Poynting's Theorem

Everything in this chapter so far has been staging. The template article promised that energy conservation would take continuity-equation form; the work article built the transfer term $\mathbf{J}\cdot\mathbf{E}$ ; the density article nominated $u=\tfrac{\varepsilon_0}{2}E^2+\tfrac{1}{2\mu_0}B^2$ ; the vector article introduced the courier $\mathbf{S}=\mathbf{E}\times\mathbf{B}/\mu_0$ on credit. This article pays every debt at once. Starting from nothing but the power delivered to charges and the two curl equations of Maxwell, a half page of vector calculus forces the energy books of electrodynamics into the chapter's template:

\frac{\partial u}{\partial t}+\nabla\cdot\mathbf{S}=-\,\mathbf{J}\cdot\mathbf{E},

with $u$ and $\mathbf{S}$ emerging from the algebra — not assumed, not chosen, but delivered, and coinciding exactly with the density and flux the earlier articles advertised. That is Poynting's theorem, and it is the cleanest demonstration in classical physics of how a global principle (energy is conserved) gets enforced point by point in a field theory. The theorem's power is best felt in audits, and this article runs two to the last decimal: a charging capacitor whose growing stockpile of field energy is fed exactly by Poynting flux walking in through the fringe of the gap, and a ramping solenoid whose $B^2/2\mu_0$ reservoir fills through its own curved surface at precisely the advertised rate. It closes with the theorem's honest fine print — the celebrated non-uniqueness of the $(u,\mathbf{S})$ pair, what physically pins down the standard choice, and the doorway this opens into the century-old Abraham–Minkowski debate about light in matter.

Key takeaways

Dot $\mathbf{E}$ into Ampère–Maxwell, use Faraday's law and one product-rule identity, and the work rate $\mathbf{J}\cdot\mathbf{E}$ reorganizes itself into $-\partial_t u-\nabla\cdot\mathbf{S}$ : Poynting's theorem, the energy continuity equation of electrodynamics.
Both $u=\tfrac{\varepsilon_0}{2}E^2+\tfrac{1}{2\mu_0}B^2$ and $\mathbf{S}=\mathbf{E}\times\mathbf{B}/\mu_0$ emerge from the derivation — the theorem certifies the chapter's density and courier in one stroke.
Integral form: power flowing in through a closed surface = rate of change of field energy inside + power delivered to charges inside. Every electromagnetic energy audit is this sentence.
Audits close exactly: a charging capacitor's $\mathrm{d}U/\mathrm{d}t$ equals the rim influx $E I d$ ; a ramping solenoid's reservoir fills through its side at $(B/\mu_0)(\mathrm{d}B/\mathrm{d}t)\pi r^2$ per length.
Fine print: $(u,\mathbf{S})$ can be shifted ( $\mathbf{S}\to\mathbf{S}+\nabla\times\mathbf{W}$ ) without changing any audit; relativity and angular-momentum bookkeeping pin the standard pair.
In linear matter the theorem reads $\partial_t\left[\tfrac12(\mathbf{E}\cdot\mathbf{D}+\mathbf{B}\cdot\mathbf{H})\right]+\nabla\cdot(\mathbf{E}\times\mathbf{H})=-\mathbf{J}_f\cdot\mathbf{E}$ — gateway to the Abraham–Minkowski question.
KEY IDEA: Maxwell's equations do not permit energy bookkeeping to fail — the work term reorganizes, identically, into a stored density plus a flux divergence.

The core insight

Multiply Faraday's and Ampère–Maxwell's laws by the right fields, add them, and the algebra forces electromagnetic energy into the universal conservation form: a stored density whose change is accounted by a flux divergence and the work done on charges.

