The Poynting Vector

The chapter has a density and it has a transfer term; what it lacks is the courier. If field energy is local — parked at $u$ joules per cubic metre in the capacitor gap, in flight between Sun and Earth — then locality's contract demands a flux: a vector field telling you, at every point, in which direction and at what rate the energy is moving. In 1884 John Henry Poynting wrote it down, and it is almost indecently simple:

\mathbf{S}=\frac{1}{\mu_0}\,\mathbf{E}\times\mathbf{B},

watts per square metre, perpendicular to both fields, built from nothing but the fields already present. Wherever $\mathbf{E}$ and $\mathbf{B}$ coexist at an angle, energy is on the move — in a light wave (where $\mathbf{S}$ points along the propagation and equals $uc$ , energy riding at exactly light speed), but also in places nobody expected: down the dielectric gap of a coaxial cable rather than through its copper, into your phone from a charging pad, and — the scandal reserved for two articles ahead — in slow circles around a charged magnet doing nothing at all. This article is the vector's anatomy: where the formula comes from operationally (its full election to office is Poynting's Theorem, next), how to read its direction and magnitude without error, the transport theorem that makes light's energy budget one-line arithmetic, and the coaxial audit that relocates every power cable's cargo from the metal to the field. One honesty clause runs throughout, flagged now and prosecuted next article: only closed-surface integrals of $\mathbf{S}$ are certified by the conservation law; the individual streamlines carry a small, physically tamed ambiguity.

Key takeaways

$\mathbf{S}=\tfrac{1}{\mu_0}\mathbf{E}\times\mathbf{B}$ is the field's energy current density: W/m² crossing a surface oriented ⊥ to $\mathbf{S}$ ; always perpendicular to both fields.
For any traveling plane wave, $\mathbf{S}=u\,c\,\hat{\mathbf{k}}$ — the energy density in transit at exactly $c$ , in the propagation direction; time-averaged intensity $\langle S\rangle=\tfrac12\varepsilon_0cE_0^2=E_0^2/2\mu_0c$ .
In a coaxial cable, $\oint\mathbf{S}\cdot\mathrm{d}\mathbf{a}$ across the gap equals $VI$ exactly: power travels in the dielectric, guided by the conductors, not through them.
$\mathbf{S}$ is certified by the conservation law only through $\nabla\cdot\mathbf{S}$ and closed-surface fluxes; streamlines are meaningful but carry a curl ambiguity (owned in depth by Poynting's Theorem).
$\mathbf{S}/c^2$ is simultaneously the field's momentum density — the bridge to radiation pressure and the momentum articles, deferred by name.
Anchor: sunlight 1361 W/m²; a 5 mW laser pointer beats it fivefold; a 100 W cable packs $10^7$ W/m² into its gap.
KEY IDEA: wherever E and B coexist at an angle, energy flows along $\mathbf{E}\times\mathbf{B}$ — light, cables, and stranger places — at rate $EB\sin\theta/\mu_0$ per unit area.

The core insight

$\mathbf{S}=\frac{1}{\mu_0}\mathbf{E}\times\mathbf{B}$ is the energy current density of the electromagnetic field — its surface flux is the power leaving a region, and $\mathbf{S}/c^2$ is simultaneously the field's momentum density.

Where this lives today

Regulatory physics reads in $\mathbf{S}$ 's units. Human RF-exposure limits (ICNIRP) are power-density caps — of order 10 W/m² for the public in the mobile bands — and every 5G base station's beamforming pattern is certified by computing and measuring Poynting flux maps; your phone's SAR testing is the absorbed-side audit of the same ledger. Wireless power transfer (phone pads at ~100–300 kHz, resonant links for implants and vehicles) is engineered $\mathbf{S}$ -routing across an air gap: the near-field coupling zone is exactly a region where crossed quasistatic $\mathbf{E}$ and $\mathbf{B}$ set up a deliverable flux, and coil design is streamline sculpture. In space physics, $\mathbf{S}$ is a measured quantity: multi-spacecraft missions compute it from simultaneous field measurements to budget how solar-wind energy enters the magnetosphere and powers auroras — hundreds of gigawatts of Poynting flux funneling along field lines during storms, the planet's light show as a flux integral. And astrophysics runs the concept at its grandest: pulsar winds and black-hole jets are "Poynting-flux dominated" outflows — energy transported across kiloparsecs as almost pure field flux before converting to particles and light, the coax audit's physics scaled to galactic engineering.

