Maxwell's Equations in Matter

The four equations of the last article are exact, everywhere, always — inside a diamond as surely as in interstellar vacuum. So why does this article exist? Because "exact" and "usable" part company inside matter. A cubic millimetre of glass contains $10^{19}$ molecules, each a nest of charges that shift when fields arrive; the microscopic Maxwell equations see every one of them, and their $\rho$ and $\mathbf{J}$ include each electron cloud's distortion. Nobody can prescribe those sources, and nobody wants to. The whole art of electromagnetism in matter — begun in the dielectrics chapter, matured in the magnetism chapter — is bookkeeping: split the sources into the part you control (free charge, free current) and the part the material contributes in response (bound charge, bound current), then hide the response inside redefined fields. This article completes that program under time dependence, and the completion springs one genuine surprise. The static chapters catalogued two bound sources: bound charge $\rho_b = -\nabla\cdot\mathbf{P}$ (E3's Bound Charges) and magnetization current $\mathbf{J}_b = \nabla\times\mathbf{M}$ (E5's Bound Currents). Let the polarization change in time and a third appears, forced by nothing less than charge conservation itself: the polarization current $\mathbf{J}_p = \partial\mathbf{P}/\partial t$ — genuinely moving charge, bound electrons shuttling as their molecules re-stretch. Miss it and the bound sources violate the continuity equation; include it and the macroscopic equations assemble themselves: $\nabla\cdot\mathbf{D} = \rho_f$ and $\nabla\times\mathbf{H} = \mathbf{J}_f + \partial\mathbf{D}/\partial t$ , with Faraday's law and the no-monopole law sailing through untouched. The reward for the bookkeeping is the physics of every transparent thing you have ever seen through: in linear media the wave machinery of the previous article re-runs with $\varepsilon_0\mu_0 \to \varepsilon\mu$ , light slows to $v = 1/\sqrt{\varepsilon\mu}$ , and the refractive index — optics' oldest number — becomes a corollary of electromagnetism. With one honest asterisk about water, which this article refuses to hide.

Key takeaways

Microscopic Maxwell is exact but intractable in matter: $10^{19}$ molecular sources per mm³. The cure is source-splitting: free (yours) versus bound (the material's response), with bound sources hidden inside $\mathbf{D} = \varepsilon_0\mathbf{E} + \mathbf{P}$ and $\mathbf{H} = \mathbf{B}/\mu_0 - \mathbf{M}$ .
Time dependence adds a third bound source: the polarization current $\mathbf{J}_p = \partial\mathbf{P}/\partial t$ — real charge motion, required by continuity of the bound charge ( $\rho_b = -\nabla\cdot\mathbf{P}$ changing means bound charge moved).
The macroscopic equations: $\nabla\cdot\mathbf{D} = \rho_f$ · $\nabla\cdot\mathbf{B} = 0$ · $\nabla\times\mathbf{E} = -\partial\mathbf{B}/\partial t$ · $\nabla\times\mathbf{H} = \mathbf{J}_f + \partial\mathbf{D}/\partial t$ . Faraday and no-monopoles are unchanged — matter offers no magnetic charge to split.
The system is open until closed by constitutive relations — $\mathbf{D} = \varepsilon\mathbf{E}$ , $\mathbf{H} = \mathbf{B}/\mu$ , $\mathbf{J}_f = \sigma\mathbf{E}$ for linear media — where all the material physics (E3, E5, and Ohm) re-enters.
In linear media the vacuum wave derivation re-runs verbatim: $v = 1/\sqrt{\varepsilon\mu} = c/n$ with $n \approx \sqrt{\varepsilon_r}$ — the refractive index derived. Caveat with teeth: $\varepsilon_r$ is frequency-dependent (water: static 80, optical 1.77), and that dispersion is a chapter of its own.
KEY IDEA: split the sources into free and bound, absorb the bound ones ( $-\nabla\cdot\mathbf{P}$ , $\nabla\times\mathbf{M}$ , and the continuity-forced $\partial\mathbf{P}/\partial t$ ) into $\mathbf{D}$ and $\mathbf{H}$ , and Maxwell's equations in matter take the vacuum form with free sources only — closed by the material's constitutive relations.

The core insight

Maxwell's equations in matter reorganize the exact vacuum equations around the free–bound source split: with $\mathbf{D} = \varepsilon_0\mathbf{E} + \mathbf{P}$ and $\mathbf{H} = \mathbf{B}/\mu_0 - \mathbf{M}$ absorbing bound charge, magnetization current, and the continuity-mandated polarization current $\partial\mathbf{P}/\partial t$ , the system reads $\nabla\cdot\mathbf{D} = \rho_f$ , $\nabla\times\mathbf{H} = \mathbf{J}_f + \partial\mathbf{D}/\partial t$ with Faraday and no-monopoles unchanged — awaiting constitutive relations to close it.

