Energy in Capacitors and Inductors
Energy grows as the square of voltage
Slide the voltage V across the capacitor and watch the stored energy. The curve is not a straight line — it keeps steepening. Set the voltage so the stored energy reaches the target of 18 mJ.
Two ways to bank up work
A capacitor stores energy in the electric field between its plates. Raising the voltage means pushing charge in, and the charge already there pushes back on the next, so the work to add more keeps growing. An inductor stores energy in the magnetic field around its coil. Raising the current means working against the coil’s opposition (its induced voltage), which is stronger the larger the current. Both store only temporarily and give it back when needed.
Why the square appears
In a capacitor the charge is proportional to the voltage (Q = CV). While raising the voltage from 0 to V, the average voltage is half of V, and the work put in is average voltage times total charge: ½·V·(CV) = ½CV². The half comes from averaging a linear rise that starts at zero; the square comes from voltage and charge growing together in proportion. An inductor follows the same logic to give ½LI².
The square magnifies small differences
The square relation means doubling the voltage quadruples the energy; tripling it makes it nine times. So to greatly increase the energy in a capacitor, raising the voltage is far more effective than raising the capacitance — and conversely, even a small overvoltage can swell the stored energy dangerously. Because this energy pours out all at once on discharge, large capacitors call for care in handling.
Back to the first screen
As you slid the voltage, the stored energy traced not a line but a curve that kept steepening, and it reached the target of 18 mJ at V = 6 V. For the same capacitance, a little more voltage raised the energy by that much squared. The capacitor’s ½CV² and the inductor’s ½LI² are the same shape, and the energy the two elements traded in transients and resonance was exactly this stored amount.