9.2 Poynting vector

Definition of Poynting vector

The Poynting vector, denoted $\vec{S}$, represents the directional energy flux density of an electromagnetic field. It tells you both the rate and direction of electromagnetic energy flow per unit area, measured perpendicular to the direction of propagation. John Henry Poynting introduced it in 1884 as the natural quantity for describing energy transport in electromagnetic fields.

Derivation from Maxwell's equations

The Poynting vector emerges directly from Maxwell's equations when you track where electromagnetic energy goes. Here's the logic of the derivation:

1. Start with Faraday's law and the Ampère-Maxwell law.

2. Dot $\vec{E}$ into Ampère's law and $\vec{H}$ into Faraday's law.

3. Subtract the two resulting expressions. The vector identity $\vec{H} \cdot (\nabla \times \vec{E}) - \vec{E} \cdot (\nabla \times \vec{H}) = -\nabla \cdot (\vec{E} \times \vec{H})$ collapses the left side into a divergence.

4. The right side yields terms for the time rate of change of stored energy and ohmic dissipation.

The quantity that naturally appears under the divergence is the Poynting vector:

$\vec{S} = \vec{E} \times \vec{H}$

This is the instantaneous Poynting vector. Note that it uses $\vec{H}$ (magnetic field intensity), not $\vec{B}$. In vacuum or linear media, $\vec{H} = \vec{B}/\mu$, so you can equivalently write $\vec{S} = \frac{1}{\mu}\vec{E} \times \vec{B}$.

Units and dimensions

• The SI unit of the Poynting vector is watts per square meter (W/m²).
• Dimensionally: $[S] = \frac{[V/m][A/m]}{1} = \frac{W}{m^2}$
• The magnitude $|\vec{S}|$ gives the instantaneous power density at a given point, meaning the power flowing through a unit area at that instant.

Direction of energy flow

The direction of $\vec{S}$ tells you where energy is going. Since $\vec{S} = \vec{E} \times \vec{H}$, energy propagates in the direction perpendicular to both field vectors. You can use the right-hand rule: point your fingers along $\vec{E}$, curl them toward $\vec{H}$, and your thumb points in the direction of energy flow.

Relationship to electric and magnetic fields

• $\vec{S}$ is always perpendicular to both $\vec{E}$ and $\vec{H}$ at every point.
• Its magnitude is proportional to the product of the field magnitudes: $|\vec{S}| = |\vec{E}||\vec{H}|\sin\theta$, where $\theta$ is the angle between them. For electromagnetic waves, $\vec{E} \perp \vec{H}$, so $\sin\theta = 1$ and $|\vec{S}| = EH$.
• In a propagating wave, $\vec{S}$ points along the wavevector $\vec{k}$, confirming that energy travels with the wave.

Poynting vector in electromagnetic waves

For electromagnetic waves, the Poynting vector is always directed along the propagation direction, and its magnitude equals the wave's intensity (power per unit area).

Plane waves

In a uniform plane wave propagating in free space, $\vec{E}$ and $\vec{H}$ are in phase, mutually perpendicular, and transverse to the propagation direction. The instantaneous Poynting vector oscillates at twice the wave frequency. Its time-averaged magnitude is:

$\langle S \rangle = \frac{1}{2}\sqrt{\frac{\varepsilon}{\mu}}\,E_0^2 = \frac{E_0^2}{2\eta}$

where $\eta = \sqrt{\mu/\varepsilon}$ is the intrinsic impedance of the medium and $E_0$ is the electric field amplitude. In vacuum, $\eta_0 \approx 377\;\Omega$.

Spherical waves

For a point source radiating spherical waves:

• $\vec{S}$ points radially outward from the source.
• Its magnitude falls off as $1/r^2$, consistent with the total power spreading over a sphere of area $4\pi r^2$.
• In the far-field region ($r \gg \lambda$), the wavefronts are locally planar, so the plane-wave relationships apply approximately.

Time-averaged Poynting vector

The instantaneous Poynting vector oscillates rapidly (at optical frequencies, for example, around $10^{15}$ Hz). What you actually measure in most experiments is the time-averaged value over one full cycle.

For time-harmonic (phasor) fields, the time-averaged Poynting vector is:

$\langle\vec{S}\rangle = \frac{1}{2}\,\text{Re}(\vec{E} \times \vec{H}^*)$

Here $\vec{H}^*$ is the complex conjugate of the phasor magnetic field. The factor of $1/2$ arises from averaging $\cos^2(\omega t)$ over a full period.

Relation to intensity

The magnitude of $\langle\vec{S}\rangle$ equals the intensity $I$ of the wave:

$I = |\langle\vec{S}\rangle| = \frac{E_0^2}{2\eta}$

For a 1 kW transmitter radiating isotropically, at a distance of 1 km the intensity would be $I = \frac{1000}{4\pi(1000)^2} \approx 8 \times 10^{-5}\;\text{W/m}^2$. This is the kind of calculation where the Poynting vector directly gives you practical numbers.

Energy conservation and Poynting's theorem

Poynting's theorem is the energy conservation law for electromagnetic fields. It tells you exactly where electromagnetic energy goes: it either flows out through a surface, gets stored in the fields, or gets dissipated as work on charges.

