Linear Dielectrics, Susceptibility and Permittivity

Look at the datasheet of any capacitor, the specification of any coaxial cable, or the process notes of any semiconductor fab, and one number keeps appearing: the dielectric constant, $\kappa$ . In Dielectric Materials and Insulators we met it as a measured fact — Faraday's observation that filling a capacitor with insulating material multiplies its capacitance. What we did not have then was the machinery to predict anything with it. That machinery is this article. The claim to be earned is startlingly economical. A dielectric is a seething crowd of some $10^{28}$ molecules per cubic metre, each distorting and reorienting in an applied field. Yet for almost every insulator in almost every laboratory field, that entire response collapses to a single dimensionless number: the electric susceptibility $\chi_e$ , defined by $\mathbf{P}=\varepsilon_0\chi_e\mathbf{E}$ . Feed that one line into the displacement field of The Electric Displacement D and the whole apparatus of vacuum electrostatics survives with a single substitution, $\varepsilon_0\to\varepsilon$ . Fields shrink by $\varepsilon_r$ , capacitances grow by $\varepsilon_r$ , and a point charge in water whispers at one-eightieth of its vacuum voice. There is one trap, and it is the deepest idea here: the $\mathbf{E}$ in the constitutive law is the total macroscopic field, including the part produced by the polarization itself. The medium responds to a field it helps create. Owning that self-consistency loop — and knowing exactly when the beautiful $\varepsilon_0\to\varepsilon$ shortcut is licensed — is what separates using dielectrics from being fooled by them.

Key takeaways

A linear dielectric responds in proportion to the field: $\mathbf{P}=\varepsilon_0\chi_e\mathbf{E}$ , with $\chi_e$ the dimensionless electric susceptibility — and $\mathbf{E}$ is the total macroscopic field inside the material, not the applied one.
Linearity makes the displacement proportional too: $\mathbf{D}=\varepsilon_0(1+\chi_e)\mathbf{E}=\varepsilon\mathbf{E}$ , defining the permittivity $\varepsilon$ and the dielectric constant $\varepsilon_r=\varepsilon/\varepsilon_0=1+\chi_e=\kappa$ .
Linearity is generic, not lucky: laboratory fields ( $\sim10^6$ V/m) are a millionth of the atomic field scale ( $\sim5\times10^{11}$ V/m), so the first Taylor term rules.
In a homogeneous linear medium, bound charge shadows free charge: $\rho_b=-\dfrac{\chi_e}{1+\chi_e}\rho_f$ — no free charge in the bulk, no bound charge in the bulk.
Consequences with numbers: a point charge in water is screened to $E_{\text{vac}}/80$ ; a filled capacitor has $C=\varepsilon_r C_0$ ; a stacked air–pyrex capacitor computes region by region with $D$ continuous.
Key idea: linearity reduces dielectric response to a single number — $\mathbf{P}=\varepsilon_0\chi_e\mathbf{E}$ makes $\mathbf{D}=\varepsilon\mathbf{E}$ , so vacuum electrostatics carries over with $\varepsilon_0\to\varepsilon=\varepsilon_0\varepsilon_r$ .

The core insight

Linearity reduces dielectric response to a single number: $\mathbf{P}=\varepsilon_0\chi_e\mathbf{E}$ makes $\mathbf{D}=\varepsilon\mathbf{E}$ , so every vacuum electrostatics result holds with $\varepsilon_0\to\varepsilon=\varepsilon_0\varepsilon_r$ .

Where this lives today

The $\varepsilon_0\to\varepsilon$ substitution is arguably the highest-leverage single line in the electronics industry. Multilayer ceramic capacitors stack hundreds of micron-thin layers of titanate ceramics with $\varepsilon_r$ in the thousands — trillions are manufactured each year, and every one is Derivation C run in a material engineered for maximal $\chi_e$ . At the opposite extreme of size, the transistor gate is a capacitor whose dielectric was, for forty years, SiO₂ ( $\varepsilon_r=3.9$ ); when its thickness shrank toward a nanometre and electrons began tunnelling through, the industry swapped in hafnium-based "high-κ" dielectrics ( $\varepsilon_r\approx25$ , a $6.4\times$ boost) to keep the capacitance while thickening the barrier. Your phone works because someone chased a bigger $\chi_e$ . And the humblest entry in the table above, water's $\varepsilon_r\approx80$ , quietly runs biochemistry: the eightyfold screening of Example 1 is what lets Na⁺ and Cl⁻ ignore each other in solution, keeps DNA's charged backbone from collapsing onto itself, and makes water the solvent in which the chemistry of life is soluble. Measuring $\varepsilon_r(T)$ remains a working molecular probe — the Debye method — separating stretchy molecules from rotating ones by how their response fades with temperature.

