Deeper Charge and Energy Budgets

Voltage-Square Energy Three-Row Scan

At fixed capacitance, stored charge follows voltage directly while stored energy follows voltage squared. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

Hold capacitance fixed and scan voltage

Keep capacitance fixed at 2 farads. Charge follows voltage directly, while stored energy follows the voltage-square column.

Q=CVU=12CV2Q=CV\qquad U=\tfrac{1}{2}CV^{2}
Fixed-capacitance scanThe checked capacitor is the middle voltage row.2 F

The square column explains the energy jump

The voltage steps are even, but the voltage-square column is not. That square column is what makes the energy column grow faster than the charge column.

CVV⋅VU2 F2 V44 J2 F4 V1616 J2 F6 V3636 J\begin{array}{c|c|c|c}C&V&V\cdot V&U\\2\ \text{F}&2\ \text{V}&4&4\ \text{J}\\2\ \text{F}&4\ \text{V}&16&16\ \text{J}\\2\ \text{F}&6\ \text{V}&36&36\ \text{J}\\\end{array}
Fixed-capacitance scanThe middle table row matches the displayed capacitor.2 F

Energy is not a linear voltage meter

Doubling voltage doubles charge for the same capacitor, but the energy route uses voltage squared. The table keeps that extra factor visible.

V↦Q linearlyV2↦UV\mapsto Q\text{ linearly}\qquad V^{2}\mapsto U
Fixed-capacitance scanThe checked diagram supports the voltage-square row.2 F