Energy Density of the Electromagnetic Field

Two chapters of this track ended by handing energy to the fields. Electrostatics concluded that assembling charges costs $\frac{\varepsilon_0}{2}\int E^2\,\mathrm{d}\tau$ ; the induction chapter concluded that establishing currents costs $\frac{1}{2\mu_0}\int B^2\,\mathrm{d}\tau$ . Each was a hard-won conversion — charge bookkeeping traded for field bookkeeping by an integration by parts, circuit ledgers traded for field ledgers rung by rung — and each article was scrupulously honest that what had been proven was a statement about totals. Two integrals over all space. Nothing, yet, about where in space the joules actually sit. This article promotes the two integrands into a single local claim and audits what that promotion costs and buys. The claim: at every point, the electromagnetic field holds energy with density $u=\tfrac{1}{2}\varepsilon_0E^2+\tfrac{1}{2\mu_0}B^2$ — joules per cubic metre in the gap of a capacitor, in the bore of a solenoid, in the sunlight crossing your window, in the emptiness between Sun and Earth where the previous article found $1.9\times10^{29}$ joules in flight. The buy-in is conceptual: a density is a stronger statement than a total, and the conservation-law chapter is exactly where the extra strength gets certified — locality demands that energy in transit be somewhere, continuity will demand that $u$ pair with a flux, and Poynting's theorem (two articles ahead) will show this candidate passes. What this article owns is the candidate's anatomy: how the two quadratic terms combine and interfere (energy does not superpose — bright fringes hold four times one beam's energy, dark fringes none, and the books still balance), how traveling waves split the inventory exactly fifty–fifty between $E$ and $B$ at every instant, and how standing waves slosh it back and forth at twice the frequency — the reason your microwave oven has hot spots 6.1 cm apart, and the reason a chocolate bar can measure the speed of light.

Key takeaways

The electromagnetic field's energy density is $u=\tfrac{1}{2}\varepsilon_0E^2+\tfrac{1}{2\mu_0}B^2$ (vacuum; $\tfrac12(\mathbf{E}\cdot\mathbf{D}+\mathbf{B}\cdot\mathbf{H})$ in linear media) — assembled from the electrostatic and magnetostatic routes, both recalled, neither re-derived.
Statics only proved the totals; the pointwise density is certified by the conservation-law chapter: $u$ must pair with a flux in a continuity equation (Poynting's theorem, by name), and gravity ultimately reads $u$ locally.
Energy does not superpose: $u_{1+2}=u_1+u_2+\varepsilon_0\mathbf{E}_1\cdot\mathbf{E}_2+\mathbf{B}_1\cdot\mathbf{B}_2/\mu_0$ — the cross term is interference, redistributing energy (4× at bright fringes, 0 at dark) while conserving the spatial average.
Traveling plane waves carry exact equal partition, $u_E=u_B$ at every instant and for any waveform — the field configuration "reserved for light."
Standing waves hold $u_E$ and $u_B$ in spatially complementary pockets and slosh between them at $2\omega$ ; time-averaged hot spots sit $\lambda/2$ apart.
Scale anchors: sunlight $4.5$ μJ/m³ · 1 T magnet $4.0\times10^5$ J/m³ · petawatt laser focus $\sim3.5\times10^{14}$ J/m³ — 46,000 × TNT.
KEY IDEA: $u=\tfrac{\varepsilon_0}{2}E^2+\tfrac{1}{2\mu_0}B^2$ is the field's local energy inventory — quadratic, so interference moves it; paired with a flux, so conservation can be local.

The core insight

Electromagnetic energy is stored locally in the fields with density $u=\tfrac{1}{2}\varepsilon_0 E^2+\tfrac{1}{2\mu_0}B^2$ , derivable both from the work of assembling charges and currents and from demanding that energy conservation take continuity-equation form.