Origins · 1 minPeter Debye and how this was discovered

Peter Debye

1884–1966 · Utrecht, Zürich, Leipzig, Cornell

Debye turned the dielectric constant from a passive material property into a molecular measuring instrument. In 1912, while the very existence of molecular dipoles was still conjecture, he separated dielectric response into an induced part (temperature-blind) and an orientational part fading as $1/T$ — so a plot of susceptibility against inverse temperature yields both the polarizability and the permanent dipole moment of a molecule (the analysis recalled from Permanent Dipoles and Orientational Polarization). Chemists ran the experiment on vapour after vapour, mapping molecular shapes decades before anyone could image one; the unit of molecular dipole moment — water's is $1.85$ debye, $6.17\times10^{-30}$ C·m — carries his name in lowercase, the physicist's quiet immortality. His 1936 Nobel Prize in Chemistry cited exactly this: dipole moments as probes of molecular structure. A theorist with an engineer's instincts, Debye kept one question in view throughout: what does the macroscopic number tell you about the molecule? — the same micro-to-macro thread this chapter follows into the Clausius–Mossotti Relation.

History

1837

— Faraday discovers specific inductive capacity: filling a capacitor multiplies its capacitance by a material constant

1850

— Mossotti connects the dielectric constant to the polarizabilities of molecular constituents

1879

— Clausius independently publishes the density–permittivity relation, completing the Clausius–Mossotti bridge

1887

— Hertz's waves travel through insulators as Maxwell's theory demands, tying ε_r to the refractive index

1912

— Debye splits dielectric response into induced and orientational parts, making ε(T) a molecular dipole-meter

1920

— Valasek finds a material (Rochelle salt) whose polarization defies linearity altogether — the first ferroelectric

2007

— Intel ships high-κ hafnium gate dielectrics, ending SiO₂'s forty-year reign in the transistor

The physical picture

Picture the dielectric as a sea of microscopic springs. Each molecule is either a stretchable charge cloud (Induced Dipoles and Atomic Polarizability) or a permanent dipole jostled by thermal motion (Permanent Dipoles and Orientational Polarization). Apply a field and the springs stretch, the dipoles lean; the material develops a polarization density $\mathbf{P}$ . For a linear medium the springs are Hookean — double the field, double the stretch — and the whole crowd's response is captured by one stiffness constant, $\chi_e$ . But here is the loop that makes dielectrics subtle. The stretched dipoles produce bound charge (the rules of Bound Charges: Surface and Volume), and that bound charge produces its own field, which opposes the applied one inside the material. The molecules do not feel the applied field; they feel the net field — applied minus the medium's pushback. So the response creates a field that modifies the response. The constitutive law $\mathbf{P}=\varepsilon_0\chi_e\mathbf{E}$ is written in terms of the total macroscopic $\mathbf{E}$ precisely so that this feedback is already settled: whatever equilibrium the tug-of-war reaches, at that equilibrium polarization and total field are proportional.

Figure

The self-consistency loop — the medium responds to a field it helps create

The macroscopic outcome of the loop is shielding. Free charge placed in or near a dielectric attracts a shadow of opposite bound charge, and the net field everywhere inside is the free-charge field divided by $\varepsilon_r$ . That single division is why a capacitor's capacitance rises by exactly $\varepsilon_r$ when filled, why ionic crystals dissolve in water ( $\varepsilon_r\approx 80$ screens the Coulomb grip between ions eighty-fold), and why the fields you compute in insulation are always gentler than the vacuum estimate. One more reading, inherited from Gauss's Law in Dielectrics: $\mathbf{D}$ is the field that never learned the material is there — its sources are free charges only. The constitutive law is the bridge back to physics: given $\mathbf{D}$ from the free charge, $\mathbf{E}=\mathbf{D}/\varepsilon$ hands you the real field that pushes on charges. And because the response is linear, superposition survives: fields of many free charges in a uniform dielectric simply add, each computed with $\varepsilon$ in place of $\varepsilon_0$ .

