Derivation of Jackson 6_37

Derivation of green Jackson 6.37 and 6.38 PhL 10.13.13 blue Jackson 6.15 and 6.16 I will do this in blue Jackson in SI units, so the equation numbers will be different 1. Statement of problem: Start with my own transmission lines equations which match Jackson's curl H = ∂D/∂t + J Maxwell curl H equation (1.1.1) curl E = - ∂B/∂t Maxwell curl E equation (1.1.2) div D = ρ 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) And make use of B = curl A E = - grad φ - ∂A/∂t . (1.3.1) and first derive these two equations: 2. Jackson's Derivation. As a hint, Jackson tells us to use just the two "inhomo" which for him are these curl H = ∂D/∂t + J div D = ρ I first rewrite these as (1/μ)curl B = ε ∂tE + J div E = ρ/ε I then insert E = - grad φ - ∂A/∂t and B = curlA to get (1/μ)curl curl A = ε∂t[- grad φ - ∂A/∂t] + J div [- grad φ - ∂A/∂t] = ρ/ε or (1/μ)curl curl A = ε [- grad ∂tφ - ∂t2A] + J div [grad φ + ∂tA] = -ρ/ε or grad div A - 2A = με [- grad ∂tφ - ∂t2A] + μJ [2 φ + ∂t div A] = -ρ/ε or 2A - grad div A =(1/c2) [grad ∂tφ + ∂t2A] - μJ [2 φ + ∂t div A] = -ρ/ε or 2A - (1/c2) ∂t2A = grad div A + (1/c2) [grad ∂tφ] [2 φ + ∂t div A] = -ρ/ε or 2A - (1/c2) ∂t2A - grad [div A + (1/c2) ∂tφ ] = - μJ [2 φ + ∂t div A] = -ρ/ε Thus I have derived the two equations shown above. Next step is to use div A = - (1/c2) ∂tφ and these equations then become 2A - (1/c2) ∂t2A = - μJ [2 φ - (1/c2) ∂t2 div A] = -ρ/ε and these are what I want, 3. An Alternate Derivation Instead of using the two "inhomo equations", use the two curl equations: curl H = ∂tD + J Maxwell curl H equation (1.1.1) curl E = - ∂tB Maxwell curl E equation (1.1.2) Rewrite these as (1/μ)curl B = ε∂tE + J curl E = - ∂tB or curl B = με∂tE + μJ curl E = - ∂tB Now in the first equation replace B = curl A to get curl curl A = με∂tE + μJ Then use the same identity used above to write this as grad div A - 2A = με∂tE + μJ Then install E = - grad φ - ∂tA to get grad div A - 2A = με [- grad ∂tφ - ∂t2A] + μJ or grad div A - 2A = με [- grad ∂tφ - ∂t2A] + μJ etc.