Section 6_5_d REVIEWED

This is the Title PhL 3.26.05 An aborted older version of 6.5 with an interesting plot including a disk. (d) The Proximity Effect At Low Frequencies Above we have shown plots of the proximity effect (Jz asymmetry) in the conductors of a transmission line made of two round cylinders. One wonders what happens to this proximity effect as ω → 0. As frequency drops, the fact that the two cylinders form a capacitor does not go away, so the asymmetric surface charge n(θ) persists and continues to have moments ηm which are non-zero. The resulting asymmetric current density is then given by (D.10.18) which says Jz(r,θ) = σ (1/4) I Rdc [ 4 + 4 Σm=1∞ (r/a)m (m+1) ηm cos(mθ) ] = J0 [ 1 + Σm=1∞ (r/a)m (m+1) ηm cos(mθ) ] where J0 ≡ σ I Rdc = I/(πa2) is the uniform current one expects in a round wire carrying a DC current I. Since the ηm are given by (6.5.4) as ηm = (-1)m e-|mξ|, we have this explicit expression for the transmission line conductor current in the ω→0 limit Jz(r,θ) = J0 [ 1 + Σm=1∞ (m+1) (-1)m e-|mξ|(r/a)m cos(mθ) ] Maple plots this function for ξ1 = - 0.75 and a = 10: and here are two views of the result: This strong asymmetry as ω → 0 is rather disturbing, since we know there is no asymmetry in a DC current flowing in an isolated round wire. The asymmetry shown arises from the "charge pumping boundary condition" Jr(a,θ) = jω n(θ) . (D.2.23) which reflects charge conservation at the conductor surface, assuming G = 0 (σ = 0) for the dielectric.