Where this lives today

The macroscopic equations are the working form of electromagnetism wherever matter matters — which is everywhere engineering happens. Every optical-design code, every fiber-optics simulation, every antenna-near-dielectric model, and every electromagnetic-compatibility analysis solves these four equations with tabulated $\varepsilon(\omega)$ , $\mu(\omega)$ , $\sigma(\omega)$ ; the multi-billion-device photonics industry is applied constitutive relations. The frontier turned the chassis into a playground: metamaterials engineer sub-wavelength structures whose effective $\varepsilon$ and $\mu$ take designer values nature never offers — including simultaneously negative pairs, yielding negative refractive index (Veselago's 1968 curiosity, realized experimentally in 2000) and the transformation-optics recipes behind cloaking prototypes and perfect-lens proposals. Reading those papers requires exactly this article's discipline: effective-medium $\varepsilon$ and $\mu$ are averaged bookkeeping, meaningful only at wavelengths long compared to the structure — the same fine print as the molecular averaging underneath. The polarization current earned its own technologies. Dielectric heating (microwave ovens, industrial RF welding, medical diathermy) is $\partial\mathbf{P}/\partial t$ driven where response lags; dielectric spectroscopy reads $\varepsilon(\omega)$ across twelve decades to fingerprint polymers, tissues, and battery electrolytes; and the terahertz gap — the hard-to-reach band between electronics and optics — is being colonized precisely by instruments that launch and detect picosecond polarization currents in nonlinear crystals. Meanwhile the auxiliaries' honesty doctrine has a live research edge: the century-old Abraham–Minkowski dispute over light's momentum in matter — is it $E/vc$ or $nE/c^2$ per photon? — turns exactly on which fields carry the mechanical bookkeeping, and its modern resolution (both, for different experimental questions) is E7-chapter physics conducted in this article's vocabulary.

Origins · 1 minHendrik Lorentz and how this was discovered

Hendrik Lorentz

1853–1928 · Leiden

Lorentz is the physicist who looked underneath this article. The macroscopic equations, as Maxwell left them, treated $\varepsilon$ and $\mu$ as God-given material constants; Lorentz's electron theory (1892–1904) derived them — modeling matter as swarms of discrete charges in vacuum, governed by the microscopic Maxwell equations plus the force law that bears his name, and obtaining the macroscopic fields by averaging. In his hands $\mathbf{P}$ became actual displaced electrons (the polarization current of Derivation A is his picture, literally), $\varepsilon(\omega)$ became the driven-oscillator response of bound charges (the Lorentz oscillator model that still anchors dispersion theory, E11), and the local-field correction distinguishing the field a molecule feels from the average (the Lorentz cavity of E3's chapter) closed the loop to the Clausius–Mossotti relation. The program's crowning surprise was not about matter at all: pushing his electron theory to moving bodies, Lorentz found the transformations under which Maxwell's equations keep their form — the Lorentz transformations — handing Einstein the formal spine of relativity. Nobel laureate (1902, with Zeeman, for magneto-optics — splitting spectral lines with $\mathbf{B}$ , pure electron-theory vindication), chair of the first Solvay Councils, and the figure of whom Einstein said he meant more to him than all the others encountered on life's journey.

History

1837

— Faraday discovers specific inductive capacity: the dielectric constant enters physics

1850

— William Thomson splits B and H for magnetized media; the auxiliary-field idea is born

1865

— Maxwell's Dynamical Theory carries D alongside E — the two-ledger structure from the start

1892

— Lorentz's electron theory begins deriving the constitutive relations from microscopic charges in vacuum

1912

— Debye's polar-molecule theory explains water's giant static permittivity and its frequency staircase

2000

— Smith and colleagues realize negative-index metamaterials: constitutive relations by design

The physical picture

Recall the accountants of chapters E3 and E5 — this article is their merger, with a clock added. The electric accountant learned that polarized matter wears effective charges wherever $\mathbf{P}$ starts, stops, or spreads ( $\rho_b = -\nabla\cdot\mathbf{P}$ , with its surface form $\sigma_b = \mathbf{P}\cdot\hat{\mathbf{n}}$ — Bound Charges, recalled whole, re-derived nowhere). The magnetic accountant learned that magnetized matter runs effective currents where $\mathbf{M}$ curls ( $\mathbf{J}_b = \nabla\times\mathbf{M}$ , surface form $\mathbf{K}_b = \mathbf{M}\times\hat{\mathbf{n}}$ — Bound Currents, likewise). Each then buried their response inside an auxiliary field — $\mathbf{D}$ cleaning a divergence, $\mathbf{H}$ cleaning a curl — and each article ended with the same honesty doctrine: the auxiliaries are bookkeeping; $\mathbf{E}$ and $\mathbf{B}$ remain the physical fields. Now start the clock, and watch a slab of dielectric while the applied field ramps. At time $t_1$ , every molecule is stretched by some amount: positive cores shifted one way, electron clouds the other — that is what $\mathbf{P}(t_1)$ means. At $t_2$ the field is stronger and every molecule is stretched further. Between the snapshots, charge physically moved: every bound electron in the slab shifted, coherently, by a fraction of an ångström. Multiply femtometres of motion by $10^{28}$ charges per cubic metre and the result is a perfectly respectable current density — as real as any conduction current, made of charges that happen to be on leashes. That is the polarization current $\mathbf{J}_p = \partial\mathbf{P}/\partial t$ , the one bound source the static chapters could never see.