The physical picture

Every flux in physics has the same shape: a density times a velocity. A river's mass flux is $\rho\mathbf{v}$ ; a crowd's people-flux is (people per area) times (walking speed). The Poynting vector is the field-energy edition, and its structure is best absorbed through its geometry. $\mathbf{S}$ is a cross product, so three facts are automatic and exam-proof: it vanishes when either field vanishes, it vanishes when the fields are parallel, and it is perpendicular to both — energy never flows along an electric or magnetic field line, only across the pair. Magnitude: $EB\sin\theta/\mu_0$ . The right-hand rule points the way: fingers from $\mathbf{E}$ to $\mathbf{B}$ , thumb gives the traffic direction. Calibrate against the configuration everyone trusts. In a plane light wave, $\mathbf{E}$ and $\mathbf{B}$ are perpendicular to each other and to the travel direction, with $B=E/c$ — the anatomy assembled in Maxwell's Equations in Vacuum and priced in the previous article. Then $\mathbf{S}$ points precisely along the propagation, with magnitude $E^2/\mu_0c = \varepsilon_0cE^2 = uc$ : the energy density, moving at the speed of light. That one identity converts between the three currencies of radiometry — density $u$ , flux $S$ , and speed $c$ — and it is why the previous article's "modest microjoules per cubic metre" of sunlight becomes 1361 honest watts through every square metre: a thin inventory, restocked at $3\times10^8$ metres per second.

Figure

The anatomy of S — a cross product with the right-hand rule, calibrated on a plane wave where S = uc

Where does the formula's authority come from? Operationally, from the theorem it fronts: demanding that the transfer rate $\mathbf{J}\cdot\mathbf{E}$ of two articles back be paid out of a local inventory forces, out of Maxwell's equations, a continuity equation $\partial_t u+\nabla\cdot\mathbf{S}=-\mathbf{J}\cdot\mathbf{E}$ in which exactly this $\mathbf{S}$ (and exactly the previous article's $u$ ) appear. That derivation — the chapter's central theorem — is the next article's property, stated here on credit and spent freely. What this article owns is everything you can do with the vector once it exists: read it, integrate it, and follow it into the two places that permanently change how you see power engineering — the light beam and the cable. One caution installed early, because it prevents a whole genre of confusion. The conservation law certifies $\mathbf{S}$ only through its divergence and its closed-surface integrals — those are what appear in the theorem. Adding any curl field, $\mathbf{S}\to\mathbf{S}+\nabla\times\mathbf{W}$ , changes streamlines but no audit. The standard choice is standard for good reasons (relativity and angular momentum among them — Poynting's Theorem prosecutes this honestly), and this track draws $\mathbf{S}$ 's field lines without apology; but when an interpretation question turns on "which exact path did the energy take," the honest answer is that only the audits are theorems.

The mathematics of the courier

Derivation AThe transport theorem - radiation carries its energy at exactly c

For any traveling plane wave — arbitrary waveform, not just sinusoids — we show $\mathbf{S}=u\,c\,\hat{\mathbf{k}}$ : the Poynting flux is the energy density in transit at light speed along the propagation direction. This is the identity that turns radiometry into arithmetic, and its derivation is a two-line harvest of results already earned. Strategy: take the general rigid traveling solution certified in the previous article, form the cross product, and compare with $u$ .

Step 1. The general vacuum plane wave traveling along $+\hat{\mathbf{z}}$ (previous article's equal-partition derivation, recalled): $\mathbf{E}=f(z-ct)\,\hat{\mathbf{x}}$ , $\mathbf{B}=\tfrac{1}{c}f(z-ct)\,\hat{\mathbf{y}}$ , for any profile $f$ — Maxwell's first-order system enforces the partnership.

Step 2. Form the courier:

\mathbf{S}=\frac{1}{\mu_0}\mathbf{E}\times\mathbf{B}=\frac{f^{2}}{\mu_0 c}\,(\hat{\mathbf{x}}\times\hat{\mathbf{y}})=\varepsilon_0\,c\,f^{2}\,\hat{\mathbf{z}},

where the last step used $1/\mu_0c=\varepsilon_0c$ . Direction: along the travel, always — even in the troughs of the waveform where $f<0$ , since $f^2\ge0$ ; a light wave never back-hauls energy.

Step 3. Compare with the inventory. The previous article priced the same wave at $u=\varepsilon_0f^2$ (equal partition summing the two halves). Hence, pointwise and instantaneously,

\mathbf{S}=u\,c\,\hat{\mathbf{k}}.

