App B retired 11_15-13 REVIEWED

Appendix B retired on 11.15.13. 11.15.13 This Appendix B has been deleted from lines doc. It was on a subject that did not have to be in an Appendix since it was treated in full in the main lines doc. I then wrote a completely unrelated Appendix B on a different subject (fields for round wire in partial waves). Appendix B: The Complex Dielectric Constant and Free Charge in a Medium B.1 A conducting parallel plate capacitor B.2 Complex ε and the loss tangent of a dielectric B.3 Continuity and free charge in a medium B.1 A conducting parallel plate capacitor In our transmission line analysis we follow the general approach used in King. One notion that King finds useful is that of a complex dielectric constant ξ which incorporates the conductivity σ of a medium along with the normal dielectric constant ε. Associated with this complex dielectric constant is a so-called "effective" surface charge neff . The term "dielectric constant" is somewhat a misnomer. As noted below, ε and ξ are not constants but in fact are strong functions of frequency ω. When small ranges of ω are considered, ε appears to be constant if all medium resonances are far away. A good way to understand ξ and neff is to consider a simple "device": a parallel plate capacitor (area A) filled with a conducting dielectric (σ and ε). The plate spacing s is assumed small so edge effects can be ignored. One can regard this device as a parallel RC circuit. Let us first imagine that the R portion is turned off and we have just the C part. Then, assuming the usual exp(jωt) time dependence, we can say that Q = CV = nA, where A is the area of a plate and n is the surface charge density per unit area. Doing a time derivative and then converting everything to the frequency domain (see (1.5.5) and Section 1.6) one gets I(ω) = jωn(ω)A, so we have a simple relationship between the current I and the area surface charge density n (arguments ω are now suppressed), I = jωnA . (B.1) If we now turn on the R part of our device, we get an extra contribution to the current, so now: I = jωnA + V/R R = s/(σA) // since R = V/IR = Es/JA = (s/A)(E/J) = (s/A)(1/σ) = s/(Aσ) or I = jωnA + VAσ/s . (B.2) As noted, s is the spacing between the plates and σ is the dielectric conductivity. We know that V = Es (B.3) where E is the electric field between the plates, and we also know that E = n/ε. This last fact follows from (1.1.3) that divD = ρ applied to a thin gaussian box containing one of the plate surfaces: ∫V dV div D = ∫S dSD // divergence theorem = Gauss's theorem, dS = surface patch => ∫V dV ρ = D A => nA = (εE)A => E = n/ε . (B.4) From (B.3) and (B.4) we find that VAσ/s = (Es)Aσ/s = EAσ = nAσ/ε so (B.1) says I = jωnA + nAσ/ε = jωnA [ 1 + ] = jωnA [ ε + ] / ε or I jωAn(ξ/εwhere ξ ≡[ε + σ/(jω)] or I = jωAneff where neff ≡ n (ξ/ε) . (B.5) By defining neff in this manner, we are able to incorporate the effect of the R part of our device into the C part. In solving a problem, we can of course work either with the true surface charge n, or with the effective quantity neff. Notice that, neff/ξ = n/ε = E (B.6) The advantage of using neff is its simple relationship (B.5) to the total current I in a situation where both R and C effects are present. Main Point: For our parallel plate RC device, I = jωneffA, which mimics the result I = jωnA when there is no resistive loss. The admittance of our conducting capacitor is given by Y ≡ = = = jωC' where C' ≡ ξ (A/s) = complex effective capacitance = (jωA/s)[ε + σ/(jω)] = (σA/s) + jω (εA/s) ≡ G + jωC where (B.7) G = σA/s = conductance = 1/R for R shown in (B.1) C = εA/s = capacitance Again, using the definition of ξ, we can absorb the R effect into the C part of our device by replacing C with the complex