The Electric Displacement D

The bound-charge articles left the working physicist with a bookkeeping complaint. Gauss's law for $\mathbf{E}$ is sourced by all charge, $\rho_f + \rho_b$ — but $\rho_b = -\nabla\cdot\mathbf{P}$ depends on the polarization, which depends on the field, which depends on the bound charge you were trying to find. Circular. The fix is one of physics's great accounting maneuvers: move the bound-charge term to the other side of the equation and give the combination a name. The result, $\mathbf{D} \equiv \varepsilon_0\mathbf{E} + \mathbf{P}$ , obeys $\nabla\cdot\mathbf{D} = \rho_f$ — a Gauss's law that sees only the charge you control, the charge on wires and plates and electrodes. For problems with symmetry, $\mathbf{D}$ falls out of a two-line flux argument before you know anything about the material at all. Then comes the part this article insists on, because generations of students have been burned by it: $\mathbf{D}$ is not "the electric field of the free charges," and the parallel between $(\mathbf{E}, \rho_{\rm total})$ and $(\mathbf{D}, \rho_f)$ — deceptive is Griffiths's word — breaks at the second glance. A field is determined by its divergence and its curl (the Helmholtz theorem, recalled from the vector-calculus chapter), and there the symmetry dies: $\nabla\times\mathbf{E} = 0$ always in statics, but $\nabla\times\mathbf{D} = \nabla\times\mathbf{P}$ , which vanishes only when the material geometry is kind. Consequently there is no Coulomb's law for $\mathbf{D}$ , no scalar potential for $\mathbf{D}$ , and no guarantee that $\rho_f = 0$ means $\mathbf{D} = 0$ — the electret sphere of the previous article, with not a single free charge anywhere, carries a thriving $\mathbf{D}$ -field whose lines close on themselves in loops. So the honest contract: $\mathbf{D}$ is a supremely useful auxiliary — the flux bookkeeper of free charge — and the physical field, the one that pushes charges and stores energy, remains $\mathbf{E}$ . This article derives the bookkeeper, works it where it shines (a symmetric insulated wire), and prosecutes the counterexample where naive trust in it collapses.

Key takeaways

Definition: $\mathbf{D} \equiv \varepsilon_0\mathbf{E} + \mathbf{P}$ — engineered so that the bound charge cancels out of Gauss's law: $\nabla\cdot\mathbf{D} = \rho_f$ , $\oint\mathbf{D}\cdot\mathrm{d}\mathbf{a} = Q_{f,\rm enc}$ .
The payoff: in symmetric problems, $\mathbf{D}$ follows from the free charge by a pillbox/cylinder/sphere flux argument before any constitutive information — the material enters only when you convert to $\mathbf{E}$ .
The catch: $\nabla\times\mathbf{D} = \nabla\times\mathbf{P}$ , generally nonzero — so $\mathbf{D}$ has no Coulomb law, no potential, and is not determined by $\rho_f$ alone (Helmholtz needs the curl too).
The counterexample to memorize: the uniformly polarized (electret) sphere has $\rho_f = 0$ everywhere yet $\mathbf{D} \neq 0$ everywhere — inside, $\mathbf{D} = +2\mathbf{P}/3$ , pointing along $\mathbf{P}$ while $\varepsilon_0\mathbf{E}$ points against.
$\mathbf{D}$ 's field lines never start or stop where there is no free charge — in the electret they form closed loops, the visual signature of a curl-carrying field.
Units housekeeping: $\mathbf{D}$ and $\mathbf{P}$ share $\mathrm{C/m^2}$ ; $\varepsilon_0\mathbf{E}$ must be added, not $\mathbf{E}$ — and $\mathbf{D}$ 's boundary behavior ( $D^\perp$ jumps by $\sigma_f$ ) is deferred by name to Boundary Conditions for D and E.
KEY IDEA: $\mathbf{D} = \varepsilon_0\mathbf{E} + \mathbf{P}$ absorbs bound charge so Gauss's law reads free charge only — but $\mathbf{D}$ inherits no curl-free guarantee, so it is a flux bookkeeper, not a field of force, and only symmetry (not $\rho_f$ alone) ever determines it.

