Voltage Squared - Electric Fields, Electrostatics & Capacitors Step by Step

Stored energy grows with voltage squared when capacitance is fixed, so doubling voltage makes four times the energy. This pattern is exact for an ideal capacitor below its limits; real devices have breakdown ratings, dielectric losses, changing capacitance, heat limits, leakage, and measurement uncertainty.

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}