Where this lives today

Antenna and RF engineering is applied Poynting's theorem, integrated over spheres: a transmitter's radiated power is $\oint\langle\mathbf{S}\rangle\cdot\mathrm{d}\mathbf{a}$ through any surrounding surface, the radiation resistance that loads the feed circuit is defined by equating that integral to $I^2R_{\rm rad}/2$ (the E9 articles cash this by name), and the near-field/far-field split of every antenna datasheet is the theorem's reactive-versus-real power decomposition: close in, $\mathbf{S}$ mostly sloshes (energy borrowed and returned each cycle — the standing-wave shuttling of the density article); far out, it departs. AC power engineering runs the same split at 50 Hz as real versus reactive power — the complex-Poynting formalism (Jackson §6.9's time-averaged theorem) is the field-level original of the power-factor bookkeeping that utilities bill by. In computation, Poynting's theorem is the standard verification oracle: FDTD and finite-element solvers check themselves by auditing $-\oint\mathbf{S}\cdot\mathrm{d}\mathbf{a}$ against $\mathrm{d}U/\mathrm{d}t+\int\mathbf{J}\cdot\mathbf{E}$ on every run — energy-ledger residuals are the first diagnostic of a broken mesh or an unstable time step. And at the frontier, laser–plasma facilities budget petawatts with the theorem (how much pulse energy crossed into the target's electrons, where, and when), while gravitational-wave astronomy uses its general-relativistic descendant: the energy LIGO reports for a black-hole merger is the integrated flux of the gravitational stress–energy through a distant sphere — Poynting's audit, translated to spacetime itself.

The physical picture

Take any region of space — the gap of a capacitor, the bore of a solenoid, a cubic metre of sunlight, the inside of a microwave oven. Three things can be true of electromagnetic energy there: it can be stored (the density $u$ ), it can be in transit (the flux $\mathbf{S}$ through the walls), and it can be changing hands (the transfer $\mathbf{J}\cdot\mathbf{E}$ to matter). Poynting's theorem is the statement that these three ledgers always, exactly, balance — not because energy conservation was imposed on electrodynamics, but because Maxwell's equations refuse to describe anything else. That refusal is the remarkable part. The derivation below starts from the transfer term alone and watches the field equations reorganize it, uninvited, into storage-plus-transit. A theory whose equations were even slightly different — an Ampère law without the displacement current, say — would fail to close the books, and that failure is cousin to the consistency argument that forced the displacement current in the first place (the E6 twin's reverse derivation, recalled by name).

Figure

The three ledgers of any region — storage, transit, transfer — and the theorem that locks them together

Structurally, the theorem is the chapter's template with its transfer slot filled: compared to charge's pristine $\partial_t\rho+\nabla\cdot\mathbf{J}=0$ , field energy carries the source term $-\mathbf{J}\cdot\mathbf{E}$ — not a violation but a named handoff, the same term crediting matter's ledger in the work article. Add the two ledgers (field energy plus mechanical energy) and the source cancels: the total obeys a sourceless continuity equation, energy conservation in the full local sense the doctrine article demanded. What should impress you most, before any derivation, is the economy. Nothing new is postulated. The ingredients are the two curl equations — already fixed by electrostatics, magnetostatics, induction, and the displacement current — plus the Lorentz force's work rate. Energy conservation with a specific, computable density and flux comes out as a theorem. When a theory hands you back more structure than you put in, you are near its load-bearing walls.

The theorem, derived and audited

Derivation APoynting's theorem from Maxwell's equations

Goal: show that the rate of energy transfer to matter, $\mathbf{J}\cdot\mathbf{E}$ , is identically equal to $-\partial_t u - \nabla\cdot\mathbf{S}$ with $u$ and $\mathbf{S}$ constructed explicitly — the local energy conservation law of electrodynamics. Strategy: solve Ampère–Maxwell for $\mathbf{J}$ , dot with $\mathbf{E}$ , and untangle the two field-derivative terms with one product rule; Faraday's law then converts the magnetic curl term into a time derivative. Watch two things: where each piece of $u$ is born, and the single sign (from Faraday's minus) that makes the flux term a divergence rather than an anti-divergence.