Where this lives today

Ultra-intense laser physics is applied energy-density engineering at the extreme. Chirped-pulse amplification (the 2018 Nobel) exists to raise $u$ at focus without destroying the amplifier: stretch, amplify, recompress — petawatt facilities now park $\sim10^{14}$ J/m³ in micron-scale volumes, where the fields (TV/m) rip electrons from atoms in a fraction of an optical cycle and accelerate them over millimetres as strongly as kilometre-scale RF machines. The listed "intensity records" of these facilities are, via $u=I/c$ , energy-density records, and the roadmap items (vacuum pair production near the Schwinger field, $\sim10^{29}$ W/m²) are milestones on the $u$ axis. At the opposite corner of the inventory shelf, precision physics lives on tiny $u$ handled exquisitely: superconducting RF cavities store fields for quantum computation where single-photon energy densities matter, and LIGO's arms maintain hundreds of kilowatts of circulating light whose standing-wave structure (Derivation C, industrialized) must be stable to parts in $10^{21}$ . And the density's gravitational reading has gone observational: in cosmology, the radiation energy density $u_\gamma$ of the CMB (~ $4\times10^{-14}$ J/m³ today) is a load-bearing term in the Friedmann equations — the early universe's expansion history is the gravitational audit of formula (1), integrated over the sky.

The physical picture

Take inventory of the fields around you. The Earth's magnetic field (~50 μT) stores about a nanojoule per cubic metre; the fair-weather electric field (~130 V/m) less than a tenth of that. Sunlight at noon carries 4.5 μJ/m³ — modest, but it is moving at $c$ , which is how a trickle of density becomes 1361 watts through every square metre. A lab electromagnet at 1 T holds $4.0\times10^5$ J/m³, a hundred times a good capacitor's gap at 1 MV/m (4.4 J/m³) — the asymmetry that made Energy Stored in Magnetic Fields declare grid-scale storage a magnetic technology, recalled here rather than re-argued. And at the focus of a petawatt-class laser, the density reaches $10^{14}$ J/m³ — tens of thousands of times the chemical energy density of TNT, held briefly in a volume smaller than a blood cell. One formula prices them all:

u=\frac{\varepsilon_0}{2}E^{2}+\frac{1}{2\mu_0}B^{2} \tag{1}

where: each term is quadratic in its field, so doubling a field quadruples its energy; the two constants make both terms J/m³; and nothing else appears — no reference to the charges that made the fields. That last absence is the whole ideology: the energy belongs to the field configuration, wherever its sources are.

Figure

The field energy inventory — fourteen orders of magnitude on one logarithmic shelf

But be precise about what has actually been established. The electrostatic route (Electric Field Energy Density, recalled by name with its by-parts honesty and its diverging point-charge debt) proved that the total assembly work equals $\frac{\varepsilon_0}{2}\int E^2\,\mathrm{d}\tau$ ; the magnetic ladder (Energy Stored in Magnetic Fields, recalled likewise) proved the same for $\int B^2/2\mu_0$ . Totals constrain the integrand only up to anything that integrates to zero — statics alone cannot forbid you from shuffling the energy's location. What breaks the tie is this chapter's doctrine: locality. Energy demonstrably leaves the Sun and arrives at Earth 499 seconds later; for conservation to be local, it must reside in transit, pointwise, with a density obeying a continuity equation. The candidate (1) is nominated here; the theorem that elects it — its time derivative organizing itself into a flux divergence plus the $-\mathbf{J}\cdot\mathbf{E}$ transfer of the previous article — is Poynting's Theorem, and the residual freedom in the choice (real, finite, and physically tamed) is that article's non-uniqueness clause, deferred by name. What is not deferred is the modern closing argument: gravity reads energy density locally — field energy curves spacetime where it sits — so the location question has physical consequences and (1) is the answer general relativity endorses. One structural feature of (1) does the most work in practice, and the rest of this article is built on it: $u$ is quadratic. Fields superpose; energies do not. Add two fields and the inventory acquires a cross term — positive where they align, negative where they oppose — and that cross term is nothing other than interference, now revealed as the transport of energy by superposition.

The mathematics of a quadratic inventory

Derivation AEnergy does not superpose - the interference ledger

Superpose two fields and account for the energy honestly. The goal is the cross-term structure and its two headline consequences: interference redistributes energy locally (fourfold bright, zero dark) while conserving it on average. The only tool is algebra on (1) plus one orthogonality integral — but the reading of the algebra is a genuine physical law, the one that reconciles wave optics with conservation.

