{C}\,dq$. Integrating from an uncharged state to final charge $Q$:
$$E = \int_0^{Q} \frac{q}{C}\,dq = \frac{Q^2}{2C}.$$
Substituting $Q = CV$ gives the three equivalent forms
$$E = \frac{Q^2}{2C} = \frac{1}{2}QV = \frac{1}{2}CV^2.$$
The factor of $\tfrac12$ appears because the voltage rises linearly from $0$ to $V$ as charge accumulates, so the average voltage during charging is $V/2$. Physically the energy resides in the electric field, with density $u = \tfrac12 \varepsilon E^2$ integrated over the volume between the plates.
History
Energy storage in capacitors dates to the Leyden jar (1745, Musschenbroek and von Kleist), the first device to hold a substantial charge. The quantitative energy expression follows from the 19th‑century definitions of capacitance and potential, and is consistent with Maxwell's identification of energy stored in the electric field itself.
Related Concepts: Inductor Energy, Capacitive Reactance, Watt's Law
Notes: Registry calculator capacitor-energy (unverified). Assumes a linear (constant‑$C$)
capacitor; ferroelectric/variable dielectrics require integrating $\int v\,dq$ directly.