Step 1. Ampère–Maxwell gives the current in terms of fields: $\mathbf{J}=\tfrac{1}{\mu_0}\nabla\times\mathbf{B}-\varepsilon_0\,\partial_t\mathbf{E}$ . Dot with $\mathbf{E}$ :

\mathbf{J}\cdot\mathbf{E}=\frac{1}{\mu_0}\,\mathbf{E}\cdot(\nabla\times\mathbf{B})-\varepsilon_0\,\mathbf{E}\cdot\frac{\partial\mathbf{E}}{\partial t}.

The last term is already half the answer: $\varepsilon_0\mathbf{E}\cdot\partial_t\mathbf{E}=\partial_t\!\left(\tfrac{\varepsilon_0}{2}E^2\right)$ — the electric inventory's growth rate, appearing unbidden.

Step 2. Untangle the curl term with the product rule for the divergence of a cross product (the E0 identity owned by Vector Identities and Product Rules, recalled by name): $\nabla\cdot(\mathbf{E}\times\mathbf{B})=\mathbf{B}\cdot(\nabla\times\mathbf{E})-\mathbf{E}\cdot(\nabla\times\mathbf{B})$ . Hence

\mathbf{E}\cdot(\nabla\times\mathbf{B})=\mathbf{B}\cdot(\nabla\times\mathbf{E})-\nabla\cdot(\mathbf{E}\times\mathbf{B}).

Why this identity and not another: it is the unique product rule that trades "E against curl B" for "B against curl E" — and Faraday's law is standing by to convert exactly the latter.

Step 3. Deploy Faraday: $\nabla\times\mathbf{E}=-\partial_t\mathbf{B}$ , so $\mathbf{B}\cdot(\nabla\times\mathbf{E})=-\mathbf{B}\cdot\partial_t\mathbf{B}=-\partial_t\!\left(\tfrac{1}{2}B^2\right)$ . The magnetic inventory's growth rate appears, with the minus sign that Faraday's law has carried since Lenz — the sign that will make the whole equation a conservation law rather than a runaway.

Step 4. Assemble. Collecting Steps 1–3 and dividing the magnetic pieces by $\mu_0$ :

\mathbf{J}\cdot\mathbf{E} =-\frac{\partial}{\partial t}\left(\frac{\varepsilon_0}{2}E^{2}+\frac{1}{2\mu_0}B^{2}\right)-\nabla\cdot\left(\frac{\mathbf{E}\times\mathbf{B}}{\mu_0}\right),

i.e., with $u$ and $\mathbf{S}$ defined by what the algebra delivered:

\frac{\partial u}{\partial t}+\nabla\cdot\mathbf{S}=-\,\mathbf{J}\cdot\mathbf{E}.

where: every term now has a certified reading — $u$ is the density whose static limits the assembly and ladder derivations found (E1 and E6 articles, recalled by name; here they are outputs), $\mathbf{S}$ is the courier of the previous article (likewise now an output), and $-\mathbf{J}\cdot\mathbf{E}$ is the transfer to matter from the work article. The theorem is the chapter's template with all three slots filled by Maxwell himself.

Step 5. Integrate over any volume $V$ with boundary $\Sigma$ and apply the divergence theorem for the working form:

-\oint_{\Sigma}\mathbf{S}\cdot\mathrm{d}\mathbf{a} =\frac{\mathrm{d}}{\mathrm{d}t}\int_V u\,\mathrm{d}\tau+\int_V\mathbf{J}\cdot\mathbf{E}\,\mathrm{d}\tau

— net inflow equals stock growth plus payments to matter. Every audit in this chapter, and every antenna power budget in E9, is an instance. Verified: dimensional audit — all terms W/m³ ✓; vacuum, source-free case reduces to $\partial_t u=-\nabla\cdot\mathbf{S}$ , which the plane wave satisfies identically ( $u$ and $S=uc$ both rigid profiles moving at $c$ ) ✓; static limit kills every term ✓; and adding matter's ledger ( $+\mathbf{J}\cdot\mathbf{E}$ , from the work article) yields a total sourceless continuity equation, matching Griffiths Eq. 8.9 and Jackson Eq. 6.108 exactly.