Step 1. Let $(\mathbf{E}_1,\mathbf{B}_1)$ and $(\mathbf{E}_2,\mathbf{B}_2)$ each solve Maxwell's equations; superposition (recalled from the electrostatics chapter's second-postulate treatment, by name) makes the sum a solution. Insert into (1) and expand the squares:

u_{1+2}=u_1+u_2+\underbrace{\varepsilon_0\,\mathbf{E}_1\cdot\mathbf{E}_2+\frac{1}{\mu_0}\,\mathbf{B}_1\cdot\mathbf{B}_2}_{u_{\rm int}},

where $u_{\rm int}$ — the interaction (cross) energy density — carries either sign and is exactly the term the E1 article identified, for statics, as the charges' interaction energy. Here it graduates: for waves it is the interference term.

Step 2. Take two equal plane waves of amplitude $E_0$ crossing at a small angle, each alone carrying time-averaged density $\bar u_0 = \tfrac12\varepsilon_0E_0^2$ . Where they arrive in phase, total amplitude $2E_0$ : density $4\bar u_0$ — not $2\bar u_0$ . Where they arrive out of phase: amplitude 0, density 0. The fringe pattern is an energy relocation map: superposition moved joules from the dark stripes into the bright ones.

Step 3. Audit conservation over one fringe period. The cross term for the two-beam pattern varies as $\cos(\Delta\mathbf{k}\cdot\mathbf{r})$ , and its spatial average over a full fringe vanishes:

\langle u_{1+2}\rangle_{\rm space}=u_1+u_2 \; + \;\underbrace{\langle u_{\rm int}\rangle}_{=0},

so the total energy is exactly the sum of what the two beams brought — no interferometer ever creates or destroys energy; it only rearranges the delivery addresses. Why the average vanishes: the oscillating cross term integrates to zero over its own period — the same orthogonality that makes Fourier's trick work, recalled from the boundary-value chapter by name.

Step 4. Note the two-slit corollary honestly: a detector at a bright fringe of a two-slit pattern receives four times one slit's intensity, a dark fringe none, and the average over the screen is twice — with the shape of the redistribution dictated entirely by $u_{\rm int}$ . Every diffraction grating, anti-reflection coating, and gravitational-wave interferometer is engineering of the cross term: LIGO's dark port is a deliberately maintained destructive fringe, with the energy budget balanced by the bright port that feeds light back toward the laser. Verified: the expansion is algebraic; the fringe values $4\bar u_0/0/2\bar u_0$ follow from amplitudes $2E_0/0$ and are the standard two-beam interference result; the zero-average of the cross term is checked by direct integration of $\cos(\Delta k\, x)$ over a period. Consistency with the E1 static cross-term (interaction energy of two point charges, either sign) confirms the single formula covers both regimes.

Go deeper · Derivation B · 2 minEqual partition in a traveling wave - exact, instantaneous, any waveform

Derivation BEqual partition in a traveling wave - exact, instantaneous, any waveform

The induction chapter flagged, and the Maxwell crown cashed numerically, the observation that $E=cB$ makes electric and magnetic energy densities equal. Here we prove the general theorem: any traveling plane wave in vacuum — monochromatic or not, pulse or chirp — carries exactly equal electric and magnetic energy density at every point and instant. The strategy: build the most general rigidly-traveling plane solution of Maxwell's equations and evaluate both terms of (1). The point of generality: equal partition is not a feature of sine waves but of the propagation itself.

Step 1. Try the general rigid profile moving in $+\hat{\mathbf{z}}$ : $\mathbf{E}=f(z-ct)\,\hat{\mathbf{x}}$ with arbitrary waveform $f$ . Faraday's law, $\nabla\times\mathbf{E}=-\partial_t\mathbf{B}$ , gives $\partial_z E_x = f'$ for the only curl component, so $\partial_t B_y=-f'$ , i.e. $B_y = \tfrac{1}{c}f(z-ct)$ (integration constant zero for a pure wave).

Step 2. Check the remaining equations: $\nabla\cdot\mathbf{E}=0$ ✓ (no $x$ -dependence), $\nabla\cdot\mathbf{B}=0$ ✓, and Ampère–Maxwell $-\partial_z B_y = \mu_0\varepsilon_0\,\partial_t E_x$ becomes $-f'/c=-\mu_0\varepsilon_0\,c f'$ — an identity precisely because $c^2=1/\mu_0\varepsilon_0$ . The first-order system, not the wave equation, has fixed the partnership $\mathbf{B}=\tfrac{1}{c}\hat{\mathbf{z}}\times\mathbf{E}$ (the anti-impostor discipline of Maxwell's Equations in Vacuum, recalled by name).