Figure

The third bound source: changing polarization is charge in motion — leashed, but moving

Why must it be exactly $\partial\mathbf{P}/\partial t$ and not something merely proportional? Because the bound charges are charges, and the previous article's theorem shows no source escapes the continuity equation. If $\rho_b = -\nabla\cdot\mathbf{P}$ changes in time, bound charge is redistributing, and the current that redistributes it is fixed — the first derivation below nails it in three lines. This is a pleasing full-circle moment for the chapter: two articles ago, continuity was saved by the displacement current; here, the same law dictates the last missing piece of matter's response. Charge conservation has become a design tool. With the source catalog complete — free $(\rho_f, \mathbf{J}_f)$ , bound $(-\nabla\cdot\mathbf{P},\ \nabla\times\mathbf{M} + \partial\mathbf{P}/\partial t)$ — the assembly is mechanical, and the result is worth previewing as architecture. Two of the four equations don't change at all: Faraday's law and $\nabla\cdot\mathbf{B} = 0$ contain no sources, so the free/bound split has nothing to grab (matter supplies no magnetic charge). The other two reorganize around the auxiliaries the static chapters already built: Gauss becomes $\nabla\cdot\mathbf{D} = \rho_f$ , and Ampère–Maxwell becomes $\nabla\times\mathbf{H} = \mathbf{J}_f + \partial\mathbf{D}/\partial t$ — with the beautiful economy that $\partial\mathbf{D}/\partial t = \varepsilon_0\,\partial\mathbf{E}/\partial t + \partial\mathbf{P}/\partial t$ swallows the vacuum displacement term and the polarization current in one symbol. The equations then run on the sources you control, awaiting only the material's constitutive confession.

The bound-source catalog, completed

The static entries, quoted from their home chapters (E3's Bound Charges; E5's Bound Currents — derivations live there):

\rho_b = -\nabla\cdot\mathbf{P}, \qquad \mathbf{J}_{\rm mag} = \nabla\times\mathbf{M} \qquad (1)

where: $\mathbf{P}$ , $\mathbf{M}$ — polarization and magnetization (dipole moments per volume); the first says bound charge appears where polarization converges or ends, the second that magnetization's curl is a genuine circulating current. The new, dynamic entry:

\mathbf{J}_p = \frac{\partial\mathbf{P}}{\partial t} \qquad (2)

where: $\mathbf{J}_p$ — the polarization current density: the literal motion of bound charge as molecules re-stretch. Note what it is not: it is not the vacuum displacement term wearing matter's clothes — $\varepsilon_0\,\partial\mathbf{E}/\partial t$ involves no charge motion anywhere, while $\mathbf{J}_p$ is charges moving, with real inertia and (in lossy media) real dissipation. The two ride together inside $\partial\mathbf{D}/\partial t$ , and confusing them is the classic error the warning box below prosecutes.

Derivation AThe polarization current — dictated by continuity, not proposed

The goal is equation (2), derived rather than asserted: show that the time-dependent bound-source catalog must include $\partial\mathbf{P}/\partial t$ , with exactly that coefficient, or bound charge is created and destroyed in violation of the continuity equation. The strategy is the audit of the previous article, pointed at the bound ledger: write down the candidate bound current, demand $\nabla\cdot\mathbf{J}_b + \partial\rho_b/\partial t = 0$ , and solve. Watch the magnetization term pass through untouched — a curl can never move net charge into a region — while the polarization term is forced, coefficient and all.

Step 1. State what must be conserved. Bound charge is charge: electrons and nuclei on molecular leashes, but countable coulombs. Whatever bound current $\mathbf{J}_b$ exists must satisfy

\nabla\cdot\mathbf{J}_b = -\frac{\partial\rho_b}{\partial t} = +\frac{\partial}{\partial t}\left(\nabla\cdot\mathbf{P}\right) = \nabla\cdot\left(\frac{\partial\mathbf{P}}{\partial t}\right),

using $\rho_b = -\nabla\cdot\mathbf{P}$ and exchanging derivatives.

Step 2. Try the static catalog alone: $\mathbf{J}_b = \nabla\times\mathbf{M}$ . Its divergence vanishes identically ( $\nabla\cdot(\nabla\times\ldots) = 0$ , the E0 identity) — so with magnetization current alone, Step 1 reads $0 = \nabla\cdot(\partial\mathbf{P}/\partial t)$ : false whenever polarization changes nonuniformly, i.e., in every capacitor being charged and every dielectric in an AC field. The static catalog, under a clock, breaks charge conservation.