The core insight

The displacement $\mathbf{D} = \varepsilon_0\mathbf{E} + \mathbf{P}$ satisfies $\nabla\cdot\mathbf{D} = \rho_f$ , letting Gauss's law run on free charge alone; but $\nabla\times\mathbf{D} = \nabla\times\mathbf{P}$ is generally nonzero, so $\mathbf{D}$ has no Coulomb law and is not determined by free charge by itself — it is an auxiliary bookkeeping field, while $\mathbf{E}$ remains the physical one.

Where this lives today

$\mathbf{D}$ is the field engineering actually specifies. Capacitor datasheets and dielectric-strength ratings, charge–voltage curves of ferroelectrics (the famous $D$ – $E$ hysteresis loops traced by Sawyer–Tower circuits), and the "displacement field" that gate engineers quote when tuning 2D-material transistors are all statements about $\mathbf{D}$ , because $\mathbf{D}$ 's flux counts the free charge electrodes must supply — the quantity circuits meter. Numerical electromagnetics is organized around the same split: finite-element solvers assemble $\nabla\cdot\mathbf{D} = \rho_f$ with constitutive models plugged in per material region, precisely the postponement strategy of worked example 1, industrialized. The concept's deepest modern echo is the one Maxwell built it for. In time-dependent fields the displacement's rate of change, $\partial\mathbf{D}/\partial t$ — vacuum term plus polarization current — completes Ampère's law and makes electromagnetic waves possible; "displacement current" carries $\mathbf{D}$ 's name into every antenna, waveguide, and optical fiber (the story is owned by the electrodynamics chapter, but the field was christened here). And the honesty doctrine has modern teeth too: in the physics of ferroelectric capacitors and negative-capacitance transistors, device electrostatics is analyzed on the $D$ – $E$ plane exactly because $\mathbf{D}$ , fixed by electrode charge, and $\mathbf{E}$ , set by material response, are independent handles — engineering that would be incoherent if $\mathbf{D}$ were merely $\mathbf{E}$ in different units.

Origins · 1 minJames Clerk Maxwell and how this was discovered

James Clerk Maxwell

1831–1879 · Aberdeen, London, Cambridge

Maxwell invented the displacement — name, concept, and consequence. Translating Faraday's picture of strained dielectric media, his 1861–62 "On Physical Lines of Force" modeled polarization as elastic displacement of charge in a medium of molecular vortices, and noticed that when the strain changes, the moving charge is a current: the displacement current, added to Ampère's law with revolutionary effect. In the 1865 "Dynamical Theory of the Electromagnetic Field" and the 1873 Treatise, the mechanical scaffolding fell away but $\mathbf{D}$ remained — one of the two electric vectors of his theory ( $\mathbf{E}$ the force intensity, $\mathbf{D}$ the flux quantity), a distinction he insisted on and posterity ratified. The payoff was the greatest unification in physics: with $\partial\mathbf{D}/\partial t$ in place, his equations propagated waves at a speed matching light's, and optics became electromagnetism. Einstein kept Maxwell's portrait on his study wall alongside Faraday's and Newton's; the equal-time verdict — "one scientific epoch ended and another began with James Clerk Maxwell" — was Einstein's too. The bookkeeping field of this article is the fragment of his vortex machinery that outlived every mechanical picture it came from.

History

1837

— Faraday's dielectric experiments picture insulation as a strained medium — displacement's physical seed

1861

— Maxwell's vortex-cell model introduces electric displacement and its current

1865

— The Dynamical Theory keeps D, drops the machinery, and delivers electromagnetic waves at the speed of light

1873

— The Treatise codifies the two electric vectors: E for force, D for flux

1884

— Heaviside recasts Maxwell's system into the four modern equations, div D = rho_f among them

1930

— Sawyer and Tower trace the first ferroelectric D–E hysteresis loops — the displacement as laboratory observable

The physical picture

Think of an accountant confronting the dielectric mess. The physical ledger — Gauss's law for $\mathbf{E}$ — mixes two kinds of entries: charges you placed (on plates, wires: the free charge) and charges the material generated in response (the bound charge). You control the first and can't directly see the second. The accountant's move is to define a new ledger line that nets out the material's response: add to $\varepsilon_0\mathbf{E}$ exactly the quantity, $\mathbf{P}$ , whose divergence is minus the bound charge. In the new ledger, material entries cancel identically, and what remains — $\nabla\cdot\mathbf{D} = \rho_f$ — answers to your inputs alone. That is all $\mathbf{D}$ is: a change of variables that hides the response charge inside the definition.

