Dilemma of Dielectric

Question

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1. What the student is really asking

The student is trying to reconcile three statements that seem to contradict one another

Statement	What it says (in the student’s words)
A “perfect dielectric”	should have infinite resistivity (i.e. zero conductivity) but the bound electrons can still be displaced to produce polarization.
A “perfect conductor”	should have infinite dielectric constant (i.e. “high dielectric”) while the free electrons are completely un‑bound and move without resistance.
General intuition	If electrons are bound they should not be called “high‑resistivity”; if they are free they should not be called “high‑dielectric”.

The question, therefore, is:

How can a material be a perfect dielectric (no conduction) yet still polarize, and why is a perfect conductor sometimes described as having an “infinite dielectric constant”? What is the precise relationship between conductivity, resistivity, and dielectric constant?

2. Step‑by‑step answer

Below we build the answer from the ground up, using the definitions that appear in Maxwell’s equations.

2.1 Basic material parameters

Symbol	Quantity	Definition	Units
σ (sigma)	Electrical conductivity	Ratio of conduction current density J to electric field E: (\mathbf J = \sigma \mathbf E) (Ohm’s law for a homogeneous medium).	S m(^{-1}) (siemens per metre)
ρ (rho)	Resistivity	Reciprocal of conductivity: (\rho = 1/\sigma).	Ω m
ε (epsilon)	Permittivity	Ratio of electric displacement D to field E: (\mathbf D = \varepsilon \mathbf E).	F m(^{-1})
ε₀	Vacuum permittivity	(\varepsilon_0 = 8.854\,187\,817\cdots\times10^{-12}\,\text{F m}^{-1}).
εᵣ (epsilon‑r)	Relative permittivity or dielectric constant	(\varepsilon_{\rm r}= \varepsilon/\varepsilon_0).
χₑ	Electric susceptibility	(\varepsilon = \varepsilon_0(1+\chi_e)).

Key point: Conductivity (σ) tells us how free charges move in response to a field (a real current).
Permittivity (ε) tells us how bound charges shift in response to a field (a displacement current). They are independent material properties.

2.2 Maxwell’s equations in matter (macroscopic form)

The two relevant equations are

[ \boxed{\mathbf J_{\rm total}= \mathbf J_{\rm conduction} + \mathbf J_{\rm displacement}} ]

[ \mathbf J_{\rm conduction}= \sigma \mathbf E \qquad\text{(Ohmic current)} ]

[ \mathbf J_{\rm displacement}= \frac{\partial \mathbf D}{\partial t} =\varepsilon\frac{\partial \mathbf E}{\partial t} \qquad\text{(due to polarization of bound charges)} ]

Thus the total current density that appears in Ampère‑Maxwell law is

[ \mathbf J_{\rm total}= \sigma\mathbf E+\varepsilon\frac{\partial \mathbf E}{\partial t}. ]

If the field is static ((\partial\mathbf E/\partial t=0)), only the conduction term survives.
If the field is time‑varying, both terms can be present.

2.3 Perfect dielectric

A perfect dielectric is defined mathematically by

[ \sigma = 0\qquad\Longleftrightarrow\qquad \rho = \infty . ]

Consequences

No conduction current can flow, no matter how large a static field you apply.
Polarization is still possible because the bound electrons (or molecular dipoles) can be displaced. This displacement is captured entirely by the permittivity ε, which remains finite (typically (\varepsilon_{\rm r}=2)–(10^3) for real dielectrics).
In a time‑varying field, the displacement current (\varepsilon\partial\mathbf E/\partial t) is the only current that can exist.

Physical picture: The electrons are “bound” to nuclei by the atomic potential. When an external field is applied, each electron cloud is shifted a tiny amount relative to its nucleus, creating a microscopic dipole. The net charge does not move from point to point, so no macroscopic conduction current appears. Energy is stored in the electric field as (\tfrac12\varepsilon E^{2}).

2.4 Perfect conductor

A perfect conductor is defined by

[ \sigma \rightarrow \infty \qquad\text{(or equivalently } \rho\rightarrow0\text{)}. ]

Consequences

Any static electric field inside the material would immediately produce an infinite conduction current, which is impossible. The only self‑consistent static solution is

[ \boxed{\mathbf E_{\text{inside}} = 0}. ]
Surface charges rearrange themselves until the interior field vanishes. The surface may support a surface charge density that produces the required external field.
The displacement term can be written as

[ \mathbf J_{\rm displacement}= \varepsilon\frac{\partial\mathbf E}{\partial t}. ]

Since (\mathbf E) inside a perfect conductor is zero (even for a time‑varying external field, the interior field remains zero at all times), the displacement current inside is also zero.

