Energy Density and the Poynting Vector

The conservation-laws chapter (E7) built the bookkeeping machinery in full generality: an energy density $u = \tfrac12(\varepsilon_0E^2 + B^2/\mu_0)$ stored in the fields, a flux $\mathbf{S} = \mathbf{E}\times\mathbf{B}/\mu_0$ carrying it, and Poynting's theorem locking the two into a continuity equation. It even proved, for any traveling waveform, that the electric and magnetic accounts hold equal balances and that the flux equals the density moving at $c$ . What it did not do — deliberately, by division of labor — is open the books of the one field configuration the rest of this chapter lives on. This article does exactly that: the complete, instantaneous energy audit of the monochromatic plane wave. The results are compact enough to memorize and rich enough to repay a careful look. The energy density is a rigid comb of $\cos^2$ sheets gliding along $\hat{\mathbf{k}}$ at $c$ , electric and magnetic halves equal at every instant and every point — not just on average. The Poynting vector is that same comb multiplied by $c$ : energy density in transit, nothing more. And the whole inventory pulses at $2\omega$ — twice the wave's frequency — about a steady mean, a factor of two with consequences from detector physics to laser safety. Everything here is instantaneous and exact; no averages are taken. That restraint is deliberate too. Real detectors cannot follow the $2\omega$ pulse of a light wave and report means instead — but which mean, and why the answer is exactly half the peak, is its own subject with its own traps, and the intensity article two doors down owns it. First the books themselves; then, separately, how the books get read.

Key takeaways

For $\mathbf{E} = E_0\cos(\mathbf{k}\cdot\mathbf{r} - \omega t)\hat{\mathbf{n}}$ in vacuum, the instantaneous energy density is $u(\mathbf{r},t) = \varepsilon_0E_0^2\cos^2(\mathbf{k}\cdot\mathbf{r}-\omega t)$ — a comb of energy sheets, spacing λ/2, riding along $\hat{\mathbf{k}}$ at $c$ .
The lock $B = E/c$ makes the electric and magnetic halves equal instantaneously: $u_E = u_B$ at every point and moment, so $u = \varepsilon_0E^2 = B^2/\mu_0$ , either account doubled.
The Poynting vector is the density in transit: $\mathbf{S} = c\,u\,\hat{\mathbf{k}}$ , pointwise, at every instant — the plane-wave instantiation of the E7 transport theorem.
Nothing about the energy oscillates at ω: $u$ and $S$ pulse at $2\omega$ , between zero and their peaks, about a mean of exactly half the peak.
Momentum rides along at $\mathbf{g} = \mathbf{S}/c^2$ — one line here, a full article (0586) next.
Key idea: in a plane wave, energy does not slosh between E and B — the two accounts are equal at every instant, and the whole inventory moves forward at c as a 2ω-pulsing comb.

The core insight

In a traveling plane wave the electric and magnetic energies are equal at every instant ( $u_E = u_B$ , by $B = E/c$ ), and the total $u = \varepsilon_0E_0^2\cos^2(\mathbf{k}\cdot\mathbf{r}-\omega t)$ rides along $\hat{\mathbf{k}}$ at $c$ with $\mathbf{S} = cu\hat{\mathbf{k}}$ — a $2\omega$ -pulsing comb of energy in transit.

Where this lives today

The instantaneous books — not just their averages — are now engineering quantities. Ultrafast optics lives on the peak, not the mean: a petawatt facility's 10²³ W/cm² focus exists for tens of femtoseconds, and it is the crest field $E_0$ (approaching and exceeding the atomic field, per Worked Example 2's scaling) that ionizes targets, accelerates electron bunches in laser-wakefield machines, and sets damage thresholds quoted on every optic in the beamline. Laser-safety standards distinguish continuous-wave exposure (average books) from pulsed exposure (peak books) for exactly the reason this article separates the comb from its mean: the same average power can be harmless or ablating depending on how the $\cos^2$ inventory is bunched. At the opposite end of the dynamic range, radio astronomy routinely detects fluxes below $10^{-26}$ W/m²·Hz — energy combs so dilute that the density in a cubic kilometre of signal is a fraction of an electronvolt — and integrates them coherently for hours, a feat possible only because the wave's phase structure (articles 0583–0584) survives across that entire twenty-plus-decade span down from the femtosecond focus. Same ledger, same three-way identity $u$ , $S = cu$ , $g = u/c$ , from the weakest pulsar to the strongest pulse ever focused.