Figure

Two ledgers for one field problem — E answers to all charge, D to free charge only

Why is the new ledger so useful? Because in the lab you set the free charge (or the voltage) and want consequences. With symmetry — plates, coaxial lines, charged spheres in layered insulation — the flux form $\oint\mathbf{D}\cdot\mathrm{d}\mathbf{a} = Q_{f,\rm enc}$ pins $\mathbf{D}$ from the free charge in two lines, before you know whether the insulation is polystyrene, water, or vacuum. All material complexity is postponed to the last step, converting $\mathbf{D}$ back to $\mathbf{E}$ (trivial once the linear-dielectric law arrives in the next article; already exact in vacuum, where $\mathbf{D} = \varepsilon_0\mathbf{E}$ identically). The division of labor is clean: $\mathbf{D}$ handles the geometry of free charge; the constitutive relation handles the material; $\mathbf{E}$ delivers the physics. And why the loud warnings? Because the analogy " $\mathbf{D}$ is to free charge what $\mathbf{E}$ is to total charge" invites an illegal inference. $\mathbf{E}$ is reconstructible from its sources by Coulomb's law because we also know its curl vanishes — the Helmholtz theorem of the vector-calculus chapter is explicit that a field needs both its divergence and its curl (plus decay) to be pinned down. $\mathbf{D}$ 's curl is $\nabla\times\mathbf{P}$ : zero for nice geometries, decidedly nonzero wherever $\mathbf{P}$ has the wrong shape — most vividly at the equator of an electret sphere, where the tangential $\mathbf{P}$ jumps from full strength to zero across the surface. There, $\mathbf{D}$ picks up circulation, its field lines close into loops (they must go somewhere, and with $\nabla\cdot\mathbf{D} = 0$ everywhere they can neither start nor end), and any attempt to compute " $\mathbf{D}$ by Coulomb integration over $\rho_f$ " returns zero — off by everything. The worked examples stage both faces: the wire where $\mathbf{D}$ is magnificent, the electret where naive $\mathbf{D}$ -trust is annihilated.

Figure

The electret sphere — E lines end on bound charge, D lines close into loops

Definition and field equations

\mathbf{D} \equiv \varepsilon_0\,\mathbf{E} + \mathbf{P} \qquad (1)

where: $\mathbf{E}$ — the macroscopic electric field (the averaged, physical field of the previous articles); $\mathbf{P}$ — the polarization; $\mathbf{D}$ — the electric displacement, units $\mathrm{C/m^2}$ (matching $\mathbf{P}$ , and matching surface charge — the units are a standing reminder that $\mathbf{D}$ 's natural habitat is flux-per-area bookkeeping). Note the $\varepsilon_0$ : it is $\varepsilon_0\mathbf{E}$ , not $\mathbf{E}$ , that joins $\mathbf{P}$ — in vacuum $\mathbf{D} = \varepsilon_0\mathbf{E}$ exactly, and forgetting the factor is the unit error the definition's asymmetric look is warning you about. The field equations it obeys, derived below:

\nabla\cdot\mathbf{D} = \rho_f, \qquad \oint_{\mathcal{S}}\mathbf{D}\cdot\mathrm{d}\mathbf{a} = Q_{f,\rm enc}, \qquad \nabla\times\mathbf{D} = \nabla\times\mathbf{P} \qquad (2)

where: $\rho_f$ , $Q_{f,\rm enc}$ — free charge density and enclosed free charge (conduction electrons, battery-delivered plate charge, embedded ions — charge with an independent existence, as opposed to the response charge bound in molecules). The first two equations are the tool; the third is the fine print that governs when the tool may be trusted alone. Together they say: $\mathbf{D}$ 's flux is pure free-charge bookkeeping; $\mathbf{D}$ 's circulation is hostage to the material geometry.

Derivation AConstructing the bookkeeper — Gauss's law with the bound charge folded in

The goal is equations (2) from what the chapter has already established, in the three lines the construction deserves, followed by the statement of scope: exactly what the new law does and does not determine. The inputs are Gauss's law for $\mathbf{E}$ with all charge (the bound-charge article's doctrine) and $\rho_b = -\nabla\cdot\mathbf{P}$ . Watch that no new physics enters — $\mathbf{D}$ is definition plus rearrangement, which is precisely why it can never contain more than $\mathbf{E}$ and $\mathbf{P}$ did.