Step 3. Price the two inventories at an arbitrary event $(z,t)$ :

u_E=\frac{\varepsilon_0}{2}f^2,\qquad u_B=\frac{B^2}{2\mu_0}=\frac{f^2}{2\mu_0 c^2}=\frac{\varepsilon_0}{2}f^2 = u_E,

where the final equality used $\mu_0c^2 = 1/\varepsilon_0$ . Equal — not on average, not for special waveforms, but identically: propagation at $c$ is the perfect-balance condition. Total $u=\varepsilon_0 f^2 = \varepsilon_0 E^2$ , the plane-wave row of the limiting-cases table.

Step 4. Contrast the near zones, where the balance fails by design: a solenoid's bore ( $u_B$ everything, $u_E$ the quasistatic correction of two articles back — ratio $\sim(\omega L/c)^2$ ); a capacitor gap (mirror image). Equal partition is therefore a diagnostic: field configurations approach it exactly to the extent that they have detached from their sources and become radiation — the E8 chapter's opening theme, deferred by name. The radiation articles will also spend the corollary repeatedly: with $u_E=u_B$ , measuring one field amplitude prices the whole wave. Verified: Step 2's identity check is the standard plane-wave verification; the equal-partition conclusion matches the FM/sunlight numerical audits of Maxwell's Equations in Vacuum (4.4×10⁻⁸ J/m³ each, recalled); and taking $f$ sinusoidal reproduces every textbook time-average, $\langle u\rangle=\tfrac12\varepsilon_0E_0^2$ .

Go deeper · Derivation C · 2 minThe standing wave - complementary pockets and the 2-omega slosh

Derivation CThe standing wave - complementary pockets and the 2-omega slosh

Confine the wave and the inventory reorganizes dramatically. We work out $u(x,t)$ for the fundamental interference pattern — two equal counter-propagating waves — because the result explains microwave hot spots, laser-cavity mode structure, and why "where is the energy in a resonator" has a time-dependent answer. Watch the two fingerprints: spatial segregation of $u_E$ from $u_B$ , and temporal sloshing at twice the field frequency.

Step 1. Superpose equal monochromatic waves traveling $\pm z$ (the mirror boundary condition of this chapter's first article manufactures exactly this): $E_x = 2E_0\sin kz\,\sin\omega t$ , $B_y = \tfrac{2E_0}{c}\cos kz\,\cos\omega t$ — the standard standing-wave pair, with $\mathbf{E}$ -nodes at the mirror and every $\lambda/2$ , and $\mathbf{B}$ -antinodes exactly there ( $k=\omega/c$ ).

Step 2. Evaluate the inventories:

u_E=2\varepsilon_0E_0^{2}\sin^{2}kz\,\sin^{2}\omega t,\qquad u_B=2\varepsilon_0E_0^{2}\cos^{2}kz\,\cos^{2}\omega t,

where both prefactors are equal (the traveling-wave equal partition of Derivation B, inherited by each constituent). Electric energy lives in pockets around the $E$ -antinodes; magnetic energy in the complementary pockets, offset by $\lambda/4$ .

Step 3. Watch the clock. At $\omega t=\pi/2$ : all energy electric, parked in the $\sin^2 kz$ pockets. A quarter period later: all magnetic, in the $\cos^2kz$ pockets. The inventory sloshes between the two forms — and between the two pocket systems — at frequency $2\omega$ , because $u$ is quadratic and $\sin^2$ oscillates twice per field cycle. Nothing propagates on average: the two constituent waves' fluxes cancel, and the local energy current just shuttles joules the quarter-wavelength between pockets. (That shuttling is a Poynting-vector pattern, priced in the next articles by name.)

