Poynting Vector in Static Field Configurations

Here is what the energy chapter has been building toward — the result that has startled every generation of physicists since 1884. Take the most mundane object in electrical technology: a warm resistive wire carrying steady current from a battery. Ask the theory where the heat's energy comes from, and follow the mathematics without flinching. The answer is not "down the wire." The Poynting vector just outside the surface points radially inward: the energy enters the wire sideways, through its curved flank, out of the empty space around it — and the audit is exact, wattage to the last decimal. The copper carries the current; the field around the circuit carries the power; the wire is merely where the delivery gets buried as heat. Poynting himself announced this in his first paper, and Heaviside spent a career insisting on it: wires are guides, not pipes. That is the first half of this article, and it is established physics you can check with a voltmeter and a compass. The second half is stranger, and this article will not pretend otherwise. Cross a static electric field with a static magnetic field — a charged capacitor inside a solenoid, a point charge beside a bar magnet — and $\mathbf{S}$ is nonzero there too, arranged in closed loops: energy apparently circulating, forever, in a configuration where nothing whatsoever happens. No heat, no motion, no change. Is the circulation real? The honest answer has three layers — the audits are silent, the bookkeeping has known flexibility, and yet the circulating pattern stores momentum and angular momentum that experiments confirm are real — and delivering those three layers without cheating is this article's closing duty. Feynman, whose Lectures made this puzzle famous, gave the verdict this track endorses: the theory forces it, the experiments back it, and "crazy" is not a counterargument.

Key takeaways

Static does not mean no flow: wherever steady $\mathbf{E}$ and $\mathbf{B}$ coexist at an angle, $\mathbf{S}=\mathbf{E}\times\mathbf{B}/\mu_0\neq0$ . In steady state Poynting's theorem reduces to $\nabla\cdot\mathbf{S}=-\mathbf{J}\cdot\mathbf{E}$ : flux converges exactly where matter is being paid.
The resistive wire: longitudinal $E$ (continuous across the surface — the boundary-conditions article) crossed with the encircling $B$ gives $\mathbf{S}$ pointing radially INWARD; the side-surface influx equals $I^2R$ exactly, and inside the wire the convergence deposits $\mathbf{J}\cdot\mathbf{E}$ pointwise.
Energy in a DC circuit travels through the fields around the conductors — launched at the battery (where $\mathbf{J}\cdot\mathbf{E}<0$ ), guided by the surface charges on the wires, buried in the resistor.
Crossed static fields with no current: $\mathbf{S}\neq0$ but $\nabla\cdot\mathbf{S}=0$ — closed-loop circulation, no audit registers anything.
The circulation's reality has teeth anyway: the same configuration stores field momentum $\mathbf{g}=\mathbf{S}/c^2$ and angular momentum, which mechanical experiments (the Feynman disk, by name) actually collect.
Honesty clause inherited from the theorem: streamlines carry bookkeeping freedom; audits, momentum, and angular momentum do not.
KEY IDEA: in steady state, energy flows through fields — into wires sideways at exactly the dissipation rate, and in closed loops around static crossed fields, where the circulation banks real momentum.

The core insight

Static fields can transport energy: wherever $\mathbf{E}$ and $\mathbf{B}$ coexist and are not parallel, $\mathbf{S}=\tfrac{1}{\mu_0}\mathbf{E}\times\mathbf{B}\neq0$ . In steady state $\nabla\cdot\mathbf{S}=-\mathbf{J}\cdot\mathbf{E}$ : energy converges on dissipation — entering resistive wires through their sides — and circulates without accumulating around dissipationless crossed fields.