Step 3. Repair minimally: add the current whose divergence Step 1 demands —

\mathbf{J}_b = \nabla\times\mathbf{M} + \frac{\partial\mathbf{P}}{\partial t},

and verify: $\nabla\cdot\mathbf{J}_b = 0 + \nabla\cdot(\partial\mathbf{P}/\partial t) = -\partial\rho_b/\partial t$ ✓. Conservation restored, identically, for any $\mathbf{P}(\mathbf{r}, t)$ and $\mathbf{M}(\mathbf{r}, t)$ whatsoever. Why the repair is also the physical truth and not just bookkeeping: track one molecule. Its dipole moment is $q\mathbf{d}(t)$ ; stretching at rate $\dot{\mathbf{d}}$ means its positive charge drifts at $\dot{\mathbf{d}}$ — a current $q\dot{\mathbf{d}}$ per molecule, and per volume ( $N$ molecules): $Nq\dot{\mathbf{d}} = \partial(Nq\mathbf{d})/\partial t = \partial\mathbf{P}/\partial t$ . The microscopic head-count and the conservation audit give the same answer independently — the strongest kind of confirmation this chapter knows. (As with the displacement current's uniqueness two articles ago, consistency fixes $\mathbf{J}_p$ up to a curl; the microscopic count closes that gap here, since the molecular motion is explicit.) *Verified: matches Griffiths §7.3.5 (Eq. 7.50) and Jackson §6.6; the bound-continuity identity checked in Step 3; the microscopic count matches the E3 definition P = Np for N dipoles per volume, differentiated.*

Go deeper · Derivation B · 2 minThe assembly — four microscopic equations become four macroscopic ones

Derivation BThe assembly — four microscopic equations become four macroscopic ones

The goal is the macroscopic system: rewrite the exact (microscopic, spatially averaged) Maxwell equations with the total sources split as $\rho = \rho_f + \rho_b$ , $\mathbf{J} = \mathbf{J}_f + \mathbf{J}_b$ , absorbing every bound term into the auxiliary fields the static chapters defined. The strategy is substitution plus the two one-line reorganizations that gave E3 its $\mathbf{D}$ and E5 its $\mathbf{H}$ — now executed simultaneously, with the clock running. Watch which equations feel the split (the two with sources) and which are immune, and watch $\partial\mathbf{D}/\partial t$ absorb two physically distinct currents in one stroke.

Step 1. Gauss's law with the split, exactly as in E3 but worth re-watching with time-dependent $\mathbf{P}$ :

\varepsilon_0\nabla\cdot\mathbf{E} = \rho_f + \rho_b = \rho_f - \nabla\cdot\mathbf{P} \qquad\Longrightarrow\qquad \nabla\cdot\underbrace{(\varepsilon_0\mathbf{E} + \mathbf{P})}_{\mathbf{D}} = \rho_f.

Nothing about the maneuver required statics; $\mathbf{D} = \varepsilon_0\mathbf{E} + \mathbf{P}$ carries over unchanged.

Step 2. Ampère–Maxwell with the complete bound catalog of Derivation A:

\frac{1}{\mu_0}\nabla\times\mathbf{B} = \mathbf{J}_f + \nabla\times\mathbf{M} + \frac{\partial\mathbf{P}}{\partial t} + \varepsilon_0\frac{\partial\mathbf{E}}{\partial t}.

Move the magnetization curl left (E5's $\mathbf{H} = \mathbf{B}/\mu_0 - \mathbf{M}$ maneuver) and gather the two time derivatives on the right:

\nabla\times\left(\frac{\mathbf{B}}{\mu_0} - \mathbf{M}\right) = \mathbf{J}_f + \frac{\partial}{\partial t}\left(\varepsilon_0\mathbf{E} + \mathbf{P}\right) \qquad\Longrightarrow\qquad \nabla\times\mathbf{H} = \mathbf{J}_f + \frac{\partial\mathbf{D}}{\partial t}.

Why the gathering is the elegant step: the vacuum displacement term (no charge moves) and the polarization current (charge genuinely moves) — different physics, one symbol. $\partial\mathbf{D}/\partial t$ is the honest generalization of the displacement current to matter, and its two-part anatomy is exactly what worked example 1 dissects with numbers.

Step 3. The other two equations: nothing to do. Faraday's law $\nabla\times\mathbf{E} = -\partial\mathbf{B}/\partial t$ and the no-monopole law $\nabla\cdot\mathbf{B} = 0$ contain no source terms — there is no magnetic charge to split into free and bound — so they pass into the macroscopic system verbatim. The asymmetry of the final system (auxiliaries on the source side only) is the direct image of nature's own asymmetry: charges exist, monopoles don't.

Step 4. Assemble, and state the system's honest incompleteness:

\nabla\cdot\mathbf{D} = \rho_f, \qquad \nabla\cdot\mathbf{B} = 0, \qquad \nabla\times\mathbf{E} = -\frac{\partial\mathbf{B}}{\partial t}, \qquad \nabla\times\mathbf{H} = \mathbf{J}_f + \frac{\partial\mathbf{D}}{\partial t}.

