The electrostatic field of conductors EDII Section 1.
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Transcript of The electrostatic field of conductors EDII Section 1.
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