Go deeper · Derivation B · 2 minThe charging capacitor - a parameter-free closure

Derivation BThe charging capacitor - a parameter-free closure

The theorem's classic audit: a circular parallel-plate capacitor (radius $R$ , gap $d$ , fringe fields idealized away) charging at constant current $I$ . We show the field energy accumulating in the gap is delivered — to the joule, at every instant — by Poynting flux entering sideways through the gap's rim, and that the closure is exact independent of every parameter. This example is also the chapter's handshake with the displacement-current article, whose gap fields it reuses by name. Watch the geometry conspiracy: three independent factors cancel to produce the exact match.

Step 1. The gap fields. Between the plates, $E=\sigma_f/\varepsilon_0$ grows at $\dot E = I/(\varepsilon_0\pi R^2)$ (the uniform-charging idealization audited in this chapter's boundary-conditions article). At the rim, the ring magnetic field of the displacement current (The Displacement Current's second derivation, recalled): $B(R)=\mu_0 I/2\pi R$ — the gap behaves as a fat wire of changing field.

Step 2. The courier at the rim: $\mathbf{E}$ axial, $\mathbf{B}$ azimuthal, so $\mathbf{S}=\mathbf{E}\times\mathbf{B}/\mu_0$ points radially inward all around the rim, with magnitude $S=EB/\mu_0 = EI/2\pi R$ . The energy is not descending the wires from the battery and squeezing through the plates; it is converging on the gap from the surrounding field, through the cylinder of area $2\pi R d$ .

Step 3. The influx: $P_{\rm in}=S\times2\pi Rd = E\,I\,d$ . where the $2\pi R$ of the rim area has cancelled the $1/2\pi R$ of the ring field — first conspiracy.

Step 4. The stock's growth: $U=\tfrac{\varepsilon_0}{2}E^2\times\pi R^2 d$ , so

\frac{\mathrm{d}U}{\mathrm{d}t}=\varepsilon_0 E\,\dot E\,\pi R^{2}d=\varepsilon_0 E\,\frac{I}{\varepsilon_0\pi R^{2}}\,\pi R^{2}d=E\,I\,d,

— the $\varepsilon_0$ 's and the plate area cancel (second and third conspiracies), leaving exactly the rim influx of Step 3. No charges live in the gap ( $\mathbf{J}\cdot\mathbf{E}=0$ there), so the theorem's ledger closes with two entries: $-\oint\mathbf{S}\cdot\mathrm{d}\mathbf{a}=\mathrm{d}U/\mathrm{d}t$ , identically, at every instant of the charge-up, for every $R$ , $d$ , and $I$ .

Step 5. Read the moral before the numbers (Worked Example 1 prices this at 100 W): the closure could not care less about geometry because it is not a coincidence — it is Derivation A evaluated on this configuration. But seeing a specific audit close teaches what the general proof cannot: the energy that ends up between the plates genuinely arrives through the side, a preview of the resistive-wire scandal that the next article owns by name. Verified: symbolic closure shown above is exact; numerically, at $E=10^5$ V/m with $I=0.5$ A, $R=5$ cm, $d=2$ mm: $S=1.6\times10^5$ W/m², influx $=100.0$ W $=\mathrm{d}U/\mathrm{d}t$ (recomputed both routes); the gap fields match the displacement-current article's worked numbers ( $\dot E=7.2\times10^{12}$ V/m·s, $B$ at rim 2.0 μT for the same parameters).

Figure

The capacitor audit — axial E crossed with the displacement current's ring B drives S inward through the rim, feeding dU/dt exactly

The fine print: how unique are u and S?