The constitutive law

Everything in this article flows from one experimental statement. For ordinary insulators in ordinary fields, polarization is proportional to the macroscopic field at the same point:

\mathbf{P}=\varepsilon_0\chi_e\mathbf{E} \qquad (1)

where: $\mathbf{P}$ — polarization density (C/m², from Polarization Density P); $\mathbf{E}$ — the total macroscopic field inside the medium (V/m); $\chi_e$ — the electric susceptibility, dimensionless because the conventional factor $\varepsilon_0=8.854\times10^{-12}$ F/m carries the units. Air at STP has $\chi_e=5.9\times10^{-4}$ ; water manages $\chi_e\approx 79$ . The one number $\chi_e$ is the material's entire electrostatic personality. Why should such a crowd of molecules obey so simple a law? Because laboratory fields are feeble on the atomic scale. The field binding an electron at one Bohr radius is $e/4\pi\varepsilon_0 a_0^2 \approx 5.1\times10^{11}$ V/m; the strongest field air can hold before sparking is $3\times10^6$ V/m — a ratio of about $6\times10^{-6}$ even at breakdown. Whatever the exact molecular response function $\mathbf{P}(\mathbf{E})$ is, we are probing only its first Taylor term, and for an isotropic material symmetry forces the even terms to vanish. Corrections enter at relative order $(E/E_{\text{atomic}})^2\sim10^{-11}$ : linearity is not an approximation you apologize for, it is eleven digits deep. (Pushing hard enough to see the next term is the business of nonlinear optics; materials that break linearity spontaneously are the business of Ferroelectrics and Spontaneous Polarization.) Insert (1) into the definition $\mathbf{D}\equiv\varepsilon_0\mathbf{E}+\mathbf{P}$ from The Electric Displacement D and the two fields lock into proportionality:

\mathbf{D}=\varepsilon_0\mathbf{E}+\varepsilon_0\chi_e\mathbf{E}=\varepsilon_0(1+\chi_e)\,\mathbf{E}\equiv\varepsilon\,\mathbf{E} \qquad (2)

where: $\varepsilon=\varepsilon_0(1+\chi_e)$ — the permittivity of the medium (F/m). This is the moment the auxiliary field earns its keep: in a linear medium $\mathbf{D}$ and $\mathbf{E}$ are parallel and interconvertible, so knowing either one is knowing both. The dimensionless version is the number on every datasheet:

\varepsilon_r\equiv\frac{\varepsilon}{\varepsilon_0}=1+\chi_e=\kappa \qquad (3)

where: $\varepsilon_r$ — relative permittivity, identical to the dielectric constant $\kappa$ measured in Dielectric Materials and Insulators. That article quoted $\kappa$ as tabulated fact; equation (3) is the promised formalism behind the table. For every ordinary material $\chi_e>0$ (the springs stretch with the field), so $\varepsilon_r\geq1$ , with equality only for vacuum.

material	$\varepsilon_r$	what it illustrates
vacuum	$1$ exactly	the reference; $\chi_e=0$ by definition
air (STP)	$1.00059$	gases barely polarize — vacuum formulas are safe in air
teflon	$2.1$	nonpolar solid: stretch response only
pyrex glass	$4.7$	typical hard insulator
silicon	$11.8$	semiconductor as high- $\varepsilon_r$ dielectric
water	$80.1$	permanent dipoles that can rotate — the solvent superpower

Combine (2) with the free-charge Gauss law $\nabla\cdot\mathbf{D}=\rho_f$ (recalled from Gauss's Law in Dielectrics, whose five-move ritual now gains a fast Move 3) and, for a homogeneous medium where $\varepsilon$ is one constant everywhere, the constant slides out of the divergence:

\nabla\cdot\mathbf{E}=\frac{\rho_f}{\varepsilon} \qquad (4)

where: homogeneity is doing real work. Equation (4) says vacuum electrostatics — every field, every potential, every capacitance formula — survives verbatim with $\varepsilon_0\to\varepsilon$ . That substitution is the entire computational payoff of this article, and it is licensed only while $\varepsilon$ is uniform over the region where the fields live. The moment $\chi_e$ jumps at an interface, pieces of bound charge appear there, and you need the matching rules of the next article, Boundary Conditions for D and E. Where does $\chi_e$ itself come from? From the molecular polarizabilities of Induced Dipoles and Atomic Polarizability — but the sum is subtler than $N\alpha$ , because each molecule feels its neighbours' fields, not the macroscopic average. That micro-to-macro bridge is owned by the Clausius–Mossotti Relation; here we take $\chi_e$ as measured and spend it.