Where this lives today

The surface-charge delivery picture has become directly visible. Scanning-probe and cold-atom electrometry now image the minute charge gradients along current-carrying conductors — the routing infrastructure of Derivation A, measured — and physics-education research uses these maps because the "energy flows in the field" account, once considered graduate esoterica, demonstrably repairs student misconceptions that the pipe picture creates. Meanwhile power engineering has always lived here in practice: transmission-line power flow is computed as field flux (the "surge impedance loading" of a line is a Poynting quantity), high-voltage insulation is designed for the region that actually carries the gigawatts — the air and polymer around the conductor — and cable ampacity splits cleanly into copper limits (current) and dielectric limits (power), the two cargoes this article separated. The static-circulation half pays off across modern physics. The angular momentum of crossed static fields is the classical skeleton of spin–momentum locking in near-field optics — evanescent fields carry transverse spin dictated by the same $\mathbf{E}\times\mathbf{B}$ bookkeeping, now routinely exploited for one-way photonic couplers. Charged, magnetized compact objects put the configuration in the sky: a rotating magnetized neutron star's exterior is a colossal static-crossed-field system whose field angular momentum budget governs pulsar spin-down. And every quantum-Hall device on a lab bench is, electromagnetically, a crossed- $\mathbf{E}\times\mathbf{B}$ platform where the drift velocity of the carriers is literally $\mathbf{S}/u$ — the circulation, harnessed.

The physical picture

Start with the steady-state form of the chapter's central theorem. With nothing changing in time, $\partial u/\partial t=0$ and Poynting's theorem (previous article, recalled) collapses to a stark routing rule:

\nabla\cdot\mathbf{S}=-\,\mathbf{J}\cdot\mathbf{E} \tag{1}

where: energy flux may converge only where charges are being paid ( $\mathbf{J}\cdot\mathbf{E}>0$ : resistors, loads) and diverge only where matter pays the field ( $\mathbf{J}\cdot\mathbf{E}<0$ : the EMF regions of batteries and generators — the source audit of the work article, recalled). Everywhere else the traffic passes through undiminished. A DC circuit, in this light, is a steady-state energy flow network in the space around the conductors: launched from the battery's volume, guided along the wires, absorbed in the load. Guided by what? This is the mechanism question, and the track has already built the answer: surface charges. A current-carrying circuit dresses itself with a whisper of static charge — gradients of surface density along the wires (the self-arranging distributions of Surface Charge and the Field of a Conductor, recalled by name) — and those charges do two jobs at once: they maintain the longitudinal $E$ inside the metal that drives the current, and they shape the exterior field so that $\mathbf{E}\times\mathbf{B}/\mu_0$ carries the load's power through the surrounding space, parallel to the cable en route and tilting into every component exactly as needed. The electrons drift millimetres per second carrying attojoules (the work article's inventory); the delivery network operates in the insulation and the air.

Figure

The DC circuit as an energy flow network — launched at the battery, guided outside the wires by surface charges, buried in the resistor

If this feels like it cannot be right — if every instinct says power flows in the copper — hold the instinct up against two facts the track has already banked. The lossless coax audit (The Poynting Vector, recalled): the delivered $VI$ rides the dielectric gap in a geometry where the metal provably transports nothing. And the electron inventory (Work Done by Electromagnetic Fields, recalled): the carriers' kinetic stock is seventeen orders of magnitude short of the job. The sideways-fed wire is not an exotic reinterpretation; it is the only account the audited theory offers, and the audit is next.

The wire, audited to the watt

Derivation AEnergy enters a resistive wire through its side - the exact audit

The chapter's most famous calculation: for a cylindrical resistor of radius $a$ , length $L$ , resistance $R$ , carrying steady current $I$ , we compute the Poynting flux through the side surface and the pointwise energy deposition inside, and close both against Joule's law exactly. Strategy: assemble the surface fields from results this track owns, cross them, integrate. Watch the boundary conditions do quiet, essential work: it is the continuity of tangential E — certified for full electrodynamics by this lane's opening article — that puts the driving field in the vacuum just outside the metal, where it can join the delivery.

Step 1. The fields at the surface. Inside the wire, Ohm demands a longitudinal field $E=V/L=IR/L$ driving the current; tangential- $\mathbf{E}$ continuity (Electrodynamic Boundary Conditions, recalled by name) forces the same longitudinal $E$ to exist just outside the surface. Encircling the wire, Ampère's law gives the azimuthal field $B=\mu_0I/2\pi a$ at the surface (recalled from the magnetostatics chapter).