Step 1. Start from the honest Gauss's law, splitting total charge into the two ledgers:

\varepsilon_0\,\nabla\cdot\mathbf{E} = \rho_{\rm total} = \rho_f + \rho_b = \rho_f - \nabla\cdot\mathbf{P}.

Step 2. Move the material's response to the left side, where it combines into a single divergence:

\nabla\cdot\left(\varepsilon_0\mathbf{E} + \mathbf{P}\right) = \rho_f \qquad\Longrightarrow\qquad \nabla\cdot\mathbf{D} = \rho_f,

with $\mathbf{D}$ defined by (1). The divergence theorem (E0) upgrades this to the integral form $\oint\mathbf{D}\cdot\mathrm{d}\mathbf{a} = Q_{f,\rm enc}$ over any closed surface — including surfaces cutting straight through dielectric, where $\mathbf{E}$ -flux audits would need the bound charge counted and $\mathbf{D}$ -flux audits need nothing but the free charge.

Step 3. Take the curl of the definition. In statics $\nabla\times\mathbf{E} = 0$ (E1's curl article), so

\nabla\times\mathbf{D} = \nabla\times\mathbf{P}.

No law of nature makes this vanish: $\mathbf{P}$ is whatever the material's state dictates. It does vanish in the friendly cases — everywhere inside a homogeneous linear dielectric in a curl-free field, and by symmetry in the standard Gauss geometries — which is exactly why the friendly cases are the ones where $\mathbf{D}$ -first methods work.

Step 4. Scope, via Helmholtz. The Helmholtz theorem (E0) reconstructs a field from both its divergence and its curl (with boundary conditions). For $\mathbf{E}$ : divergence known from all charge, curl known to be zero — reconstructible, hence Coulomb's law. For $\mathbf{D}$ : divergence known from free charge, curl equal to $\nabla\times\mathbf{P}$ — unknown until you know the material's polarization, which is the thing you were trying to avoid computing. So $\rho_f$ alone underdetermines $\mathbf{D}$ ; the missing information is supplied either by symmetry (which forces the curl-carrying part to vanish) or by a constitutive relation plus boundary conditions (the next articles). There is, correspondingly, no "Coulomb's law for $\mathbf{D}$ " — no formula $\mathbf{D}(\mathbf{r}) = \frac{1}{4\pi}\int\rho_f\,\hat{\boldsymbol{\mathscr{r}}}/\mathscr{r}^2\,\mathrm{d}\tau'$ — and integrating that illegal formula is the canonical way to get dielectric problems silently wrong.

Go deeper · Derivation B · 2 minThe deceptive parallel, prosecuted — what D is not

Derivation BThe deceptive parallel, prosecuted — what D is not

The goal is to make the warnings quantitative on the cleanest possible counterexample: the uniformly polarized sphere of the previous article — an object with zero free charge everywhere, whose $\mathbf{D}$ -field we can write down exactly. The strategy is to assemble $\mathbf{D}$ from the known $\mathbf{E}$ and $\mathbf{P}$ , then interrogate it: where do its lines go, what is its circulation, and what would the naive free-charge-only reasoning have predicted? This derivation is short on algebra and long on verdicts; its numbers are re-used in worked example 2.

Step 1. Assemble $\mathbf{D}$ inside. The electret sphere has $\mathbf{E}_{\rm in} = -\mathbf{P}/3\varepsilon_0$ , so

\mathbf{D}_{\rm in} = \varepsilon_0\mathbf{E}_{\rm in} + \mathbf{P} = -\frac{\mathbf{P}}{3} + \mathbf{P} = +\frac{2}{3}\,\mathbf{P}.

Note the sign spectacle: $\varepsilon_0\mathbf{E}$ points against the polarization, $\mathbf{D}$ points with it — the two fields disagree about direction inside matter, an impossibility if $\mathbf{D}$ were "basically $\mathbf{E}$ rescaled."

Step 2. Assemble $\mathbf{D}$ outside: $\mathbf{P} = 0$ there, so $\mathbf{D}_{\rm out} = \varepsilon_0\mathbf{E}_{\rm out}$ — $\varepsilon_0$ times the pure dipole field.

