Finding a meaning for Chapter 2 REVIEWED

Finding a Meaning for Chapter 2 PhL 11.28.13 Reviewed with some red line on 11.30.13. Moved bulk App D stuff to Actions 1 doc. This is a fairly complicated subject, be prepared! History 1. In Chapter 2 I have this rather nice treatment of the round wire which then shows the skin effect. The treatment is made under a certain set of assumptions regarding the various fields and symmetry. I had always been a little uneasy one of the conditions (weak z dependence), but it has really implicitly been there since 1991 when I first wrote Chapter 2. I think I confused E(x,ω) with E(x,y,ω) which are different functions. 2. Somewhere along the way I wrote Appendix D doing a "general treatment" of the round wire, but after writing it, I ignored it as just so much math fiddling. 3. Doc 2.5 and the Exterior Mistake. As part of my cleanup of lines doc, I then decided to clean up Chapter 2 a bit. My first action on 11.26 was to write a final section for Chapter 2 which computed the fields outside the wire, so they I would have a complete solution within my set of assumptions. I would even match the boundary conditions and set all the constants that way. I went ahead and wrote this up in " Section 2_5 on round wire solution.doc" [ renamed "Exterior solution for round wire as Section 2_5 REVIEWED.doc" ] (call it Doc 2.5). In this working doc, I first start off playing with things, then I start more seriously where it says Back up in bold. I assumed that E = E(r) and B = B(r) [ as in my stated Chap 2 assumptions list] , and I plugged these forms into my E and B Helmholtz wave equations which had a generic unspecified β2 constant. I was not doing region 1 and region 2 yet, just a generic region with β. I then assumed that this general form was applicable inside and out (region 2 and 1) and I solved the resulting ODE in each region. In the interior region, I replicated the results of Chapter 2, not surprising since Chap 2 assumed the same things I assumed here. But then I went ahead and solved the exterior region assuming that the Helmholtz equation for that region simply had β12 in place of β22. That led to H1(1)(β1r) as the exterior solution, and I even made some plots. I thought all was well and done. 4. But then on 11.27 I realized that something was not right. I wanted to interpret the Chap 2 problem as described there (with assumptions) as some kind of long wavelength wave going down the wire. In Doc 2.5, I never once thought about the z dependence of anything, I just said "there is no z dependence". 5. Sisyphus Paradox and Exterior Mistake Identified. Next, on 11.27 I wrote "Sisyphus paradox for 11_27.13.doc" where I summarized my exterior solution from Doc 2.5 My only Paradox here is that I could not come up with a "physical situation". I tried to assume exp(-jkz) z-dependence, and I then got (22D + β12 - k2 )E0(x,y,ω) = 0. That is because ∂z2 in 2 generates a term -β12 inside the Helmholtz equation I then realized that probably a true wave solution must have exp(-jβ1z) dependence in both regions 1 and 2, in order for it to really be a "travelling wave", where β1 is of course the dielectric value. I argued that at r = a when you match BC's, you will find that the internal solution must have this exact same z dependence. I noticed that I had already made this assumption in the long-forgotten Appendix D. The implication here was that the exterior Helm equation is really (22D + 0)E0(x,y,ω) = 0 whereas in Doc 2.5 I had incorrectly used (22D + β12)E0(x,y,ω) = 0. 6. Doc 2.5a Repair, but then Maxwell's violated. I saw this as a minor error of my Doc 2.5 which I would then correct. So my next doc was "attempted rewrite of section 2_5.doc" ( call it Doc 2.5a) where I intended to simply correct this exterior solution, rematch BC's, and be done with it. Along the way I noted that the interior solution of Doc 2.5 would not be much altered because I would be replacing β22 by β22 - β12 but since |β2| >> β1, this really made no difference in the interior solution. That meant the interior work of Chapter 2 is still OK. I started working in Doc 2.5a. With the usual assumptions of Chap 2 that E = E(r) and B = B(r) I set out to solve 2E = 0 and 2B = 0 in the exterior region. I came up with simple exterior solutions of the form E(r) = A + B lnr and B(r) = Dr-1. They differ because in cyl coordinates there is an extra -r-2 term in the Helm equation for B. I then thought it would be good to see if these solutions satisfied the Maxwell equations. When I impose div E = 0 the result is then 0 = -jβ1[A + B lnr] which is a contradiction. Since E(r) has to match large E(a), we cannot just have A and B = 0. This was the first sign of trouble. The div B = 0 equation was OK. I next looked at curl E = - jωB and found that it could be satisfied if B = jωD, so that one was OK. Finally, when I looked at curl B = μ1 jω ε1E, that implied that 0 = μ1 jω ε1[A + Blnr] which is very similar to the div E = 0 problem just quoted. So