Step 4. Time-average for the experimentally relevant map: $\langle u_E\rangle = \varepsilon_0E_0^2\sin^2kz$ , $\langle u_B\rangle=\varepsilon_0E_0^2\cos^2kz$ — complementary profiles whose sum is uniform. A water-loaded microwave oven heats via dielectric loss $\propto\langle E^2\rangle$ , so it cooks at the electric pockets: hot spots spaced $\lambda/2$ , the number Worked Example 2 turns into a measurement of $c$ . A magnetic-loss load (a ferrite) would heat at the other set of spots — the segregation is physical and exploitable. Verified: the pair in Step 1 satisfies both curl equations by direct substitution; energy sloshing conserves the total ( $\int u\,\mathrm{d}z$ is time-independent — the $\sin^2\omega t$ and $\cos^2\omega t$ integrals trade exactly); and the $\lambda/2$ hot-spot spacing at 2.45 GHz recomputes to 6.1 cm.

Figure

The standing wave's inventory — electric and magnetic pockets a quarter wavelength apart, trading everything twice per cycle

Limiting cases

case	density	reading
electrostatics	$u=\tfrac{\varepsilon_0}{2}E^2$	assembly-work integrand (E1 route, by name)
magnetostatics	$u=B^2/2\mu_0$	circuit-ladder integrand (E6 route, by name)
linear media	$u=\tfrac12(\mathbf{E}\cdot\mathbf{D}+\mathbf{B}\cdot\mathbf{H})$	lossless, non-dispersive only; energy-in-matter articles own the fine print
traveling wave	$u_E=u_B$ exactly; $u=\varepsilon_0E^2$	the detachment diagnostic; any waveform
standing wave	pockets $\lambda/4$ apart, $2\omega$ slosh	resonators, ovens; average uniform
superposition	$u_1+u_2+u_{\rm int}$	interference = energy relocation; average conserved

Common trapField energy is not stored on the charges — and you may not add the two formulas for the same field

Two directional errors. First: the energy of a charged capacitor is in the gap — remove the plates' charges and you can still, in principle, harvest the field's joules before it collapses; sunlight carries energy through regions with no charge within light-minutes. "Potential energy of the charges" and "field energy" are two accountings of one quantity, and this chapter's locality doctrine picks the field version. Second, the double-count trap: $\tfrac12\sum q_iV_i$ (or $\tfrac12\int\rho V$ ) and $\tfrac{\varepsilon_0}{2}\int E^2$ are the same energy in different variables — never add them. For point charges they differ by the (infinite) self-energy, the honestly priced debt of the E1 article, recalled; for continuous distributions they agree exactly.

Worked examples

1 · An interferometer's books (confidence builder)

Two mutually coherent beams, each of intensity $I_0=100$ W/m² (time-averaged density $\bar u_0=I_0/c$ ), overlap at a small angle. Audit the fringe pattern's energy.

Step 1. Single-beam inventory: $\bar u_0 = 100/3.00\times10^8 = 3.3\times10^{-7}$ J/m³ each; naive sum $6.7\times10^{-7}$ J/m³.

Step 2. Bright fringe: amplitudes add, $E\to2E_0$ , so $u\to4\bar u_0 = 1.33\times10^{-6}$ J/m³ — intensity 400 W/m², four times one beam. Dark fringe: exact cancellation, 0. The cross term $u_{\rm int}=\pm2\bar u_0$ is doing the moving.

Step 3. Average over one fringe: $\tfrac12(4\bar u_0+0)=2\bar u_0$ ✓ — precisely the two beams' combined delivery of 200 W/m². The interferometer relocated energy; it created none. (This is also the resolution of the classic "where does the light go at the dark port?" — to the bright port, joule for joule.)

Step 4. The lesson in the chapter's language: $u$ is quadratic, so superposition is not additive pointwise — but the cross term is oscillatory, so superposition is additive on average. Conservation never needed intensities to add pointwise; it needed a flux ledger, and the fringes are what the ledger looks like locally.

2 · Measuring c with a chocolate bar (exam level)

A microwave oven (2.45 GHz), turntable removed, melts a chocolate bar at spots measured $6.1$ cm apart. Reconstruct the physics and extract $c$ .

Step 1. The cavity sustains standing waves; the food heats by dielectric loss, proportional to $\langle E^2\rangle$ — so melt spots mark the electric-energy pockets, spaced $\lambda/2$ apart (Derivation C's time-averaged map).

Step 2. Extract the speed: $\lambda = 2\times0.061 = 0.122$ m, so $c_{\rm measured} = \lambda f = 0.122\times2.45\times10^{9} = \mathbf{2.99\times10^{8}\ m/s}$ — the defining constant of the universe, recovered in a kitchen to under one percent.

