the applied Ja issue REVIEWED

The applied Ja issue PhL 10.13.13 Comments 12/5/13. I was battling here with two issues. The first is the historical Ja "applied current" bugaboo which I think I have permanently banished from the Kingdom. The second issue had to do with whether β is real or complex. This was all before I had the multiple region concept where we apply a King gauge for region 1 to all regions so then in the end when the dielectric region current is absorbed you get a complex β working in all regions! That thing really was a breakthrough for me. This issue is directly related to the Ja applied current issue. I kept getting a homo wave equation for A with no driving current, so the Helm integral was 0 and I had no King like solution, so I was just making up Ja in order to have some driving current. I then had an elaborate iteration scheme based on that. It was horrible. This matter just seems hazy to me and I want it to be clean and not hazy. Scenario #1: Suppose I were solving the twin-lead problem where I know Jz will be a bit off center in each conductor. I could start off by assuming that Jz was uniform, and this would be my "applied" value. Using that Jz, I could compute H both inside and outside my one conductor then I could add these H's for both conductors to get a total H and then a total B field. From that I could somehow get the E field, and from that I could somehow get the Ez field inside the conductor, and I would find that it was not uniform, so Jz is then not be uniform. I would compute this new Jz and I could compare it to my starting Jz and I would see a difference. Perhaps then I could start with a modified Jz that was off-center and then redo the whole calculation and in that way iterate until I get a final solutions. If the first correction is very small, then my initial ansatz for Jz was reasonable and gives a reasonable result for the fields. Now in my opening section the original uniform Jz would be called Ja, z. The correction I guess would be called J, and then JT = J + Ja. So do I have any evidence of J as being a "small correction" in anything I have done? In the above scenario, one way to compute B would be to compute vector potential A using the special gauge choice shown in 1.3.4, which then gives 2 - με ∂t2 - μσ ∂t)A = - μJa (1.3.5) (2 - με ∂t2 - μσ ∂t)φ = - (1/ε)ρT . (1.3.6) where now we see Ja as a "driving" inhomo term. I end up soon with A(x,ω) = R = |x - x'| (1.5.9) φ(x,ω) = R = |x - x'| write as surface integral? (1.5.10) So there you are. Use the Ja and do the integral and you get A. But this is only a particular solution, and in theory you can add to this any solution of 2 - με ∂t2 - μσ ∂t)A = 0 What do I know about possible solutions of this homo equation? Let's first go to the ω domain where we then have ( 2 + β2) A(x,ω) = - μJa(x,ω) (1.5.1) ( 2 + β2) φ(x,ω) = - (1/ε)ρT(x,ω) (1.5.2) β2 ≡ ω2μ ξ (1.5.3) ξ ≡ ε - jσ/ω . (1.5.4) Then the question becomes: what do we know about solutions to ( 2 + β2) A(x,ω) = 0 ( 2 + β2) φ(x,ω) = 0 The first is a messy vector Helmholtz equation! But now go back to φ(x,ω) = I already argue that if this includes ALL charge density, then it is the complete answer and in the Laplace case the only homo adder solution is 0. Do I know anything about the solution to this problem: ( 2 + β2) φ(x,ω) = 0 and φ(∞) = 0 Well consider instead this equation which agrees with the above almost everywhere ( 2 + β2) φ(x,ω) = δ(r) This gives φ = eiβr/4π r and it drops off as in (1.5.8). But what about generic homo solutions? φ = -(β2/2)x2 ∂xφ = -β2x ∂x2φ = -β2 So in theory you could have this solution floating around, but it blows up at ∞. Perhaps the same kind of thing happens with the vector Helmholtz equation. Consider this example from 3D electrostatics. Suppose I take two spheres with charges +1 and -1 which have radius a and are distance b apart. I could start by assuming a uniform σ distribution on each sphere. I could then use that to compute the E field of each sphere and then add these up (field of two point charges!). I could then use ρ = div E to compute the actual σ on those two spheres. It would be non-uniform. But then for that particular E field, I know exactly what surface charge creates it, so I at least have the solution of a certain problem. If the error is small (and it would be if the spheres were far apart), then I have an approximate solution to the problem of the field of two charged spheres. What is the exact solution to this problem? Have I ever done this? No. I found a paper that gives the exact solution in terms of an infinite series of image charges, fine. I stored it. The problem also appears as a Problem in blue Jackson page 86. The first image charge is sphere center and corresponds to my simple "applied" solution noted above. I am wondering now if there might be a better notation for things. 