Electric Polarization and the Displacement Field

We inherit the machinery of this article ready-built from electrostatics. The companion aq-em-polarization-density-p has already established what the polarization field $\mathbf{P}$ is — the electric dipole moment per unit volume — and proved that a non-uniform $\mathbf{P}$ leaves behind real charge: a volume density $\rho_b=-\nabla\cdot\mathbf{P}$ and a surface density $\sigma_b=\mathbf{P}\cdot\hat{\mathbf{n}}$ . The companion aq-em-gausss-law-in-dielectrics has defined the displacement field $\mathbf{D}=\varepsilon_0\mathbf{E}+\mathbf{P}$ so that its divergence sees only the free charge, $\nabla\cdot\mathbf{D}=\rho_f$ . This article does not re-derive those; it recalls them by name and asks what they mean for light. The optical reframing is a shift of emphasis that changes everything. In electrostatics $\mathbf{P}$ is a passive response and $\mathbf{D}$ a clever piece of bookkeeping. In optics $\mathbf{P}$ is the star of the show: the oscillating induced polarization is a distribution of accelerating charge, and accelerating charge radiates. The wave you measure after light crosses a slab of glass is the original wave plus the field re-radiated by $\mathbf{P}$ — and it is that secondary radiation, lagging the driver, that slows light to $c/n$ and bends it at surfaces. Polarization is not the medium's reaction to the wave; at optical frequencies it is a co-author of the wave. The static definitions carry over unchanged into the frequency domain: replace $\mathbf{P}$ by $\mathbf{P}(\omega)$ and $\chi_e$ by the complex $\chi_e(\omega)$ , and $\mathbf{D}=\varepsilon_0\mathbf{E}+\mathbf{P}$ , $\nabla\cdot\mathbf{D}=\rho_f$ hold at every frequency. That is why $\mathbf{D}$ — not $\mathbf{E}$ — is the natural field for wave propagation in dielectrics, and why the whole of linear optics is a story about $\mathbf{P}(\omega)$ .

Key takeaways

Recalled from electromagnetism: $\mathbf{P}$ = dipole moment per unit volume; a polarized medium carries bound charge $\rho_b=-\nabla\cdot\mathbf{P}$ , $\sigma_b=\mathbf{P}\cdot\hat{\mathbf{n}}$ (aq-em-polarization-density-p); the displacement $\mathbf{D}=\varepsilon_0\mathbf{E}+\mathbf{P}$ satisfies $\nabla\cdot\mathbf{D}=\rho_f$ (aq-em-gausss-law-in-dielectrics).
The split $\mathbf{D}=\varepsilon_0\mathbf{E}+\mathbf{P}$ is a division of labour: $\varepsilon_0\mathbf{E}$ is the vacuum part, $\mathbf{P}$ is the material's own contribution.
A uniformly polarized slab has no volume bound charge ( $\rho_b=0$ ) but surface sheets $\sigma_b=\pm P$ ; their field inside is the depolarizing field $E_{\text{dep}}=-P/\varepsilon_0$ .
Optics' reframing: the time-harmonic $\mathbf{P}(\omega)$ is a radiating source — its re-radiated field, lagging the driver, is what slows light to $c/n$ and bends it at interfaces.
With $\mathbf{D}$ , only free charge sources the field; in a glass slab more than half of $\mathbf{D}$ is the polarization $\mathbf{P}$ itself.
Key idea: polarization $\mathbf{P}$ is the dipole moment per unit volume and, in optics, a radiating source; the displacement $\mathbf{D}=\varepsilon_0\mathbf{E}+\mathbf{P}$ repackages its bound-charge response so that $\nabla\cdot\mathbf{D}=\rho_f$ depends only on free charge.

The core insight

Polarization $\mathbf{P}$ is the dipole moment per unit volume; the displacement $\mathbf{D}=\varepsilon_0\mathbf{E}+\mathbf{P}$ repackages the medium's bound-charge response so that $\nabla\cdot\mathbf{D}=\rho_f$ sees only free charge. In optics the oscillating $\mathbf{P}$ is itself a radiating source, and its re-radiated field is what slows and bends light.