Derivation A delivered a pair $(u,\mathbf{S})$ , but delivery is not uniqueness. Add to $\mathbf{S}$ any curl, $\mathbf{S}'=\mathbf{S}+\nabla\times\mathbf{W}$ , and $\nabla\cdot\mathbf{S}'=\nabla\cdot\mathbf{S}$ : the theorem holds verbatim, every closed-surface audit is unchanged, and no experiment reading energy budgets can tell the pairs apart. More general reshuffles trade terms between $u$ and $\mathbf{S}$ simultaneously. This is the non-uniqueness the previous article flagged, and the honest hierarchy is: the theorem is exact; the standard pair is a choice; the choice is heavily pinned. Three pins. Relativity: in the covariant formulation (E10, by name) $u$ and $\mathbf{S}/c^2$ sit inside the stress–energy tensor, and demanding the standard symmetric tensor — the one general relativity couples to gravity — selects the Poynting pair; energy location has gravitational consequences, and GR endorses this bookkeeping. Angular momentum: the field's angular-momentum accounting (this chapter's closing articles, by name) balances experiments like the Feynman disk only with $\mathbf{g}=\varepsilon_0\mathbf{E}\times\mathbf{B}=\mathbf{S}/c^2$ — the momentum shadow of the standard courier. Simplicity with locality: the standard pair is built pointwise from the fields alone, with no nonlocal integrals and no preferred frame. Within classical electrodynamics you may regard alternatives as gauge-like curiosities; the moment gravity or momentum enters, the standard pair is the physical one. In matter the same manipulation run on the macroscopic equations (Maxwell's Equations in Matter, recalled by name) yields the linear-media theorem $\partial_t\left[\tfrac12(\mathbf{E}\cdot\mathbf{D}+\mathbf{B}\cdot\mathbf{H})\right]+\nabla\cdot(\mathbf{E}\times\mathbf{H})=-\mathbf{J}_f\cdot\mathbf{E}$ — free sources only on the right, bound-charge energy folded into the density (with the lossless, non-dispersive fine print the energy-in-matter articles own, E3 and E5, by name). And here a genuine century-long argument opens: in matter, the momentum companion to this energy theorem admits two natural forms — Abraham's and Minkowski's — that disagree about how much momentum light carries in glass. Both sides are internally consistent; the resolution (both are right, bookkeeping different splits of field-plus-matter momentum) took until the 2010s to be experimentally anatomized, and the momentum articles of this chapter open that file by name. Poynting's theorem is where the debate's energy half is settled and its momentum half begins.

Limiting cases

case	theorem reduces to	reading
vacuum, source-free	$\partial_t u=-\nabla\cdot\mathbf{S}$	pure transport; plane waves satisfy it identically
statics	$0=0$	inventories frozen, no traffic, no transfer
steady currents	$\nabla\cdot\mathbf{S}=-\mathbf{J}\cdot\mathbf{E}$	flux converges exactly on dissipation — next article's wire
ideal cable	$\nabla\cdot\mathbf{S}=0$ in the gap	courier passes through, nothing deposited
linear media	$\tfrac12\partial_t(\mathbf{E}\cdot\mathbf{D}+\mathbf{B}\cdot\mathbf{H})+\nabla\cdot(\mathbf{E}\times\mathbf{H})=-\mathbf{J}_f\cdot\mathbf{E}$	lossless/non-dispersive fine print applies
field + matter total	sourceless continuity	full local energy conservation, doctrine satisfied

Common trapThe J·E term is power to charges — not necessarily heat, and not a leak

Two habitual misreadings of the right-hand side. First: $\mathbf{J}\cdot\mathbf{E}$ is the rate energy leaves the field ledger for the matter ledger — whether it becomes heat (resistors), kinetic energy (beams), or chemical potential run backwards (charging a battery: $\mathbf{J}\cdot\mathbf{E}<0$ , matter paying the field) is matter's business, sorted by the two-fates split of the work article. Writing " $=$ Joule heating" unconditionally is wrong everywhere interesting. Second: a nonzero source term does not mean field energy "isn't conserved" — the identical term with opposite sign sits in matter's energy equation, and the sum is sourceless. The theorem is a transfer invoice, not a leak report.