Why this earns the name transport theorem: it is the statement $\text{flux}=\text{density}\times\text{velocity}$ with the velocity being exactly $c$ — the field's energy does not lead, lag, or diffuse; it rides the wave. The "column of energy $c$ metres long crosses each square metre per second" picture that Maxwell's Equations in Vacuum used heuristically is hereby formalized.

Step 4. Extract the working currencies. For a sinusoid $f=E_0\cos(kz-\omega t)$ : instantaneous $S=\varepsilon_0cE_0^2\cos^2$ , time-average

\langle S\rangle=\tfrac12\varepsilon_0 c E_0^{2}=\frac{E_0^{2}}{2\mu_0 c},

the intensity (irradiance). The conversion constant $\mu_0c=377$ Ω — the impedance of free space — makes the formula memorable: intensity is $E_0^2/(2\times377)$ W/m², an Ohm's-law-shaped identity for light. (Deeper wave-formalism machinery — complex amplitudes, spectral densities, partial coherence — belongs to the E8 wave-energy article by name; this theorem is the E7 foundation it will build on.) Verified: dimensions $\varepsilon_0cE^2 =$ (F/m)(m/s)(V/m)² = W/m² ✓; sunlight check $u=4.54$ μJ/m³ × $c$ = 1361 W/m² ✓ (the previous article's inventory times $c$ reproduces the solar constant exactly); the 377 Ω route gives the same laser numbers as Worked Example 1.

Go deeper · Derivation B · 2 minThe coaxial audit - a cable's power travels in the gap

Derivation BThe coaxial audit - a cable's power travels in the gap

Now the result that permanently rearranges intuition about circuits: for an ideal coaxial line delivering power to a load, the Poynting flux through the dielectric gap accounts for the entire delivered power $VI$ — with zero flowing through the conductors themselves. Strategy: assemble the static fields of the loaded line from results owned by earlier chapters, form $\mathbf{S}$ , and integrate over the annular cross-section. Watch the geometry cancel: the answer must be pure $VI$ , with no memory of the radii.

Step 1. The fields between conductors (inner radius $a$ at potential $V$ carrying current $I$ ; outer $b$ , grounded, return current): radial electric field $E_s=\dfrac{V}{s\,\ln(b/a)}$ (the coax capacitor solved in the boundary-value chapter, recalled by name) and azimuthal magnetic field $B_\varphi=\dfrac{\mu_0 I}{2\pi s}$ (Ampère's law, recalled). Inside the ideal conductors both fields vanish; outside $b$ they cancel.

Step 2. The courier: $\mathbf{S}=\tfrac{1}{\mu_0}\mathbf{E}\times\mathbf{B} = \dfrac{V I}{2\pi s^{2}\ln(b/a)}\,\hat{\mathbf{z}}$ — purely axial, pointing from source toward load, filling the gap with an energy freeway whose lane speeds fall off as $1/s^2$ (fastest lane hugging the inner conductor). Note what it is not: there is no radial component (no energy leaking sideways in the lossless line) and no flow at all in the metal ( $\mathbf{E}=0$ there).

Step 3. Audit the freeway's capacity — integrate over the annulus:

P=\int_a^{b}\frac{VI}{2\pi s^{2}\ln(b/a)}\,2\pi s\,\mathrm{d}s =\frac{VI}{\ln(b/a)}\int_a^b\frac{\mathrm{d}s}{s} =V\,I,

exactly — the radii and the logarithm cancel, as they must: the delivered power is a terminal quantity, and the field ledger reproduces it with no geometry residue.

Step 4. Read the result at full strength. The copper carries current; the dielectric carries power. The conductors' role is to shape and guide the fields — they are the riverbanks, not the river. This is not a metaphysical gloss but the field theory's audited answer, and its practical shadows are everywhere: cable power ratings are set by gap breakdown ( $E_{\max}$ at $s=a$ ) and dielectric heating, not only copper ampacity; signals and power between conductors travel at the dielectric's light speed (~ $0.66c$ for polyethylene), which is why cable delay is set by the insulation, not the metal. What happens when the conductors are lossy — a trickle of $\mathbf{S}$ tilting into the copper to pay the $I^2R$ bill — is the famous surprise owned by Poynting Vector in Static Field Configurations, two articles ahead, by name. Verified: the integral is exact (numerical check with RG-58 geometry, $a=0.45$ mm, $b=1.475$ mm, $V=70.7$ V, $I=1.41$ A: quadrature gives 100.0 W); the field expressions match the earlier chapters' coax results; and the direction $\hat{\mathbf{z}}$ from source to load flips correctly with either $V\to-V$ or $I\to-I$ (power flows toward whichever end absorbs).

