Problem with the curl B Maxwell Equation REVIEWED

Problem with the curl B Maxwell Equation and low ω limit of theory PhL 10.4.14 I had a paradox here where it seemed that when I use the low-ω E and low-ω B fields, the Maxwell curl B equation was not being satisfied. But then I realized that in the limit of small x which those fields represent, it really was being satisfied for small x, an interesting little puzzle of the day. Comment: Only today did I first think about checking the Maxwell curl B equation after taking the low-ω limits of the round wire E and B fields. As I show below, this curl B equation is violated. I will first show why this is the case, then try to find the cause of the problem. I do know that before the low-ω limit is taken, this curl B equation is OK because I verify it with Maple in "B field verify 9_20". At that point everything is given in terms of Bessel functions. I have spent a lot of time looking for "something wrong" in my derivation of the low-ω limit of the round wire theory, and finally I think I have found something wrong. I first latched on to this problem in "Infinite B fields directly from (D.4.13).doc", but I will move that work to this doc now. Review of verifying the curl B equation before the limit is taken. I obtain the B fields from the curl E equation and then I show that the combined E and B fields satisfy the curl B equation, all in B field verify 9_20.mws. So, how did this work out right for the full E and B fields?? I did that in D.5 and "B field verify 9_20". In that case I had k, β2 and β'2 all floating around. Looking at that mws file, you see that, in verifying the equation curl B = j (β2/ω) E, the curl B components are functions of β' and k. And the E components are functions of β' and k as well. On the right side which says j (β2/ω) E I replace β2 with β'2 - k2. Thus, in verifying curl B = j (β2/ω) E I have each side being a function of β' and k, so at least it stands a chance of working our right, and as Maple shows, it does work out right. Problem verifying after the low-x limit is taken I first take the low x limits of the E fields. In doing this I use these low-x limits of the fm type functions, fm = (r/a)m (m+1) (2/β'a) f0 = 4/(aβ') gm = (r/a)m+1 + (r/a)m-1 g0 = 2 (r/a) hm = (r/a)m+1 - (r/a)m-1 h0 = 0 x = β'r xa = β'a In doing these limits, I assume a priori that xa = β'a << 1, and that in turn forces x = β'r << 1. For the moment, I defer the claim that this small xa limit aligns with the small ω limit and/or the small k limit. Let's just think of this as a small-x limit ! Notice that β' only appears in two places above, and there are no k appearances. When these fm etc are used in the E field expressions, I get Ez(r,m) = (1/2) ηm CV Rdc (ω/k) (r/a)|m| (|m|+1) (7.4.1) Er(r,m) = (j/4) ηm CV Rdc (ωa) [(r/a)|m|+1 + (r/a)|m|-1] Eθ(r,m) = (1/4) ηm CV Rdc (ωa) [(r/a)|m|+1 - (r/a)|m|-1] Ez(r,0) = CV Rdc (ω/k) Er(r,0) = (j/2) CV Rdc (ωr) Eθ(r,0) = 0 // low ω E fields A key point is that only k appears, there are no β' appearances (and of course no β appearances). At this point there are two different methods of obtaining the corresponding B fields. In the first method, I work directly with the above expressions and compute curl E of curl E = -jωB. This is my approach in Chapter 7, and here is what I find: Bz(r,m) = (j/2) ηmCVRdc (r/a)m (m+1) Br(r,m) = (1/4) ηmCVRdc (r/a)m-1 (1/ak)[ -2m(m+1) + k2(a2-r2)] Bθ(r,m) = (j/4) ηmCVRdc (r/a)m-1 (1/ak) [ -2m(m+1) + k2(r2+a2)] m > 0 Bz(r,0) = 0 Br(r,0) = 0 Bθ(r,0) = (j/2) CV Rdc kr m = 0 Not surprisingly, since these B fields are derived from the E fields, only k appears. In the second method of obtaining the B fields, I work directly from the "full" B field expressions which are these Bz(r,m) = (j/4) (a/ω) ηm I Rdc (k β' em ) Br(r,m) = - (1/4) (a/ω) ηm I Rdc ( r-1m β' fm + k2 hm ) Bθ(r,m) = (j/4) (a/ω) ηm I Rdc ( k2 gm - β'2fm [ (m/x) - Jm+1(x)/Jm(x)] ) and I take the small x limit just as I did for the E fields. I show in "infinite B fields directly..." that doing this direct reduction from the full B fields gives the same low-x B fields (at least for m > 0) as obtained by the first method. Now here is a simple statement of the paradoxical problem. I have E and B fields which in the low-x limit depend only on k. The curl operation can only introduce factors of k due to the ∂z → -jk fact which of course follows from the famous e-jkz ansatz. Therefore, curlB and E are each functions only of k, and not of β or β'. But then I have to verify this Maxwell curl B equation: curl B = μ J + μ jωεE = μ(σ + jωε) E = μ(jω)( ε - jσ/ω) E = jω μξ E (D.5.1) = j (β2/ω) E . // see (1.5.1c) which I write simply as curl B = j (β2/ω) E . Since curl B on the left and E on the right are functions only of k, it is impossible that this equation be satisfied! So that is our little conundrum addressed in this doc! Statement: Assuming the e-jkz ansatz, I obtain full E and B fields in terms of Bessel functions, and I show that these full fields satisfy all four Maxwell equations including the curl B equation. My problem is that when I take the small x limit of the E and B fields, it seems that the curl B equation is no longer satisfied. The fact that things work with the full fields tells me that the e-jkz ansatz is not the problem. Not enough terms in the small x limit? Somehow this seems a likely culprit. Especially when you combine this with the derivatives which appear in the curl operator. I addressed this issue at one time just for the E fields and concluded that extra terms made no difference and could be ignored, but I did not do that for the B fields. My previous work could have been wrong of course! Let's put this idea on hold for a moment. Treating β2 for small x ? We know that β'2 = β2 - k2 so that β2 = β'2 + k2 Now one argument might be that in our small-x limit, we are assuming small β', so in that case β2 = k2 in our limit. Then the curl B equation becomes this: curl B = j (k2/ω) E and this equation does have a chance of being valid. Let's check it. I did this in "B for small omega" and it all works just fine. My paradox has gone away. The point is that you only expect curl B = j (β2/ω) E to be valid for small x, not in general, and for small x we have β ≈ k. Does this conflict with my use of k(ω) ? In order to make things work for the curl B equation, I had to say that β2 = β'2 + k2 ≈ k2 dropping terms of order β'2 which is small In the low-x limit we must have β' be small. We do not need to have k or β be small. Here are two ways that we can have β' be small 1) β and k need not be small, but we must have β ≈ k in order for β' to be small 2) β and k are both small, so in that case β'2 = β2 - k2 will be small. In this case, we do not need to have β ≈ k in order that β' be small. In this second case, consider curl B = j (β2/ω) E = j (β'2/ω) E + j (k2/ω) E We can certainly neglect the first term since we are in low-x. But in this case, the second term is also negligible. I don't think there are any problems here.