electromagnetic energy density
Empty space threaded by an electric or magnetic field is not truly empty — it holds energy, energy you can extract from a charged capacitor or a current-carrying inductor. The electromagnetic energy density is how much of that energy is stored in each cubic metre of field.
The energy density is u = (epsilon_0/2) E^2 + (1/(2 mu_0)) B^2, a sum of an electric part growing as the square of E and a magnetic part growing as the square of B. Integrate u over all space and you recover the total energy it took to assemble the charges and currents. Two familiar special cases confirm it: the energy stored in a capacitor is the volume integral of (epsilon_0/2)E^2, giving the usual (1/2)CV^2, and the energy in an inductor is the integral of B^2/(2 mu_0), giving (1/2)LI^2.
This density is the u that appears in Poynting's theorem, du/dt + div S = -J·E, tying stored energy to energy flow and to work on charges. For a plane wave the electric and magnetic contributions are exactly equal, so the energy is shared fifty-fifty between E and B, and the density travels along with the wave at speed c. That the energy lives in the FIELD, spread through space, rather than in the charges, is a genuinely non-obvious claim — it is what lets energy leave the source entirely and radiate to infinity.
Between the plates of a parallel-plate capacitor holding field E, the stored energy per unit volume is (epsilon_0/2)E^2. Multiply by the volume between the plates and you recover exactly the textbook (1/2)CV^2 — the same energy, viewed as residing in the field rather than on the plates.
u = (epsilon_0/2)E^2 + (1/(2 mu_0))B^2: energy stored per unit volume of electric and magnetic field.
The energy is attributed to the FIELD throughout space, not to the charges — a conceptual leap that only fields, not action-at-a-distance, make sensible. In a linear medium the vacuum constants are replaced by epsilon and mu, giving u = (1/2)(E·D + B·H).