Worked examples

1 · The capacitor audit, in watts (confidence builder)

Run Derivation B's closure with numbers: $R=5.0$ cm, $d=2.0$ mm, $I=0.50$ A, at the instant $E=1.0\times10^5$ V/m.

Step 1. Rim magnetic field: $B=\mu_0I/2\pi R = 2.0\times10^{-6}$ T (the displacement-current article's own number for this geometry, recalled).

Step 2. Courier at the rim: $S=EB/\mu_0 = 10^5\times2.0\times10^{-6}/1.257\times10^{-6} = \mathbf{1.6\times10^{5}\ W/m^2}$ , radially inward — a hundred and sixty kilowatts per square metre converging on a bench-top gap.

Step 3. Influx through the rim cylinder ( $2\pi Rd = 6.28\times10^{-4}$ m²): $P_{\rm in} = 1.59\times10^5\times6.28\times10^{-4} = \mathbf{100\ W}$ . Check against $EId = 10^5\times0.5\times0.002 = 100$ W ✓.

Step 4. Stock growth: $\mathrm{d}U/\mathrm{d}t = \varepsilon_0E\dot E\,\pi R^2d$ with $\dot E = I/\varepsilon_0\pi R^2 = 7.2\times10^{12}$ V/m·s: $= 8.854\times10^{-12}\times10^5\times7.2\times10^{12}\times\pi\times0.0025\times0.002 = \mathbf{100\ W}$ ✓. The ledger closes to the watt — and closes at every instant, since both sides scale as $E(t)$ .

2 · Filling a solenoid's reservoir through its own side (exam level)

The track's standard solenoid ( $n=1000$ turns/m, radius $r=1.5$ cm — the geometry of Self-Inductance and Energy Stored in Magnetic Fields, recalled) carries $I=2.0$ A being ramped at $\mathrm{d}I/\mathrm{d}t=100$ A/s. Audit the energy flow per unit length.

Step 1. Interior field $B=\mu_0nI = 2.5\times10^{-3}$ T, growing at $\dot B=\mu_0n\,\dot I = 0.126$ T/s. The changing flux induces a circumferential electric field at the surface — the Faraday-loop technology of Induced Electric Fields, recalled: $E_{\rm ind}=(r/2)\dot B = 9.4\times10^{-4}$ V/m, wrapped around the solenoid against the current's ramp.

Step 2. Courier at the surface: $\mathbf{E}$ azimuthal, $\mathbf{B}$ axial → $\mathbf{S}=\mathbf{E}\times\mathbf{B}/\mu_0$ points radially inward, magnitude $S=E_{\rm ind}B/\mu_0 = 1.9$ W/m². Influx per unit length: $P=S\times2\pi r = \mathbf{0.178\ W/m}$ .

Step 3. Stock growth per unit length: $\mathrm{d}/\mathrm{d}t\left[(B^2/2\mu_0)\pi r^2\right]=(B\dot B/\mu_0)\pi r^2 = (2.51\times10^{-3}\times0.126/1.257\times10^{-6})\times\pi\times2.25\times10^{-4} = \mathbf{0.178\ W/m}$ ✓ — exact closure, the magnetic mirror of Example 1.

Step 4. Circuit cross-check: the back-EMF power per length is $(LI\,\mathrm{d}I/\mathrm{d}t)$ per length $= (\mu_0n^2\pi r^2)\,I\,\dot I = 1.257\times10^{-6}\times10^6\times\pi\times2.25\times10^{-4}\times2.0\times100 = 0.178$ W/m ✓ — the power supply's extra work against the back-EMF (the circuit-side story of Energy Stored in Magnetic Fields) is identical to the Poynting influx through the solenoid's flank. Circuit theory and field theory are keeping the same books; Poynting's theorem is why they must.