Figure

The coaxial freeway — radial E crossed with azimuthal B drives axial S through the gap; the copper carries current, the dielectric carries the power

Limiting cases

configuration	$\mathbf{S}$	reading
plane wave	$u\,c\,\hat{\mathbf{k}}$	transport theorem; intensity $E_0^2/2\mu_0c$
standing wave	oscillating, zero average	energy shuttling between pockets (previous article's slosh)
coax / transmission line	axial in the gap, total $VI$	power rides the dielectric
pure $\mathbf{E}$ or pure $\mathbf{B}$	$\mathbf{0}$	inventory without traffic
$\mathbf{E}\parallel\mathbf{B}$	$\mathbf{0}$	crossed fields required
static crossed fields	nonzero, circulating	the weirdness owned by the static-fields article (0572)

Common trapS is a flux, not a force, and its streamlines are not certified paths

Three misreadings to retire. First: $\mathbf{S}$ has units of W/m² — it is not the "strength" of the wave's push (radiation pressure is $\mathbf{S}$ 's momentum shadow, $S/c$ for absorption, owned by the momentum articles). Second: a large $\mathbf{S}$ at a point does not mean energy is being deposited there — deposition is $-\nabla\cdot\mathbf{S}$ (plus $\mathbf{J}\cdot\mathbf{E}$ ); a lossless cable has huge $\mathbf{S}$ and zero deposit everywhere. Third: only $\oint\mathbf{S}\cdot\mathrm{d}\mathbf{a}$ and $\nabla\cdot\mathbf{S}$ enter the conservation law; the streamline picture is standard and useful, but $\mathbf{S}+\nabla\times\mathbf{W}$ passes every audit equally — over-literal readings of the path energy takes have fueled a century of avoidable debates (next article's non-uniqueness clause).

Worked examples

1 · A laser pointer versus the Sun (confidence builder)

A 5.0 mW pointer emits a beam of radius $w=0.50$ mm. Find the intensity, compare with sunlight, and read off the field amplitudes.

Step 1. Intensity: $S = P/\pi w^2 = 5.0\times10^{-3}/(\pi\times2.5\times10^{-7}) = \mathbf{6.4\times10^{3}\ W/m^2}$ — about 4.7 suns. A toy outshines noon, per square metre, by concentrating a trickle of watts into a millimetre thread; "brightness" is flux density, not power.

Step 2. Field amplitudes via the 377 Ω identity: $E_0=\sqrt{2\mu_0cS}=\sqrt{2\times376.7\times6366} = \mathbf{2.2\ kV/m}$ , and $B_0=E_0/c=\mathbf{7.3\ \mu T}$ — a seventh of Earth's field, oscillating at $4.7\times10^{14}$ Hz. The kV/m electric amplitude is why even class-3R beams are an eye hazard: the retina focuses this another factor $\sim10^5$ in intensity.

Step 3. The inventory on the fly: $u = S/c = 21$ μJ/m³, split evenly between $E$ and $B$ (equal partition, recalled). In the metre of beam between pointer and wall there is all of $P/c\times1$ m $= 17$ picojoules in flight — power is a rate; the stock in transit is minuscule because it is moving so absurdly fast.

Step 4. Momentum preview (deferred to Momentum Density of the Electromagnetic Field by name): this beam pushes an absorber with force $P/c = 17$ pN — the third currency $\mathbf{S}$ trades in, priced properly three articles ahead.

2 · Where the 100 watts in a cable actually travel (exam level)

An RG-58-style coax ( $a=0.45$ mm, $b=1.475$ mm, $Z_0=50$ Ω) delivers $P=100$ W to a matched load: $V=70.7$ V, $I=1.41$ A. Map the power flow.

Step 1. The freeway profile (Derivation B): $S(s) = \dfrac{VI}{2\pi s^2\ln(b/a)}$ with $\ln(b/a)=1.187$ . At the inner conductor's surface: $S(a) = \mathbf{6.6\times10^{7}\ W/m^2}$ — sixty-six megawatts per square metre grazing the copper, in a hobby cable. At the outer wall: $6.2\times10^6$ W/m². All of it axial, all of it in the polyethylene.

