Where field energy lives and flows
The fields themselves store energy, with energy density u, and they transport it, with a flux given by the Poynting vector \mathbf{S}. Poynting's theorem is nothing but local energy conservation: at any point, the rate the stored field energy falls, plus the net energy streaming out, equals the rate the fields do work on charges.
Electromagnetic energy density (electric plus magnetic) and the Poynting vector, the energy flux.
Poynting's theorem — a continuity equation for energy, coupling the field to the work done on charges.
Worked example: energy flows in through the sides of a wire
Take a straight cylindrical wire of radius a and length L carrying steady current I, with resistance R and voltage V=IR across it. Let us find the Poynting flux into its surface and see where the Joule heat actually comes from.
Step 1. The electric field along the surface is parallel to the wire, E_\parallel = V/L = IR/L. Step 2. The magnetic field circles the wire; at the surface B = \mu_0 I/(2\pi a). Step 3. \mathbf{S}=\tfrac{1}{\mu_0}\mathbf{E}\times\mathbf{B} points radially inward, with magnitude S = \tfrac{1}{\mu_0}\cdot\tfrac{IR}{L}\cdot\tfrac{\mu_0 I}{2\pi a} = \dfrac{I^2 R}{2\pi a L}. Step 4. Multiply by the side area 2\pi a L: the total power flowing in is S\cdot 2\pi a L = I^2 R.
Fields carry momentum too
If the field carries energy, relativity insists it also carries momentum. The momentum stored per unit volume is \mathbf{g}=\varepsilon_0\,\mathbf{E}\times\mathbf{B}=\mathbf{S}/c^2. The flow of that momentum — the stress in the field — is bookkept by the Maxwell stress tensor T_{ij}, whose divergence gives the electromagnetic force per unit volume on the charges.
The Maxwell stress tensor: the flux of the i-component of field momentum across a surface facing the j-direction.
Electromagnetic momentum density — the field itself stores momentum, tied to the energy flux by a factor 1/c².
Worked example: the push of sunlight
Because it carries momentum, light exerts pressure. A beam of intensity I (power per area) fully absorbed delivers a radiation pressure P=I/c; if perfectly reflected, the momentum reverses and the pressure doubles to $2I/c$.
Radiation pressure for full absorption and full reflection — a direct consequence of field momentum.
Put in numbers. Sunlight above the atmosphere has intensity I\approx 1361\ \mathrm{W/m^2} (the solar constant). Absorbed, that gives P \approx 1361/(3\times10^8) \approx 4.5\times10^{-6}\ \mathrm{Pa} — about a ten-billionth of atmospheric pressure. Tiny, but utterly real: it is the physics behind solar sails and a genuine perturbation on the orbits of small bodies in the solar system.
Where this leads next
You now hold classical electromagnetism as a complete field theory. Three roads lead onward. First, the covariant road: package \mathbf{E} and \mathbf{B} into one field tensor F^{\mu\nu} so that Lorentz invariance becomes manifest — the Radiation & Covariant EM track. Second, radiation: solve \Box A^\mu=\mu_0 J^\mu for accelerating charges and watch them shine. Third, quantization: promote the field to an operator and its excitations become photons — the beginning of quantum electrodynamics.