Derivation AThe hidden-charge theorem — bound charge shadows free charge

The goal is the cleanest structural fact about homogeneous linear dielectrics: the bound volume charge is not free to appear wherever it likes — it materializes only where free charge sits, in a fixed negative proportion. The strategy is three substitutions: write $\rho_b$ in terms of $\mathbf{P}$ , use linearity to trade $\mathbf{P}$ for $\mathbf{E}$ , then use the macroscopic Gauss law to trade $\mathbf{E}$ 's divergence for $\rho_f$ . Watch where homogeneity is invoked — the theorem dies without it.

Step 1. Start from the bound-charge rule of Bound Charges: Surface and Volume, valid for any material:

\rho_b=-\nabla\cdot\mathbf{P}

Step 2. Substitute the constitutive law (1) and pull the constant $\chi_e$ through the divergence:

\rho_b=-\nabla\cdot\big(\varepsilon_0\chi_e\mathbf{E}\big)=-\varepsilon_0\chi_e\,\nabla\cdot\mathbf{E}

Why this needs homogeneity: if $\chi_e$ varied with position, the product rule would leave an extra term $-\varepsilon_0\mathbf{E}\cdot\nabla\chi_e$ — bound charge generated wherever the material's response changes, free charge or not. Interfaces are the extreme case of that term.

Step 3. Eliminate $\nabla\cdot\mathbf{E}$ using equation (4):

\rho_b=-\varepsilon_0\chi_e\cdot\frac{\rho_f}{\varepsilon_0(1+\chi_e)}=-\frac{\chi_e}{1+\chi_e}\,\rho_f=-\Big(1-\frac{1}{\varepsilon_r}\Big)\rho_f

Step 4. Read the result physically. Wherever $\rho_f=0$ — which is the bulk of essentially every problem, since free charge lives on electrodes and embedded conductors — the bound volume charge vanishes identically, no matter how strongly polarized the material is. All the medium's response is exiled to surfaces ( $\sigma_b$ ) and to the immediate location of embedded free charge, where it cancels the fraction $(1-1/\varepsilon_r)$ of it. Water hides $98.75\%$ of a charge placed in it; pyrex ( $\varepsilon_r=4.7$ ) hides $78.7\%$ . Two limits anchor the result: for vacuum, $\chi_e=0$ gives $\rho_b=0$ — nothing to polarize; as $\varepsilon_r\to\infty$ the fraction $\to1$ and the medium cancels free charge completely, which is exactly the conductor behaviour of Conductors in Electrostatic Equilibrium, reached here as a limit of dielectrics.

Figure

Screening in a linear dielectric — the medium hides all but one part in epsilon-r of an embedded charge

Go deeper · Derivation B · 1 minPoint charge embedded in an infinite linear dielectric

Derivation BPoint charge embedded in an infinite linear dielectric

The goal is the flagship of the $\varepsilon_0\to\varepsilon$ substitution, and the example Gauss's Law in Dielectrics explicitly deferred here: a free point charge $q$ at the origin of an unbounded homogeneous linear dielectric, solved end to end — field, polarization, and the screening cloud. The route is the displacement ritual of that article: symmetry gives $\mathbf{D}$ from free charge alone, and the constitutive law — the new ingredient — converts to $\mathbf{E}$ in one line. The thing to watch is the bound charge: a neutral medium must hide equal and opposite amounts somewhere.

Step 1. Spherical symmetry (point free charge, isotropic homogeneous medium) certifies $\mathbf{D}=D(r)\hat{\mathbf{r}}$ . Run $\oint\mathbf{D}\cdot\mathrm{d}\mathbf{a}=Q_{f,\text{enc}}$ on a concentric sphere of radius $r$ :

4\pi r^2 D = q \quad\Rightarrow\quad \mathbf{D}=\frac{q}{4\pi r^2}\hat{\mathbf{r}}

Why this is remarkable: $\mathbf{D}$ is byte-for-byte the vacuum field of $q$ — the medium has not entered at all yet.