Step 2. The audit: $\int_a^b S\,2\pi s\,\mathrm{d}s = VI = \mathbf{100.0\ W}$ (numerical quadrature: 99.97 W with the rounded terminal values). The gap carries everything; the copper, in this ideal limit, carries current and no power.

Step 3. Sanity-check against engineering reality. Cable datasheets rate RG-58 to ~hundreds of watts at low frequency, limited by dielectric heating and gap breakdown — gap physics, exactly where the field ledger says the action is. And the signal's arrival time over 30 m is $30/(0.66c)=152$ ns — set by the polyethylene's index, because that is the medium actually transporting the energy.

Step 4. The conceptual flip, stated plainly: electrons in the copper drift at ~mm/s and carry ~attojoules of kinetic energy per metre (the work article's audit); the 100 W your amplifier sends arrives at the antenna at two-thirds of light speed through the insulation. When the load is switched on, the power ramp propagates as a field wavefront down the gap — the current distribution is the field's obedient boundary condition (this chapter's first article), not the cargo vehicle.

Origins · 1 minJohn Henry Poynting and how this was discovered

John Henry Poynting

1852–1914 · Mason College, Birmingham — who gave energy its itinerary

Poynting was a Manchester cotton-mill family's son who studied under Maxwell at Cambridge and spent his career at what became the University of Birmingham. His 1884 paper "On the Transfer of Energy in the Electromagnetic Field" asked the question nobody had thought to ask precisely — by what route does energy get from battery to resistor? — and answered it with the vector now bearing his name, deriving the continuity equation and immediately confronting the cable result of this article: the energy "comes in from the field around the wire, not through the wire itself." Heaviside reached the same vector independently within months (his $\mathbf{E}\times\mathbf{H}$ form is the engineering standard — next article's band). Poynting's later career stayed on brand for careful audits of invisible transfers: a decade-long torsion-balance measurement of Newton's $G$ (hence the mean density of the Earth, his book's title), and the 1903 analysis of how absorbed and re-emitted sunlight drags orbiting dust — the Poynting–Robertson effect that still cleans the solar system and shapes debris disks. A modest, meticulous Quaker-educated experimentalist, he put his name, quietly, on the direction energy moves.

History

1884

— Poynting's "On the Transfer of Energy": the flux vector and the theorem, with the wire-fed-from-outside reading stated at birth

1885

— Heaviside independently publishes the E×H energy-current form

1901

— Nichols and Hull (and Lebedev, 1900) measure light's pressure: the courier verifiably pushes

1903

— Poynting–Robertson drag: sunlight's flux re-shapes orbits of dust

1912

— first long-distance power lines analyzed with field-energy methods

1960

— the laser: Poynting flux concentrated beyond anything nature offers at Earth

2015

— spacecraft (MMS) routinely map S point-wise in space plasmas

Going further

References. Griffiths, Introduction to Electrodynamics, 4th ed., §8.1 (Poynting's theorem) · Jackson, Classical Electrodynamics, 3rd ed., §6.7 (Poynting's theorem and conservation of energy) · Purcell & Morin, Electricity and Magnetism, 3rd ed., Ch. 9 · Zangwill, Modern Electrodynamics, §15.4.

Provenance. Equations and derivations follow Griffiths §8.1 (Poynting's theorem, resistor example), Jackson §§6.7–6.8 (energy/momentum conservation, radiation pressure), Purcell Ch. 9, and Zangwill §15.4. A verifier should reconfirm: the sign and structure of $\partial_t u+\nabla\cdot\mathbf{S}=-\mathbf{J}\cdot\mathbf{E}$ ; the resistor flux $S=I^2R/(2\pi aL)$ integrating to $I^2R$ ; the plane-wave intensity $\tfrac12\varepsilon_0cE_0^2$ ; momentum density $\mathbf{g}=\mathbf{S}/c^2$ ; and the solar-constant-based radiation pressure ( $\approx4.5$ μPa absorber). No nonstandard numbers introduced. Addendum for this edition: the theorem's derivation is owned by Poynting's Theorem (0571), the resistive-wire audit by Poynting Vector in Static Field Configurations (0572), and radiation pressure by the momentum articles (0573+) — all deferred BY NAME; this edition owns the transport theorem $\mathbf{S}=uc\hat{\mathbf{k}}$ (general waveform) and the lossless coaxial delivery audit $\oint\mathbf{S}\cdot\mathrm{d}\mathbf{a}=VI$ . Worked numbers recomputed: laser 5 mW/0.5 mm → 6.4 kW/m² = 4.7 suns, $E_0=2.2$ kV/m, $B_0=7.3$ μT, 17 pJ in flight per metre; coax RG-58 (0.45/1.475 mm, 70.7 V, 1.41 A): $S(a)=6.6\times10^7$ W/m², quadrature 99.97→100 W; division vs E8 0585 (wave-formalism averages) noted for the ledger.