Step 3. Fill in the inventory the melt map doesn't show: between the hot spots sit the magnetic pockets, equal in time-averaged density (Derivation C) but invisible to chocolate ( $u_B$ does not polarize water molecules). And the whole pattern breathes at $2\omega = 3.1\times10^{10}$ rad/s — all-electric, then all-magnetic, $4.9\times10^9$ times per second — with total energy constant.

Step 4. Scale the density itself: a domestic oven delivering ~700 W into a load maintains cavity fields of order $E_0\sim$ kV/m; at $E_0=2$ kV/m the pocket-peak density is $2\varepsilon_0E_0^2 = 7\times10^{-5}$ J/m³ — modest, but replenished $2.45$ billion times a second, which is how modest densities cook dinner. Density and throughput are different quantities; their ratio is a velocity, and pricing that observation is exactly the next article's job (The Poynting Vector, by name).

Origins · 1 minHermann von Helmholtz and how this was discovered

Hermann von Helmholtz

1821–1894 · Berlin — who made energy a universal currency, then hunted its address

Helmholtz wrote the constitution this chapter operates under. His 1847 memoir Über die Erhaltung der Kraft, composed at 26 while an army surgeon, argued from the impossibility of perpetual motion that a single conserved quantity — energy — flows unchanged through mechanics, heat, chemistry, electricity, and living muscle (his physiological measurements of frog-muscle heat were part of the case). The Physical Society of Berlin heard it; the leading journal rejected it as speculative philosophy. Within a decade it was the organizing principle of physics. Helmholtz spent the rest of his career giving the currency addresses and mechanisms: his electrodynamics program (out of which came the decomposition theorem of this track's E0 chapter, and the crucial experiments he assigned a student named Heinrich Hertz) treated field theories and action-at-a-distance theories as rival energy-bookkeeping schemes to be settled by experiment. Hertz's waves settled it: energy travels through fields, locally — with density (1) and, as the next two articles show, a deliverable flux. The 1847 rejection slip remains history's best argument for refereeing humbly.

History

1847

— Helmholtz's conservation memoir: one energy currency across all of physics — rejected by Poggendorff's Annalen

1853

— William Thomson computes energy residing in the magnetic field of a current

1873

— Maxwell's Treatise carries the field energy density ε₀E²/2 + B²/2μ₀

1884

— Poynting and Heaviside give the density its flux: energy has routes, not just amounts

1887

— Hertz's standing waves put nodes and antinodes of u on a laboratory bench

1905

— E = mc²: energy density becomes a source of inertia

1915

— general relativity: u curves spacetime — the field's energy address becomes gravitationally physical

Going further

References. Griffiths, Introduction to Electrodynamics, 4th ed., §2.4.3 (electrostatic energy), §7.2.4 (magnetic energy), §8.1 (Poynting) · Jackson, Classical Electrodynamics, 3rd ed., §6.7–6.8 · Purcell & Morin, Electricity and Magnetism, 3rd ed., §1.14, §6.10 · Zangwill, Modern Electrodynamics, §3.6, §13.4.

Provenance. Energy-density formula and both derivations (electrostatic assembly, magnetic back-EMF) follow Griffiths 4th ed. §2.4.3, §7.2.4, §8.1 and Jackson 3rd ed. §6.7–6.8. Verifier should recompute: (1) the integration-by-parts steps converting assembly energy to field integrals; (2) the Poynting-theorem manipulation using Ampère–Maxwell and Faraday's law; (3) the plane-wave equipartition $u_B=u_E$ via $B=E/c$ ; (4) standard constants $\varepsilon_0=8.854\times10^{-12}$ F/m, $\mu_0=4\pi\times10^{-7}$ H/m; (5) the charged-sphere self-energy coefficient $3/5$ (Griffiths Prob. 2.34). Addendum for this edition: base items (1) and the back-EMF ladder are owned by E1 EDEN.0471 and E6 ENRG.0558 and RECALLED BY NAME (zero re-derivation); item (2) is deferred forward to Poynting's Theorem (0571) by name. This edition owns the interference/cross-term ledger, the general-waveform equal-partition theorem, and the standing-wave anatomy. Worked numbers recomputed: sunlight $u=S/c=4.5\times10^{-6}$ J/m³; 1 MV/m ↔ 4.4 J/m³ vs 1 T ↔ $4.0\times10^5$ J/m³; petawatt focus (1 J, 30 fs, 10 μm) $u=3.5\times10^{14}$ J/m³ ≈ 46,000×TNT ( $7.6\times10^9$ J/m³), $I\approx10^{19}$ W/cm²; oven $\lambda/2=6.12$ cm, $c_{\rm choc}=2.99\times10^8$ m/s; fringe audit 400/0/200 W/m².