Count: twelve unknown field components ( $\mathbf{E}, \mathbf{B}, \mathbf{D}, \mathbf{H}$ ), eight equations. The system is open — necessarily, because it has so far assumed nothing about the material. Closure comes from the constitutive relations, where E3's and E5's physics re-enters: for linear media, $\mathbf{D} = \varepsilon\mathbf{E}$ , $\mathbf{H} = \mathbf{B}/\mu$ , plus Ohm's $\mathbf{J}_f = \sigma\mathbf{E}$ for conductors — with everything honest about those relations (frequency dependence, anisotropy, hysteresis, nonlinearity) owned by Linear Dielectrics, Linear and Nonlinear Magnetic Media, and the dispersion chapters. The macroscopic equations are a chassis; the material supplies the engine. Verified: matches Griffiths §7.3.5 (Eqs. 7.55–7.56) and Jackson §6.6; the D and H definitions are E3/E5's, unchanged; the equation-versus-unknown count matches Zangwill §2.2's discussion; boundary conditions across interfaces (D-perp jumps by σf, H-parallel by Kf, E-parallel and B-perp continuous) follow from these four by pillbox-and-loop arguments owned by Electrodynamic Boundary Conditions, next in this chapter — deferred by name.

Figure

The assembled macroscopic system: free sources drive D and H, matter's response hides in the definitions

The reward: light in matter

Feed the closed linear system ( $\mathbf{D} = \varepsilon\mathbf{E}$ , $\mathbf{H} = \mathbf{B}/\mu$ , no free sources) back through the previous article's wave derivation. Every step re-runs verbatim with $\varepsilon_0 \to \varepsilon$ , $\mu_0 \to \mu$ :

\nabla^2\mathbf{E} = \varepsilon\mu\,\frac{\partial^2\mathbf{E}}{\partial t^2} \qquad\Longrightarrow\qquad v = \frac{1}{\sqrt{\varepsilon\mu}} = \frac{c}{n}, \qquad n = \sqrt{\frac{\varepsilon\mu}{\varepsilon_0\mu_0}} \approx \sqrt{\varepsilon_r} \qquad (3)

where: $n$ — the refractive index, the number optics has measured since Snell; the last approximation uses $\mu \approx \mu_0$ , excellent for transparent media (glasses and water are feebly magnetic — E5's susceptibility tables). Electromagnetic waves propagate inside matter slower by exactly the factor optics always said, and three centuries of lens-making become a corollary of the assembled equations. For glass, $\varepsilon_r \approx 2.25$ at optical frequencies gives $n = 1.5$ ✓. And now the asterisk, stated where students can see it. Try water: the static dielectric constant is $\varepsilon_r \approx 80$ , predicting $n \approx 8.9$ — while measured optical $n = 1.33$ . The formula is not wrong; the input is. Constitutive relations are frequency-dependent: $\varepsilon_r(\omega)$ for water is 80 when fields change slowly enough for whole molecules to rotate and align, but at $5\times10^{14}$ Hz the sluggish rotors cannot follow at all, and only the nimble electron clouds respond — $\varepsilon_r(\text{optical}) \approx 1.77$ , whose square root is 1.33 ✓. Matter has internal clocks, and $\varepsilon(\omega)$ is their transcript — the dispersion physics (why prisms disperse, why absorption lines exist, Kramers–Kronig's deep bookkeeping) is owned by the E8 and E11 chapters (Absorption and Dispersion), and equation (3) hands them their starting point.

Limiting cases

Case	Result	Meaning
vacuum ( $\mathbf{P} = \mathbf{M} = 0$ )	$\mathbf{D} = \varepsilon_0\mathbf{E}$ , $\mathbf{H} = \mathbf{B}/\mu_0$ : previous article recovered	matter equations contain the vacuum ones exactly
statics ( $\partial_t = 0$ )	E3's $\nabla\cdot\mathbf{D} = \rho_f$ and E5's $\nabla\times\mathbf{H} = \mathbf{J}_f$ , separately	the two accountants' chapters, reunited
linear media, no free sources	waves at $v = c/n$ , $n \approx \sqrt{\varepsilon_r(\omega)}$	refraction derived; dispersion honest via $\varepsilon(\omega)$
conductor ( $\sigma$ large)	$\mathbf{J}_f = \sigma\mathbf{E}$ dominates $\partial\mathbf{D}/\partial t$	skin effect, screening — the E8 conductor-waves story
interfaces	$D_\perp$ jumps by $\sigma_f$ , $H_\parallel$ by $K_f$ ; $E_\parallel$ , $B_\perp$ continuous	derived by pillbox/loop in Electrodynamic Boundary Conditions (next)
ferroics (hysteresis)	no single-valued $\varepsilon$ or $\mu$ exists	constitutive closure fails honestly — E3/E5 own the story
$\varepsilon(\omega)$ , $\mu(\omega)$ engineered	metamaterials: $n$ by design, even $n < 0$	the constitutive chassis as a technology platform

Common trapdD/dt bundles two different physics — and D and H remain bookkeeping, not fields of force