Where this lives today

Every high-permittivity capacitor and every gate dielectric in a microchip is an application of $\mathbf{D}=\varepsilon_0\mathbf{E}+\mathbf{P}$ : the free charge sets $\mathbf{D}$ , the material's polarization lowers $\mathbf{E}$ , and the capacitance rises by $\varepsilon_r$ — which is why a modern transistor can pack a huge field response into a nanometre-thin layer of hafnium oxide. Ferroelectrics like barium titanate, with $\varepsilon_r\sim10^3$ , push the polarization contribution of $\mathbf{D}$ to over 99.9%. In optics the same $\mathbf{P}$ , now oscillating at $10^{14}\ \mathrm{Hz}$ , is the source term in every simulation of light in matter. Nonlinear optics is written entirely as an expansion of $\mathbf{P}$ in powers of $\mathbf{E}$ , so that second-harmonic generation and the electro-optic effect are the second-order polarization $\mathbf{P}^{(2)}$ radiating new frequencies. When a laser pulse doubles its colour in a crystal, it is $\mathbf{P}$ — this article's field — doing the emitting.

The physical picture

Switch on a field across a slab of neutral atoms and every electron cloud drifts slightly against its nucleus, turning each atom into a tiny dipole. Deep inside the slab the positive end of one dipole abuts the negative end of its neighbour and the charges cancel; only at the two faces, where there is no neighbour to cancel, does a sheet of uncancelled charge survive — the bound surface charge $\sigma_b=\mathbf{P}\cdot\hat{\mathbf{n}}$ . This is the macroscopic fingerprint of $\mathbf{P}$ , and the electromagnetism owner article derives it in full. Our job is to read it optically.

Figure

A uniformly polarized slab: interior dipole charges cancel, leaving surface sheets σ_b = ±P whose field inside is the depolarizing field E_dep = −P/ε₀

The bound sheets set up their own field opposing the applied one, so the net $\mathbf{E}$ inside a dielectric is reduced. The displacement $\mathbf{D}$ is constructed to ignore this: it counts only the free charge an experimenter placed. In a capacitor $\mathbf{D}$ is pinned by the plate charge and is the same with or without dielectric, while $\mathbf{E}$ drops by $\varepsilon_r$ . The identity $\mathbf{D}=\varepsilon_0\mathbf{E}+\mathbf{P}$ is a clean division of labour — $\varepsilon_0\mathbf{E}$ is the vacuum field the same charge would make, $\mathbf{P}$ is what the material adds — and in a dense optical medium that material contribution is not a small correction but the majority of $\mathbf{D}$ .

Figure

The punchline: inside glass (ε_r = 2.25) the displacement splits as D = ε₀E + P with ε₀E = 44% and P = 56% — more than half of D is the material's own polarization

The core equations

Everything begins with the polarization, the dipole moment averaged over a volume small on the macroscopic scale but large enough to contain many atoms. This coarse-graining, and the bound charge it implies, are owned by aq-em-polarization-density-p; we quote the results.

\mathbf{P}(\mathbf{r})=\lim_{\Delta V\to 0}\frac{1}{\Delta V}\sum_{i\in\Delta V}\mathbf{p}_i,\qquad \rho_b=-\nabla\cdot\mathbf{P},\qquad \sigma_b=\mathbf{P}\cdot\hat{\mathbf{n}} \qquad (1)

where: $\mathbf{P}$ — polarization ( $\mathrm{C\,m^{-2}}$ ); $\mathbf{p}_i$ — dipole moment of the $i$ -th atom; $\rho_b,\sigma_b$ — bound volume and surface charge densities; $\hat{\mathbf{n}}$ — outward surface normal. A non-uniform polarization piles up net charge; a uniform one shows charge only at surfaces. To isolate the charge we control, electromagnetism defines the displacement field and, from the total-charge Gauss law, obtains its source (this is the content of aq-em-gausss-law-in-dielectrics).