However, if we insist on keeping the relationship (\mathbf D = \varepsilon\mathbf E) while letting (\mathbf E\to0) but keeping (\mathbf D) finite (because surface charge may be finite), we can think of the effective permittivity as

[ \varepsilon_{\text{eff}} = \frac{D}{E}\;\; \xrightarrow[E\to0]{}\;\; \infty . ]

Hence textbooks sometimes say a perfect conductor has an “infinite dielectric constant”. It is a limit rather than a literal material property: the static electric field is forced to zero, so the ratio (D/E) diverges.
In practice, real metals have a very large but finite σ (≈ 10⁷ S m(^{-1})) and a finite ε that is dominated by the conduction term at low frequencies. At optical frequencies the conduction term becomes comparable to the displacement term, leading to a complex permittivity.

2.5 Why “high dielectric constant” ≠ “good conductor”

Property	High εᵣ (dielectric constant)	High σ (conductivity)
Microscopic origin	Strong polarizability of bound charges (large χₑ)	Large density of free carriers (electrons or holes)
Energy storage	Large (\tfrac12\varepsilon E^{2}) → good capacitor material	Not relevant; energy dissipated as Joule heating
Loss (for AC)	Determined by loss tangent (\tan\delta = \sigma/(\omega\varepsilon)). A high εᵣ can still have tiny σ → low loss.	Large σ → large conduction loss (heat).
Example	Barium titanate (εᵣ≈ 5000, σ≈10⁻⁸ S m(^{-1}))	Copper (σ≈ 5.8 × 10⁷ S m(^{-1}), εᵣ≈ 1)

Thus a material can have a very large dielectric constant while still being an excellent insulator because its conductivity remains essentially zero.

2.6 Summarising the two “perfect” limits

Quantity	Perfect dielectric	Perfect conductor
Conductivity σ	0 (no free‑charge flow)	∞ (free‑charge flow unlimited)
Resistivity ρ	∞	0
Permittivity ε (static)	Finite (material‑dependent)	∞ (limit (D/E) as (E\to0))
Electric field inside (static)	Can be non‑zero (field penetrates)	0
Polarization P	Finite (bound charges shift)	Not defined (bound charges negligible compared with free charges)
Energy storage	(\tfrac12\varepsilon E^{2}) (capacitor)	No static field → no stored electric energy (magnetic energy may dominate)

The apparent “conflict” in the Wikipedia wording stems from mixing two different limits:

“High dielectric constant” refers to large εᵣ (strong bound‑charge response).
“Perfect conductor” refers to σ→∞, and the mathematical limit ε→∞ follows from the condition E=0 inside.

3. Final answer (concise)

A perfect dielectric has zero conductivity (σ = 0, ρ = ∞) but a finite permittivity ε. The bound electrons can be displaced, producing polarization and storing electric energy, but no free charge moves; thus the material is an ideal insulator.
A perfect conductor has infinite conductivity (σ → ∞, ρ → 0). In the static case the interior electric field must be zero; mathematically this forces the ratio D/E (the permittivity) to diverge, so one may say it has an “infinite dielectric constant”. The material’s response is dominated by free‑charge motion, not by bound‑charge polarization.
“High dielectric constant” and “high conductivity” are independent properties. A material can possess a huge εᵣ while still having negligible σ, and vice‑versa.

4. Common mistakes (and how to avoid them)

Mistake	Why it’s wrong	How to correct it
Confusing εᵣ with σ. Thinking that a large dielectric constant automatically means the material conducts electricity.	εᵣ measures how strongly bound charges polarize; σ measures how easily free charges move.	Always ask: Is the current due to free carriers (σ) or to dipole displacement (ε)?
Saying “electrons move in a dielectric”. Interpreting polarization as a flow of charge through the bulk.	Polarization is a local shift of bound charge, not a net transport of charge from one side to the other.	Visualise each atom as a tiny dipole; the centre‑of‑charge of the whole sample does not move.
Equating “infinite ε” with a real material property. Believing a material can literally have infinite permittivity.	The “infinite ε” of a perfect conductor is a limit that results from forcing E=0 inside; no real material has ε = ∞.	Remember that “∞” appears only in the ideal mathematical model, not in measured values.
Thinking that a perfect dielectric can store unlimited charge.	The charge that can be stored is limited by the breakdown field of the material; ε only tells how much energy per unit field is stored.	Distinguish **

Original question: Dilemma of Dielectric on Physics Stack Exchange, licensed CC BY-SA.