The physical picture

Take the plane wave of article 0583 and paint each point of space with its energy content. The picture that appears is a second comb, riding on the first: wherever $|\mathbf{E}|$ crests — twice per wavelength, at the positive and negative peaks of the cosine — the energy density crests too, and wherever the fields cross zero, the energy density is exactly zero. Because energy goes as the square of the field, the energy comb has half the spacing of the field comb: sheets of energy every λ/2, all positive, gliding rigidly forward at $c$ with the phase. The surprise for circuit-trained intuition is what does not happen. Nothing sloshes. In an LC oscillator, energy shuttles between capacitor and inductor, electric when one is full, magnetic when the other is. In the traveling plane wave the amplitude lock $B = E/c$ — with $c^2 = 1/\mu_0\varepsilon_0$ — forces $\tfrac12\varepsilon_0E^2 = B^2/2\mu_0$ identically: the two accounts are equal at every instant, both peaking together, both vanishing together. The E7 energy article proved this equal partition for any traveling waveform; the monochromatic wave inherits it as the cleanest special case. Where the fields are zero, there is genuinely no energy at that point at that instant — the inventory is one λ/2 ahead and behind, moving in.

Figure

The energy comb — u rides the squared field at half the wavelength spacing, equal E and B accounts at every instant, the whole inventory in transit at c

Once the comb is drawn, the Poynting vector stops being a formula and becomes a tautology: energy density $u$ moving rigidly at speed $c$ is a flux $cu$ through any surface it crosses — the E7 Poynting article proved $\mathbf{S} = uc\hat{\mathbf{k}}$ as a general transport theorem for traveling waves, and the plane wave realizes it pointwise, sheet by sheet. What this article adds to the E7 statement is the explicit anatomy: exactly how $u$ and $\mathbf{S}$ are distributed in space, exactly how they pulse in time, and exactly what the numbers are for real beams — from a broadcast wave's $10^{-13}$ J/m³ to a focused femtosecond pulse's $10^{11}$ J/m³, a dynamic range of twenty-four decades carried by one pair of formulas.

The books, opened

The chapter needs the plane wave's $u$ and $\mathbf{S}$ as explicit functions of space and time, and the honest way to get them is to feed the lawful wave into the E7 machinery and watch the structure collapse. Both derivations are short; their brevity is inherited rigor, not hand-waving — every hard theorem was proved in E7, and this article is the cash withdrawal.

Derivation AThe instantaneous energy ledger of a plane wave — equal partition, collapsed to one term

The goal is the exact, instantaneous energy density $u(\mathbf{r},t)$ of the monochromatic plane wave, exhibiting the equal split between electric and magnetic accounts and the $2\omega$ pulse structure. The strategy: insert the locked pair $\mathbf{E} = E_0\cos\varphi\,\hat{\mathbf{n}}$ , $\mathbf{B} = \tfrac{E_0}{c}\cos\varphi\,(\hat{\mathbf{k}}\times\hat{\mathbf{n}})$ — with $\varphi = \mathbf{k}\cdot\mathbf{r}-\omega t + \delta$ — into the E7 density $u = \tfrac12\varepsilon_0E^2 + B^2/2\mu_0$ , whose derivation and status (certified by Poynting's theorem, recalled by name) belong to the E7 energy article. Watch two things: how the amplitude lock forces the two terms equal before any averaging, and how the trigonometric identity at the end splits the result into a steady part plus a pure $2\omega$ oscillation — the decomposition the complex-notation article's lemma predicted term for term.

Step 1. The electric account. $u_E = \tfrac12\varepsilon_0E^2 = \tfrac12\varepsilon_0E_0^2\cos^2\varphi$ .

Step 2. The magnetic account. $u_B = \dfrac{B^2}{2\mu_0} = \dfrac{E_0^2\cos^2\varphi}{2\mu_0c^2} = \tfrac12\varepsilon_0E_0^2\cos^2\varphi$ , using $c^2 = 1/\mu_0\varepsilon_0$ to convert $1/\mu_0c^2 = \varepsilon_0$ . Why this is exact and instantaneous: the lock $B = E/c$ holds at every point and time (Faraday's law's doing, article 0583), so the equality $u_B = u_E$ is pointwise — no averaging, no "on balance." The E7 energy article proved equal partition for arbitrary traveling profiles $f(z-ct)$ ; the cosine is the special case, and the mechanism (the lock) is identical.