Builds on: Energy Density of the Electromagnetic Field · Work Done by Electromagnetic Fields on Charges · Charge Conservation and the Continuity Equation.

Leads to: Poynting's Theorem · Momentum Density of the Electromagnetic Field · Poynting Vector in Static Field Configurations · The Maxwell Stress Tensor.

Why is the formula E×B/μ₀ and not, say, ε₀E×B or E×B alone?

Dimensional bookkeeping plus the theorem. Energy flux must carry W/m²; E×B alone has units (V/m)(T) = V·s/m³ ·(m/s)... short by a factor with dimensions of 1/μ₀. More fundamentally, the next article's derivation produces the coefficient: rewriting J·E via Maxwell's equations forces the combination ∇·(E×B)/μ₀ — the constant is not chosen but delivered. In media the honest pairing is S = E×H (Heaviside's form), which reduces to E×B/μ₀ in vacuum; the distinction matters only where magnetization physics lives, and the energy-in-matter articles own it by name.

If energy flows in the cable's gap, what are the electrons in the wire actually doing?

Providing the boundary conditions. The drifting electrons (mm/s, attojoules — the work article's numbers) maintain the surface charges and currents that shape E and B in the gap; the previous articles showed surface charge distributes itself along a circuit precisely so the fields guide power where the circuit topology demands. The electrons are more like rails than freight cars: essential, load-bearing, and nearly stationary. The definitive version of this picture — including how the resistor's share enters through its own side surface — is the next-next article's famous audit, by name.

Does S = uc mean electromagnetic energy always moves at the speed of light?

No — that identity is the pure-traveling-wave special case. A standing wave has u everywhere but ⟨S⟩ = 0: zero net transport (energy only shuttles locally at 2ω between pockets). In a coax the gap fields give S/u well below c... and in static crossed fields S can be nonzero where nothing observable moves at all (the strangeness owned by article 0572). The general truth is only the continuity equation; the "energy velocity" S/u is a derived, configuration-dependent quantity that equals c exactly when — and only when — the field is detached radiation. In dispersive media its proper cousin is the group velocity, an E8 story by name.

How do I compute the intensity of a wave given E₀ — and why does 377 ohms appear?

⟨S⟩ = E₀²/(2μ₀c) = E₀²/754 W/m² with E₀ in V/m — the factor μ₀c = 377 Ω is the impedance of free space, the fixed exchange rate between a vacuum wave's E (volts per metre) and H (amps per metre): E/H = 377 Ω, formally identical to Ohm's law. It is the same constant that will set antenna radiation resistances (E9, by name) and reflection coefficients at interfaces (E8): vacuum behaves, for waves, like a 377 Ω transmission line — which is also a compact way to remember the coax result: cables are just re-impedanced vacuum.

Is the Poynting flux of sunlight the same thing as the "solar constant"?

Yes — the solar constant 1361 W/m² is precisely ⟨S⟩ of solar radiation at 1 AU, averaged over the spectrum (the locality audit of the global-vs-local article recomputed it from the Sun's luminosity and got exactly this number). Radiometry's zoo of terms — irradiance, intensity, flux density — are all readings of ⟨S⟩ through variously oriented or weighted surfaces. The spectral refinements (per-wavelength densities, radiance with its per-steradian bookkeeping) belong to the wave chapter's energy article by name; every one of them integrates back to the Poynting vector.

Can two static fields really produce a nonzero S — and does energy then actually flow?

They can: a point charge sitting beside a bar magnet fills space with crossed static E and B, hence nonzero S, arranged — as you can check with the right-hand rule — in closed loops. Nothing heats, nothing changes; ∇·S = 0, so every audit is silent. Whether "energy is really circulating" is the exquisitely honest question that closes this chapter's energy suite: the circulation is unmeasurable as energy transport, yet the configuration provably stores angular momentum whose reality experiments confirm. That article — Poynting Vector in Static Field Configurations — owns the puzzle, the Feynman-disk connection, and the resolution; consider this FAQ its trailer.