Step 2. Convert with the constitutive law (2):

\mathbf{E}=\frac{\mathbf{D}}{\varepsilon}=\frac{q}{4\pi\varepsilon r^2}\hat{\mathbf{r}}=\frac{1}{\varepsilon_r}\,\mathbf{E}_{\text{vac}}

The physical field — the one a test charge feels, the one that stores energy — is the vacuum field divided by $\varepsilon_r$ . In water, an eightyfold whisper.

Step 3. Harvest the polarization:

\mathbf{P}=\varepsilon_0\chi_e\mathbf{E}=\frac{\chi_e}{1+\chi_e}\,\frac{q}{4\pi r^2}\hat{\mathbf{r}}

Radial, pointing away from a positive $q$ : every molecular dipole aims its negative end inward at the intruder.

Step 4. Locate the bound charge. In the bulk, $\rho_f=0$ so the hidden-charge theorem gives $\rho_b=0$ — the polarized bulk is rigorously neutral. The only place left is the origin itself. Surround $q$ with a shrinking sphere; the bound charge inside is the inward flux of $\mathbf{P}$ :

q_b=-\oint\mathbf{P}\cdot\mathrm{d}\mathbf{a}=-\frac{\chi_e}{1+\chi_e}\,q=-\Big(1-\frac{1}{\varepsilon_r}\Big)q

A screening cloud of opposite sign hugs the free charge at molecular range, and the net charge seen from any finite distance is $q+q_b=q/\varepsilon_r$ — which is precisely why Step 2's field is the vacuum field of $q/\varepsilon_r$ . The books balance globally too: the companion $+|q_b|$ sits on the medium's outer boundary, pushed off to infinity with it. Two checks close the loop: setting $\varepsilon_r=1$ returns $q_b=0$ and $\mathbf{E}=\mathbf{E}_{\text{vac}}$ ; and applying ordinary Gauss's law to the net charge $q+q_b$ reproduces the field of Step 2, confirming the free-plus-bound accounting is self-consistent.

Go deeper · Derivation C · 1 minFilled capacitor — why capacitance scales by exactly the dielectric constant

Derivation CFilled capacitor — why capacitance scales by exactly the dielectric constant

The goal is the law that made dielectrics famous, $C=\varepsilon_r C_0$ , derived so that you can see where the enhancement physically lives: in bound surface charge that cancels part of the plates' field. Take two large plates of area $A$ carrying fixed free charge $\pm\sigma_f A$ , separation $d$ , the gap completely filled with a linear dielectric $\varepsilon_r$ . The strategy: get $D$ from free charge (it cannot know about the filling), convert to $E$ , integrate to $V$ , divide.

Step 1. Planar symmetry and the displacement ritual give, between the plates,

D=\sigma_f

independent of the material — the same value the empty capacitor would have.

Step 2. The constitutive law converts:

E=\frac{D}{\varepsilon}=\frac{\sigma_f}{\varepsilon_0\varepsilon_r}

At fixed free charge, the field is $\varepsilon_r$ times weaker than the empty-gap value $\sigma_f/\varepsilon_0$ .

Step 3. The voltage follows by integrating the uniform field across the gap:

V=Ed=\frac{\sigma_f d}{\varepsilon_0\varepsilon_r}

Step 4. Divide charge by voltage:

C=\frac{\sigma_f A}{V}=\frac{\varepsilon_0\varepsilon_r A}{d}=\varepsilon_r\,C_0

Same free charge, $\varepsilon_r$ times less voltage, hence $\varepsilon_r$ times more capacitance. An $8.85$ pF air capacitor ( $A=1\ \text{cm}^2$ , $d=0.1$ mm) becomes $41.6$ pF filled with pyrex.