Step 3. Audit the naive prediction. Free charge: none, anywhere. " $\mathbf{D}$ is sourced by free charge" (naively) $\Rightarrow \mathbf{D} = 0$ everywhere. Reality: $|\mathbf{D}_{\rm in}| = 2P/3 \neq 0$ , and a whole exterior dipole pattern besides. The flux law is not violated — $\oint\mathbf{D}\cdot\mathrm{d}\mathbf{a} = 0$ over every closed surface, as direct integration of the dipole-plus-uniform pattern confirms — but zero flux through every surface only means the field is divergence-free, not that it vanishes. The information gap is exactly the curl.

Step 4. Find the curl and the loops. $\nabla\times\mathbf{D} = \nabla\times\mathbf{P}$ is zero in the bulk (uniform $\mathbf{P}$ ) and zero outside ( $\mathbf{P} = 0$ ) — the curl lives entirely on the surface, as a delta-function sheet where the tangential component of $\mathbf{P}$ (largest at the equator, $P\sin\theta$ ) jumps to zero. Take the circulation of $\mathbf{D}$ around a closed loop threading the sphere along its axis and returning far outside: the interior leg contributes $D_{\rm in}\times 2R = (2P/3)(2R) = 4PR/3$ . The exterior return leg contributes $+2PR/3$ . so the total loop circulation is $4PR/3 + 2PR/3 = 2PR$ . Nonzero circulation, from a field with zero divergence sources: $\mathbf{D}$ 's field lines can neither begin nor end, so they close into loops — down the inside of the sphere, out the polar cap, around the outside, back in the other pole, like the $\mathbf{B}$ -field of a bar magnet (an analogy that is in fact exact: $\mathbf{D}$ here maps onto $\mathbf{B}$ of a uniformly magnetized sphere). Consequences, itemized: no scalar potential for $\mathbf{D}$ (circulation $\neq 0$ ); no Coulomb integral (it would give 0); and no shortcut past the material — to find $\mathbf{D}$ you first needed $\mathbf{E}$ and $\mathbf{P}$ , the honest solution of the previous article. The flux of $\mathbf{D}$ over any concentric sphere vanishes because the dipole pattern outside carries equal inward and outward flux; the total circulation gives $2PR=4.0\times10^{-6}\,\mathrm{C/m}$ for the example sphere; and the $\mathbf{B}$ -field analogy is the standard uniformly-magnetized-sphere correspondence, previewed here and developed fully in the magnetism chapter.

Limiting cases

Case	Result	Meaning
vacuum	$\mathbf{D} = \varepsilon_0\mathbf{E}$ exactly	the ledgers coincide; nothing to hide
symmetric free charge + homogeneous medium	$\mathbf{D}$ from flux alone	the tool at full power (worked example 1)
linear dielectric (next article)	$\mathbf{D} = \varepsilon\mathbf{E}$	conversion becomes one multiplication
electret, $\rho_f = 0$	$\mathbf{D} \neq 0$ , loops	free charge alone underdetermines $\mathbf{D}$
inside electret sphere	$\mathbf{D} = +2\mathbf{P}/3$ , $\varepsilon_0\mathbf{E} = -\mathbf{P}/3$	the two fields point opposite ways
tangential $\mathbf{P}$ jump at a surface	curl sheet: $\nabla\times\mathbf{D} \neq 0$	where the no-potential verdict is earned
interface, free charge $\sigma_f$	$D^\perp$ jumps by $\sigma_f$ ; $D^\parallel$ jumps with $P^\parallel$	full story deferred to Boundary Conditions for D and E

Common trapThere is no Coulomb law for D — and "no free charge" never means "no D"

The deceptive parallel claims its victims in two standard ways. First: computing $\mathbf{D}$ by integrating $\rho_f$ with a Coulomb kernel, or equivalently assuming $\rho_f = 0 \Rightarrow \mathbf{D} = 0$ . The electret sphere is the two-line refutation: no free charge anywhere, yet $\mathbf{D} = 2\mathbf{P}/3$ inside and a full dipole pattern outside. A divergence alone determines nothing — Helmholtz demands the curl, and $\mathbf{D}$ 's curl belongs to the material. Second: treating $\mathbf{D}/\varepsilon_0$ as "the field with the dielectric ignored" — inside the electret, $\mathbf{D}$ and $\mathbf{E}$ point in opposite directions; no rescaling relates them. The safe operating envelope: use $\oint\mathbf{D}\cdot\mathrm{d}\mathbf{a} = Q_f$ when symmetry pins the direction and uniformity of $\mathbf{D}$ on your Gaussian surface (plane, cylinder, sphere, with media respecting the symmetry) — and outside that envelope, solve for $\mathbf{E}$ and $\mathbf{P}$ honestly and assemble $\mathbf{D}$ , never the reverse.

