Electromagnetic Power & Poynting Vector

Poynting-vector energy flow around a battery and resistor circuit. — Field-energy flow around a simple DC circuit: the Poynting vector points from the source through surrounding fields toward the load. Image source: Wikimedia Commons, Chetvorno / King of Hearts.

30-second intuitionThe Poynting vector tells you where electromagnetic energy is flowing. In a transmission line, waveguide, antenna beam or optical system, useful power is transported by fields through space—not by an abstract voltage alone.

Instantaneous power flow

\[\boxed{\mathbf S=\mathbf E\times\mathbf H}\]

with units W/m$^2$.

The direction of $\mathbf S$ gives local electromagnetic energy-flow direction.

Time-average sinusoidal power

For peak-value phasors,

\[\boxed{\langle\mathbf S\rangle=\frac12\operatorname{Re}\{\mathbf E\times\mathbf H^*\}.}\]

For a uniform lossless plane wave,

\[\langle S\rangle=\frac{|E|^2}{2\eta}=\frac{\eta|H|^2}{2}.\]

Energy density

For a linear nondispersive medium,

\[u_E=\frac12\mathbf E\cdot\mathbf D,\] \[u_H=\frac12\mathbf B\cdot\mathbf H.\]

Poynting theorem

A differential energy-conservation form is

\[\boxed{\nabla\cdot\mathbf S+\frac{\partial u}{\partial t}+\mathbf J\cdot\mathbf E=0.}\]

Interpretation:

$\nabla\cdot\mathbf S$: net field-energy flow out;
$\partial u/\partial t$: change in stored field energy;
$\mathbf J\cdot\mathbf E$: power transferred from field to matter/charges.

This is the electromagnetic analog of an energy bookkeeping equation.

Complex Poynting vector

In sinusoidal steady state,

\[\mathbf S_c=\frac12\mathbf E\times\mathbf H^*.\]

Its real part represents average power flow; the imaginary part is associated with reactive/stored energy exchange under common conventions.

Worked example — field strength from power density

For a free-space plane wave carrying $1$ W/m$^2$ average power,

\[|E|=\sqrt{2\eta_0\langle S\rangle} \approx\sqrt{2\times376.73}\approx27.4\ \text{V/m}\]

for peak field amplitude.

If using RMS electric field instead,

\[E_{rms}=\sqrt{\eta_0\langle S\rangle}\approx19.4\ \text{V/m}.\]

Always state whether field amplitudes are peak or RMS.

Antennas and links

In the far field, integrating radial Poynting flux over a sphere gives radiated power:

\[P_{rad}=\int\!\!\int \langle S_r\rangle r^2d\Omega.\]

This leads naturally to radiation intensity, directivity and gain.

Measurement

Power flow is inferred using power meters, calibrated antennas/probes, directional couplers, field sensors and calorimetric methods. A local E-field measurement only converts to plane-wave power density when the plane-wave impedance relation is justified.

Common misconception

Near-field warningIn a reactive near field, $E/H$ need not equal $377\ \Omega$, and $E^2/377$ is not automatically the correct local power density.

Engineering reality

Where the energy goesLoss appears through $\mathbf J\cdot\mathbf E$ in conductors/materials, radiation carries energy away, and resonators repeatedly exchange electric and magnetic stored energy before dissipation or extraction.

References

J. D. Jackson, Classical Electrodynamics.
R. F. Harrington, Time-Harmonic Electromagnetic Fields.
S. Ramo, J. R. Whinnery and T. Van Duzer, Fields and Waves in Communication Electronics.

Advanced topic — reactive near-field power and energy accounting

For time-harmonic phasors with a consistent RMS or peak convention, the complex Poynting vector separates the real power flux from a quadrature term associated with reactive energy exchange. With peak phasors,

\[\mathbf S_c=\frac12\mathbf E\times\mathbf H^*.\]

Its real part gives time-averaged energy flux. The imaginary part should not be interpreted as an independently transported average power; near-field energy is repeatedly stored and returned.

For a closed surface enclosing sources and lossy objects, apply the frequency-domain Poynting theorem to distinguish radiation, conductor/dielectric dissipation and local storage. In dispersive materials, electromagnetic stored-energy density needs a treatment compatible with material dispersion, rather than indiscriminately using (\tfrac14(\epsilon

^2+\mu

^2)).

Advanced exercise: integrate complex power over spheres surrounding a Hertzian dipole and compare the radiated real power with the radius-dependent reactive contribution.

Further reading: R. F. Harrington, Time-Harmonic Electromagnetic Fields.