Energy in Capacitors and Inductors
Energy grows as the square of voltage
The graph shows the energy stored in a capacitor as a function of the voltage applied to it. Raise the voltage V and you can see the curve is not a straight line but keeps getting steeper. Set the voltage at which the stored energy reaches the target of 18 mJ marked on the vertical axis.
Two ways to bank up work
A capacitor stores energy in the electric field between its plates. Raising its voltage means pushing charge in, and because the charge already there repels each new charge, the work needed keeps growing. An inductor stores energy in the magnetic field around its coil. Increasing its current means working against the voltage the coil induces, and the larger the current, the stronger that pushback. In both, the storage is temporary, and the energy is handed back when needed.
Why the square appears
In a capacitor the charge is proportional to the voltage (Q = CV). While the voltage rises from 0 to V, the average voltage is half of V, and the work put in is the average voltage times the total charge, ½·V·(CV) = ½CV². The half comes from an average that started at 0 and rose in a straight line, and the square comes from voltage and charge growing together in proportion. By the same reasoning, an inductor stores ½LI².
The square magnifies small differences
The square law means that doubling the voltage quadruples the energy, and tripling it multiplies the energy by nine. The capacitor in the plot is 1 mF, so the energy of 2 mJ at 2 V becomes 18 mJ at 6 V, nine times as much. That is why raising the voltage is far more effective than raising the capacitance when you want to store much more energy. Conversely, even a small overvoltage can make the stored energy dangerously large, and since that energy pours out all at once on discharge, large capacitors must be handled with care.
Back to the first screen
As you raised the voltage, the stored energy traced not a straight line but an ever steeper curve, and it reached the target of 18 mJ at V = 6 V. With the same capacitance, a little more voltage made the energy grow as fast as its square. The capacitor's ½CV² and the inductor's ½LI² have the same form, and the energy the two elements traded back and forth in transients and resonance was exactly this stored amount.