Energy Density in the Electric Field

A charged capacitor stores energy, (1/2) C V², but that formula doesn't say where the energy is. The field picture answers that: the energy sits in the space between the plates, wherever the electric field exists. Nothing physical travels across the gap, yet energy is there the whole time, held in the field itself. This is a shift in thinking from charges acting on each other at a distance to the field as a real thing that carries energy on its own.

To turn (1/2) C V² into an energy per volume, use a parallel-plate capacitor with area A and separation d. Its capacitance is C = epsilon₀ A / d, and the field between the plates is uniform with magnitude E = V/d, so V = Ed. Substitute both into (1/2) C V²: (1/2)(epsilon₀ A/d)(Ed)² = (1/2) epsilon₀ E² (Ad). The quantity Ad is just the volume of the gap, so dividing total energy by that volume leaves u = (1/2) epsilon₀ E², the energy per cubic meter stored wherever a field of strength E exists. This result isn't special to capacitors; any electric field, from any source, stores this much energy per unit volume.

This connects directly to how a capacitor discharges. When you let charge flow off the plates, the field between them collapses, and the energy that was stored in that field converts to kinetic energy, light, or heat in the circuit. The punch you feel from a discharging capacitor is literally the field energy cashing out. It also explains the extension case: stretching the plates apart increases the volume holding the field, so even though energy density depends only on E, the total stored energy changes with geometry in ways that depend on whether charge or voltage is held fixed.