Step 2. The courier just outside: $\mathbf{E}$ axial, $\mathbf{B}$ azimuthal — $\mathbf{S}=\tfrac{1}{\mu_0}\mathbf{E}\times\mathbf{B}$ points radially inward (check with the right hand: for conventional current up and $B$ counterclockwise seen from above, $\hat{\mathbf{z}}\times\hat{\boldsymbol{\varphi}}=-\hat{\mathbf{s}}$ ), with magnitude

S=\frac{EB}{\mu_0}=\frac{(IR/L)\,(\mu_0 I/2\pi a)}{\mu_0}=\frac{I^{2}R}{2\pi a L}.

Step 3. Integrate over the side surface (area $2\pi aL$ ):

P_{\rm in}=S\times 2\pi a L=I^{2}R,

exactly — every geometric factor cancels, and the wire's entire heat budget arrives through its flank. Note what does not contribute: the end caps ( $\mathbf{S}$ has no axial component at the surface, since the radial $E$ of the surface charge crossed with azimuthal $B$ gives axial flow outside in the feed region, not through the resistor's ends at this order) — the delivery is genuinely lateral.

Step 4. Push inside for the pointwise ledger. Within the wire, $B(s)=\mu_0Is/2\pi a^2$ (Ampère on an interior loop, uniform current density), so $S(s) = E\,I s/2\pi a^{2}$ inward — the flux fades linearly as it penetrates, because it is paying as it goes. Take the divergence in cylindricals: $\nabla\cdot\mathbf{S} = -\tfrac{1}{s}\tfrac{\partial}{\partial s}(s\,S) = -\,\dfrac{EI}{\pi a^{2}} = -\,J\,E$ , where $J=I/\pi a^2$ : equation (1) verified pointwise — the convergence of the energy traffic at every interior radius equals the local Joule deposit exactly. By the centre, the flux has spent itself completely.

Step 5. Complete the circuit's books with the source: over the battery's volume, $\int\mathbf{J}\cdot\mathbf{E}\,\mathrm{d}\tau=-\mathcal{E}I$ (the work article's source audit, recalled), so by (1) the flux diverges from the battery at rate $\mathcal{E}I$ — the network launches there, passes along the wires' exteriors (where $\mathbf{J}\cdot\mathbf{E}\approx0$ in good copper: near-lossless guidance with a slight inward tilt paying the wires' own small $I^2r$ ), and converges on the load. One theorem, one circuit, books closed at every point of space. Why the direction is robust: reversing either $I$ or the battery polarity flips both $\mathbf{E}$ and $\mathbf{B}$ together, so $\mathbf{S}=\tfrac{1}{\mu_0}\mathbf{E}\times\mathbf{B}$ stays inward at the resistor — dissipation demands it, and the algebra confirms it. The interior divergence check is exact symbolic algebra, and Worked Example 1 prices the full closure numerically (nichrome, 35.0 W both routes). at the resistor ( $E$ and $B$ flip together), as dissipation demands.

Figure

The wire's cross-section — energy dives in through the flank and spends itself linearly on the way to the centre

The strange half: circulation with nothing happening

Now remove all dissipation and keep the crossed fields. A parallel-plate capacitor, charged and disconnected, sits inside a solenoid carrying a persistent current: uniform static $\mathbf{E}$ down, uniform static $\mathbf{B}$ sideways, nothing moving, nothing warming, nothing changing — and $\mathbf{S}=\mathbf{E}\times\mathbf{B}/\mu_0$ is uniform and nonzero in the overlap volume, with return paths through the fringing fields closing every streamline into loops. A point charge on the axis of a magnetic dipole dresses the pair in an azimuthal $\mathbf{S}$ that circulates around the axis endlessly. Equation (1) is satisfied trivially — $\mathbf{J}\cdot\mathbf{E}=0$ , $\nabla\cdot\mathbf{S}=0$ — so no closed surface anywhere registers any transport. The energy chapter's instruments all read zero. And yet the formula insists: energy is circulating.

Go deeper · Derivation B · 2 minCrossed static fields - circulation, its silence, and its momentum

Derivation BCrossed static fields - circulation, its silence, and its momentum

We work the capacitor-in-solenoid configuration honestly: establish the circulating $\mathbf{S}$ , prove every energy audit is silent, and then show the configuration nonetheless holds a nonzero field momentum — the quantity that converts the circulation from philosophy into physics. Strategy: direct computation, then the momentum bookkeeping stated with its ownership boundaries (this chapter's remaining articles own the full machinery).