Step 3. The total, in both useful forms:

u(\mathbf{r},t) = \varepsilon_0E_0^2\cos^2(\mathbf{k}\cdot\mathbf{r}-\omega t+\delta) = \varepsilon_0E^2(\mathbf{r},t).

The second form is worth engraving: for a plane wave you may compute the total energy density as if the field were purely electric, doubled — $u = \varepsilon_0E^2$ , no ½ — or equally as $u = B^2/\mu_0$ . One field's instantaneous value determines the entire inventory.

Step 4. The pulse structure. The half-angle identity $\cos^2\varphi = \tfrac12(1 + \cos2\varphi)$ splits the ledger:

u = \underbrace{\tfrac12\varepsilon_0E_0^2}_{\text{steady mean}} + \underbrace{\tfrac12\varepsilon_0E_0^2\cos\left(2\mathbf{k}\cdot\mathbf{r}-2\omega t+2\delta\right)}_{2\omega\text{ ripple, zero mean}}.

Why the frequency doubling matters physically: the energy content of the wave contains no component at ω at all — squaring maps both the crest and the trough of the field to energy maxima, so the inventory beats twice per field cycle. A detector fast enough to follow the energy of green light would need $2f \approx 1.1\times10^{15}$ Hz bandwidth; none exists, which is why every real instrument reports the steady term — the intensity article (0587) takes it from here, by name.

Go deeper · Derivation B · 1 minThe flux is the density in transit — S = cu, pointwise

Derivation BThe flux is the density in transit — S = cu, pointwise

The goal is the exact Poynting vector of the plane wave and the pointwise identity $\mathbf{S} = c\,u\,\hat{\mathbf{k}}$ , tying this article's two protagonists into one object. The strategy is a direct cross product — the E7 Poynting article owns the flux formula $\mathbf{S} = \mathbf{E}\times\mathbf{B}/\mu_0$ and its credentials (only divergences and closed-surface integrals are certified; recalled, not re-argued) — followed by a one-line comparison with Derivation A. Watch the geometry do the work: the triad makes the direction automatic, and the amplitude lock makes the magnitude collapse to $cu$ .

Step 1. The cross product. With $\mathbf{E} = E_0\cos\varphi\,\hat{\mathbf{n}}$ and $\mathbf{B} = \tfrac{E_0}{c}\cos\varphi\,(\hat{\mathbf{k}}\times\hat{\mathbf{n}})$ :

\mathbf{S} = \frac{\mathbf{E}\times\mathbf{B}}{\mu_0} = \frac{E_0^2\cos^2\varphi}{\mu_0c}\,\hat{\mathbf{n}}\times(\hat{\mathbf{k}}\times\hat{\mathbf{n}}) = \frac{E_0^2}{\mu_0c}\cos^2\varphi\ \hat{\mathbf{k}},

using the BAC–CAB expansion $\hat{\mathbf{n}}\times(\hat{\mathbf{k}}\times\hat{\mathbf{n}}) = \hat{\mathbf{k}}(\hat{\mathbf{n}}\cdot\hat{\mathbf{n}}) - \hat{\mathbf{n}}(\hat{\mathbf{n}}\cdot\hat{\mathbf{k}}) = \hat{\mathbf{k}}$ (transversality kills the second term). Why the direction never wavers: $\cos^2 \geq 0$ , so $\mathbf{S}$ points along $+\hat{\mathbf{k}}$ at every instant — even as the fields themselves reverse sign twice per cycle. Both fields flip together (in phase), and the cross product of two sign flips is no flip. A plane wave never momentarily "flows backward."

Step 2. Compare with Derivation A. Since $1/\mu_0c = c\varepsilon_0$ :

\mathbf{S} = c\,\varepsilon_0E_0^2\cos^2\varphi\ \hat{\mathbf{k}} = c\,u(\mathbf{r},t)\,\hat{\mathbf{k}}.