\mathbf{D}=\varepsilon_0\mathbf{E}+\mathbf{P},\qquad \nabla\cdot\mathbf{D}=\rho_f \qquad (2)

where: $\mathbf{D}$ — electric displacement field ( $\mathrm{C\,m^{-2}}$ ); $\mathbf{E}$ — macroscopic electric field ( $\mathrm{V\,m^{-1}}$ ); $\rho_f$ — free (externally placed) charge density; $\varepsilon_0=8.854\times10^{-12}\ \mathrm{F\,m^{-1}}$ . The bound charge $-\nabla\cdot\mathbf{P}$ has been folded into $\mathbf{D}$ , so only $\rho_f$ remains as a source. For weak fields the polarization is linear in the field, which closes the system and — read at optical frequency — supplies the refractive index. This is the electrostatic constitutive relation of aq-em-linear-dielectrics-susceptibility-and-permittivity; optics promotes $\chi_e$ to $\chi_e(\omega)$ .

\mathbf{P}=\varepsilon_0\chi_e\mathbf{E},\qquad \mathbf{D}=\varepsilon\mathbf{E}=\varepsilon_0(1+\chi_e)\mathbf{E},\qquad n=\sqrt{1+\chi_e} \qquad (3)

where: $\chi_e$ — electric susceptibility (dimensionless); $\varepsilon_r=1+\chi_e$ — relative permittivity; $n$ — refractive index at the optical frequency. The proportion of $\mathbf{D}$ carried by the material is $P/D=1-1/\varepsilon_r$ , which climbs toward one for high-index media. The optical content is the reinterpretation of $\mathbf{P}$ as a radiating source. Because an oscillating dipole moment is an accelerating charge, the induced $\mathbf{P}(\omega)$ radiates a secondary field; the observed wave is driver plus re-radiation.

\mathbf{P}(\mathbf{r},t)=\text{oscillating dipole density}\;\Rightarrow\;\text{radiates }\mathbf{E}_{\text{rad}},\qquad \mathbf{E}_{\text{total}}=\mathbf{E}_{\text{driver}}+\mathbf{E}_{\text{rad}} \qquad (4)

where: $\mathbf{E}_{\text{rad}}$ — field re-radiated by the induced polarization; the phase lag of $\mathbf{E}_{\text{rad}}$ behind the driver is what makes the net wave travel at $c/n<c$ . This is the physical origin of the refractive index, elaborated in The Refractive Index and The Lorentz Oscillator Model.

Derivation ABound charge and the depolarizing field of a slab

The aim is to make the abstract $\sigma_b=\mathbf{P}\cdot\hat{\mathbf{n}}$ concrete on the simplest geometry, and to extract from it the depolarizing field — the self-field a polarized body sets up inside itself. We take a slab of thickness $d$ with uniform polarization $\mathbf{P}=P\hat{\mathbf{z}}$ across it, use the bound-charge results recalled from aq-em-polarization-density-p, and add the elementary field of two charged sheets. The payoff, $E_{\text{dep}}=-P/\varepsilon_0$ , is the field that partly cancels an applied field inside a dielectric and the reason $\mathbf{E}$ drops while $\mathbf{D}$ does not.

Step 1. No interior volume charge: $\rho_b=-\nabla\cdot\mathbf{P}=-\partial_z P=0$ because $P$ is constant. Why: bound volume charge counts the pile-up of a varying polarization; a uniform $\mathbf{P}$ has none, so all the charge is pushed to the boundaries.

Step 2. Surface charge on each face: $\sigma_b=\mathbf{P}\cdot\hat{\mathbf{n}}=+P$ on the top ( $\hat{\mathbf{n}}=+\hat{\mathbf{z}}$ ) and $-P$ on the bottom ( $\hat{\mathbf{n}}=-\hat{\mathbf{z}}$ ). Why: the outward normal flips between the two faces, so the same $\mathbf{P}$ deposits opposite charge sheets — a parallel-plate arrangement of bound charge.

Step 3. Field of the two sheets. Infinite sheets of $\pm\sigma_b$ separated by $d$ make a uniform interior field $E_{\text{dep}}=\sigma_b/\varepsilon_0=P/\varepsilon_0$ directed from $+$ to $-$ , i.e. opposite to $\mathbf{P}$ : $E_{\text{dep}}=-P/\varepsilon_0$ . Why: the depolarizing field always opposes the polarization that made it, which is why a dielectric weakens the field it sits in and why $\mathbf{P}$ saturates rather than running away.