I was then in this situation: The Helm equations are derived from the Max equations, but two of the Max equations are violated. I was forced to conclude that one or more of my ansatz assumptions was not viable. One assumption was the exp(-jβ1z) z dependence, the others were E = E(r) and B = B(r). [ I think this was a correct thing to conclude.] I then started to flail inside Doc 2.5a. I could say that 0 = -jβ1[A + B lnr] is OK in some sense when ω is very small since β12 = ω2μ1ε1. This would also "fix" the other violation 0 = μ1jωε1[A + Blnr]. But then the question is "how small" must ω be to have OK violations? I then deferred this "how small" question and did Plan A which uses the exterior solution just discussed and matches the boundary conditions. While doing this, I rewrote the Chapter 2 first summary box because I realized that both B(a) and E(a) are determined by the interior solution alone, given the current I in the wire. I then got a final answer for the Plan A, and I just ignore the violations. I wonder if there are violations both for interior and exterior solutions? In Plan B I tried adding an Er field to see if that would quench my violations. This complicated the analysis a bit, but in the end I got the exact same violation as with not having this extra term, since it turns out that this extra term does not affect the violation for the curl B equation. In Plan C I bit the bullet and said I should really allow all 6 field components. But then I suddenly realized that I have done all this before, and that was in Appendix D! So then I have come full circle in this little history. My next actions will be these: (1) see if I can estimate how small ω has to be in the simpler model of Doc 2.5a (2) do the same thing in the full solution of Appendix D. I don't really know where to start for item (1) so I guess I will attempt item (2). [ I now know that E = E(r) and B = B(r) is over-restrictive and if you expect to satisfy Maxwell's equations, you must allow more general forms for E and B. ] Fri Nov 29,2013. 7. Regarding the Meaning Issue. I think I understand now. The wire in isolation does not allow the exact Maxwell equation solution I tried to impose on it with my ansatz. [ but neither does the coax situation ]. That is WHY the two Max equations are violated [well, maybe related to WHY] . You have to find a physically realizable situation in order to have all Maxwell's equations respected [true]. The appropriate situation here is that the round wire is the center conductor of a large diameter coaxial cable where the current return occurs on a metal cylindrical surface far out from the central wire. In this situation, the capacitance of the cable is C = 2πε ln(b/a), the inner wire radius, if b >> a. In the true wave situation, then, charge must exist on the round wire surface which my ansatz ignores [correct]. This charge generates an internal and external radial Er field which the ansatz model does not include. [correct, and this will affect divE = 0. ] When I added such a possible Er field in Plan B of Doc 2.5a, I found Er = Fr-1 for my outside solution which seems very reasonable. However, adding this Er field to the picture failed to cure the problem that div E ≠ 0. So adding just this extra field, though very reasonable, does not seem to make Maxwell happy. I need to return to this point soon. Meanwhile, the "meaning" of the wire as a large coax center conductor has the extra advantage of realizing the assumed axial symmetry. This means m = 0 in Appendix D. This then suggests the following solution method using Appendix D. Recall that App D solves only for the field E using the same method of Doc 2.5a (assumes k = βd for all z dependence). If App D is correct, then it will produce a certain m = 0 solution for E. Then I can compute B from curlE = -jωB, and THEN maybe I have a solution that makes Max happy! Recall that App D makes a special point of being sure that div E = 0 is respected. My full result is in a box (D.4.6). 8. What does App D have to say for m = 0 ? Here is the m=0 version of that E field solution: E Fields Inside a Round Wire x = β'r β'2 = β2- βd2 ≈ β2 β = ω ξ =[ε + σ/(jω)] ≈ σ/(jω) conductor xa = β'a βd = ω ξd =[εd + σd/(jω)] ≈ εd dielectric E(r,φz,t) = ej(ωt-βz) E(r,φ) (D.1.2) E = Er + Eφ + Ez (for any arguments) E(r,φ) = E(r,0) + 2!Syntax Error, I[Re{E(r,m)}cos(mφ) - Im{E(r,m)} sin(mφ)] = real (D.1.4) where: Ez(r,0) = (1/2) I Rdc [] // = (2.2.22) for βd = 0 η0 = 1 Er(r,0) = (j/2) (aβd) I Rdc [ ] Rdc = Eφ(r,0) = 0 (D.4.6) Note: (1) Chapter 2 assumes ( jωε + σ) ≈ σ in its development. (2) Chapter 2 assumes βd on the outside and β on the inside where β = (j - 1) We can compare the Ez solution to the Chapter 2 solution E(r) = - (jω/β) [μI/(2πa)] (2.2.30) Ez(r,0) = (1/2) I Rdc [] above box m = 0 = (1/2) I β'a Now assuming β' = β, I do know that in the conductor β2 = ω2μξ ≈ ω2μ[σ/jω] = -jωμσ which then says β = (1/β)[ -jωμσ] = (-jω/β) μσ (-jω/β) = β/(μσ) If I use this in the m=0 box