Step 5. Find the mechanism. The dielectric face against the positive plate carries

\sigma_b=\mathbf{P}\cdot\hat{\mathbf{n}}=-\varepsilon_0\chi_e E=-\Big(1-\frac{1}{\varepsilon_r}\Big)\sigma_f

Each plate's free charge is locally shadowed by opposite bound charge; a pillbox through plate-plus-face encloses net $\sigma_f/\varepsilon_r$ , and that reduced effective sheet is what generates Step 2's reduced field. Nothing mystical: the capacitance boost is bound charge doing partial cancellation at zero cost to you. As a check, $\varepsilon_r=1$ recovers $C_0=\varepsilon_0A/d$ with $\sigma_b=0$ . Energy bookkeeping of the same system — does filling raise or lower stored energy? the answer depends on what is held fixed — is taken up in Energy Stored in Dielectric Systems.

Limiting cases

Each word in "linear isotropic homogeneous dielectric" is a separate assumption, and each fails somewhere interesting.

case	result	meaning
vacuum / dilute gas	$\chi_e\to0$ , $\varepsilon_r\to1$	air's $\chi_e=5.9\times10^{-4}$ justifies every "ignore the air" you have ever committed
$\varepsilon_r\to\infty$	$\mathbf{E}\to0$ inside, $q_b\to-q$	the medium becomes a conductor: perfect screening as a limit of dielectric response
inhomogeneous, $\varepsilon(\mathbf{r})$	$\rho_b\neq-\frac{\chi_e}{1+\chi_e}\rho_f$ ; extra term $\propto\mathbf{E}\cdot\nabla\chi_e$	bound charge grows wherever the response varies; interfaces demand Boundary Conditions for D and E
anisotropic crystal	$P_i=\varepsilon_0\chi_{ij}E_j$ , $\mathbf{D}\nparallel\mathbf{E}$	one number becomes a tensor — the closing article Anisotropic Dielectrics and the Permittivity Tensor
high frequency	$\varepsilon(\omega)$ , complex	dipoles lag the drive; dispersion and absorption, deferred to the electromagnetic-waves chapter
strong field / ferroelectric	higher-order terms, hysteresis	nonlinear optics; spontaneous $\mathbf{P}$ with no linear law at all — Ferroelectrics and Spontaneous Polarization

Common trapThe chi in the law multiplies the total field, not the applied one

It is tempting to read $\mathbf{P}=\varepsilon_0\chi_e\mathbf{E}$ as "polarization = $\chi_e$ × the field I switched on." Wrong field. $\mathbf{E}$ is the net macroscopic field inside the material — the applied field already diminished by the bound charges' pushback. Use the applied field and you overstate $\mathbf{P}$ , sometimes catastrophically: in water the internal field is eighty times smaller than the free-charge field. When you must express the response to the applied field (a capacitor at fixed plate charge, a slab in a set external field), solve the self-consistency first — that is exactly what Derivations B and C did, and what the depolarization-factor machinery of Dielectric Sphere in a Uniform Field systematizes for curved bodies. And one level deeper: even the total macroscopic $\mathbf{E}$ is not the field a single molecule feels — that distinction belongs to Local Field and the Lorentz Cavity.

Worked examples

Example 1 — a charge dropped into water (confidence builder). A $q=1.0$ nC free charge sits deep inside a large tank of water, $\varepsilon_r=80.1$ . Find the field at $r=10$ cm, and the bound charge screening $q$ .

Step 1. Vacuum reference: $E_{\text{vac}}=q/4\pi\varepsilon_0r^2=(8.988\times10^9)(1.0\times10^{-9})/(0.10)^2=898.8$ V/m.

Step 2. In the dielectric, Derivation B gives $E=E_{\text{vac}}/\varepsilon_r=898.8/80.1=11.2$ V/m. The medium swallows nearly $99\%$ of the field.

Step 3. The screening cloud: $q_b=-(1-1/\varepsilon_r)q=-(1-1/80.1)(1.0\ \text{nC})=-0.9875$ nC pressed up against the charge. Net visible charge: $q/\varepsilon_r=12.5$ pC.

Step 4. Sanity check via polarization: $P(0.1\ \text{m})=\varepsilon_0\chi_e E=(8.854\times10^{-12})(79.1)(11.2)=7.86\times10^{-9}$ C/m²; its inward flux over the $10$ cm sphere is $-4\pi r^2P=-0.9875$ nC. The books balance.

Example 2 — the air–pyrex sandwich (exam level). A parallel-plate capacitor has $A=4.0$ cm², plate gap $d=2.0$ mm, held at $V=100$ V. A pyrex slab ( $\varepsilon_r=4.7$ ) of thickness $t=1.5$ mm lies on the bottom plate; the remaining $0.5$ mm is air ( $\varepsilon_r\approx1$ ). Find the fields, all charges, and the capacitance.