capacitance C'(ξ/ε)C. Finally, = jωC' => jωC' = => C' = ≡ (B.8) which relates complex capacitance C' to the total effective charge qeff and the voltage V. B.2 Complex ε and the loss tangent of a dielectric. The normal dielectric constant in fact has a real and imaginary part, usually expressed as: ε = ε' - jε" . (B.9) Thus, we can write our complex dielectric constant ξ as ξ = [ε - jσ/ω] = ε'ε0 - j(σ + ωε"ε0)/ω] (B.10) The cause of ε" is a time-lag loss mechanism in the dielectric. In polar dielectrics, this is due to friction which keeps the polar molecules from instantly following the E field at high frequencies. In non-polar dielectrics (those used in coax cables) this is mainly due to the classical resonance absorption mechanisms, such as molecular vibrational modes. The standard classical (non-quantum) theory for ε due to vibrating electrons in a medium yields this general form for ε(ω): [ Jackson p 310 (7.51) ] = 1 + Σn = 1 + Σn (B.11) where Σn is a sum over the possible resonances of electrons in the medium, ωn are the resonant frequences, γn the corresponding damping factors, and fn the relative "oscillator strengths". Here m is the electron mass, e the electron charge, and N is the density of electrons per m3. Each resonance is being modeled by a damped driven harmonic oscillator. The main point of (B.11) is that in general ε(ω) always has an imaginary part, which comes from the jγn damping term. In general ε" is small except in the region of a resonance (in the above, at ω such that ω ≈ ωn). We can conveniently incorporate both loss mechanisms (conduction of dielectric and absorption) by defining σeff to be the grouping which appears in (B.10), σeff ≡ (σ + ωε"ε0) (B.12) so then ξ = [ε'ε0 - jσeff/ω] = ε'ε0 [ 1 - jσeff/(ωε'ε0)] = ε'ε0( 1 - j tanL) (B.13) where we have defined tanL to be tanL ≡ = . (B.14) This last quantity tanL is called the "loss tangent" (or dissipation factor) of the dielectric because it is (minus) the ratio of the imaginary and real parts of ξ, as (B.13) shows, and there is a "loss angle" in the complex ξ plane which is θL = tan-1(tanL) = tan-1[σeff/(ωε'ε0)]. If θL= 0, there is no loss. This tanL is a function of ω explicitly as shown, and implicitly since in general σ, ε' and ε" are functions of ω. B.3 Continuity and free charge in a medium This section is unrelated to the above two sections and is just placed here for convenience. Consider the continuity equation (1.1.8) inside a medium, div J = -∂ρ/∂t, where J = σE is the conduction current in the medium, and ρ is the free charge. Using (1.1.7, 6 and 3), div J = σ divE = σρ/ε) so we get: + ρ(t) = 0 [jω + ] ρ(ω) = 0 for ρ = ρ0 ejωt time dependence (B.15) The ω-domain equation says that the only monochrome ejωt solution is ρ(ω) = 0 => ρ(t) = 0. The t-domain equation has this transient solution ρ(t) = ρ(0) exp(- t/τ) τ = ε/σ  (B.16) This says that if there is free charge out in the middle of a conducting medium, it will run away and park on the boundaries. In our conducting capacitor, this happens with time constant τ despite the fact that the capacitor plates are driving the medium with ejωt time dependence. Any existing charge density will dissipate in time τ. Examples: copper σ = 5.81 x 107 mho/m ε ≈ ε0 = 8.85 x 10-12 farad/m => τ ≈ 1.5 x 10-18 sec polyethylene σ ~ 10-14 mho/m ε ≈ 2.3 ε0 = 8.85 x 10-12 farad/m (ρ > 1016 ohm-cm) => τ ≈ 2000 sec ≈ 34 minutes To summarize, there is no steady-state sinusoidal free charge density out in the middle of a medium, conducting or otherwise. Even if free charge could somehow exist at t = 0 inside a medium, after t >> τ free charge is only allowed on boundaries. One can of course have a current J in a medium, but the charge is balanced so there is zero net charge density, ρ = 0.