result I get Ez(r,0) = (1/2) I (1/β) [ -jωμσ] a = I (1/β) [ -jωμ] = - (jω/β) [μI/(2πa)] and I am VERY HAPPY to report that this then exactly matches the Chapter 2 result. The conclusions at this juncture are: (1) The Ez field for the coaxial center conductor in my "meaning model" exactly agrees with the result of a detailed calculation in Appendix D. (2) Only fields Ez and Er exist inside the wire (for m = 0) (3) The Er field also has a complex Bessel form as shown. Er(r,0) = (j/2) (aβd) I Rdc [ ] and this is indeed "small" in the transmission line limit, a new and useful fact. BUT, below I will show that this is the wrong thing to say about smallness! Discussion of the m = 0 solution: We get these fields Ez(r,0) = (1/2) (aβ') I Rdc [ J0(β'r)/ J1(β'a) ] Er(r,0) = (j/2) (aβd) I Rdc [J1(β'r)/ J1(β'a)] Eφ(r,0) = Cφ0 J1(β'r) Two of the components are fully nailed down! I think that part of the calculation is OK, though we may still have Maxwell equation violations I suppose. For the round wire in Chapter 2 I would just set the constant Cφ0 = 0 as ansatz. Then the Er seems to be "pumping charge" to the surface of the wire. I think my boundary condition Er(r=a,m) = (jω/σ) Nm is probably still valid, so I then have (jω/σ) N0 = (j/2) (aβd) I Rdc In this scenario with no other partial waves active, n = N0 from (D.1.6) so this equation is actually telling me the physical surface charge on the conductor : n = N0 = (σ/2ω) (aβd) I Rdc = (σ/2ω) (aβd) I = (βd/ω) I /(2πa) = I /(2πa) n(φ,z,t) = ej(ωt-βz) n(φ) so this really is looking like my model for the Chapter 2 round wire -- center of a transmission line. It cannot operate without having this surface charge! This charge density is controlled by I and the exterior region quantity . The total charge per length on the round wire is Q = I = I/vd Coul/m = Coul/sec * sec/m check So that is an interesting fact all in itself. Q = I/vd. In Chapter 2 I ignored this charge, and I also ignored the Er field and we see it is very small since (aβd) << (aβ') or βd << β' which is certainly true. [ do not need to invoke "transmission line limit"] Can I relate this to capacitance? I know C = 2πεdln(b/a) per unit length for a coax center round wire. Suppose the inner conductor has n = I /(2πa) charge density which is then Q = I per unit length of wire. Then Q = CV so ΔV = Q/C = I / 2πεdln(b/a) and all I can say is that the ratio of the radii tells you ΔV, so no great realization here. This charge Q is of course going to affect the exterior Er field for my coax problem. What about Eφ(r,0) = Cφ0 J1(β'r) ? I see easily from (D.1.18) why the Eφ field decouples from the other fields, and that is why it has a separate constant. If Cφ0 > 0, the is an electromotive force around the wire circumference which I guess exists just under the surface and it would drive a Jφ current. This is basically an independent torsional wave solution that the round wire can support. Even if I = 0 and the other fields went away, you could still have this torsion wave going down the wire. One way maybe to drive such a wave would be to put a little axial cut in the wire at one end and somehow apply and AC voltage to it. Maybe you apply a Bz magnetic AC field at the end and the Faraday's law might drive it. I hunted for torsion wave and azimuthal wave on line, the latter appears but not in my context. Now what about the m> 0 modes? Ez(r,m) = Czm Jm(β'r) = Km (β'/βd)(1/2j) Jm(β'r) (D.1.27) Er(r,m) = am x-1 Jm(x) + Jm+1(x) . (D.2.11) jEφ(r,m) = - am x-1 Jm(x) + ( + ) Jm+1(x) x = β'r . (D.2.15) Now for general m, the Eφ(r,m) field is coupled to the other fields and you don't get that torsion wave freedom you had with m = 0. The Eφ field just "comes out" as you see it here. All three components are now interrelated. 9. Am I done with the Meaning for Chapter 2? Let's try to spin an answer. The round wire is the center conductor of a coaxial cable, and we are only interested in the fields inside this round wire. We assume this coax cable is a transmission line which carries a wave having ej(ωt-βz) time and z dependence. This is our ansatz. For a fat coax cable if we are far from the round wire, we can imagine a wave with ej(ωt-βz) where βd is the dielectric's wave number. This is the nature of a plane longitudinal wave through the dielectric. At the round wire surface the dielectric wave has this ej(ωt-βz) dependence and through that boundary it "rubs off" on what goes on inside the round wire. That is our ansatz. 10. Decision: The purpose of the current document was to "find a meaning for Chapter 2", and that task has now been carried out. Only the m = 0 part of the Appendix D analysis matters. I just added more text at the front of Chapter 2 to put the round wire into its proper "coaxial cable wave context". This was really my Big Problem and I think it is now solved, so I will close this doc down. The issue of this am and Km coefficients I will put off for a future day. You cannot solve all problems at once or you get nowhere fast!