jackson paradox REVIEWED

Paradox with Jackson PhL 10.13.13 The point here is that you get different Az wave equations if you select different gauges. I think I am happy with all this stuff nowadays. 1. Statement of the Paradox (a) On the one hand, in lines doc, I have this derivation of the potential wave equation. I have copied the derivation but have set Ja = 0 everywhere. ______________________________________________ Start with (1.1.1) curl H = ∂D/∂t + J Replace H by B/μ = curlA/μ to get (1/μ) curl curl A = ∂D/∂t + J or (1/μ) curl curl A = ε ∂E/∂t + σ E Now use the identity curl curl A = grad divA - 2A // verified against vector identities doc and we get grad divA - 2A = με ∂E/∂t + μσ E Next replace on the right E = - grad φ - ∂A/∂t to get grad divA - 2A = με [- grad ∂tφ - ∂t2A ] + [ μσ (- grad φ - ∂A/∂t) ] or just changing signs, -grad divA + 2A = με [grad ∂tφ + ∂t2A ] + [ μσ ( grad φ + ∂A/∂t) ] or 2A - με ∂t2A - μσ ∂tA = grad divA + με grad ∂tφ + μσ grad φ or ( 2 - με ∂t2 - μσ ∂t)A = grad divA + με grad ∂tφ + μσ grad φ or ( 2 - με ∂t2 - μσ ∂t)A = grad [ divA + με ∂tφ + μσ φ ] (1.3.3) ______________________________________________ Then if I use my King gauge condition 1.3.4 div A = - με ∂tφ - μσ φ . (1.3.4) I then end up with ( 2 - με ∂t2 - μσ ∂t)A = 0. If after doing the above I take the limit σ → 0, I get ( 2 - με ∂t2)A = 0 div A = - με ∂tφ . (b) On the other hand, green Jackson page 180 (6.38) tells me that ( 2 - με ∂t2)A = -4πJ // gaussian units These results conflict with each other, hence the paradox. And I have derived the Jackson results in a separate document. 2. Discussion of paradox This is a very subtle little deal here. I got the impression in the Jackson case that we had σ = 0 so no conductivity in our medium. If that were true, then J = σE = 0 and so J = 0 and then there is no paradox, both methods agree that ( 2 - με ∂t2)A = 0. In order to have a J, you have to have something that conducts, like a wire! Inside that wire, σ > 0. When you apply Ampere's Law you do ∫JdA over a surface that includes a cross section of a wire, and THAT is where J is located. Fact 1: So my first misconception is that in the Jackson world we had σ = 0. That is just not so. He is assuming ε0 and μ0 but he is not assuming σ = 0! Fact 2: Therefore, our two gauge conditions are NOT the same: div A = - με ∂tφ - μσ φ my gauge condition (King) div A = - με ∂tφ his gauge condition (Lorenz) Both are "legal" gauge conditions. If you use the first, you obtain a version of A,φ that does not see real currents as a driving term! If you use the second, you obtain a different version of A,φ that DOES see the current as a driving term! Somehow this same mechanism causes ρ to appear in the potential equation. So, I guess I need to rewrite my derivations with the proper gauge conditions! And don't replace J by σE so early, just leave it as J.