The electrostatic field of conductors

EDII Section 1

Matter in an electric field

Variations on atomic or molecular scales

Miicroscopic potential

Average potential

“Macroscopic” Electrodynamics

Take spatial average over interatomic length scales.

Actual microscopic field

The length scale for averaging depends on the problem

Conductors: Those media for which an electric current (flow of charge) is possibleElectrostatics: Stationary state of constant energy.

The electrostatic electric field inside a conductor is zero.A non-zero field would cause current, in which case the state would not be stationary due to dissipation.

Any charges in a conductor are at the surface. Otherwise there would be non-zero field inside. Charges on the surface are distributed so that E = 0 inside.

What we can know about Electrostatics of Conductors?

1. We can find E in the vacuum outside.2. We can find the surface charge distribution.

That’s it.

Far from the surface:0

Average potential

Actual microscopic potential

Surface

Medium Vacuum

Exact microscopic field equations in vacuum

We will set <h>r = 0, because we assume no macroscopic net currents in electrostatics

Now take spatial average < >r

Spatially averaged fields

These are the usual equations for constant E-field in vacuum

f is a “potential function”

Laplace’s equation

Boundary conditions on conductor surface:Curl E = 0 both inside and outside

For a homogeneous surface

and

are finite

Curl E = 0

Finite, so

is finite across the boundary

is continuous across the boundary.Same for Ex.

Since E = 0 inside a conductor, Et =0 just outside.

E is perpendicular to the surface every point.

Surface of a homogeneous conductor is an equipotential of the electrostatic field.

No change in f along the surface

Normal component of E field and surface charge density are proportional

Derivative along the outward normal

Only non-zero on the outside surface

Total charge on the conductor is the integral of the surface charge density

Whole surface

Theorem

The potential f(x,y,z) can take max or min values only at the boundaries of regions where E is non-zero (boundaries of conductors) .

Consequence

• A test charge e cannot be in stable equilibrium in a static field since ef has no minimum anywhere.

Proof. Suppose f has a maximum at point A not on a boundary of a region with non-zero E.

Surround A with a surface. Then on the

surface at all points, and

Contradiction!

But Gauss

Laplace