The flux is the density times $c$ , at every point, at every instant — the comb in transit. The E7 transport theorem stated this for arbitrary traveling waveforms from the continuity equation; here it emerges by direct multiplication, with the $\cos^2$ profile visible. Peak flux $S_{\max} = c\varepsilon_0E_0^2$ , mean (when the time comes to take it) exactly half.

Step 3. The momentum trailer. The E7/EM-15 momentum article established $\mathbf{g} = \mathbf{S}/c^2$ as the field's momentum density; for the plane wave that reads $\mathbf{g} = (u/c)\hat{\mathbf{k}}$ , pointwise — the same comb again, divided by $c$ . One line here; the pushes, pressures, sails, and comet tails it finances are the next article's entire subject (0586, by name).

The two results deserve to be seen side by side, because together they say the plane wave has one energy story told three ways:

u = \varepsilon_0E_0^2\cos^2\varphi, \qquad \mathbf{S} = c\,u\,\hat{\mathbf{k}}, \qquad \mathbf{g} = \frac{u}{c}\,\hat{\mathbf{k}} \qquad (1)

where: one function of one phase variable — the comb — carries the density, the flux, and the momentum, scaled by $1$ , $c$ , and $1/c$ respectively. Read it physically: measure any one of the three at a point and instant and the other two follow with no further information about the wave. This rigidity is special to the single traveling plane wave; the warning box below patrols its boundary.

Figure

Twenty-four decades on one shelf — plane-wave energy densities from a broadcast whisper to a femtosecond focus, all priced by the same comb

Limiting cases

Case	Result	Meaning
at a field crest ( $\cos\varphi = \pm1$ )	$u = \varepsilon_0E_0^2$ , $S = c\varepsilon_0E_0^2$ : both peak	energy maxima at both signs of the field — the 2ω origin
at a field zero	$u = 0$ , $S = 0$ exactly, there, then	real zeros; inventory is λ/2 away, arriving in T/4
average over a cycle (preview)	$\langle u\rangle = \tfrac12\varepsilon_0E_0^2$ , $\langle S\rangle = \tfrac12c\varepsilon_0E_0^2$	owned by 0587; the ½ is the DC term of Step 4
average over one wavelength at fixed t	same ½ factors	spatial and temporal means agree — the comb's symmetry
arbitrary traveling profile $f(z-ct)$	$u_E = u_B$ , $S = cu$ still exact	E7's general theorems; monochromatic case is the display model
superposition of two waves	$u \neq u_1 + u_2$ in general	energy is quadratic — E7's interference ledger, recalled
standing wave	$u_E$ , $u_B$ in quadrature; $\langle\mathbf{S}\rangle = 0$	the anti-case: E7 energy article owns its anatomy

Common trapS = cu is a single-traveling-wave identity — export it to a superposition and the books go wrong

The comb picture is so clean that it begs to be applied everywhere; its license is narrow. $\mathbf{S} = cu\hat{\mathbf{k}}$ and $u = \varepsilon_0E^2$ presume ONE traveling plane wave, whose fields are locked by $B = E/c$ in the right-handed triad. Superpose a second wave and both identities die: in a standing wave (incident plus reflected), the electric and magnetic energies fall into quadrature — they do slosh, LC-style, twice per cycle — the energy density has nodes that never move, and the average flux is zero while $u$ is decidedly not; using $S = cu$ there predicts energy transport that does not exist. In an interference region of two crossed beams, $u$ carries cross terms (the E7 interference ledger), and $\varepsilon_0E^2$ no longer equals $\tfrac12\varepsilon_0E^2 + B^2/2\mu_0$ pointwise. The discipline: identities (1) are properties of the elementary mode, not of light in general. For anything built from several modes, return to the primitive formulas $u = \tfrac12(\varepsilon_0E^2 + B^2/\mu_0)$ and $\mathbf{S} = \mathbf{E}\times\mathbf{B}/\mu_0$ with the total fields, and let the cross terms speak.

Worked examples

1 · The energy books of a broadcast wave (confidence builder)

Near an FM tower the field amplitude is $E_0 = 0.10$ V/m at 100 MHz. Open the instantaneous books: peak $u_E$ , peak $u_B$ , peak total $u$ , peak $S$ , and the energy content of a one-square-metre, one-wavelength-thick slab of the wave.

