1. The electrostatic field of conductors EDII Section 1

2. Matter in an electric field Variations on atomic or molecular scales Miicroscopic potential Average potential

3. “Macroscopic” Electrodynamics Take spatial average over interatomic length scales. Actual microscopic field

4. The length scale for averaging depends on the problem

5. Conductors: Those media for which an electric current (flow of charge) is possible Electrostatics: 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.

6. 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.

7. Far from the surface:0 Average potential Actual microscopic potential Surface Medium Vacuum

8. Exact microscopic field equations in vacuum We will set r = 0, because we assume no macroscopic net currents in electrostatics Now take spatial average r

9. Spatially averaged fields These are the usual equations for constant E-field in vacuum  is a “potential function” Laplace’s equation

10. Boundary conditions on conductor surface: Curl E = 0 both inside and outside For a homogeneous surface and are finite

11. Curl E = 0 Finite, so is finite across the boundary is continuous across the boundary. Same for E x. Since E = 0 inside a conductor, E t =0 just outside. E is perpendicular to the surface every point.

12. Surface of a homogeneous conductor is an equipotential of the electrostatic field. No change in  along the surface

13. Normal component of E field and surface charge density are proportional Derivative along the outward normal Only non-zero on the outside surface

14. Total charge on the conductor is the integral of the surface charge density Whole surface

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

16. Consequence A test charge e cannot be in stable equilibrium in a static field since e  has no minimum anywhere.

17. Proof. Suppose  has a maximum at point A not on a boundary of a region with non-zero E. Surround A with a surface. Thenon the surface at all points, and Contradiction! But Gauss Laplace