Worked examples

1 · The insulated wire — D at full power (confidence builder)

A long straight wire carries free line charge $\lambda = 20\,\mathrm{nC/m}$ , surrounded by a rubber insulation sleeve out to radius $a = 1.0\,\mathrm{cm}$ (properties unknown). Find $\mathbf{D}$ everywhere, $\mathbf{E}$ outside the rubber, and state honestly what remains unknown.

Step 1. Symmetry first. Cylindrical symmetry forces every field — $\mathbf{E}$ , $\mathbf{P}$ , hence $\mathbf{D}$ — to point radially with magnitude depending only on $s$ ; this is what licenses the flux argument (and guarantees the curl-carrying part vanishes).

Step 2. Flux audit with a coaxial cylinder of radius $s$ , length $L$ : $\oint\mathbf{D}\cdot\mathrm{d}\mathbf{a} = D\,(2\pi sL) = Q_{f,\rm enc} = \lambda L$ , so

\mathbf{D} = \frac{\lambda}{2\pi s}\,\hat{\mathbf{s}} \qquad\text{— for all } s, \text{ inside the rubber and out.}

Numbers: $D = 6.4\times10^{-7}\,\mathrm{C/m^2}$ at $s = 5.0\,\mathrm{mm}$ (mid-rubber), $1.6\times10^{-7}$ at $s = 2.0\,\mathrm{cm}$ (outside). The material never entered — the rubber's bound charges are exactly the entries $\mathbf{D}$ 's ledger was built to hide.

Step 3. Convert where conversion is free. Outside the rubber, vacuum: $\mathbf{E} = \mathbf{D}/\varepsilon_0 = \lambda/2\pi\varepsilon_0s\,\hat{\mathbf{s}} = 1.8\times10^{4}\,\mathrm{V/m}$ at $s = 2.0\,\mathrm{cm}$ — identical to a bare wire's field: from outside, the neutral insulation is electrostatically invisible (its bound-charge sheets at inner and outer surfaces cancel in the enclosed total).

Step 4. The honest remainder. Inside the rubber, $\mathbf{D}$ is known but $\mathbf{E} = (\mathbf{D} - \mathbf{P})/\varepsilon_0$ is not — we lack the constitutive information connecting $\mathbf{P}$ to $\mathbf{E}$ . $\mathbf{D}$ does the geometry; the material science is genuinely postponed, not skipped — one article later, "linear dielectric, $\varepsilon_r$ given" turns Step 4 into a single division.

2 · The electret sphere — D at its worst (exam level)

For the uniformly polarized sphere ( $P = 1.0\times10^{-4}\,\mathrm{C/m^2}$ , $R = 2.0\,\mathrm{cm}$ ; fields from the previous article), evaluate $\mathbf{D}$ inside and out, test the naive free-charge logic, and exhibit the circulation.

Step 1. Inside: $\mathbf{D}_{\rm in} = \varepsilon_0(-\mathbf{P}/3\varepsilon_0) + \mathbf{P} = +\tfrac23\mathbf{P}$ : magnitude $6.7\times10^{-5}\,\mathrm{C/m^2}$ , pointing along the polarization — while $\varepsilon_0\mathbf{E}_{\rm in} = -3.3\times10^{-5}\,\mathrm{C/m^2}$ points against it. Two "electric fields," antiparallel.

Step 2. Outside: $\mathbf{D} = \varepsilon_0\mathbf{E}_{\rm dipole}$ — at the north pole surface, $\varepsilon_0(2P/3\varepsilon_0) = +6.7\times10^{-5}\,\mathrm{C/m^2}$ radially out. Note $D^\perp$ is continuous across the pole ( $2P/3$ both sides): no free surface charge, no $\mathbf{D}$ -flux jump — the boundary rule previewed here and derived in Boundary Conditions for D and E.