Step 1. Strategy: $D$ is normal to the layers and, with no free charge at the air–pyrex interface, the same constant in both regions (the ritual's pillbox: all $D$ -flux from the plates' $\sigma_f$ ). Let $D=\sigma_f$ , unknown for now.

Step 2. Fields region by region from $E=D/\varepsilon$ : $E_{\text{air}}=\sigma_f/\varepsilon_0$ and $E_{\text{slab}}=\sigma_f/4.7\varepsilon_0$ . The voltage constraint stitches them: $E_{\text{air}}(d-t)+E_{\text{slab}}\,t=V$ , i.e. $\dfrac{\sigma_f}{\varepsilon_0}\Big[(d-t)+\dfrac{t}{\varepsilon_r}\Big]=V$ . The bracket is an effective vacuum gap $d_{\text{eff}}=0.5+1.5/4.7=0.8191$ mm.

Step 3. Numbers: $E_{\text{air}}=V/d_{\text{eff}}=100/(8.191\times10^{-4})=122.1$ kV/m; $E_{\text{slab}}=122.1/4.7=26.0$ kV/m. Voltage split: air $61.0$ V, pyrex $39.0$ V (sums to $100$ V ✓). Note the air gap — the weaker dielectric — eats the larger field: the standard failure mode of voids in real insulation.

Step 4. Charges: $\sigma_f=\varepsilon_0E_{\text{air}}=1.081$ μC/m²; bound charge on the slab faces $|\sigma_b|=(1-1/4.7)\sigma_f=0.851$ μC/m² in magnitude (Derivation C's signed $\sigma_b=\mathbf{P}\cdot\hat{\mathbf{n}}$ is negative on the face toward the positive plate). Check: $(\sigma_f-\sigma_b)/\varepsilon_0=26.0$ kV/m $=E_{\text{slab}}$ ✓.

Step 5. Capacitance: $C=\varepsilon_0A/d_{\text{eff}}=(8.854\times10^{-12})(4.0\times10^{-4})/(8.191\times10^{-4})=4.32$ pF, with $Q=CV=0.432$ nC. Bracketing sanity: empty gap $1.77$ pF, fully filled $8.32$ pF — the sandwich sits between, at $2.44\times$ empty. $C=4.32$ pF.

Going further

References. Griffiths, Introduction to Electrodynamics, 4th ed., §4.4 · Jackson, Classical Electrodynamics, 3rd ed., §4.3–4.5 · Purcell & Morin, Electricity and Magnetism, 3rd ed., Ch. 10 · Zangwill, Modern Electrodynamics, Ch. 6 Key results collected here: $\varepsilon_r=1+\chi_e$ and $\mathbf{D}=\varepsilon\mathbf{E}$ ; screened point-charge bound charge $q_b=-(1-1/\varepsilon_r)q$ ; Clausius–Mossotti relation $\frac{\varepsilon_r-1}{\varepsilon_r+2}=N\alpha/3\varepsilon_0$ ; capacitance scaling $C=\varepsilon_r C_0$ ; $\varepsilon_0=8.854\times10^{-12}\,\mathrm{F/m}$ ; water $\varepsilon_r\approx 80$ , air $\varepsilon_r\approx 1.00059$ .

Builds on: Polarization Density P · The Electric Displacement D · Gauss's Law in Dielectrics · Induced Dipoles and Atomic Polarizability Leads to: Boundary Conditions for D and E · Dielectric Sphere in a Uniform Field · Dielectric Materials and Insulators

Does chi multiply the field I apply, or the field that ends up inside the material?

The field that ends up inside — total, self-consistent, including the bound charges' own contribution. This is a real practical distinction: apply a fixed external field to a water-filled gap and the internal field settles eighty times lower; $\mathbf{P}$ is $\chi_e\varepsilon_0$ times that. If you need response-to-applied-field, solve the feedback loop first (as the filled-capacitor derivation did). The constitutive law is written with the total field precisely because that form stays local and geometry-independent.

What is the difference between chi, epsilon, epsilon-r and kappa? Four symbols feels like three too many.