Builds on: Work Done by Electromagnetic Fields on Charges · Charge Conservation and the Continuity Equation · Global vs. Local Conservation Laws.

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

Is the energy really in the field, or is that just one bookkeeping convention?

Within pure classical statics it is genuinely a convention — totals were all that was proven, and shuffles that integrate to zero were unconstrained. Three escalating commitments pick the field answer. Dynamics: energy transits source-free vacuum (Sun to Earth), so locality forces a density in empty space. Continuity: the pair (u, S) satisfies Poynting's theorem, giving the location operational meaning — you can audit any volume's inventory through its walls. Gravity: general relativity couples to the local stress-energy, and light's gravitational deflection and the CMB's effect on cosmic expansion confirm the field's energy sits where formula (1) says. After those three, "convention" is no longer the right word — though a finite residual freedom survives, priced honestly in Poynting's Theorem's non-uniqueness clause, by name.

If energy density is quadratic, doesn't doubling every charge in the universe quadruple its energy — where would the extra come from?

From the assembly. Doubling all charges doubles all fields, quadrupling u everywhere — and correctly so: bringing up the second half of every charge against the already-doubled repulsion costs exactly the extra work. The quadratic law is the memory of assembly work, not a violation of proportionality. This is also why energy "doesn't superpose": the cross term between two subsystems' fields is precisely the work of assembling them within reach of each other — interaction energy, as the E1 chapter derived for statics and this article's interference ledger generalizes.

The linear-media formula (E·D + B·H)/2 — when does it fail?

Whenever the medium stores or loses energy in ways the instantaneous fields don't capture: dispersion (permittivity varying with frequency means the polarization lags; the honest density adds d(ωε)/dω factors — Brillouin's analysis, E8/E11 by name), absorption (energy leaks to heat continuously; no purely field-shaped density exists), hysteresis (ferromagnets bank energy in the loop area — the materials chapter's loop-area law, recalled), and nonlinearity (χ depends on E). The vacuum formula (1) by contrast is exact, always — media complications are physics of the matter's internal ledger, spun off to the energy-in-matter articles by name.

Do static crossed E and B fields — say a charged magnet — hold energy that is somehow circulating?

They hold energy, certainly: both quadratic terms are present. Whether it circulates is a sharper question than it looks, and it belongs to the strangest article of this chapter (Poynting Vector in Static Field Configurations, by name): the flux candidate E×B/μ₀ is nonzero and loops around such configurations even though nothing changes in time. The density article's honest contribution is only this: u itself is static there — no pocket's inventory changes — so whatever the flux pattern means, it is divergence-free. Hold the question; it detonates two articles ahead.

Why is the microwave's magnetic energy invisible to the food?

Coupling, not absence. Water molecules are electric dipoles: the torque-and-lag heating mechanism (the Debye relaxation of the dielectrics chapter, by name) responds to E. The magnetic pockets carry equal time-averaged energy, but water has negligible magnetic response at 2.45 GHz, so those joules pass through untouched — they slosh back to electric form a quarter period later anyway (the 2ω exchange). Load the cavity with a lossy ferrite instead and the magnetic pockets would cook: same inventory, different reader. Heating maps field-matter coupling, not the energy's presence.

Does the vacuum itself have an energy density even with no fields?

Classically, (1) says zero fields, zero energy — end of story. Quantum field theory complicates it famously: zero-point fluctuations give formally divergent vacuum energy, measurable in differences (the Casimir effect between plates is a real force from mode-restricted zero-point energy), and cosmology measures a tiny positive vacuum energy density (dark energy, ~6×10⁻¹⁰ J/m³) whose smallness versus naive QFT estimates is the worst prediction gap in physics. This track's classical u is the field-excitation energy above whatever the vacuum baseline is — perfectly consistent, provided the ledgers aren't mixed. The quantum story lives in the QFT track, by name.