Two standing confusions, both inherited from the auxiliaries' convenience. First: inside $\partial\mathbf{D}/\partial t = \varepsilon_0\,\partial\mathbf{E}/\partial t + \partial\mathbf{P}/\partial t$ live one term that moves no charge (the vacuum displacement term — pure field structure, present in empty space) and one that moves real charge (bound electrons shuttling — with inertia, and with dissipation whenever the response lags the field; that lag is dielectric loss, the $\tan\delta$ on every capacitor datasheet and the operating principle of your microwave oven). Calling the whole of $\partial\mathbf{D}/\partial t$ "displacement current in matter" is standard shorthand; forgetting the bundle's anatomy makes microwave heating, dielectric loss, and worked example 1's 75/25 split unintelligible. Second: the honesty doctrine of E3 and E5 survives the merger untouched. Forces on charges are $q(\mathbf{E} + \mathbf{v}\times\mathbf{B})$ — always $\mathbf{E}$ and $\mathbf{B}$ , never $\mathbf{D}$ and $\mathbf{H}$ ; the auxiliaries obey no Coulomb or Biot–Savart law of their own ( $\mathbf{D}$ 's curl and $\mathbf{H}$ 's divergence are hostage to the material); and energy/momentum bookkeeping in matter (the Abraham–Minkowski subtleties of E7's chapter) demands the physical fields. The macroscopic equations are the most useful bookkeeping in applied physics — read them as bookkeeping.

Worked examples

1 · The dielectric-filled capacitor: dissecting dD/dt (confidence builder)

The charging capacitor of the displacement-current article ( $R = 5.0$ cm plates, wire current $I = 0.50$ A) is now filled with a linear dielectric, $\varepsilon_r = 4.0$ . Find the total gap current density, its split between vacuum term and polarization current, and the magnetic field at the rim.

Step 1. The total is fixed by the free charge arriving: $\partial D/\partial t = \partial\sigma_f/\partial t = I/\pi R^2 = \mathbf{64\ A/m^2}$ — unchanged from the vacuum case, because $\nabla\cdot\mathbf{D} = \rho_f$ ties $\mathbf{D}$ to the free charge regardless of the filling (that is the whole point of $\mathbf{D}$ ).

Step 2. The anatomy. In the linear dielectric, $E = D/\varepsilon = D/4\varepsilon_0$ , so the vacuum term is $\varepsilon_0\,\partial E/\partial t = \tfrac14\,\partial D/\partial t = \mathbf{16\ A/m^2}$ — and the polarization current carries the rest: $\partial P/\partial t = \partial D/\partial t - \varepsilon_0\partial E/\partial t = \mathbf{48\ A/m^2}$ . Three-quarters of the gap current is now genuinely moving charge — bound electrons shuttling inside the dielectric as it polarizes. In general the split is $1/\varepsilon_r$ field-structure to $(1 - 1/\varepsilon_r)$ real motion: fill with water at low frequency ( $\varepsilon_r = 80$ ) and the gap current is 99% leashed charge.

Step 3. The magnetic field doesn't care about the anatomy — only the total enters $\nabla\times\mathbf{H} = \mathbf{J}_f + \partial\mathbf{D}/\partial t$ : at the rim, $H = I/2\pi R$ , and with $\mu \approx \mu_0$ , $B = \mu_0 I/2\pi R = \mathbf{2.0\ \mu T}$ — identical to the vacuum-gap answer. The splice of the displacement-current article survives the filling: wire current continues seamlessly as gap current, whatever the gap is made of.

Step 4. The physical dividend of the split: the polarization current, unlike the vacuum term, can dissipate. Drive this capacitor with AC at a frequency where the dielectric's response lags (near a molecular relaxation), and the 48 A/m² of real charge motion does work against molecular friction — dielectric heating. Your microwave oven runs at 2.45 GHz precisely to sit in water's lossy relaxation band: it is a machine for pumping $\partial\mathbf{P}/\partial t$ through wet food.

2 · Light through glass and the water paradox (exam level)

(a) Crown glass has $\varepsilon_r = 2.25$ at optical frequencies, $\mu \approx \mu_0$ . Find the wave speed and the wavelength inside the glass for 600 nm (vacuum) light. (b) Water has static $\varepsilon_r = 80$ : what index does naive substitution predict, what is measured, and what resolves the discrepancy?

Step 1. (a) Index: $n = \sqrt{\varepsilon_r} = \sqrt{2.25} = 1.50$ . Speed: $v = c/n = 3.00\times10^8/1.50 = \mathbf{2.0\times10^8\ m/s}$ — light loses a third of its speed crossing an air–glass surface, and regains it on exit, with no paradox: the E8 chapter's microscopic reading (incident wave plus coherent re-radiation by the polarization currents of Derivation A) is where the slowing physically lives. Note who does the slowing: the very $\partial\mathbf{P}/\partial t$ this article added to the catalog.

Step 2. (a) Wavelength: frequency is fixed by the source (the field oscillation cannot skip cycles at the boundary), so $\lambda_{\rm glass} = v/f = \lambda_0/n = 600/1.50 = \mathbf{400\ nm}$ . Six hundred-nanometre orange light has a four-hundred-nanometre wavelength inside the lens focusing it onto your retina — wavelength, not frequency, is the medium-dependent quantity, the fact behind every thin-film and interference calculation to come.

