Conductor rules
(E2.4.1) E2.4 · Conductors in Electrostatic Equilibrium →Field zero inside; excess charge on the outer surface; field just outside perpendicular to the surface with magnitude σ/ε₀; charge densest where the surface curves most sharply.
| local surface charge density | C/m² | |
| field just outside the surface | N/C | |
| permittivity of free space | 8.854×10⁻¹² C²/(N·m²) |
A conductor in electrostatic equilibrium — nothing flowing.
Applies only to a conductor in electrostatic equilibrium with nothing flowing — a current-carrying conductor has a non-zero field inside it.
If any field remained inside, the free electrons would keep accelerating — that is not equilibrium. So:
Draw a gaussian surface just inside the outer skin. Zero field means zero flux, so it encloses no net charge — any excess must be on the surface:
A field component along the surface would push surface charge sideways, so only the perpendicular part can survive.
Now use a small box with one face just outside and one just inside. The inner face contributes nothing (E = 0 there), so all the flux leaves through the outer face:
Equal potential fixes the charge ratio:
Surface density scales as charge over area:
The smaller sphere carries five times the surface density, and therefore the stronger field — the principle of the lightning rod.
Note the factor 2 against the isolated sheet σ/2ε₀ — here the flux only escapes one way, because the field inside the metal is zero.
Van de Graaff generators, precipitators that pull soot from flue gas, photocopiers and laser printers all live off these four rules — charged surfaces, and the field peaking at sharp points. Faraday cages live off the zero interior field.