Voltage Squared - Capacitors & Stored Charge Step by Step

Stored energy is proportional to voltage squared at fixed capacitance. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

Start with the base energy

At 6 volts, the stored energy is 36 joules.

U_{\text{base}} = 36\ \text{J}

Double voltage

Raise the voltage to 12 volts while keeping capacitance the same.

V_{\text{changed}} = 12\ \text{V}

Energy scales with voltage squared

Because voltage is squared, doubling voltage multiplies stored energy by 4. The new energy is 144 joules.

U_{\text{changed}} = 4\cdot 36\ \text{J} = 144\ \text{J}

Squaring voltage bends the pattern

Hold capacitance fixed. Doubling voltage from three to six makes four times the energy; doubling again from six to twelve does the same.

\begin{array}{c|c|c|c}C&V&Q&U\\2\ \text{F}&3\ \text{V}&6\ \text{C}&9\ \text{J}\\2\ \text{F}&6\ \text{V}&12\ \text{C}&36\ \text{J}\\2\ \text{F}&12\ \text{V}&24\ \text{C}&144\ \text{J}\\\end{array}