Step 4. Consistency with $\mathbf{D}$ . Inside, $D=\varepsilon_0 E+P$ ; with the free charge fixed, $D$ is unchanged, and the induced $P$ lowers $E$ by exactly the depolarizing amount. Why: $\mathbf{D}$ tracks only free charge, so it is blind to the bound sheets — the whole reason it is the convenient field for interface problems and wave propagation. *Verified: $\rho_b=0$ , $\sigma_b=\pm P$ , and $E_{\text{dep}}=P/\varepsilon_0=1.13\times10^5\ \mathrm{V\,m^{-1}}$ for $P=1\ \mu\mathrm{C\,m^{-2}}$ in verify_aq-optics-electric-polarization-displacement-field.py.*

Go deeper · Derivation B · 1 minWhy P is a radiating source, and how it slows light

Derivation BWhy P is a radiating source, and how it slows light

The goal is to justify the optical reframing that distinguishes this article from its electrostatic parent: that the polarization is not merely a passive bookkeeping density but an active emitter whose radiation constitutes the refracted wave. We follow the field across a thin dielectric layer and show that the transmitted wave is the incident wave plus a re-radiated wave $90^\circ$ out of phase, whose sum is a wave retarded exactly as $c/n$ demands. This is the physical mechanism behind the index; the quantitative $\chi(\omega)$ is owned by the Lorentz-oscillator article.

Step 1. Drive the medium. An incident field $E_0\cos\omega t$ induces $P=\varepsilon_0\chi E_0\cos\omega t$ , so each slab of thickness $\delta z$ carries an oscillating dipole sheet. Why: the polarization tracks the field in a linear medium, so the driver writes an oscillating dipole density into the slab.

Step 2. The sheet radiates. An oscillating polarization sheet radiates a field proportional to $\dot{P}\propto -\sin\omega t$ — one quarter-cycle ( $90^\circ$ ) behind the driver. Why: radiation from an accelerating charge depends on the time-derivative of the current $\partial_t P$ , which for a cosine driver is a sine, i.e. a $90^\circ$ phase lag.

Step 3. Add driver and re-radiation. A carrier plus a small quadrature component is, to first order, the same carrier shifted in phase: $\cos\omega t - \epsilon\sin\omega t\approx\cos(\omega t+\epsilon)$ . A phase advance accumulating with distance is precisely a wave slowed to $c/n$ with $n=1+\tfrac12\chi$ for small $\chi$ . Why: a uniform phase shift per unit length is indistinguishable from a reduced phase velocity — the refracted wave is the interference of the original and the polarization's own radiation.

Step 4. Read off the index. Matching the accumulated phase to $\exp[i(n-1)\omega z/c]$ gives $n-1\propto\chi$ , recovering $n=\sqrt{1+\chi}\approx1+\tfrac12\chi$ for a dilute medium. Why: it identifies the refractive index as an interference effect — light is never "in" the glass travelling slowly; it is continuously rebuilt from delayed re-radiation. *Verified: the dilute expansion $n=\sqrt{1+\chi}\approx1+\chi/2$ and the $90^\circ$ quadrature phase are checked in verify_aq-optics-electric-polarization-displacement-field.py.*

Limiting cases

Case	Result	Meaning
Vacuum	$\mathbf{P}=0$ , $\mathbf{D}=\varepsilon_0\mathbf{E}$	No dipoles; $\varepsilon_r=1$ , $n=1$ ; $\mathbf{D}$ and $\mathbf{E}$ carry the same content
Uniform slab	$\rho_b=0$ , $\sigma_b=\pm P$ , $E_{\text{dep}}=-P/\varepsilon_0$	Bound charge only on the faces; the self-field opposes $\mathbf{P}$
Linear isotropic	$\mathbf{D}=\varepsilon_0(1+\chi_e)\mathbf{E}$ , $P/D=1-1/\varepsilon_r$	Glass, water: material carries a fixed fraction of $\mathbf{D}$
Dispersive optical	$\mathbf{P}(\omega)=\varepsilon_0\chi_e(\omega)\mathbf{E}(\omega)$	Complex, frequency-dependent; $\mathbf{P}$ radiates → dispersion + absorption
Anisotropic crystal	$D_i=\varepsilon_0\varepsilon_{ij}E_j$	$\varepsilon$ a tensor, $\mathbf{D}\nparallel\mathbf{E}$ — birefringence

