Poynting vector

The Poynting vector is the local energy flux of an electromagnetic field:

$\mathbf{S}=\mathbf{E}\times \mathbf{H}{\text{(W/m}}^2\text{)}.$

It points in the direction of energy flow, with magnitude equal to power per unit area through a surface perpendicular to that direction. At every point in space and time, $\mathbf{S}$ tells you “energy is moving this way, at this rate.”

This is the EM version of “intensity” — power per area. Sunlight at Earth has $|\mathbf{S}|\approx 1361$ W/m² (the solar constant). Wi-Fi radiation a meter from a router is a few μW/m². The local Poynting vector encodes how energy redistributes itself as fields propagate.

Why $\mathbf{E}\times \mathbf{H}$

The Poynting vector falls out of Maxwell’s equations by a manipulation that derives an energy continuity equation. Starting from Faraday’s and Ampère’s laws and combining:

$\nabla \cdot (\mathbf{E}\times \mathbf{H})=-\mathbf{E}\cdot \mathbf{J}-\mathbf{E}\cdot \frac{\partial \mathbf{D}}{\partial t}-\mathbf{H}\cdot \frac{\partial \mathbf{B}}{\partial t}.$

The right side has three pieces:

• $-\mathbf{E}\cdot \mathbf{J}$ — rate of work done by the field on currents (resistive dissipation: this is Joule heating).
• $-{\partial }_t\frac{1}{2}\mathbf{D}\cdot \mathbf{E}$ — rate of change of electric energy density.
• $-{\partial }_t\frac{1}{2}\mathbf{B}\cdot \mathbf{H}$ — rate of change of magnetic energy density.

So we have Poynting’s theorem:

$\nabla \cdot \mathbf{S}+\frac{\partial }{\partial t}\left(\frac{1}{2}\mathbf{D}\cdot \mathbf{E}+\frac{1}{2}\mathbf{B}\cdot \mathbf{H}\right)=-\mathbf{E}\cdot \mathbf{J}.$

Read as an energy-conservation equation: divergence of energy flux + rate of change of stored energy density = $-$ rate of work done on currents (dissipation). The form is the same as the Charge continuity equation — same physics, different conserved quantity (energy instead of charge).

In a plane wave

For a Plane wave propagating in $+\hat{\mathbf{z}}$ with $\mathbf{E}=\hat{\mathbf{x}}{E}_0\cos (\omega t-\beta z)$ and $\mathbf{H}=\hat{\mathbf{y}}(E_0/\eta )\cos (\omega t-\beta z)$:

$\mathbf{S}=\mathbf{E}\times \mathbf{H}=\hat{\mathbf{z}}\frac{E_0^2}{\eta }{\cos }^2(\omega t-\beta z).$

Direction: along $\hat{\mathbf{z}}$ — same as the propagation direction. Magnitude oscillates at twice the wave frequency (because ${\cos }^2$ has half the period).

Time-averaged Poynting vector (using $\overline{{\cos }^2}=\frac{1}{2}$):

$⟨\mathbf{S}⟩=\hat{\mathbf{z}}\frac{E_0^2}{2\eta }{\text{(W/m}}^2\text{)}.$

This is the time-average intensity of the wave — the steady power per unit area that a sensor would measure. In free space with ${\eta }_0=377$ Ω, a plane wave with peak field $E_0=1$ V/m carries average intensity $1.33\times 10^{-3}$ W/m².

Phasor form

In sinusoidal steady state, the time-average Poynting vector is given by

$⟨\mathbf{S}⟩=\frac{1}{2}\text{Re}[\tilde{\mathbf{E}}\times {\tilde{\mathbf{H}}}^{\ast }],$

where $\tilde{\mathbf{E}},\tilde{\mathbf{H}}$ are the phasor fields and ${}^{\ast }$ is complex conjugate. This is the standard tool in transmission line and antenna analysis.

The same factor of $\frac{1}{2}$ appears here as in $|\tilde{V}{|}^2/(2Z)$ for time-average power in circuit analysis — both come from $\overline{{\cos }^2(\omega t)}=\frac{1}{2}$.

What it means physically

The Poynting vector identifies where and in what direction electromagnetic energy is moving. Some non-intuitive consequences:

1. Energy doesn’t flow along the wire in a DC circuit. It flows through the space around the wire, guided by the fields outside. The wire is just the boundary; the energy is in the field. For a battery driving a resistor, $\mathbf{S}$ in the surrounding space points from battery to resistor.

2. Antennas radiate via $\mathbf{S}$. The radiation pattern of an antenna is the angular distribution of $|\mathbf{S}|$ at far distance. Maximizing radiation in a chosen direction means maximizing $\mathbf{S}$ there.

3. Reflection and transmission at interfaces are governed by matching $\mathbf{S}$ (and tangential $\mathbf{E},\mathbf{H}$) on each side. Power conservation says the incident $\mathbf{S}$ equals reflected plus transmitted $\mathbf{S}$ in lossless media.

Limitations

The Poynting expression $\mathbf{S}=\mathbf{E}\times \mathbf{H}$ gives the correct energy flux for any closed surface integral, but the local identification of $\mathbf{S}$ as “the energy flow at a point” is ambiguous — you can add any divergence-free vector to $\mathbf{S}$ without changing the physics. The conventional choice $\mathbf{E}\times \mathbf{H}$ is the one that respects causality and falls out of the relativistic stress-energy tensor.

For engineering purposes (computing radiated power, power flow in transmission lines, etc.), this subtlety doesn’t matter — $\mathbf{S}=\mathbf{E}\times \mathbf{H}$ always gives the right answer for measurable quantities.