Step 1. Fields in the overlap region: $\mathbf{E}=E\,(-\hat{\mathbf{z}})$ between the plates, $\mathbf{B}=B\,\hat{\mathbf{x}}$ from the solenoid. The courier: $\mathbf{S}=\tfrac{1}{\mu_0}\mathbf{E}\times\mathbf{B} = \tfrac{EB}{\mu_0}\,\hat{\mathbf{y}}$ — a uniform river of apparent energy flow, transverse to both fields, threading the capacitor. Outside the overlap, the fringing fields bend the streamlines around and close them: $\nabla\cdot\mathbf{S}=0$ globally (statics, no dissipation), so the pattern is a closed circulation.

Step 2. Audit silence, proven not assumed: for any closed surface, $\oint\mathbf{S}\cdot\mathrm{d}\mathbf{a}=\int\nabla\cdot\mathbf{S}\,\mathrm{d}\tau=0$ ; and $\partial u/\partial t=0$ everywhere means no inventory anywhere changes. The theorem certifies only divergences and closed-surface fluxes (the standing honesty clause of the previous two articles) — and every one of them is zero. Within pure energy bookkeeping, the circulation is unfalsifiable: you could add a curl field to cancel it and no energy measurement would object.

Step 3. Enter momentum, where the freedom dies. The field carries momentum density $\mathbf{g}=\varepsilon_0\,\mathbf{E}\times\mathbf{B}=\mathbf{S}/c^{2}$ — the identification argued from radiation pressure and relativity in Momentum Density of the Electromagnetic Field (next article, deferred by name; here we spend only its existence). Our uniform river then banks a net momentum

\mathbf{p}_{\rm field}=\int \varepsilon_0\,\mathbf{E}\times\mathbf{B}\;\mathrm{d}\tau = \varepsilon_0\,E\,B\,V_{\rm overlap}\;\hat{\mathbf{y}},

pointing sideways through a system in which nothing moves. This is no longer reshuffle-away-able: momentum bookkeeping (unlike streamline aesthetics) is anchored by conservation experiments, and the same $\mathbf{E}\times\mathbf{B}$ combination is forced.

Step 4. The reality check the chapter will cash: discharge the capacitor (or quench the solenoid) and the collapsing fields exert real impulses on the circuitry — the stored $\mathbf{p}_{\rm field}$ and, in axially symmetric versions, the stored angular momentum $\mathbf{L}_{\rm field}=\int\mathbf{r}\times\mathbf{g}\,\mathrm{d}\tau$ are handed to matter and show up on torque meters. That is the Feynman disk paradox — charges on a disk, a dying solenoid, and the disk turns — owned in full by its own article at this chapter's close (by name), with the subtle mechanical counterpart (Hidden Momentum in Magnetic Systems, by name) that squares the static system's total momentum with relativity. The charge-plus-monopole angular momentum that quantized Dirac's charge (Divergence of B's Thomson route, recalled by name) is this same circulating bookkeeping at its most consequential.

Step 5. State the three-layer verdict cleanly. Layer one: all energy audits of static circulation are silent — zero net transport, provably. Layer two: the streamline picture retains the theorem's known bookkeeping freedom — "which loop the joules take" is not an observable. Layer three: the momentum and angular momentum of the identical field pattern are observable, forced, and experimentally collected — so the crossed-field configuration is genuinely, physically loaded, whatever language one prefers for the loading. The chapter endorses Feynman's summary: the theory that never fails its audits is telling you something with the circulation; the strangeness is a fact about nature, not an error to be edited out. $\nabla\cdot\mathbf{S}=0$ by direct computation (uniform vector field; fringing closure follows from $\nabla\cdot\mathbf{S}=-\mathbf{J}\cdot\mathbf{E}=0$ globally); the momentum integral is priced numerically in Worked Example 2; the disk/hidden-momentum/Thomson cross-references are ownership pointers, not derivations, per chapter memory.