Common trap“D is the real field and bound charge is a fiction”

It is exactly backwards. $\mathbf{E}$ is the field that exerts force on a charge ( $\mathbf{F}=q\mathbf{E}$ ); it is the physically fundamental field. $\mathbf{D}=\varepsilon_0\mathbf{E}+\mathbf{P}$ is the accounting construct, defined so that its divergence sees only free charge. And the bound charges $\rho_b=-\nabla\cdot\mathbf{P}$ , $\sigma_b=\mathbf{P}\cdot\hat{\mathbf{n}}$ are entirely real — genuine displaced electrons and nuclei that produce a measurable field and can be probed directly. What is "not really there" is any implication that $\mathbf{D}$ is a force field: it is not, and $\mathbf{D}\parallel\mathbf{E}$ only in isotropic linear media. In optics the reality of the bound charge is the whole point, because its acceleration is what radiates the refracted wave.

Worked examples

1 · Bound charge and depolarizing field of a polarized slab (confidence builder)

A dielectric slab is uniformly polarized to $P=1.0\ \mu\mathrm{C\,m^{-2}}$ across its thickness. Find the bound charge densities and the depolarizing field inside.

Step 1. Volume bound charge: $\rho_b=-\nabla\cdot\mathbf{P}=0$ , since $\mathbf{P}$ is uniform. All the bound charge lives on the two faces.

Step 2. Surface bound charge: $\sigma_b=\mathbf{P}\cdot\hat{\mathbf{n}}=\pm P=\mathbf{\pm 1.0\ \mu C\,m^{-2}}$ , positive on the face the polarization points out of, negative on the opposite face.

Step 3. Depolarizing field: the two sheets make a uniform interior field $E_{\text{dep}}=\sigma_b/\varepsilon_0=(1.0\times10^{-6})/(8.854\times10^{-12})=\mathbf{1.13\times10^{5}\ V\,m^{-1}}$ , directed opposite to $\mathbf{P}$ . This is the field that partially cancels any applied field inside the dielectric. σ_b = ±1.0 µC m⁻², E_dep = 1.13 × 10⁵ V m⁻¹.

2 · How much of D is the material? (exam level)

A glass of $\varepsilon_r=2.25$ sits in a capacitor whose fixed free charge gives $D=2.0\ \mu\mathrm{C\,m^{-2}}$ . Split $\mathbf{D}$ into its vacuum and material parts and find the field $E$ .

Step 1. $\mathbf{D}$ is fixed by free charge alone, $D=\sigma_f=2.0\ \mu\mathrm{C\,m^{-2}}$ , unchanged by the glass — that is the definition of $\mathbf{D}$ .

Step 2. Field in the glass: $E=D/(\varepsilon_0\varepsilon_r)=(2.0\times10^{-6})/(8.854\times10^{-12}\times2.25)=\mathbf{1.0\times10^{5}\ V\,m^{-1}}$ , so the vacuum part is $\varepsilon_0E=0.89\ \mu\mathrm{C\,m^{-2}}$ (44% of $D$ ).

Step 3. Material part: $P=D-\varepsilon_0E=D(1-1/\varepsilon_r)=2.0\times(1-1/2.25)=\mathbf{1.11\ \mu C\,m^{-2}}$ — 56% of the displacement. In a dense dielectric the polarization is the majority of $\mathbf{D}$ , and at optical frequency it is precisely this majority contribution that oscillates and radiates the refracted wave. ε₀E = 0.89 (44%), P = 1.11 µC m⁻² (56%).

Physics of the portrait

The displacement field is the quiet workhorse of electromagnetism-in-matter: an auxiliary construct that hides an unmanageable swarm of bound charges and leaves clean equations behind — and, in optics, the field whose oscillation carries the wave.

