Electromagnetic fields can store and transport energy. This is quantified by the energy density of the electric and magnetic fields and the Poynting vector, which describes the flow of electromagnetic energy through space.
In this section, we derive these quantities from first principles using Maxwell’s equations and the Lorentz force law.
Suppose we have some charge and current configuration which, at time t, produces fields E and B. In the next instant, dt, the charges move. How much work dW is done by the electromagnetic forces in the interval dt?
By the Lorentz force law, the work done on a charge q is:
F⋅dℓ=q(E+v×B)⋅vdt=qE⋅vdt
In terms of charge and current densities, we substitute q→ρdτ and ρv→J, so the total work rate over a volume V is:
dtdW=∫VE⋅Jdτ
This tells us that E⋅J is the power delivered per unit volume. We now express this in terms of the fields alone, using Ampère-Maxwell’s law:
J=μ01∇×B−ε0∂t∂E
Substituting into the power density:
E⋅J=E⋅(μ01∇×B−ε0∂t∂E)
Now we pause for a minute and look at the following vector identity:
∇⋅(E×B)=B⋅(∇×E)−E⋅(∇×B).
Replace the first term on the R.H.S with Faraday’s law, and rearrange to get
E⋅(∇×B)=−B⋅∂t∂B−∇⋅(E×B).
This looks nearly identical to the first term on the R.H.S of our working equation (except for the 1/μ0 factor).
This is Poynting’s theorem, the work energy theorem of electrodynamics. The first integral on the R.H.S is the
total energy stored in the fields. The second term evidently represents the rate at which energy is transported out of V, across the boundary surface S, by the electromagnetic fields. Poynting’s theorem says, then, that the work done on
the charges by the electromagnetic forces is equal to the decrease in energy remaining in the fields, less the energy
that flowed out through the surface.
The energy density stored in the electromagnetic fields is thus:
u=21(ε0E2+μ01B2)
While the energy per unit time, per unit area, transported by the fields is called the Poynting vector:
S=μ01E×B
This gives us the conservation law known as Poynting’s theorem:
∂t∂u+∇⋅S=−E⋅J
It states that the rate at which work is done on charges is equal to the decrease in field energy inside the volume, plus the outward flux of electromagnetic energy.
The electromagnetic field stores energy in both the electric and magnetic components. The energy density expressions differ depending on the unit system used. Below are the forms of the electric field energy densityuE and magnetic field energy densityuB in three common systems.
In the Gaussian (centimeter–gram–second) system, ε0 and μ0 are absorbed into the definitions of the fields. The speed of light c relates electric and magnetic quantities.
Electric field:
uE=8πE2
Magnetic field:
uB=8πB2
So the total energy density is:
u=8π1(E2+B2)
In this system, E and B have the same units: statvolts/cm.
Poynting’s theorem describes the local conservation of electromagnetic energy. Field energy is stored in both E and B, and the Poynting vector tracks how energy flows through space — essential in understanding wave propagation, radiation, and energy loss mechanisms in both laboratory and astrophysical settings.