Limiting cases

configuration	$\mathbf{S}$ pattern	audit reading
resistive wire	inward through the flank, fading linearly inside	$\oint=I^2R$ ; pointwise $\nabla\cdot\mathbf{S}=-JE$
ideal (lossless) wire	axial in the gap only; none enters the metal	guidance without payment
battery volume	outward	source: $-\mathcal{E}I$ exported
crossed static fields, no current	closed loops	all energy audits zero; momentum nonzero
$\mathbf{E}\parallel\mathbf{B}$ static	$\mathbf{S}=0$	loaded but trafficless
superconducting wire	$E_\parallel=0$ : no inward component	zero dissipation, zero delivery needed — consistent

Common trapStatic does not mean no flow — and "along the wire" is the wrong picture twice over

The two instincts this article exists to retire. First: setting $\partial_t=0$ silences the storage term of Poynting's theorem, not the flux — steady circuits are steady flows, with $\nabla\cdot\mathbf{S}=-\mathbf{J}\cdot\mathbf{E}$ routing power from sources to sinks through the surrounding space. Second: the energy of a DC circuit does not travel inside the copper like water in a pipe — it travels in the fields around the conductors and enters each component through its sides, the wires' surface charges acting as the routing infrastructure. The pipe picture fails quantitatively (electron kinetic inventory: attojoules), topologically (the lossless coax carries everything in the gap), and experimentally (the audits of this article close only the field-route books). Keep the copper as the guide; put the cargo in the field.

Worked examples

1 · A heater wire, fed through its skin (confidence builder)

A nichrome wire, radius $a=1.0$ mm, length $L=1.0$ m, resistance $R=1.4$ Ω, carries $I=5.0$ A. Audit the energy delivery.

Step 1. Joule budget (the circuit route): $P=I^2R=\mathbf{35.0\ W}$ .

Step 2. Surface fields: $E=IR/L=7.0$ V/m longitudinal, continuous across the surface (boundary conditions); $B=\mu_0I/2\pi a=1.0\times10^{-3}$ T azimuthal at the flank.

Step 3. The courier: $S=EB/\mu_0 = 7.0\times1.0\times10^{-3}/1.257\times10^{-6} = \mathbf{5.6\times10^{3}\ W/m^2}$ , radially inward — five and a half kilowatts per square metre diving into the flank of a heater wire. Over the side area $2\pi aL=6.28\times10^{-3}$ m²: $P_{\rm in}=5570\times6.28\times10^{-3}=\mathbf{35.0\ W}$ ✓ — the field route and the circuit route agree to the joule.

Step 4. The interior fade: at half radius, $S(a/2)=EIs/2\pi a^2\big|_{s=a/2} = 2.8$ kW/m² — half the surface value, the linear spend-down of Derivation A, with the deposit rate $J E = (I/\pi a^2)(E) = 1.1\times10^{7}$ W/m³ uniform throughout. The heat appears uniformly in the metal precisely because the influx fades linearly. Every glowing toaster coil is running this audit, silently, correctly.

2 · Loading the vacuum: momentum in a bench-top crossed field (exam level)

A parallel-plate capacitor holds $E=1.0\times10^5$ V/m over an overlap volume $V=1.0$ litre inside a solenoid providing perpendicular $B=0.10$ T. Everything is static. Price the circulation.

Step 1. The apparent energy river: $S = EB/\mu_0 = 10^5\times0.10/1.257\times10^{-6} = \mathbf{8.0\times10^{9}\ W/m^2}$ — eight gigawatts per square metre of formal energy flux, in a system doing absolutely nothing. (Let the number complete its rhetorical job: streamline magnitudes, unaudited, can be arbitrarily theatrical.)

Step 2. Every energy audit: zero. $\nabla\cdot\mathbf{S}=0$ , no surface collects anything, no inventory changes, nothing warms. The gigawatts circulate in closed loops through the fringing fields, indistinguishable — by any energy measurement — from a bookkeeping mirage.