Step 1. Peak electric account: $u_E^{\max} = \tfrac12\varepsilon_0E_0^2 = \tfrac12(8.854\times10^{-12})(0.10)^2 = \mathbf{4.43\times10^{-14}\ J/m^3}$ .

Step 2. Peak magnetic account: with $B_0 = E_0/c = 3.34\times10^{-10}$ T, $u_B^{\max} = B_0^2/2\mu_0 = (3.34\times10^{-10})^2/(2\times1.257\times10^{-6}) = 4.43\times10^{-14}$ J/m³ — equal to the electric account, as the lock demands, with no averaging invoked.

Step 3. Peak total and peak flux: $u^{\max} = \varepsilon_0E_0^2 = 8.85\times10^{-14}$ J/m³; $S^{\max} = cu^{\max} = 2.998\times10^8\times8.85\times10^{-14} = \mathbf{2.65\times10^{-5}\ W/m^2}$ , along $\hat{\mathbf{k}}$ , pulsing at $2f = 200$ MHz between zero and this value.

Step 4. The slab. At 100 MHz, $\lambda = c/f = 3.00$ m. The energy per m² in one wavelength of comb is the spatial mean density times the thickness — and the mean of $\cos^2$ over a full wavelength is exactly ½ by the comb's symmetry: $W = \tfrac12\varepsilon_0E_0^2 \times \lambda = 4.43\times10^{-14}\times3.00 = \mathbf{1.33\times10^{-13}\ J}$ . A tenth of a picojoule per square metre per wavelength — and that slab replenishes itself $10^8$ times per second, which is how a whisper-thin density sustains a usable power flux. Thin stock, fast restock: the E7 Poynting article's slogan, now with the comb's numbers in it.

2 · The energy density frontier — a femtosecond pulse audited (exam level)

A titanium–sapphire amplifier delivers $1.0$ mJ in a 25 fs pulse, focused to a spot of radius $10\ \mu$ m. Estimate the peak flux, the peak energy density, the peak fields, and compare them to atomic benchmarks. (Treat the focus as a plane wave over the spot — the local-plane-wave license of article 0583.)

Step 1. Peak power: $P = W/\tau = 10^{-3}/25\times10^{-15} = \mathbf{4.0\times10^{10}}$ W — forty gigawatts, several power grids' worth, for twenty-five femtoseconds.

Step 2. Peak flux: spot area $A = \pi(10^{-5})^2 = 3.14\times10^{-10}$ m², so $S = P/A = 4.0\times10^{10}/3.14\times10^{-10} = \mathbf{1.27\times10^{20}\ W/m^2}$ ( $1.27\times10^{16}$ W/cm² in laser-lab units).

Step 3. Peak energy density, by the transport identity run backwards: $u = S/c = 1.27\times10^{20}/2.998\times10^8 = \mathbf{4.2\times10^{11}\ J/m^3}$ — the energy content of a good rifle round, in every cubic millimetre of focus, while the pulse is there.

Step 4. The fields, from $u = \varepsilon_0E_0^2$ at the crest: $E_0 = \sqrt{S/(c\varepsilon_0)} = \sqrt{1.27\times10^{20}/(2.998\times10^8\times8.854\times10^{-12})} = \mathbf{2.2\times10^{11}\ V/m}$ , and $B_0 = E_0/c = \mathbf{730\ T}$ .

Step 5. Benchmarks. The field binding hydrogen's electron is $E_{\text{at}} = e/4\pi\varepsilon_0a_0^2 = 5.1\times10^{11}$ V/m: this tabletop pulse reaches 43% of the atomic field — enough to rip electrons out by suppressing the Coulomb barrier (tunnel ionization), the working regime of attosecond science. The 730 T magnetic crest is sixteen times any sustained laboratory magnet. Both numbers follow from one energy comb, read at its peak. Answers: $S = 1.3\times10^{20}$ W/m², $u = 4.2\times10^{11}$ J/m³, $E_0 = 2.2\times10^{11}$ V/m, $B_0 = 730$ T.

