Old section 3.4

Old section 3.4 retired 10.11.14 3.4 Size of E fields in conductor and dielectric; conservation of total current at a boundary We know from (1.1.48) that the following E field condition applies at a boundary between two media, where n refers to the normal component, ξ1En1 = ξ2En2 // frequency domain (1.1.48) or (ε1 + σ1/jω) En1 = (ε2 + σ2/jω) En2 . (3.4.1) Let 1 = dielectric and 2 = conductor. From (3.3.8) we set ξ1 ≈ ε1 (at least for f < 10 GHz), and from (2.2.3) we set ξ2 ≈ σ2 /jω for f << 109 GHz (for polyethylene and copper), so (3.4.1) then reads ε1 En1 ≈ (σ2/jω) En2 => ratio = ≈ . (3.4.2) We can look at some typical numbers, ε1 = 2.3 ε0 (polyethylene) (3.4.3) σ2 = 5.81 x 107 mho/m (copper) ε0 = 8.85 x 10-12 farad/m so = ≈ (1/2)1018/f ≈ 109/[2f(GHz)] (3.4.4) At a high frequency of f ≈ 500GHz the ratio in (3.4.2) is ~ 106, and at lower frequencies the ratio only increases. Thus, we arrive at these useful facts: Fact 1: The total current in a dielectric is dominated by displacement current, while that in a conductor is dominated by conduction current. (3.4.5) Fact 2: At a boundary between a good dielectric and a good conductor, the normal E field is at least 1 million times larger in the dielectric than it is in the conductor for frequencies under 500 GHz. (3.4.6) Fact 3: This large jump in En at the boundary must be supported by a significant surface charge density n on the boundary since, according to (1.1.47), n = ε1En1 - ε2En2 ≈ ε1En1. (3.4.7) Imagine now a tiny patch of area (bordered in red) on the surface between a conductor and a dielectric, Fig 3.2 Defining a total current Jtot,n ≡ jωεEn + σEn, as in (2.2.1), we have shown that this total current flows right through the area patch but changes its nature from mostly conduction current on one side to mostly displacement current on the other side. In the next section, we identify the normal direction with the local radial direction. Then the total current passing through a tiny square patch like that in Fig 3.2 can be regarded as being "fed" by the radial current Jr just inside the conductor where Jr = σEr. This current feeds the surface charge on the boundary which in turn creates a large E field and thus a large displacement current in the dielectric. We sometimes refer to this mechanism as "charge pumping".