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contradiction bug REVIEWED

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Short note by Phil dated 9.25.13, in the Transmission Lines chapter 2 (round wire) folder. It takes the Bessel-function field solutions E(r) and B(r) for a round wire and evaluates B/E at small frequency using the J0 and J1 expansions, getting a real ratio of μσr/2. It concludes the contradiction came from an incorrect fact assuming a rigid 90 degree phase relation between complex E and B.

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Contradiction Bug PhL 9.25.13 Resolved. I thought there was a rigid 90 degree relation between complex E and B, but not so. I have shown in lines that for the round wire, we have the following fields E(r) = E(a) [J0(βr) / J0(βa)] B(r) = - (β/jω) E(a) [J1(βr) / J0(βa)] The ratio of these fields is given by B(r)/E(r) = - (β/jω) [J1(βr) / J0(βr)] β = ej3π/4 I now evaluate this in two different ways for small ω. Method 1. I use these expansions from Spiegel p 136 J0(x) ≈ 1 + order(x2) x = βr = r ej3π/4 = small for small ω J1(x) ≈ x/2 + order(x3). Thus, we have [J1(βr) / J0(βr)] ≈ (βr/2) and so B(r)/E(r) ≈ - (β/jω) [(βr/2)] = -β2r/ [2jω] = - (-jωμσ)r/[2jω] = (μσ)r/[2] = μσr/2. This ratio then has zero phase for small ω. If μ = 1 and σ = 1 and r = 1 then B(r)/E(r) = 1/2 Maple agrees with this calculation of 1/2 so no contradiction yet. Maple can plot the phase of the ratio: This shows that the phase of the ratio B(r)/E(r) is not constant in ω. Method 2. I show in Fact (I.3.7) of "complex E and B..." that this ratio must be a constant. [ but this Fact is wrong! ] Therefore we have a contradiction.