Poynting Vector and Power Flow

Before energy can flow, it must be somewhere. The fields themselves hold it: an electric field stores energy in proportion to \(E^2\), and a magnetic field in proportion to \(H^2\). These densities, first met in electrostatics (Chapter 8) and magnetostatics (Chapter 14), now stand side by side:

For a plane wave the two are not just comparable — they are equal at every instant, because \(E/H=\eta=\sqrt{\mu/\varepsilon}\) forces \(\tfrac12\varepsilon E^2=\tfrac12\mu H^2\). Energy sloshes equally between the electric and magnetic forms as the wave advances, the field-theory echo of an LC circuit trading energy between capacitor and inductor.

Take the dot product of Maxwell's curl equations with \(\vec{H}\) and \(\vec{E}\), subtract, and apply a vector identity. Out drops a bookkeeping statement of breathtaking generality — Poynting's theorem, the energy-conservation law for electromagnetic fields:

Read it left to right: the net power flowing into the closed surface \(S\) (the minus sign makes inflow positive) equals the rate at which energy is stored in the fields, plus the rate at which it is dissipated as ohmic heat \(\sigma E^2\). No energy appears or vanishes — it only flows, accumulates, or turns to heat. The same statement in differential form is \(-\nabla\cdot(\vec{E}\times\vec{H})=\partial w/\partial t+\sigma E^2\).

The surface integrand in the theorem is the quantity we have been chasing. The Poynting vector is the power crossing unit area, and — crucially — it points in the direction that power flows:

The instantaneous \(\vec{\mathcal{P}}\) pulses at twice the wave frequency; what a meter — or an engineer — cares about is the average over a cycle. Using the half-real-part rule for products of phasors from Chapter 17, the average power density is the real part of the complex Poynting vector:

The split mirrors AC circuit theory exactly. The real part is the average power that actually travels and can be delivered to a load; the imaginary part is reactive power that merely sloshes back and forth, stored and returned each cycle without net transport. The conjugate \({}^{*}\) on \(\vec{H}_s\) is what encodes the phase angle between the fields — and in a lossy medium that angle is precisely the \(\theta_\eta\) of Chapter 19.

Apply the rule to the plane wave \(\vec{E}_s=E_0e^{-\alpha z}e^{-j\beta z}\hat{a}_x\), whose companion \(\vec{H}_s\) lags by \(\theta_\eta\) and has amplitude \(E_0/|\eta|\). The cross product lands along \(\hat{a}_z\), and the result captures everything at once:

Three features deserve a pause. The power falls as \(e^{-2\alpha z}\) — twice as fast as the field, because power goes as amplitude squared. The factor \(\cos\theta_\eta\) is a power factor identical to \(\cos\phi\) in AC circuits: a lossy medium's field-phase lag throttles the deliverable power. And in the lossless limit \(\alpha=0\), \(\theta_\eta=0\), it reduces to the clean free-space result:

Where does the missing power go? Straight into heat. The average power dissipated per unit volume in a conducting medium is set by the same half-real-part rule applied to \(\vec{E}\) and the conduction current \(\sigma\vec{E}\):

The two pictures must agree, and they do. Differentiating the average power density gives \(-d\mathcal{P}_{\text{avg}}/dz=2\alpha\,\mathcal{P}_{\text{avg}}\): the power lost per metre is exactly the heat deposited per cubic metre, \(\tfrac12\sigma E_0^2 e^{-2\alpha z}\). Attenuation is not the wave "running out of steam" — it is energy being handed, joule by joule, to the medium. That closed loop between \(\alpha\), \(\eta\), and dissipation is Poynting's theorem made local.

For each item, decide whether you need an energy density, a power density, a total power, or a balance check — then reach for the matching relation. Difficulty rises down the list.