Step 3. The momentum, which is not a mirage: $\mathbf{g}=\mathbf{S}/c^2 = 8.0\times10^9/(9.0\times10^{16}) = 8.9\times10^{-8}$ kg·m⁻²·s⁻¹, so $p_{\rm field}=gV = \mathbf{8.9\times10^{-11}\ kg\,m/s}$ — the momentum of a dust grain drifting a millimetre per second, stored sideways in a static litre of field. Small, but forced: discharge the capacitor through the field region and impulses of exactly this budget appear on the hardware (with the relativistic mechanical subtlety — the system's total momentum — squared away by Hidden Momentum in Magnetic Systems, deferred by name).

Step 4. Scale honestly: the momentum is small because $c^2$ divides the flux — field momentum only becomes prominent where $S$ is astronomical (radiation pressure on solar sails, next articles) or where angular momentum integrates over favorable geometry (the Feynman disk, where it is measurable with undergraduate apparatus). The lesson survives the smallness: static crossed fields are loaded, and the loading obeys exactly the $\mathbf{E}\times\mathbf{B}$ bookkeeping whose energy shadow so unsettled us in Step 1.

Origins · 1 minRichard Feynman and how this was discovered

Richard Feynman

1918–1988 · Caltech — who told students the truth about the crazy energy flow

Volume II of the Feynman Lectures (1964) did something textbooks had quietly avoided for eighty years: it worked this article's physics in front of students and refused to sand off the strangeness. On the wire: energy flows in "from the field outside... however strange it may seem." On static crossed fields, his capacitor-and-magnet example: the theory says energy circulates in closed loops, and "you may think this is nuts — but it is not nuts," because the same bookkeeping stores the angular momentum that experiment collects. He then armed the objection with its sharpest test case, the charged-disk-and-dying-solenoid thought experiment universally called the Feynman disk paradox (this chapter's closing article), whose resolution requires the static field's angular momentum to be real. Feynman's broader habit — auditing every beautiful formalism against what can actually be measured, then trusting the audited theory over intuition — is the exact method this article inherited: streamlines held lightly, audits held absolutely, and nature allowed to be strange where the audits back her.

History

1884

— Poynting's founding paper already contains the scandal: the wire is fed "from the medium around it"

1885

— Heaviside makes field-borne energy flow the organizing doctrine of circuit physics

1893

— Hertz's textbook treatment normalizes the sideways-fed conductor for a generation

1949

— Sommerfeld's Electrodynamics presents the DC circuit's complete field-energy map

1964

— Feynman Lectures II confront the static circulation honestly and stage the disk paradox

1967

— Shockley and James pose the hidden-momentum puzzle that completes the static bookkeeping

2010

s — momentum-of-light experiments (Abraham–Minkowski program) anatomize field momentum in matter

Going further

References. Griffiths, Introduction to Electrodynamics, 4th ed., §8.1.2 (Poynting's theorem) and Problem 8.1 (resistive wire); §8.2 (momentum) · Jackson, Classical Electrodynamics, 3rd ed., §6.7 (Poynting's theorem, energy and momentum) · Purcell & Morin, Electricity and Magnetism, 3rd ed., Ch. 9 (energy flow in circuits) · Zangwill, Modern Electrodynamics, §15.4 (electromagnetic energy and the Poynting vector) · Feynman Lectures on Physics, Vol. II, §27 (field energy and momentum). All equations are standard results. Key checks: the resistive-wire surface flux is $|\mathbf{S}|=VI/(2\pi aL)$ , integrating to $P_{\rm in}=VI=I^2R$ ; the steady-state reduction $\nabla\cdot\mathbf{S}=-\mathbf{J}\cdot\mathbf{E}$ follows directly from Poynting's theorem; and the crossed-field momentum density is $\mathbf{g}=\mathbf{S}/c^2$ . Numerical constants ( $\mu_0$ , $c$ ) are CODATA standard values. Worked numbers: nichrome wire ( $a=1$ mm, $L=1$ m, $R=1.4$ Ω, $I=5$ A) gives $E=7.0$ V/m, $B=1.0$ mT, $S=5.6\times10^3$ W/m², with closure 35.0 W both routes and interior $\nabla\cdot\mathbf{S}=-JE$ confirmed symbolically; crossed-field litre ( $10^5$ V/m × 0.1 T) gives $S=8.0\times10^9$ W/m² and $p=8.9\times10^{-11}$ kg·m/s.

