Conductors, shielding and field mapping

Copper offers roughly 8.5 × 10²⁸ free electrons in every cubic metre, none of them tied to one atom. Switch on a field inside that metal and every one of them is pushed, and the only way the pushing stops is for charge to pile up until the field it makes cancels the field that moved it. So E = 0 inside a conductor is not a property of the substance, it is the end state of a motion — and it arrives fast. The characteristic time is τ = ε₀ρ, which for copper's resistivity 1.7 × 10⁻⁸ Ω m is 1.5 × 10⁻¹⁹ s. Two consequences follow at once. Draw any closed surface inside the material: E = 0 everywhere on it, so no flux crosses it, and Gauss's law turns that into zero enclosed charge — which drives every excess charge out onto a surface. And a wire carrying a steady current is not in this state at all, because there a field of order 10⁻² V m⁻¹ inside the metal is exactly what keeps the electrons drifting.

Nothing about a metal blocks a field. Superposition still holds term by term: the external source's field is present at every point inside the copper, as strong as it was before the metal arrived, and the induced surface charge contributes a second field that is equal and opposite there. Zero is a sum. Take a wide slab in a uniform external field of 5.0 × 10⁴ N C⁻¹. Its two faces acquire sheets of −σ and +σ, and a pair of sheets of density σ makes a field σ/ε₀ in the gap between them and nothing outside, so setting σ/ε₀ = E₀ gives σ = ε₀E₀ = 8.85 × 10⁻¹² × 5.0 × 10⁴ = 4.4 × 10⁻⁷ C m⁻². On a 10 cm × 10 cm face that is 4.4 nC — a countable amount of charge that really had to move. The same picture says what shielding cannot do: those induced charges push back on whatever made the external field, so the source still feels the metal.

Just outside the metal the field has no component along the surface. If it had one, that component would drag surface charge sideways and the conductor would not be in equilibrium — so field lines always meet a conductor at 90°, and the surface is an equipotential. The magnitude comes from a thin pillbox straddling a patch of surface, one face just outside and one just inside. The inside face sits in dead metal and passes no flux, the sides pass none if the box is thin enough, so the whole flux EA leaves through the outer face and Gauss's law gives EA = σA/ε₀, that is E = σ/ε₀. The factor deserves a pause. An isolated charged sheet in free space gives σ/2ε₀ on each side; a conductor's surface gives twice that, because the patch's own σ/2ε₀ is joined by another σ/2ε₀ from all the rest of the conductor's charge — the same contribution that cancels the patch's field on the inside. So a meter reading 2.3 × 10⁴ N C⁻¹ just off the metal is reporting σ = 2.0 × 10⁻⁷ C m⁻² on the patch beneath it.

Hollow the conductor out and the empty cavity is dead. No field line can cross it: such a line would leave the wall at one place and land at another, and closing the path back through the metal, where E = 0, would give a non-zero net push around a closed loop — something an electrostatic field never does. So the cavity wall stays bare and the cavity holds no field, whatever storm of charge rages outside. Now put +5.0 nC in the cavity of a shell whose own net charge is −2.0 nC. A closed surface drawn inside the metal must again enclose nothing, so the wall takes −5.0 nC, piled up to match wherever the charge sits. Conservation then leaves −2.0 + 5.0 = +3.0 nC on the outer surface, and that layer is plainly visible from outside: at 10 cm from the centre a test charge feels the field of 3.0 nC. Shielding is one-way for an isolated shell. Earth it and the outer layer drains away, the external field goes with it, and the screening finally works both ways.

Surface charge only spreads evenly when the surface is symmetric. Join two distant spheres of radii R₁ and R₂ with a wire and they become one conductor, so charge flows until the potentials match: Q₁/R₁ = Q₂/R₂. Since σ = Q/(4πR²), that means σ ∝ 1/R, or σ₁R₁ = σ₂R₂, and with E = σ/ε₀ the surface field is strongest where the radius is smallest. A 1.0 mm bump on a dome sitting at 9.0 kV sees roughly 9.0 kV ÷ 1.0 mm = 9.0 × 10⁶ V m⁻¹, three times the 3.0 × 10⁶ V m⁻¹ at which air breaks down, so it sprays charge into the air long before the smooth part of the dome is in any difficulty. That is why high-voltage hardware is built round and polished, why a lightning rod is a spike, and why the charge distribution on a real conductor has to be solved for rather than assumed.

Fields with no symmetry are measured, not derived. The bench method paints two electrodes onto resistive paper, drives a small steady current between them, and hunts with a probe for points at the same voltage; joining those points draws one equipotential contour. Three rules then read the map. Every electrode surface is itself one contour, because the metal is an equipotential. Field lines cross the contours at right angles and run from high V to low. And the spacing gives the magnitude, |E| ≈ ΔV/Δs measured along the shortest path between neighbours: contours drawn every 1.00 V and found 4.0 mm apart mean |E| ≈ 250 V m⁻¹, while the same contours crowded to 1.0 mm mean 1.0 × 10³ V m⁻¹. Inside a closed conducting electrode the probe finds one single voltage everywhere, with no contours at all to cross — the shielded interior showing up as a flat patch on a real measurement.