old section 3_1 REVIEWED

Old Section 3.1 PhL 1.2.14 3.1 Why is there no charge inside a conductor? There can be charge inside a conductor, but only if the conductor is excited at an extremely high frequency. A charge density ρ inside the conductor implies an electric field E according to div E = ρ/ε. This E field then causes a current J = σE which attempts to drain the charge off to the surface. Here is a simple way to estimate the time constant for this process. Imagine a one dimensional conductor (dimension x) with some charge density ρ inside. Then from div E = ρ/ε0 we get ∂xEx = ρ/ε0 . (3.1.1) The electric field so generated causes a current Jx = σ Ex so apply ∂x to get ∂xJx = σ ∂x Ex . (3.1.2) But this current drains the charge away according to div J = -∂tρ, which says ∂xJx = -∂tρ . (3.1.3) Combining these three equations we get ∂tρ = - (σ/ε0) ρ . (3.1.4) This implies that ρ decays exponentially with time constant T = ε0/σ . (3.1.5) For copper, σ = 5.81 x 107 mho/m, and ε0 = 8.85 x 10-12 F/m, so T = 1.52 x 10-19 sec. Thus, any process which occurs inside a conductor at a frequency much less than 1019 Hz always allows plenty of time for any interior charge to move to the surface. This is 1010 GHz, far beyond the operating frequency of a transmission line. We may therefore conclude that: Fact 1: In a transmission line, charge exists only on the surface of conductors.