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Transcript
The electrostatic field of conductors EDII Section 1 Some external electric field Continuous medium with ions and electrons Microscopic potential Variations on atomic or molecular length scales Average potential “Macroscopic” Electrodynamics Spatially average the actual microscopic field over atomic length scales. Actual microscopic field The length scale for averaging depends on the problem For these devices, we average over scales at least 10 nm, but less than 10 microns. Definitions Conductor: Medium in which an electric current (flow of charge) is possible Electrostatics: Stationary state of constant energy. Theorem The electrostatic electric field inside a conductor is zero. A non-zero field would cause current. Then there would be dissipation of energy. Then the state of the conductor would not be stationary. Theorem Any excess charges in a conductor must reside 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? What we can know about Electrostatics of Conductors? 1. We can find electric field in the vacuum outside of them. 2. We can find the surface charge distribution on them. That’s it. Surface Far from the surface, potential = 0 and Average potential Actual microscopic potential Medium Vacuum Exact microscopic field equations in vacuum Now take spatial average < >r We set <h>r = 0, because in electrostatics there can be no macroscopic net currents as sources of magnetic field. Spatially averaged fields These are the usual equations for constant E-field in vacuum Laplace’s equation f is a “potential function” 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 conductor surface every point. No change in f along the surface Surface of a homogeneous conductor is an equipotential of the electrostatic field. Normal component of E field and surface charge density are proportional Unit area. Infinitesimal thickness. Non-zero only on the outside surface of the integration volume Derivative along the outward normal 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. Then Surround A with a surface. on the surface at all points, and But Gauss Contradiction! Laplace