Conductors, insulators, and polarization

A cubic centimetre of copper holds about 8.5 × 10²² conduction electrons, one per atom, each belonging to the crystal rather than to any particular ion. That is 1.4 × 10⁴ C of mobile charge sitting in a piece of metal you can hold. Charging that same centimetre cube to 1 μC takes 6.2 × 10¹² electrons away, seven parts in a hundred thousand million of what is there, which is why a charged conductor is still, chemically and mechanically, ordinary copper. In an insulator every electron is bound to its own molecule and cannot go further than a fraction of a molecular width. The two names label the ends of one continuum: resistivity runs from 1.7 × 10⁻⁸ Ω m for copper to about 10¹⁶ Ω m for fused quartz, twenty-four decades, with semiconductors, salt water and damp air spread across the middle.

A mobile carrier feels F = qE. If any macroscopic field survived inside the conducting material, those carriers would still be accelerating and the arrangement would not be static. So electrostatic equilibrium in a conductor means exactly one thing: E = 0 at every point inside the metal. Two results follow with no further machinery. Excess charge cannot rest in the interior, because charge in the interior would source a field there, so it collects on the outer surface — spray 25 nC onto a copper shell of outer radius 5.0 cm and it settles over 3.14 × 10⁻² m² as a surface density of 0.80 μC m⁻². And the field just outside must meet the surface at a right angle, since any component along the surface would drive the surface charge sideways and the arrangement would still be changing.

Excess charge placed inside a conductor drains to the surface with a characteristic time τ = ε₀ρ, set by the material's resistivity alone. For copper, ρ = 1.68 × 10⁻⁸ Ω m gives τ = 1.5 × 10⁻¹⁹ s. That answer should make you suspicious: it is far shorter than the mean time between electron collisions, roughly 2.5 × 10⁻¹⁴ s, so the smooth continuum picture has outrun its own assumptions. The honest statement is that redistribution in a metal is over within about 10⁻¹⁴ s. Feed the same formula fused quartz, ρ ≈ 10¹⁶ Ω m, and τ = 8.9 × 10⁴ s, roughly a day — one expression covering both ideal materials, which differ only by a number. In practice a charged insulator leaks far sooner than that through a film of adsorbed water on its surface, which is why electrostatics demonstrations fail on a humid afternoon.

Two mechanisms, and they behave differently. Induced polarization stretches: the electron cloud of an atom slides bodily against its nucleus, giving p = αE. A hydrogen atom, with α/(4πε₀) = 0.667 × 10⁻³⁰ m³, in a field of 1.0 × 10⁶ V m⁻¹ takes on p = 7.4 × 10⁻³⁵ C m, a displacement of 4.6 × 10⁻¹⁶ m — about nine millionths of the atom's own radius. That tininess is why the response is linear. Orientational polarization turns: a molecule that is already lopsided, such as hydrogen chloride with p₀ = 3.6 × 10⁻³⁰ C m, is not stretched but swung round. In the same field the alignment energy p₀E = 3.6 × 10⁻²⁴ J is dwarfed at 300 K by kT = 4.1 × 10⁻²¹ J, so the mean alignment is only ⟨cos θ⟩ ≈ p₀E/(3kT) = 2.9 × 10⁻⁴ — yet even that leaves ⟨p⟩ = 1.0 × 10⁻³³ C m along the field, fourteen times the hydrogen atom's stretched moment. Heating weakens the turning and leaves the stretching alone, which is how the two are told apart in the laboratory.

Define the polarization P as the dipole moment per unit volume, P = np for n dipoles per cubic metre. Inside a uniformly polarized slab the positive end of each dipole sits beside the negative end of its neighbour and the pair contributes nothing; only the two faces are left uncancelled, carrying bound surface densities σ_b = P and −σ_b. Run that definition backwards to feel how small the underlying shift is. Suppose a 1.0 cm² face of such a slab is measured to carry 1.0 nC: σ_b = P = 1.0 × 10⁻⁹ C ÷ 1.0 × 10⁻⁴ m² = 1.0 × 10⁻⁵ C m⁻². With n = 1.0 × 10²⁸ molecules per cubic metre that is p = P/n = 1.0 × 10⁻³³ C m apiece, one electronic charge moved 6.2 × 10⁻¹⁵ m, about two parts in a hundred thousand of a molecular width. Two points to keep. The faces are equal and opposite, so the slab is still exactly neutral — polarization redistributes, it does not create. And bound charge cannot be drawn off down a wire, which is the operational difference between it and the free charge on a conductor.

Grounding means connecting a conductor to something so large that any charge your experiment handles leaves its state unmeasurably changed. The Earth carries about −4.6 × 10⁵ C spread over 5.1 × 10¹⁴ m² of surface; handing it 1 μC alters that by two parts in a million million. A grounded conductor can therefore take in or give up mobile charge freely, and its carriers settle wherever the force on them vanishes. Two consequences. Grounding works on conductors only: touch a wire to one corner of a charged plastic rod and you drain that corner, because nothing elsewhere in the rod can travel to the wire. And grounding does not automatically neutralize — with another charged body held nearby, the carriers settle into a distribution that carries net charge, and what is left is decided by the order in which the wire and the neighbour are removed.