Builds on: The Poynting Vector · Poynting's Theorem · Energy Density of the Electromagnetic Field · Work Done by Electromagnetic Fields on Charges.

Leads to: Momentum Density of the Electromagnetic Field · Hidden Momentum in Magnetic Systems · The Maxwell Stress Tensor · Angular Momentum of the Electromagnetic Field.

Is the sideways energy flow experimentally testable, or just a consistent story?

Testable in every part that the theorem certifies. The surface fields it requires (longitudinal E outside the wire, the azimuthal B) are measured directly; the surface-charge gradients that shape the exterior field have been imaged; the audits close (flank influx = I²R, gap flux = VI) using only measured quantities; and the transient version is decisive — switch on a circuit and the power front travels through the insulation at the cable's dielectric light speed, arriving at the load before any signal conducted "through the metal" could (standard time-domain reflectometry fact). What remains conventional is only streamline-level detail below the resolution of any audit — the honesty clause carried since the theorem article.

How fast does a DC circuit set up this whole delivery network?

At field speed, in a few round trips. Close the switch and a field wavefront races down the transmission-line structure of the circuit at the dielectric's light speed, charging the wires' surfaces as it goes; reflections at the load and source settle the surface-charge distribution — and with it the steady S-pattern — within a few L/c transit times (nanoseconds for bench circuits). The electrons' drift settles on the much slower momentum-relaxation clock, but by then their marching orders (the fields) are already in steady state. "DC" is the long-time limit of a fully electromagnetic setup transient.

Could we not redefine S to abolish the static circulation, since only divergences are certified?

For energy alone, yes — that freedom is real and this track has never hidden it. The reason physics declines: the same E×B pattern, with the same spatial arrangement, is forced upon you as momentum density g = ε₀E×B by radiation-pressure experiments and relativistic consistency (next article), and as angular-momentum density r×g by the Feynman-disk family. A bookkeeping that erases the energy circulation while keeping the momentum it must keep becomes incoherent — the pins discussed in Poynting's Theorem. So the standard account keeps the circulation and pays the interpretive bill: closed-loop energy flow with silent audits, backed by loud momentum.

Does any energy at all flow along the inside of the copper?

A little, and inward-spiralling. Inside the wire the axial E coexists with the interior azimuthal B, giving the fading inward S of Derivation A — so within the metal, energy moves radially, being spent as it goes; the axial transport inside good copper is negligible because the axial component of S there involves the tiny radial E of space-charge corrections. In the ideal-conductor limit the interior flow vanishes entirely with the interior fields. The joules that heat the wire's centre entered at its surface moments (nanoseconds) earlier and spiralled in. The metal is not even a minor freight route; it is the destination.

What is the relationship between this article's circulation and the Feynman disk paradox?

Trailer to feature. The disk paradox: charges fixed on a free-spinning disk, a solenoid on its axis carrying current; the current is switched off; the dying flux induces an azimuthal E that torques the charges — and the disk rotates, apparently conjuring angular momentum from a static system. The resolution is exactly this article's Step-4 physics: the original static configuration already stored angular momentum in its circulating E×B pattern, and the switch-off merely transfers the field's L to the disk, conserving the total. The paradox article (closing this chapter, by name) does the full calculation; this article's job was to make the initial storage credible — which the momentum bookkeeping, unlike the silent energy audits, genuinely certifies.

Why does a superconducting wire receive no Poynting influx — where does its current's energy come from?

It needs none, and the bookkeeping agrees perfectly. With zero resistance, E∥ = 0 at and inside the surface: the inward component of S (∝ E∥) vanishes identically, and correspondingly J·E = 0 — no dissipation to pay for. The persistent current, once established, is a zero-maintenance flow; energy was invested only during establishment (the ramping audit of Poynting's Theorem — influx through the flank while dI/dt ≠ 0, banked as B²/2μ₀). A superconducting cable delivering power to a load still routes the load's watts through the field around itself — the gap's axial S is alive and well — but takes no delivery cut. The limiting-case table's consistency, realized in hardware in every MRI magnet.