One physical fact, four dialects. $\chi_e$ (dimensionless) measures the response: $\mathbf{P}=\varepsilon_0\chi_e\mathbf{E}$ . $\varepsilon=\varepsilon_0(1+\chi_e)$ (F/m) is bundled for use in $\mathbf{D}=\varepsilon\mathbf{E}$ and Gauss's law. $\varepsilon_r=\varepsilon/\varepsilon_0$ strips the units back off, and $\kappa$ is the same number in its engineering costume. Given any one, you have all: pyrex's $\kappa=4.7$ means $\chi_e=3.7$ and $\varepsilon=4.16\times10^{-11}$ F/m. Only habit decides which appears in a given book.

Why is epsilon-r never less than 1? Could some exotic material push the field up instead of screening it?

For ordinary matter in static equilibrium, no. $\chi_e<0$ would mean dipoles aligning against the field — energetically uphill, an unstable arrangement that would spontaneously flip. So statics gives $\varepsilon_r\geq1$ : dielectrics always screen. The loophole is frequency: in the dispersive regime $\varepsilon_r(\omega)$ can dip below 1 and even go negative (plasmas, metals near their plasma frequency), because a driven oscillator can respond out of phase. That is dynamics, not statics — deferred to the waves chapter.

If D = εE lets me forget the material, why did Gauss's Law in Dielectrics insist D is not determined by free charge alone?

Both statements survive, because they answer different questions. The flux law pins $\mathbf{D}$ only in the three symmetric geometries; in general $\mathbf{D}$ has a curl-carrying part the free charge does not fix. What linearity plus homogeneity adds is a different rescue: since $\mathbf{E}$ is curl-free, $\mathbf{D}=\varepsilon\mathbf{E}$ is curl-free too when $\varepsilon$ is one constant — the pathological ingredient (the electret's frozen-in $\mathbf{P}$ ) is banned by the constitutive law. Break homogeneity, and pieces of the warning come right back at every interface.

Where does the positive partner of the screening charge go in the embedded-point-charge problem?

Global neutrality is honest but easy to lose track of. The medium contributes $q_b=-(1-1/\varepsilon_r)q$ hugging the charge; the balancing $+(1-1/\varepsilon_r)q$ sits as bound surface charge on the outer boundary of the dielectric. For a genuinely infinite medium that boundary is at infinity, where the finite total charge spreads to zero surface density and produces no field — which is why the derivation could ignore it. In any real tank, it is physically present on the walls; enclose the whole tank and you see the full $q$ again.

Water has epsilon-r of 80 — why don't capacitor makers just fill everything with water?

Because $\varepsilon_r$ is only one column of the datasheet. Water's screening comes from mobile, rotatable polar molecules — the same freedom that dissolves ions makes real water conductive, so a water capacitor leaks its charge in moments. Its response also collapses at high frequency (rotation cannot follow GHz drive), and dissolved impurities lower its breakdown strength. Engineering dielectrics optimize a bundle: high $\chi_e$ , negligible conduction, high breakdown field, low loss. Titanate ceramics and polymer films win that contest; water wins the one evolution entered it in.

The hidden-charge theorem says bound charge appears only where free charge is. But the dielectric between capacitor plates is clearly polarized — where is its bound charge?

On the surfaces. The theorem governs volume bound charge $\rho_b$ ; surface bound charge $\sigma_b=\mathbf{P}\cdot\hat{\mathbf{n}}$ is separate bookkeeping and very much alive — it is the whole mechanism of the capacitance boost. A uniformly polarized slab has $\rho_b=0$ in its interior (uniform $\mathbf{P}$ has zero divergence) while its two faces carry $\mp\sigma_b$ . When you hear "no bound charge in the bulk," hear "the action is at the boundaries."

Is the linear law instantaneous? What happens if the field oscillates fast?

Static $\chi_e$ assumes the molecules finish responding before you look. Stretch response (electron clouds) keeps up to optical frequencies, but orientational response (rotating molecules like water's) lags beyond gigahertz: the polarization falls out of phase, $\varepsilon$ becomes frequency-dependent and complex, and the imaginary part is absorption — your microwave oven is water's dielectric lag turned into dinner. Statics is the $\omega\to0$ edge of that story; the full $\varepsilon(\omega)$ belongs to the electromagnetic-waves chapter.