scraps appendix M

This is the Title PhL 3.26.05 Note that page numbering is turned on in this template. scraps from Appendix M work Anomaly: [ no longer relevant! ] For a conducting dielectric (σd> 0 => ωd> 0) the ratio |k/β| for small ω has the form |k/β| = → = = (M.13) which becomes infinite at ω = 0. The issue here is that as ω→0 we have β→0 but k→ -j ≠ 0. Since the dielectric conducts current, there must be a DC loss going down the transmission line in z and this is associated with Im(k) = - , so it is expected that k 0. We then expect our 10-3 inequality rule of (M.12) to be violated when 10-6 < RdcGdc/ (ωμσ) or ω < 106 RdcGdc/ (μσ) ≈ 106 RdcGdc/73 . // copper conductors For the Belden 8281 cable of Appendix R, we find that Rdc ≈ .036, Gdc = ωdC ≈ 0.8 x 10-14 so our condition for violation becomes ω < 106 (.036)( 0.8 x 10-14/73 ) = 3.9 x 10-12 sec-1 It is not unreasonable that at such ultra-low frequencies near DC the transverse current Jr can be large relative to Jz. The network model is then that of Fig K.4. One could solve the DC magnetostatics problem for an infinite transmission line with round conductors to determine Jr and Jz and compare their magnitudes. Since a practical transmission line is not operated at such ultra low frequencies, we ignore the anomaly and state the claim (M.1) as if it were valid all the way down to ω = 0. NOTE in red: In (D.11.14) I claim to know Jz and Jr as ω→0 for each partial wave. The ratio seems to be this Jr/Jz ≈ aG / = a For Belden 8281 center wire this is Jr/Jz = a1 = .00039 * = .7 x 10-11 which disagrees with the above discussion!! I do state there that k ≈ -j which agrees with above. In box (D.11.7) the ratio Er/Ez is clearly just (ak). What happened to (k/β) ? Aha! It is those Bessel functions that I ignored above !!!!