ELC-08

Capacitor stored energy

U = ½ C V² = ½ Q V.

Reading speed

Energy storageMaxwell

Governing equation

U=\tfrac12 C V^2=\tfrac12 Q V

$V \in \left[0.1, 1000\right]$

where

C: Capacitance (µF)
V: Voltage (V)
U: Energy (J)
Q: Charge (mC)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

00:0 / 00:08

①

Governing law — red

U=\tfrac12 C V^2=\tfrac12 Q V

$V \in \left[0.1, 1000\right]$

$V = 0.1$ $U = 5\times 10^{-7}$

Along the curve

②

Domain of the variable — blue

V_{\min} = 0.1,\quad V_{\max} = 1000

$V \in \left[0.1, 1000\right]$

$V = 0.1$ $V = 0.1$

Along the curve

③

Response curve — green

U=\tfrac12 C V^2=\tfrac12 Q V

$V \in \left[0.1, 1000\right]$

$V = 0.1$ $U = 5\times 10^{-7}$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}U}{\mathrm{d}V}

$V \in \left[0.1, 1000\right]$

$V = 6.389$ $U^{\prime} = 6.39\times 10^{-4}$

Along the curve

Lecture brief

Historical brief

Ohm (1827), Kirchhoff (1845) and Maxwell’s circuit reduction still run every board: RLC transients, transformers, skin effect and three-phase power. The sheets are those network laws, not a SPICE deck. This sheet (ELC-08 — Capacitor stored energy) is the form associated with Maxwell. Working symbols: $C$ , $V$ $\rightarrow$ $U$ , $Q$ . The work to assemble charge on a capacitor is the integral V dq = C V dV, hence ½ C V².

Purpose

Purpose: compute

U

Q

from

C

V

in Electrical via

U=\tfrac12 C V^2=\tfrac12 Q V

U = ½ C V² = ½ Q V. Use it when a real electrical question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

C = 100.000\,\mathrm{\mu F}

V = 24.000\,\mathrm{V}

, the governing relation

U=\tfrac12 C V^2=\tfrac12 Q V

yields

U = 0.0288\,\mathrm{J}

Q = 2.4000\,\mathrm{mC}

. Linear capacitor, quasistatic. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Capacitance C100 µFVoltage V24 V

Outputs

Energy U0.0288 J
Charge Q2.4000 mC

Reading speed

Watch on YouTube

Free library

Full library

Free PDF / open book

YouTube channels

00:0 / 00:08

Narration of this film

Linear capacitor, quasistatic.

The work to assemble charge on a capacitor is the integral V dq = C V dV, hence ½ C V².

Reading speed

Watch on YouTube