1.1 Maxwell's Equations in a Medium Our working set of equations is the following: curl H = ∂D/∂t + J = ∂D/∂t + [ ΔJ + Ja ] Maxwell curl H equation (1.1.1) curl E = - ∂B/∂t Maxwell curl E equation (1.1.2) div D = ρ = [ Δρ + ρa ] Maxwell div D equation (1.1.3) div B = 0 Maxwell div B equation (1.1.4) B = μH magnetic permeability μ (1.1.5) D = εE electric permeability ε (dielectric constant) (1.1.6) J = σE Ohm's Law (σ = conductivity) (1.1.7) div(J) = - ∂ρ/∂t Equation of Continuity (1.1.8) Suppose I do the above where now J and ρ are the "total" quantities and the Δ items are "corrections" in the sense of the above spheres discussion. What implications would this have? Let's cut and paste and edit a few things. Section 1.1 (e) The symbols Ja and ρa stand for certain "applied" current and charge densities, and in terms of these symbols, the total current and charge density are given by J = ΔJ + Ja and ρ = Δρ + ρa. These applied current and charge densities only exist in a certain approximation philosophy as outlined below. In the exact world in which Maxwell's equations apply everywhere, one has Ja = 0 and ρa = 0, and in this case ΔJ = 0 and Δρ = 0. In the approximation model one regards the applied Ja (and corresponding ρa) as a current density which is "applied to" or "driven into" the conductors of a radiating antenna or a transmission line, as if this could be done outside the realm of Maxwell's equations. From the assumed applied Ja and ρa one computes the fields E and B using Maxwell's equations and then hopes that these fields in turn produce J and ρ on the conductors which are consistent with the assumed applied Ja and ρa. [ this seems pretty reasonable in this new notation. ] div D = ρ ∫V ρ dV = ∫S D dA (1.1.13) div E = ρ/ε (1/ε)∫V ρ dV = ∫S E dA if ε is constant in space (1.1.14) div(J) = - ∂tρ -∂t[∫V ρ dV] = ∫S J dA (1.1.16) curl H = ∂D/∂t + J H ds = ∫S [∂t D+J] dA (1.1.18) (1/μ) curl B = ε ∂E/∂t + J B ds = μ ∫S [ε ∂t E + J] dA (1.1.19) These certainly look nicer. Section 1.2 In my verification doc, I have now rederived (1.2.1) and (1.2.2) with these results (either pair is OK) (2 - με ∂t2)H = - curl J (2 - με ∂t2)E = μ∂tJ + (1/ε) grad ρ // driven wave equations (2 - με ∂t2 - μσ∂t)H = 0 (2 - με ∂t2 -μσ∂t)E = (1/ε) grad ρ // damped wave equations if J = σE I think I will only put the first form, and I can put in a Jackson reference if I want. But actually I need the second form later, In the ω domain we have (2 + με ω2)H = - curl J (2 + με ω2)E = μ∂tJ + (1/ε) grad ρ // driven wave equations (2 + με ω2 - jμσω)H = 0 (2 + με ω2 -jμσω)E = (1/ε) grad ρ // damped wave equations if J = σE So I might want to define β02 = μεω2 β2 = μεω2 -jμσω = μω2 [ε -jσ/ω] = μω2 [ε +σ/jω] = what I have been using all along Thus, we can write (2 + β02)H = - curl J (2 + β02)E = μ∂tJ + (1/ε) grad ρ // driven wave equations (2 + β2)H = 0 (2 + β2)E = (1/ε) grad ρ // damped wave equations if J = σE inside a conductor this last item will have ρ = 0, then it will find use ahead. Section 1.3 These too I have rederived in the verifications doc, with these results (2 - με ∂t2)A = grad [με ∂tφ + divA ] - μJ (1.3.2) 2φ + ∂t (div A) = - (1/ε)ρ (1.3.3) Then use of this gauge condition divA = - με ∂tφ (1.3.4) gives these results again with Jackson refs: (2 - με ∂t2)A = - μJ (1.3.5) (2 - με ∂t2)φ = - (1/ε)ρ (1.3.6) Relativity note Cleans up just fine, but have to do all the words! Section 1.4 retarded potentials Just remove the T subscripts Section 1.5 Change example to match the equation! I see big trouble brewing here now! The equation after the above repair is this: (2 - με ∂t2)φ = - (1/ε)ρ There is no damping term, so my whole β2 discussion will be wrong !!!!! Yikes!! I will just edit right here and see what happens: (2 + μεω2) φ(x,ω) = - (1/ε)ρ(x,ω) or (2 + β2) φ(x,ω) = - (1/ε)ρ(x,ω) where β2 is the following complex "Helmholtz parameter" [of operator (2 + β2) ] β2 = μεω2 = ω2/v2 Holy cow, this is going to be a huge change. How are we going to get that 3/4π special phase in the round wire situation? Let's hold right here in the middle of Section 1.5 Jump ahead to 2.1 on the round wire. When I get to (2.1.12) I could quote my new OK, enough goofing around here. I really need to do a MASSIVE edit on the entire lines doc! Be sure to do a very good backup save!! I have to read every single paragraph.