# What is a conductor

1. Jul 24, 2014

### Greg Bernhardt

Definition/Summary

A material is a conductor if it contains charges (eg electrons ions or "holes") which are free to move.

In equilibrium, the charge inside a conductor is zero, and the density of charge at the surface is greater where the radius of curvature is less (and in the direction of increasing surface component of any applied field).

In equilibrium, the electric field inside a conductor is zero (the charges rearrange themselves to cancel out any external field), the electric field at the surface is perpendicular to the surface and has magnitude $\sigma/\varepsilon$ (surface charge density divided by ambient permittivity), and the electric potential is the same over the whole surface.

In particular, a nearby charge will cause opposite charge to accumulate on the near surface (and like charge on the far surface) so as to cancel the field inside the conductor: this charge will then attract the original charge.

An electric potential difference (a voltage) applied externally across a conductor causes an electric field inside it, and a flow of charge (a current).

Conductance (the reciprocal of resistance) is current per voltage, the rate of flow of charge per potential difference. Its SI unit is the siemens ($S$). Conductivity of a material is conductance times distance per area ($S/m$).

Equations

Extended explanation

"The electric field inside a conductor is zero":

Often quoted, but not true: it only applies in equilibrium.

When a charge is first brought near a conductor, the free charge inside the conductor rearranges itself extremely quickly to cancel out the field … while this is happening, of course there is a field!

(And of course there is an electric field inside every current-carrying wire.)

Surface charge density:

Charge accumulates at the more "pointy" parts of the surface.

This is because, at any given distance from any given point, there is less surface on the side where the radius of curvature is smaller, so the charge density needs to be greater on that side to maintain equilibrium.

At radius of curvature r, the amount of charge in an arc between fixed distances s and s + ds (measured in 3D space, not along the surface) will be less for greater r (by a factor approximately cos(s/r), the radius of the arc), and its component along the surface will also be less (by the less significant factor cos(s/2r)).

An applied field will also have an effect: but only the component parallel to the surface matters, since the charge cannot leave the surface.

Electric displacement field:

The electric displacement field (in coulombs per square metre) in equilibrium is normal (perpendicular) to the surface, and has magnitude equal to the surface charge density; dividing by the permittivity of the surrounding medium gives the electric field:
$D = \sigma,\ \ E = D/\varepsilon = \sigma/\varepsilon$​

Lightning rod:

The point of a lightning rod has an extremely small radius of curvature. A small external charge (ionised air) will cause a much greater accumulation of charge there than on any nearby less pointy (and/or less conducting) object: this charge will attract the external charge more than the other object will.

[Does any of the above need to be changed to take account of susceptibility and the polarisation field?]

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