Origins · 1 minJames Clerk Maxwell and how this was discovered

James Clerk Maxwell

1831–1879 · Cambridge and Glenlair — who costed sunlight's fields before anyone could measure them

Maxwell did not stop at predicting the waves; he opened their books. In the Treatise (1873) he used his theory's energy density — the same $\tfrac12\varepsilon_0E^2 + B^2/2\mu_0$ this article spends — together with contemporary measurements of the solar constant to compute the actual field strengths in sunlight, arriving at an electromotive intensity of about 600 volts per metre near Earth (the modern value for $E_0$ is close to 1,000 V/m; his input solar data, not his method, carried the error). It was a characteristic Maxwell move: a theory is not finished until it prices something. He performed the same audit for momentum, predicting radiation pressure of order $10^{-6}$ Pa for sunlight — "a very small pressure," he noted, which "might perhaps be detected by a delicate apparatus" — a challenge Lebedev, Nichols and Hull answered experimentally two decades after his death, and the next article inherits. Maxwell died at 48, before Hertz's spark; he never saw a measured electromagnetic wave, only the ledgers he had already balanced for them.

History

1847

— Helmholtz's Über die Erhaltung der Kraft: energy conservation as a universal audit, fields not yet on the books

1853

— William Thomson expresses electrostatic energy as a field-volume integral: energy gets an address in space

1873

— Maxwell's Treatise computes the field strengths of sunlight from the solar constant — the first plane-wave energy audit

1884

— Poynting's theorem: storage, transit, transfer locked into one continuity equation (E7's chapter)

1888

— Hertz's receiver sparks: wave energy demonstrably delivered across a room

1960

— Maiman's laser concentrates the comb: energy densities beyond any natural source at Earth

2019

— ELI and petawatt facilities push focused pulses past 10²² W/cm² — field crests a thousand times the atomic field

Going further

References. Griffiths, Introduction to Electrodynamics, 4th ed., §9.2.3 · Jackson, Classical Electrodynamics, 3rd ed., §7.1 · Purcell & Morin, Electricity and Magnetism, 3rd ed., §9.6 · Zangwill, Modern Electrodynamics, §16.3. The key results for the monochromatic plane wave are $u = \varepsilon_0E^2$ , $\mathbf{S} = cu\hat{\mathbf{z}}$ , and $\mathbf{g} = u\hat{\mathbf{z}}/c$ . The equal partition $u_B = u_E$ follows pointwise from $B = E/c$ and $c^2 = 1/\mu_0\varepsilon_0$ ; the flux $\mathbf{S} = cu\hat{\mathbf{k}}$ follows by direct cross product with the BAC–CAB identity; and the DC + 2ω split follows from $\cos^2\varphi = \tfrac12(1+\cos2\varphi)$ . Worked numbers: FM 0.10 V/m gives $u_E^{\max} = u_B^{\max} = 4.43\times10^{-14}$ J/m³, $u^{\max} = 8.85\times10^{-14}$ J/m³, $S^{\max} = 2.65\times10^{-5}$ W/m², and a one-wavelength slab (λ = 3.00 m) holds $1.33\times10^{-13}$ J/m²; the femtosecond example gives $P = 4.0\times10^{10}$ W, $S = 1.27\times10^{20}$ W/m², $u = 4.2\times10^{11}$ J/m³, $E_0 = 2.19\times10^{11}$ V/m (43% of the atomic field $5.14\times10^{11}$ V/m), and $B_0 = 730$ T.

Builds on: Monochromatic Plane Waves in Vacuum · The Electromagnetic Wave Equation · Complex Notation for Electromagnetic Waves.

Leads to: Intensity and Time-Averaged Quantities · Momentum and Radiation Pressure · The Fresnel Equations · Reflection and Transmission at Normal Incidence.

If the fields vanish twice per cycle at my location, does the wave's energy vanish there too? Where does it go?

It genuinely vanishes there, then — and it hasn't "gone" anywhere: the sheets of the energy comb adjacent to you (λ/2 ahead and behind) carry it, and the whole comb is moving at $c$ , so a crest arrives a quarter period later. Energy is conserved as a continuity statement (E7's theorem), not as a fixed value at each point. Integrate $u$ over any full wavelength and the total is constant in time; watch one point and you see the 2ω pulse. Nothing oscillates between electric and magnetic form at your point — both accounts read zero together and peak together.

How can u_E equal u_B when B₀ is ten orders of magnitude smaller than E₀?