Origins · 1 minelectric displacement field and how this was discovered

electric displacement field

the auxiliary field D = ε₀E + P · electromagnetism of dielectric media

The electric displacement field $\mathbf{D}$ was Maxwell's device for taming matter. Faced with a dielectric holding an astronomical number of induced dipoles, he defined a field whose sources are only the free charges an experimenter controls, $\nabla\cdot\mathbf{D}=\rho_f$ , absorbing all the bound charge into the combination $\varepsilon_0\mathbf{E}+\mathbf{P}$ . The name "displacement" survives from his mechanical aether model, in which $\mathbf{D}$ measured the elastic displacement of a charged medium; the mechanics is gone but the field remains indispensable. Its subtle status — a bookkeeping construct, not a force field — has confused students for a century, yet its payoff is enormous: with $\mathbf{D}$ , every capacitor and every dielectric interface becomes solvable without tracking induced charge, and the boundary condition $D_\perp$ continuous is one of the four that generate the Fresnel equations. In optics $\mathbf{D}$ takes on a second life as the field that naturally carries $\mathbf{P}(\omega)$ into the wave equation, so that the medium's radiating polarization — not an abstraction but a real oscillating charge density — becomes the origin of the refractive index itself.

History

1837

— Faraday introduces "specific inductive capacity", the first quantitative measure of a dielectric's effect

1850

— Mossotti models a dielectric as conducting spheres, linking macroscopic response to microscopic structure

1873

— Maxwell formalises the displacement field

\mathbf{D}

in his Treatise

1879

— Clausius derives what becomes the Clausius–Mossotti relation for the density dependence of

\varepsilon_r

1900

— Drude's electron theory recasts

\mathbf{P}

as the response of mobile and bound electrons

Going further

Texts. Griffiths, Introduction to Electrodynamics, 4th ed., ch. 4 (fields in matter) · Hecht, Optics, 5th ed., ch. 3 · Jackson, Classical Electrodynamics, 3rd ed., §4.3–4.5 · Feynman, Lectures on Physics, Vol. II, ch. 32 (the index of refraction, the radiating- $\mathbf{P}$ picture) · Saleh & Teich, Fundamentals of Photonics, 3rd ed., ch. 5.

Provenance. The polarization $\mathbf{P}$ , the bound-charge results $\rho_b=-\nabla\cdot\mathbf{P}$ , $\sigma_b=\mathbf{P}\cdot\hat{\mathbf{n}}$ , and the displacement field $\mathbf{D}=\varepsilon_0\mathbf{E}+\mathbf{P}$ with $\nabla\cdot\mathbf{D}=\rho_f$ are recalled by name from the electromagnetism owners aq-em-polarization-density-p and aq-em-gausss-law-in-dielectrics, which own their derivation; this article owns the slab depolarizing field and the optical reframing of $\mathbf{P}$ as a radiating source (the Feynman/Ewald–Oseen picture). A verifier should confirm: $\rho_b=0$ , $\sigma_b=\pm P$ for a uniform slab; $E_{\text{dep}}=P/\varepsilon_0=1.13\times10^5\ \mathrm{V\,m^{-1}}$ for $P=1\ \mu\mathrm{C\,m^{-2}}$ ; $P/D=1-1/\varepsilon_r=0.556$ for $\varepsilon_r=2.25$ ; $n=\sqrt{1+\chi}\approx1+\chi/2$ . $\varepsilon_0=8.854\times10^{-12}\ \mathrm{F\,m^{-1}}$ is CODATA.

Prerequisites. Electric Polarization and the Displacement Field builds on electrostatics and Gauss's law · the electric dipole · the divergence theorem. Leads to. Dielectric Susceptibility and Permittivity · Maxwell's Equations in Dielectric Media · The Refractive Index · The Lorentz Oscillator Model.

If D isn't a force field, why do we bother with it?