Step 3. (b) Naive water: $n = \sqrt{80} = 8.9$ ; measured: $n = 1.33$ . The factor-of-seven failure is not in equation (3) but in feeding it a DC number for an optical question: $\varepsilon_r = 80$ is the response of water molecules rotating to follow a slow field. At $5\times10^{14}$ Hz the rotors (relaxation time ~10 ps, natural response ceiling ~ $10^{11}$ Hz) are effectively frozen; only electronic polarization survives, $\varepsilon_r(\text{opt}) = n^2 = 1.77$ .

Step 4. Read the lesson forward: $\varepsilon_r(\omega)$ for water descends a staircase — ~80 (DC through RF, rotors following) → ~5 (infrared, rotors frozen, atomic stretching alive) → 1.77 (optical, electrons only) — and each step is a class of internal motion dropping out. The staircase is dispersion; its slopes near resonances are absorption; and the machinery relating the two (Lorentz oscillator models, Kramers–Kronig) is the E11 chapter's, launched from exactly this example. Constitutive relations are not constants; they are matter's frequency response — the single most consequential fine-print in the macroscopic equations.

Going further

References. Griffiths, Introduction to Electrodynamics, 4th ed., §7.3.5–7.3.6 · Jackson, Classical Electrodynamics, 3rd ed., §6.6, §I.5 · Zangwill, Modern Electrodynamics, §2.2, Ch. 6, Ch. 13 · Purcell & Morin, Electricity and Magnetism, 3rd ed., Ch. 10–11.

Provenance. All four matter equations and the definitions $\mathbf{D}=\varepsilon_0\mathbf{E}+\mathbf{P}$ , $\mathbf{H}=\mathbf{B}/\mu_0-\mathbf{M}$ follow Griffiths §7.3 and Jackson §6.6; a verifier should re-derive $\rho_b=-\nabla\cdot\mathbf{P}$ (Griffiths §4.2), $\mathbf{J}_b=\nabla\times\mathbf{M}+\partial\mathbf{P}/\partial t$ , the bound-continuity check, and the four boundary conditions (Griffiths §7.3.6, Jackson §I.5). Constants: $\varepsilon_0=8.854\times10^{-12}$ F/m, $\mu_0=4\pi\times10^{-7}$ H/m (pre-2019 exact), and $n=\sqrt{\varepsilon_r}=1.5$ for $\varepsilon_r=2.25$ are standard textbook values. Addendum for this edition: $\rho_b = -\nabla\cdot\mathbf{P}$ and $\mathbf{J}_{\rm mag} = \nabla\times\mathbf{M}$ are owned by E3 Bound Charges and E5 Bound Currents (recalled by name, not re-derived); boundary conditions deferred by name to Electrodynamic Boundary Conditions (0563); this edition owns the polarization-current derivation, the assembly, the constitutive closure, and the $c/n$ reward; worked numbers recomputed — filled capacitor ( $R$ = 5 cm, $I$ = 0.5 A, $\varepsilon_r$ = 4): $\partial D/\partial t$ = 64 A/m² splitting 16 (vacuum term) + 48 (polarization current) A/m², $B_{\rm rim}$ = 2.0 μT unchanged; glass $n$ = 1.50, $v$ = 2.0×10⁸ m/s, 600 → 400 nm; water $\sqrt{80}$ = 8.9 vs measured 1.33, $\varepsilon_r(\text{opt}) = 1.33^2 = 1.77$ .

Builds on: The Displacement Current · The Continuity Equation · Faraday's Law of Induction · Ohm's Law and Conductivity

Leads to: Waves in Linear Dielectric Media · Energy Stored in Magnetic Fields · Boundary Value Problems in Dielectrics

Are the macroscopic equations approximations, or exact like the vacuum ones?

Exact in one precise sense, approximate in another — and knowing which is which prevents real confusion. The equations themselves are exact consequences of microscopic Maxwell plus the definitions of $\mathbf{P}$ , $\mathbf{M}$ , $\mathbf{D}$ , $\mathbf{H}$ : no approximation enters the assembly of Derivation B. What is approximate is the usefulness package around them: the spatial averaging that makes smooth $\mathbf{P}(\mathbf{r})$ and $\mathbf{M}(\mathbf{r})$ meaningful (valid at scales large compared to molecules — fails inside a molecule, and gets subtle in nanostructures and metamaterials near their structural scale), and above all the constitutive relations, which are models of real matter with regimes of validity (linearity, frequency range, field strength). The professional summary: exact bookkeeping, approximate closure. When macroscopic electromagnetism "fails," it is essentially always the constitutive model, not the four equations, that broke.

Is the polarization current really distinct from the vacuum displacement term? Both live in dD/dt.

Physically distinct by every test that could distinguish them. The polarization current is charge in motion: it has carriers (bound electrons and ions), it dissipates energy when the motion lags the drive (dielectric loss — measurable as heat), it exhibits inertia (contributing to $\varepsilon(\omega)$ 's frequency dependence), and it produces magnetic field as moving charge. The vacuum term $\varepsilon_0\,\partial\mathbf{E}/\partial t$ moves nothing, heats nothing, and exists in perfect emptiness. They share a symbol because the bookkeeping is cleanest that way — $\mathbf{D}$ was engineered so free sources stand alone — and because both source $\nabla\times\mathbf{H}$ identically. Worked example 1's split (25/75 at $\varepsilon_r = 4$ ) is not a formal decomposition: run the capacitor at a lossy frequency and calorimetry finds heat proportional to exactly the 75% share. One symbol, two kinds of physics — the article's warning box exists because the shorthand invites forgetting that.