Step 3. The naive test. $\rho_f = 0$ everywhere; "therefore $\mathbf{D} = 0$ " — refuted at every point of space. Verify the flux law survives: any sphere concentric with the electret has $\oint\mathbf{D}\cdot\mathrm{d}\mathbf{a} = 0$ (inside: uniform field, equal flux in and out; outside: dipole pattern, zero net flux) ✓ — zero divergence, not zero field.

Step 4. The circulation. Loop: up the polar axis inside ( $2R$ long), return along an exterior path. Inside leg: $\int \mathbf{D}_{\rm in} \cdot \mathrm{d}\boldsymbol{\ell} = (2P/3)(2R) = +2.67\times10^{-6}\,\mathrm{C/m}$ . Return leg: $\int \mathbf{D}_{\rm out} \cdot \mathrm{d}\boldsymbol{\ell} = +2PR/3 = +1.33\times10^{-6}\,\mathrm{C/m}$ . Total loop circulation $= 2PR = +4.0\times10^{-6}\,\mathrm{C/m} \neq 0$ : $\mathbf{D}$ is not conservative, admits no potential, and its lines — forced by $\nabla\cdot\mathbf{D} = 0$ to be endless — close into loops through the sphere and around it, a bar magnet's portrait drawn in electrostatics. (The analogy is exact: this $\mathbf{D}$ is the $\mathbf{B}$ -field pattern of a uniformly magnetized sphere, a correspondence the magnetism chapter will earn.)

Going further

References. Griffiths, Introduction to Electrodynamics, 4th ed., §4.3 · Jackson, Classical Electrodynamics, 3rd ed., §4.3–4.4 · Purcell & Morin, Electricity and Magnetism, 3rd ed., §10.10–10.11 · Zangwill, Modern Electrodynamics, §6.5. Key constants and worked numbers: $\varepsilon_0 = 8.854\times10^{-12}\,\mathrm{F/m}$ . The electret interior field follows from $\mathbf{E}_{\rm in} = -\mathbf{P}/3\varepsilon_0$ , giving $D_{\rm in} = 2P/3 = 6.7\times10^{-5}\,\mathrm{C/m^2}$ vs $\varepsilon_0 E_{\rm in} = -3.3\times10^{-5}\,\mathrm{C/m^2}$ , and circulation $(2P/3)(2R) = 2.7\times10^{-6}\,\mathrm{C/m}$ . Wire results: $D = 6.4\times10^{-7}\,\mathrm{C/m^2}$ at 5 mm, $1.6\times10^{-7}\,\mathrm{C/m^2}$ at 2 cm, $E = 1.8\times10^4\,\mathrm{V/m}$ outside.

Builds on: Bound Charges: Surface and Volume · Polarization Density P · Uniformly Polarized Sphere · Helmholtz Theorem · Gauss's Law in Differential Form

Leads to: Gauss's Law in Dielectrics · Linear Dielectrics, Susceptibility and Permittivity · Boundary Conditions for D and E · Dielectric Sphere in a Uniform Field

Is D a physical field or just a mathematical convenience?

A convenience with an exact definition — and the question has a sharp answer: forces on charges are $q\mathbf{E}$ , energy density involves $\mathbf{E}$ , and the microscopic world averages to $\mathbf{E}$ ; nothing pushes on anything "with $\mathbf{D}$ ." Yet convenience is not disparagement: $\mathbf{D}$ 's flux is what electrodes and meters actually account (free charge), which is why engineers specify it and why it earns its place in the macroscopic Maxwell equations. The healthy attitude: $\mathbf{E}$ is the field of physics, $\mathbf{D}$ the field of bookkeeping — and good books are kept for good reasons.

Why is there no factor of one-third or geometry in the definition D = ε0 E + P?

Because the definition is local and universal — it must cancel $\rho_b = -\nabla\cdot\mathbf{P}$ out of Gauss's law at every point of every problem, and the divergence operator fixes the combination uniquely: only $\varepsilon_0\mathbf{E} + \mathbf{P}$ , with coefficient exactly 1 on $\mathbf{P}$ , does it. Geometry factors like the sphere's $\tfrac13$ belong to solutions (how big $\mathbf{E}$ and $\mathbf{P}$ come out for a given shape), never to the definition. When a $\tfrac13$ appears next to a $\mathbf{P}$ in later articles (Lorentz cavity, Clausius–Mossotti), it is the polarized-sphere geometry resurfacing — not a modification of $\mathbf{D}$ .