Because it makes dielectric problems solvable. The field that pushes charges around is $\mathbf{E}$ , but $\mathbf{E}$ is sourced by all the charge, free and bound, and the bound charge depends on $\mathbf{E}$ itself — a self-consistent tangle. Defining $\mathbf{D}=\varepsilon_0\mathbf{E}+\mathbf{P}$ so that $\nabla\cdot\mathbf{D}=\rho_f$ breaks the tangle: you compute $\mathbf{D}$ from the free charge you placed, then get $\mathbf{E}=\mathbf{D}/\varepsilon$ from the material's $\varepsilon_r$ . In a capacitor $\mathbf{D}$ is fixed by the plate charge regardless of what dielectric you slide in — an enormous simplification. It is a tool, and a superb one, but not the thing a test charge feels.

What does it mean that the polarization "radiates"?

An oscillating dipole is an accelerating charge, and accelerating charge emits electromagnetic radiation. When light drives a dielectric, the induced polarization $\mathbf{P}(\omega)$ is a dense array of oscillating dipoles, and each radiates a little wave. The total field beyond the medium is the original driving wave plus the sum of all these re-radiated wavelets. Because the re-radiation lags the driver by $90^\circ$ , the sum is the same wave shifted in phase — and a phase shift that accumulates with distance is exactly a wave travelling slower than $c$ . This is the Ewald–Oseen picture, and it is why the refractive index is fundamentally an interference effect, not light "moving through molasses".

Is bound charge really the same kind of charge as free charge?

Physically, yes — both are electrons and nuclei, and both source $\mathbf{E}$ identically. The only distinction is bookkeeping: "free" charge is what you can move around with a wire and a battery (conduction electrons, ions you deposit); "bound" charge is displaced electrons that snap back when the field is removed. A bound-charge sheet on a polarized slab produces a perfectly measurable field and will, for instance, attract a nearby test charge. The labels reflect controllability, not a difference in the physics of the charge, which is why $\mathbf{D}$ (tracking free charge) is convenient while $\mathbf{E}$ (tracking all charge) is fundamental.

Why does E drop inside a dielectric but D does not?

Because they answer different questions. $\mathbf{D}$ is fixed by the free charge, which you did not change when you inserted the dielectric, so $\mathbf{D}$ is unchanged. $\mathbf{E}$ , however, is sourced by all the charge, and the newly induced bound sheets set up a depolarizing field $-P/\varepsilon_0$ that opposes the applied field — so $\mathbf{E}$ falls by the factor $\varepsilon_r$ . The relation $\mathbf{E}=\mathbf{D}/(\varepsilon_0\varepsilon_r)$ packages exactly this: same $\mathbf{D}$ , smaller $\mathbf{E}$ , the difference being the material's own polarization $\mathbf{P}$ .

How does this connect to birefringence?

Through the constitutive relation becoming a tensor. In an isotropic medium $\mathbf{P}=\varepsilon_0\chi\mathbf{E}$ is a simple proportionality and $\mathbf{D}\parallel\mathbf{E}$ . In an anisotropic crystal the electrons are easier to displace along some axes than others, so $\chi$ — and hence $\varepsilon$ — is a tensor, $D_i=\varepsilon_0\varepsilon_{ij}E_j$ , and $\mathbf{D}$ need not be parallel to $\mathbf{E}$ . The wave then sees a direction-dependent index, and a single ray splits into two with different phase velocities. Every wave plate, calcite crystal, and Nicol prism runs on the fact that $\mathbf{P}$ responds anisotropically — the tensor generalisation of this article's scalar $\chi_e$ .

Where does the frequency dependence of χ come from physically?

From the inertia and damping of the charges that make up $\mathbf{P}$ . A permanent molecular dipole (as in water) reorients slowly and contributes a huge response only at low frequency; ionic displacements respond up to infrared; bound electrons respond up to the ultraviolet. As the driving frequency climbs past each mechanism's natural response rate, that mechanism drops out and $\chi(\omega)$ steps down — which is why water's static $\varepsilon_r$ is $80$ but its optical $\varepsilon_r$ is only $1.77$ . The detailed shape of $\chi(\omega)$ , with resonances and their associated absorption, is the Lorentz oscillator model; this article establishes only that $\mathbf{P}$ , and hence $\chi$ , is what carries that frequency dependence into the wave.