Origins · 1 minOliver Heaviside and how this was discovered

Oliver Heaviside

1850–1925 · London and Devon — the self-taught operator who found the energy current independently

Heaviside left school at sixteen, learned physics as a telegraph operator, and re-derived the deep structure of Maxwell's theory from a rented room. In 1885 — working independently of Poynting's 1884 paper — he published the energy-flux theorem in his own form, $\mathbf{E}\times\mathbf{H}$ , the version engineering still prefers in media, discovered en route to his life's project: recasting Maxwell's twenty quaternion-flavored equations into the four vector equations every student now learns. "Energy current" was, for Heaviside, not a corollary but an organizing principle — he insisted energy moves through the field, wires being merely its guides, and fought the establishment (and the Post Office engineers) for decades over the consequences, from inductive loading of telephone lines (his analysis, resisted, then adopted, multiplied cable reach) to the ionospheric layer bearing his name. Denied formal position most of his life, elected FRS in 1891, he grew reclusive, signing letters "W.O.R.M." Physics kept the vector notation, the telegrapher's equations, the step function, and the habit — begun with this theorem — of asking where, exactly, the energy goes.

History

1884

— Poynting derives the theorem and the flux vector: energy transfer localized for the first time

1885

— Heaviside, independently, publishes the E×H energy-current form amid his vectorization of Maxwell

1889

— Lodge's "identity of energy" popularizes continuous energy tracking as physical doctrine

1893

— Föppl's textbook carries the theorem to the continent (a young Einstein reads it)

1908

— Minkowski's field momentum in media opens the Abraham–Minkowski debate at the theorem's momentum flank

1909

— Abraham's rival momentum density: the century's politest unresolved argument begins

2010

— Barnett's resolution: kinetic vs canonical momentum — both bookkeepings right, questions different

Going further

References. Griffiths, Introduction to Electrodynamics, 4th ed., §8.1.2 · Jackson, Classical Electrodynamics, 3rd ed., §6.7 · Purcell & Morin, Electricity and Magnetism, 3rd ed., §9.6 · Zangwill, Modern Electrodynamics, §15.4.

Provenance. Differential and integral forms verified against Griffiths Introduction to Electrodynamics 4th ed. §8.1.2 (Eq. 8.9, 8.11) and Jackson Classical Electrodynamics 3rd ed. §6.7 (Eq. 6.108–6.111). A verifier should recompute the curl identity step, confirm dimensional consistency ( $\mathbf{J}\cdot\mathbf{E}$ in W/m³), and check the resistive-wire example ( $\oint\mathbf{S}\cdot\mathrm{d}\mathbf{a}=-VI=-I^2R$ ) as in Purcell & Morin 3rd ed. §9.6. Constants $\varepsilon_0=8.854\times10^{-12}$ F/m and $\mu_0=1.257\times10^{-6}$ H/m are CODATA standard values. Time-averaged complex Poynting vector as in Jackson Eq. 6.132 and Zangwill §15.4. Addendum for this edition: the base's resistive-wire example is RESCOPED forward to Poynting Vector in Static Field Configurations (0572) by name — this edition owns the theorem's derivation (E0 identity recalled from VID.0454), the charging-capacitor closure (numbers cross-checked against DISPJ.0559's shared geometry: $\dot E=7.2\times10^{12}$ V/m·s, rim $B=2.0$ μT, closure 100 W both routes at $E=10^5$ V/m), the ramping-solenoid audit (0.178 W/m three routes: field influx, $B\dot B\pi r^2/\mu_0$ , and $LI\dot I$ per length with the SELFL/ENRG solenoid geometry), the non-uniqueness clause with its three pins, and the linear-media form with the Abraham–Minkowski doorway (momentum articles by name).

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

Leads to: Global vs. Local Conservation Laws · Momentum Density of the Electromagnetic Field · The Maxwell Stress Tensor · Conservation of Linear Momentum in Electrodynamics · Poynting Vector in Static Field Configurations.

