Closure Audits

Stored Energy Cross-Check Audit

one-half C V squared equals one-half Q V

Stored capacitor energy is checked both ways, with three exact rows showing the voltage-square pattern.

Example

Stored capacitor energy is checked both ways, with three exact rows showing the voltage-square pattern. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

Stored energy has two equivalent routes

A capacitor with 2 farads at 4 volts has 8 coulombs and stores 16 joules.

U=12CV2U=12QVU=\tfrac{1}{2}CV^{2}\qquad U=\tfrac{1}{2}QV
Stored energy closureThe checked capacitor value is used in both energy routes.2 F

The capacitance-voltage route shows the square

Hold capacitance at 2 farads and scan three voltages. Energy does not rise linearly with voltage because voltage is squared.

CVQU2 F2 V4 C4 J2 F4 V8 C16 J2 F8 V16 C64 J\begin{array}{c|c|c|c}C&V&Q&U\\2\ \text{F}&2\ \text{V}&4\ \text{C}&4\ \text{J}\\2\ \text{F}&4\ \text{V}&8\ \text{C}&16\ \text{J}\\2\ \text{F}&8\ \text{V}&16\ \text{C}&64\ \text{J}\\\end{array}
Stored energy closureThe middle table row is the checked capacitor.2 F

The charge-voltage route gives the same rows

The same rows can be read through one-half Q V after charge is computed from Q equals C V. The energy column must match.

QVU4 C2 V4 J8 C4 V16 J16 C8 V64 J\begin{array}{c|c|c}Q&V&U\\4\ \text{C}&2\ \text{V}&4\ \text{J}\\8\ \text{C}&4\ \text{V}&16\ \text{J}\\16\ \text{C}&8\ \text{V}&64\ \text{J}\\\end{array}
Stored energy closureThe middle table row is the checked capacitor.2 F

Voltage doubling gives four times the energy

Compare voltage factors against the energy column. From two volts to four volts, voltage doubles and energy is four times larger; from two volts to eight volts, voltage is four times larger and energy is sixteen times larger.

Vvoltage-square factorU2 V14 J4 V416 J8 V1664 J\begin{array}{c|c|c}V&\text{voltage-square factor}&U\\2\ \text{V}&1&4\ \text{J}\\4\ \text{V}&4&16\ \text{J}\\8\ \text{V}&16&64\ \text{J}\\\end{array}
Stored energy closureThe middle table row is the checked capacitor.2 F

Both routes must close before the lesson is trusted

The helper rejects mismatched charge or energy before any diagram is rendered. The table is therefore not free text: the charge, voltage, capacitance, and energy all close under the same exact unit algebra.

2⋅16 J=8 C⋅4 V2\cdot 16\ \text{J} = 8\ \text{C}\cdot 4\ \text{V}
Stored energy closureThe same capacitor closes both energy routes.2 F
capacitors Check stored energy through both one-half C V squared and one-half Q V.