If D handles free charge and E handles everything, when would I ever compute with E directly in a dielectric problem?

Whenever the question is physical rather than bookkeeping: forces, energies, potentials, breakdown margins are all $\mathbf{E}$ -questions. The efficient workflow in symmetric problems is a relay — flux law gives $\mathbf{D}$ from $\rho_f$ ; constitutive law converts to $\mathbf{E}$ ; physics is extracted from $\mathbf{E}$ . In unsymmetric problems the relay reverses or collapses: you solve a boundary-value problem for $V$ (hence $\mathbf{E}$ ) with the dielectric's response built into the boundary/matching conditions, and $\mathbf{D}$ is assembled afterwards if a free-charge audit is wanted. The electret is the extreme case: there $\mathbf{D}$ is only computable from $\mathbf{E}$ and $\mathbf{P}$ , never the reverse.

Does the continuity of D-perp at the electret's surface mean D is always better behaved than E at boundaries?

It means $\mathbf{D}$ and $\mathbf{E}$ carry complementary boundary information, and the full rules — $D^\perp$ jumps by $\sigma_f$ (only free charge), $E^\parallel$ continuous always — are derived in Boundary Conditions for D and E. At the electret's pole, $D^\perp$ sails through smoothly (no free charge) while $E^\perp$ jumps by $\sigma_b/\varepsilon_0$ ; at the equator, $E^\parallel$ is continuous while $\mathbf{D}$ 's tangential part jumps with $P^\parallel$ — the very jump that carries the curl sheet. Neither field is globally "smoother"; each is smooth across exactly the charge its ledger ignores.

What exactly counts as free charge — is an ion in a crystal free or bound?

The split is operational, not metaphysical: free charge is whatever you account independently of the polarization model — conduction electrons, battery-delivered plate charge, ions implanted where they sit. Bound charge is whatever your $\mathbf{P}$ -field already encodes. An ion sitting in a lattice can be booked either way: include its displacement in $\mathbf{P}$ (ionic polarization — the standard choice for dielectric response) and it is bound; treat it as an embedded charge (a dopant, a trapped charge in an oxide) and it is free. The physics — total charge, total field — is identical under either booking; only the split of $\rho$ and the corresponding $\mathbf{D}$ change. Consistency is the sole rule: one ledger per problem, no double counting.

Why do many books write D = εE from the start — is that not the definition?

That is the constitutive relation for linear media, one article ahead, wearing the definition's clothes. $\mathbf{D} = \varepsilon_0\mathbf{E} + \mathbf{P}$ is always true — it defines $\mathbf{D}$ . In a linear dielectric, $\mathbf{P} = \varepsilon_0\chi_e\mathbf{E}$ additionally holds, whereupon $\mathbf{D} = \varepsilon_0(1 + \chi_e)\mathbf{E} = \varepsilon\mathbf{E}$ — a material law, valid for that class of materials only. Electrets and ferroelectrics break it flagrantly ( $\mathbf{P} \neq 0$ at $\mathbf{E}_{\rm applied} = 0$ ; our electret has $\mathbf{D}$ antiparallel to... rather, not proportional to $\mathbf{E}$ inside). Keeping definition and constitutive law in separate mental files is precisely what lets you handle the materials that make the news.

Is the D-field's bar-magnet pattern in the electret a coincidence?

No — it is a precise duality. For the magnetized sphere, $\mathbf{B}$ satisfies $\nabla\cdot\mathbf{B} = 0$ and carries curl from the magnetization; for the electret, $\mathbf{D}$ satisfies $\nabla\cdot\mathbf{D} = 0$ (no free charge) and carries curl from the polarization. Same differential structure, same sphere geometry, same solution: uniform $+\tfrac23$ of the moment density inside, dipole outside, closed loops throughout. The dual pair runs the other way too: the electret's $\mathbf{E}$ (curl-free, sourced by bound charge) mirrors the magnet's $\mathbf{H}$ . The magnetism chapter builds this correspondence properly; meeting it here inoculates against surprise when $\mathbf{H}$ turns out to be magnetism's bookkeeping field with exactly this article's fine print.