The comparison "B is smaller than E" is a units artifact — volts per metre and teslas are different currencies, and the exchange rate in the energy formula is enormous: $u_B = B^2/2\mu_0$ divides by $\mu_0 \sim 10^{-6}$ , while $u_E = \tfrac12\varepsilon_0E^2$ multiplies by $\varepsilon_0 \sim 10^{-11}$ . Run the numbers for any plane wave and the two accounts land exactly equal; the lock $B = E/c$ conspires with $c^2 = 1/\mu_0\varepsilon_0$ to make it an identity. In Gaussian units, where E and B share dimensions, the wave has $|E| = |B|$ outright and the equality is visible without arithmetic. What is legitimately smaller is the magnetic force on slow charges — by $v/c$ — but force bookkeeping and energy bookkeeping are different ledgers.

Is the 2ω oscillation of the energy actually measurable, or just formalism?

Measurable, wherever detectors are fast enough or the effect is nonlinear. At optical frequencies no photodetector follows $2\omega \sim 10^{15}$ Hz directly — that is precisely why detectors report the mean (article 0587). But at radio frequencies the $2\omega$ ripple in absorbed power is trivially visible on an oscilloscope (rectify a 1 MHz carrier and the 2 MHz ripple is right there). Optically, the ripple's reality shows indirectly: the ponderomotive force on electrons in a focused pulse, second-harmonic generation in crystals (driven by the field's square), and the "quiver energy" of ionized electrons all trace to the instantaneous $E^2$ comb, not merely its average. The books pulse; slow readers see the mean.

Why is the total energy density ε₀E² — I memorized u = ½ε₀E² from electrostatics?

Both are right; they answer different questions. $\tfrac12\varepsilon_0E^2$ is the electric energy density, always. In a plane wave there is an exactly equal magnetic account riding along ( $u_B = u_E$ , by the lock), so the total is twice the electric share: $u = \varepsilon_0E^2$ . The convenience is real — measure only $E(t)$ and you know the whole inventory — but label it honestly: it is "electric share, doubled," valid only where the equal-partition lock holds (traveling waves). In a capacitor there is no magnetic partner and the total truly is $\tfrac12\varepsilon_0E^2$ ; in a standing wave the split is position- and time-dependent and neither shortcut applies.

Does the S = cu identity mean electromagnetic energy always travels at exactly c?

For a single plane wave in vacuum, yes — the comb translates rigidly at $c$ and the flux-to-density ratio is exactly $c$ everywhere. As a general principle, no: $|\mathbf{S}|/u \leq c$ always, with equality only for pure unidirectional radiation. A standing wave has $u > 0$ with zero average flux (ratio zero); near-field regions of antennas and reactive zones of circuits have energy largely parked, shuttling locally; in dispersive media the transport speed becomes the group velocity (article 0598). The E7 Poynting article's FAQ makes the same point from the general side; this article supplies the extreme case where the bound is saturated.

For a laser pulse rather than an infinite wave, do these formulas still work?

Yes, with the two standard licenses. Spatially: within the beam's Rayleigh range the field is locally a plane wave, so $u = \varepsilon_0E^2$ and $S = cu$ hold point by point across the spot (Worked Example 2 leaned on this). Temporally: a pulse is a superposition of nearby frequencies, but because they all travel at the same $c$ in vacuum (non-dispersion), the envelope translates rigidly and the comb picture simply acquires an envelope — $u$ is the $\cos^2$ comb modulated by the pulse shape, still moving at $c$ , still satisfying the pointwise identities. The license expires in matter, where dispersion makes envelope and carrier travel at different speeds — the phase/group velocity article (0598) owns that split.

If energy density can be relocated by the E7 non-uniqueness freedom, is the comb picture even real?

The freedom is real but far smaller than it sounds, and the comb survives it. Poynting's theorem certifies $u$ and $\mathbf{S}$ only up to certain redefinitions (adding a curl to $\mathbf{S}$ , shuffling $u$ by a divergence — the E7 Poynting's-theorem article owns the fine print). But every proposed alternative must reproduce the same audits: the same energy through every closed surface, the same totals in every region. For a plane wave, those audits pin the physics that matters — the energy per wavelength slab, the power through any plane, the momentum delivered to any absorber — and the standard $u$ , $\mathbf{S}$ are the unique local pair built algebraically from the fields with no preferred direction beyond the wave's own. The comb is the canonical telling of a story whose measurable content no retelling can change.