Why is there no "bound charge" version of the magnetic Gauss law or Faraday's law?

Because those two equations have no source slots to split. The free/bound program operates on sources: $\rho$ splits, $\mathbf{J}$ splits, and the two equations containing them reorganize. But $\nabla\cdot\mathbf{B} = 0$ says magnetic field lines end nowhere — there is no magnetic charge, free or bound, so nothing to catalog; and Faraday's law couples fields to fields with no source at all. Deeper version: in the covariant language of E10, the sourced pair descends from $\partial_\mu F^{\mu\nu} = \mu_0 J^\nu$ (matter modifies $J^\nu$ ), while the unsourced pair is the Bianchi identity — a statement about the field's geometric structure that matter cannot touch. If monopoles were ever found, magnetizable monopole-matter would give the magnetic Gauss law its own bound sources and a " $\mathbf{P}$ -like" companion — a symmetry nature has so far declined.

Which fields do I match at a boundary, and where do those rules come from?

The four matching rules — $D_\perp$ jumps by the free surface charge $\sigma_f$ ; $H_\parallel$ jumps by the free surface current $K_f$ ; $E_\parallel$ and $B_\perp$ are always continuous — come from running the four macroscopic equations through infinitesimal pillboxes (for the divergences) and loops (for the curls) straddling the interface. Note the pattern: the auxiliaries jump by free sources only (their design brief), while the physical fields' continuous components are source-blind. The systematic derivations, the dielectric–dielectric and conductor cases, and the workhorse applications (Snell's law and Fresnel coefficients are boundary conditions applied to waves) belong to Electrodynamic Boundary Conditions, immediately next in this chapter — this article stops at the doorstep deliberately.

If E and B are the real fields, why do practitioners work almost entirely in D and H?

Because practitioners control free sources, and the auxiliaries are the fields whose equations run on free sources alone. A capacitor designer sets free charge on plates ( $\mathbf{D}$ 's domain); a magnet engineer sets winding currents ( $\mathbf{H}$ 's domain — every $B$ – $H$ curve on a datasheet has the controllable variable on the horizontal axis); an RF engineer launches free currents in antennas. The auxiliaries quarantine the material's unknown response into constitutive relations, which is exactly where measured material data enters. The division of labor is clean and worth stating once: solve with $\mathbf{D}$ and $\mathbf{H}$ (they see your knobs), interpret forces and energy with $\mathbf{E}$ and $\mathbf{B}$ (they push the charges). Both chapters of this track's matter physics (E3, E5) end on that doctrine, and it survives time dependence unamended.

What determines how epsilon depends on frequency — can I estimate the staircase myself?

To useful accuracy, yes: match field frequency against the response times of the material's internal motions. Orientational polarization (whole polar molecules rotating): picosecond-and-slower — alive up to ~GHz–THz in liquids, dead beyond (water's 80 lives here). Ionic/atomic polarization (nuclei stretching in bonds): infrared, ~ $10^{13}$ Hz. Electronic polarization (clouds deforming): ultraviolet resonances, ~ $10^{15-16}$ Hz — the only survivor at optical frequencies, whence glass's modest 2.25 and water's 1.77. Each mechanism contributes while the drive is slower than its response and drops out (through a lossy resonance or relaxation band) when the drive outruns it: $\varepsilon(\omega)$ is a staircase descending toward 1, with absorption at every step edge. The quantitative machinery — Debye relaxation for rotors, Lorentz oscillators for resonances, Kramers–Kronig gluing the real and imaginary parts — is the E11 chapter's (Absorption and Dispersion); the estimate above gets the staircase's floors right for most materials you will meet.

Do the macroscopic equations still enforce charge conservation the way the vacuum ones do?

Yes — twice over, and checking it is a nice exercise in the chapter's own methods. Take the divergence of $\nabla\times\mathbf{H} = \mathbf{J}_f + \partial\mathbf{D}/\partial t$ and use $\nabla\cdot\mathbf{D} = \rho_f$ : out falls $\nabla\cdot\mathbf{J}_f + \partial\rho_f/\partial t = 0$ — free charge conserved by itself, identically (the previous article's audit, now running on the macroscopic chassis). Meanwhile Derivation A verified the bound ledger separately: $\nabla\cdot\mathbf{J}_b + \partial\rho_b/\partial t = 0$ by construction of $\mathbf{J}_p$ . Two independently conserved ledgers, free and bound — which is physically right: dielectric response shuffles bound charge locally but never converts it into free charge (ionization would be a new, nonlinear entry in the constitutive relations, beyond the linear regime). The bookkeeping is airtight at every level — as, after the last article, it had better be.