Is Poynting's theorem an independent law of nature or a consequence of Maxwell's equations?

A strict consequence — that is its glory. Given Maxwell's equations and the Lorentz force, the theorem is an algebraic identity; no new postulate enters. Its content as physics is the interpretation: reading u as stored density and S as flux converts an identity into a local conservation law, and the doctrine article's locality arguments justify exactly that reading. The deeper frame: Noether's theorem derives energy conservation from time-translation symmetry of the field theory, and the (u, S) pair is precisely the Noether current — so the theorem is also the electromagnetic instance of the symmetry-conservation bridge, machinery the field-theory track owns by name.

Why does the derivation dot E — not B — into a Maxwell equation?

Because energy enters and leaves through E alone. The work article proved the transfer term is J·E, with no magnetic counterpart (v×B ⊥ v), so a bookkeeping of transfer must start there; J appears in exactly one Maxwell equation, Ampère–Maxwell, and dotting it with E is the only move that brings the transfer term face to face with field derivatives. The apparent asymmetry of the derivation is the physical asymmetry of the Lorentz force. The result, satisfyingly, is symmetric anyway: u weighs both fields equally, and S needs both.

What is "reactive power" in this language?

The part of the Poynting traffic that commutes without emigrating. Near any AC device, some flux flows out during one quarter-cycle and back the next — energy borrowed by the fields and returned, time-averaging to zero. The complex-Poynting formalism splits ⟨S⟩ into a real part (net transport: billable watts, radiated power) and an imaginary part (the commuting reserve: volt-amperes-reactive), and the utility's power-factor penalty is a charge for making the grid carry the commuters. In antenna terms the same split is near-field storage versus far-field radiation, with the crossover at r ~ λ/2π. The full circuit-level treatment belongs to the AC-circuits gap chapter, and the wave-level version to E8/E9, by name.

Does the theorem hold in materials, where the medium itself stores energy?

Yes, in two honest flavors. Microscopically, the vacuum theorem is exact everywhere — matter is just more charges inside J·E. Macroscopically, running the derivation on the averaged equations gives the E×H form with density (E·D + B·H)/2, valid verbatim for linear, lossless, non-dispersive media; dispersion and absorption require the Brillouin corrections and loss terms that the wave-in-matter articles own by name. The one genuinely deep wrinkle is momentum, not energy: Abraham and Minkowski's rival momentum densities in media coexist because "field momentum" versus "field-plus-matter-response momentum" are different, individually consistent splits — anatomized experimentally only in the last two decades, and opened properly by this chapter's momentum articles.

If u and S can be redefined, is the energy's location physically meaningless?

No — bounded, not meaningless. The reshuffling freedom is real within pure electrodynamics, but it is not unlimited: the alternatives must still satisfy the theorem, and the standard pair is uniquely selected once you demand (i) the relativistic packaging that gravity couples to — GR reads energy location, and light bends spacetime where the Poynting bookkeeping says its energy is — and (ii) consistency of angular-momentum audits (the Feynman-disk file, two articles ahead). What remains genuinely conventional is finer-grained than any experiment: streamline-level narratives in regions where no audit can follow. The honest posture: state audits as theorems, tell streamline stories with the standard pair, and flag the distinction when it matters — which this track does.

How does the theorem handle a battery charging — where does chemical energy enter the books?

Through the sign of J·E, with no new terms needed. Inside a charging battery, the external supply drives current from + to − internally, along the interior field: J·E > 0 there... while inside a discharging battery J·E < 0 (matter pays the field — the source audit of the work article). The chemical ledger itself sits outside electrodynamics: the theorem's matter side receives J·E and hands it to chemistry as increased reaction free energy, minus I²r of heat. Poynting's theorem tracks energy only while it is electromagnetic; the interfaces where it enters and leaves the field are exactly the J·E ≠ 0 regions, and the audits close at every one of them.