Integral form

The integral form states: the net power flowing out through a closed surface equals the rate of decrease of stored electromagnetic energy inside, minus the power dissipated by the fields doing work on charges:

$\oint_S \vec{S} \cdot d\vec{A} = -\frac{d}{dt} \int_V (u_E + u_H)\, dV - \int_V \vec{J} \cdot \vec{E}\, dV$

• $u_E = \frac{1}{2}\varepsilon|\vec{E}|^2$ is the electric energy density.
• $u_H = \frac{1}{2}\mu|\vec{H}|^2$ is the magnetic energy density.
• $\vec{J} \cdot \vec{E}$ is the power density delivered to (or extracted from) charges. For ohmic conductors, this is $\sigma|\vec{E}|^2$, representing resistive loss.

Differential form

Applying the divergence theorem converts the integral form into a local (point-by-point) statement:

$\nabla \cdot \vec{S} = -\frac{\partial u}{\partial t} - \vec{J} \cdot \vec{E}$

where $u = u_E + u_H$. This says: if $\nabla \cdot \vec{S} < 0$ at some point, energy is accumulating there (stored energy increasing) or being absorbed by currents. If $\nabla \cdot \vec{S} > 0$, energy is flowing away from that point.

Applications of Poynting vector

Power transmission in waveguides

To find the total power carried by a waveguide mode, you integrate the Poynting vector over the guide's cross-sectional area:

$P = \int_A \langle\vec{S}\rangle \cdot d\vec{A}$

This is how you determine how much power a rectangular or circular waveguide actually delivers. The field distributions (which depend on the mode) determine where the power density is concentrated across the cross section.

Radiation pressure and solar sails

Electromagnetic waves carry momentum as well as energy. The radiation pressure on a perfectly absorbing surface is:

$P_{\text{rad}} = \frac{\langle S \rangle}{c}$

For a perfectly reflecting surface, the pressure doubles to $2\langle S \rangle / c$. At Earth's orbit, solar intensity is about $1361\;\text{W/m}^2$, giving a radiation pressure of roughly $4.5\;\mu\text{Pa}$. That's tiny, but over a large solar sail area it produces measurable thrust for spacecraft propulsion.

Electromagnetic shielding effectiveness

Shielding effectiveness (SE) is often quantified by comparing the Poynting vector magnitudes outside and inside a shield:

$\text{SE (dB)} = 10\log_{10}\frac{|\vec{S}_{\text{incident}}|}{|\vec{S}_{\text{transmitted}}|}$

This ratio captures how well the enclosure attenuates the incoming electromagnetic energy, which is essential for protecting sensitive electronics from interference.

Poynting vector vs. energy density

These two quantities are related through Poynting's theorem but describe different things:

Their connection: $\nabla \cdot \vec{S} = -\partial u/\partial t - \vec{J}\cdot\vec{E}$. In a region with no charges and no time variation of stored energy, the Poynting vector is divergence-free, meaning energy flows through without accumulating.

Poynting vector in dispersive media

In dispersive media, $\varepsilon(\omega)$ and $\mu(\omega)$ depend on frequency, which complicates the energy flow picture. The standard expression $\vec{S} = \vec{E} \times \vec{H}$ still gives the instantaneous energy flux, but interpreting the stored energy requires care.

Modifications for frequency-dependent permittivity and permeability

For a narrowband signal in a weakly dispersive, low-loss medium, the time-averaged energy density generalizes to the Brillouin formula:

$\langle u \rangle = \frac{1}{4}\frac{d(\omega\varepsilon')}{d\omega}|\vec{E}|^2 + \frac{1}{4}\frac{d(\omega\mu')}{d\omega}|\vec{H}|^2$

where $\varepsilon'$ and $\mu'$ are the real parts of the permittivity and permeability. The time-averaged Poynting vector itself remains $\langle\vec{S}\rangle = \frac{1}{2}\text{Re}(\vec{E}\times\vec{H}^*)$, but the energy velocity ($v_e = \langle S \rangle / \langle u \rangle$) now differs from both the phase and group velocities. This distinction matters in regions of anomalous dispersion where the group velocity can exceed $c$ or become negative, while the energy velocity remains physically meaningful.

Experimental measurement techniques

Near-field scanning optical microscopy (NSOM)

NSOM maps the Poynting vector in the near-field region with sub-wavelength resolution. A sharp probe tip or sub-wavelength aperture scans across the sample surface, collecting evanescent field components that conventional optics can't resolve. By measuring both amplitude and phase of the field components, you can reconstruct the local energy flow pattern. This technique is particularly valuable for studying energy flow in photonic crystals, plasmonic structures, and nano-optical devices.

Electromagnetic field probes

At RF and microwave frequencies, dedicated field probes measure $\vec{E}$ and $\vec{H}$ simultaneously at a point in space. These probes use small dipole or loop antennas, Hall effect sensors, or electro-optic crystals depending on the frequency range. Once you have both field components, you compute $\vec{S} = \vec{E} \times \vec{H}$ directly. This approach gives real-time Poynting vector measurements and works across a broad range of frequencies and field strengths, making it the standard method for EMC testing and antenna characterization.