Field energy density

Electric fields carry energy. Wherever an electric field exists, there is an associated energy density u = ½ε₀E² — joules per cubic metre, proportional to the square of the field strength. Integrate that density over all space and you get the total energy it took to assemble whatever charge distribution is sourcing the field. This is where the energy stored in a capacitor actually lives: not on the plates, but in the field between them.

The idea that the field itself stores energy was revolutionary. In the Newtonian action-at-a-distance picture, there was nothing between two charges; energy was just a property of the pair. Faraday and Maxwell said no — the energy is spread through the space around the charges, every cubic metre holding its share of ½ε₀E². The total is the same as the old accounting, but the location is completely different. This picture turned out to be essential once electromagnetic waves were discovered: a light wave propagating through vacuum carries energy in exactly this form, even though there are no charges at all in the space it occupies.

Field energy density is the electric companion of the analogous magnetic expression, u_B = B²/(2μ₀). In a general electromagnetic field both contribute, and their sum is the local energy density of the field. The flow of that energy through space is described by the Poynting vector.