new Section B_7 REVIEWED

New Section B.7 PhL 1.16.14 Section B.7 Application of the Jm Theorem to a round wire with uniform Jz. 1 (a) The Az(c) term 1 (b) The Az(m) term 3 (c) Adding the two terms and checking boundary conditions 4 (d) Plots of Az and Bθ and Hθ 5 Section B.7 Application of the Jm Theorem to a round wire with uniform Jz. Assuming the transmission line limit of small β so e-jβR ≈ 1, we start with (B.6.1) Az(x) = ∫ dV' [μ2 Jcz(x') + μ0 Jmz(x')] . R = |x - x'| (B.6.1) (B.7.1) but we go at once to the 2D solution [22DA(x) = - μ2Jc(x)] limit to get -Az(x) = ∫ dA' [μ2 Jcz(x') + μ0 Jmz(x')] // ln(1/R) = - The two currents are given by Jcz(x') = Jcz = I/(πa2) // uniform (B.1.10) Jmz(x') = Kz δ(r-a) with Kz = - ( - ) Hθ (B.7.2) (a) The Az(c) term The first term in ** is then -Az(c)(r,θ) = ∫ dA' = !Syntax Error, Ir' dr'!Syntax Error, Idθ' = !Syntax Error, Ir' dr'!Syntax Error, Idθ' ln(r'2 +r2-2rr' cos(θ-θ') ) = !Syntax Error, Ir' dr'!Syntax Error, Idx ln(r'2 +r2 - 2rr' cosx ) = !Syntax Error, Ir' dr' Q(r',r) (B.7.3) where we have defined the integral Q(r',r) ≡ !Syntax Error, Idx ln [r'2 +r2-2rr' cosx] (B.7.4) The round wire geometry is shown in this drawing, where R2 comes from the law of cosines, Fig B.7 The integral Q(r',r) may be evaluated using GR7 p 531 4.224, with a = r'2 +r2 and b = -2rr' and a2-b2 = (r'2-r2)2 so that = | r'2-r2 | .The condition a > |b| > 0 is met since (r±r')2 > 0 => r2+r'2 > ±2rr' which says a > ±b so a > |b|. Thus, Q(r',r) = !Syntax Error, Idx ln [r'2 +r2-2rr' cosx)] = π ln [ ] = = 2π . (B.7.5) We then have -Az(c)(r,θ) = !Syntax Error, I dr' r' Q(r',r) = !Syntax Error, I dr'r' = !Syntax Error, I dr' r' { lnr' θ(r'>r) + lnr θ(r'<r) } = [ θ(r<a) !Syntax Error, I dr' r' lnr' + lnr !Syntax Error, I dr' r' ] = { θ(r<a) (1/2){a2lna - r2lnr - (a2-r2)/2} + (1/2)lnr [min(a,r)]2 } (B.7.6) where Maple says . We then write out Az(c)(r) in its two regions Az(c)(r>a) = - { (1/2) a2lnr } = - lnr (B.7.7) Az(c)(r<a) = - { (1/2){a2lna - r2lnr - (a2-r2)/2} + (1/2) r2 lnr} = { (a2-r2)/2 - a2lna } (b) The Az(m) term From (B.7.1) and (B.7.2), -Az(m)(x) = μ0∫ dA' Jmz(x')] = !Syntax Error, Ir' dr'!Syntax Error, Idθ' [- ( - ) Hθ(r') ] δ(r'-a) ln(R2) = - Hθ(a) (μ2- μ1)!Syntax Error, Idθ' ln(a2 +r2-2ra cos(θ-θ')) = - Hθ(a) (μ2- μ1) !Syntax Error, Idx ln(a2 +r2-2ra cos(x)) = - Hθ(a) (μ2- μ1) Q(a,r) = - Hθ(a) (μ2- μ1) 2π = a Hθ(a) (μ1- μ2) (B.7.8) From Ampere's law (1.1.37) we have (ignoring displacement current inside the conductor) H ds = 2πaHθ(a) = ∫S J dS = I => Hθ(a) = and then -Az(m)(x) = (μ1-μ2) . (B.7.9) Then Az(m)(r>a) = (μ2-μ1) lnr Az(m)(r<a) = (μ2-μ1) lna (B.7.10) (c) Adding the two terms and checking boundary conditions Az(r>a) = - lnr + (μ2- μ1) lnr = - lnr Az(r<a) = { (a2-r2)/2 - a2lna } + (μ2-μ1) lna = { (a2-r2)/(2a2) - lna } + (μ2-μ1) lna = - { μ2 (r2-a2)/(2a2) + μ1 lna } Here then are the finally results for the potential Az = Az(c) + Az(m), Az(r>a) = - μ1lnr Az(r<a) = - { μ1 lna + μ2 } (B.7.11) As a check, we calculate the B and H fields implied by these potentials B = curl A = [ r-1∂θAz - ∂zAθ] + [∂zAr - ∂rAz] + [ r-1∂r(rAθ) - r-1∂θAr ] = [- ∂rAz] => Bθ = -∂rAz(r) Then Bθ(r>a) = μ1 (1/r) Bθ(r<a) = μ2 = μ2 (r/a2) (B.7.12) so the H fields are then Hθ(r>a) = (1/r) Hθ(r<a) = (r/a2) (B.7.13) which agrees with Ampere's Law applied in these two regions: 2πr Hθ(r>a) = I => Hθ(r>a) = (1/r) 2πr Hθ(r<a) = I(πr2/πa2) => Hθ(r>a) = (r/a2) . (B.7.14) Next, we check our two boundary conditions required by (B.6.0) : value at r = a: Az(r>a)|r=a – Az(r<a) |r=a = - μ1lna - ([ μ2 - μ1 lna ]) = 0 OK slope at r = a: [∂rAz = -Bθ so use(B.7.12) ] ∂rAz(r>a) |r=a = - μ1 (1/a) ∂rAz(r<a) |r=a = - μ2(a/a2) = - μ2(1/a) ∂rAz(r>a) |r=a – ∂rAz(r<a) |r=a = - (1/a) - [- (1/a)] = 0 OK Thus we have shown for the round conductor with uniform Jz that "the Jm theorem" works. By adding the bogus surface current term, we generate the correct homogeneous solution which when added to the Helmholtz integral provides the correct total solution which meets both boundary conditions. (d) Plots of Az and Bθ and Hθ Maple provides plots of Az from (B.7.11), Bθ from (B.7.12) and Hθ from (B.7.13) for this round wire situation. Parameters are set to I = 1, a = 2, μ1 = 2 (dielectric), μ2 = 3 (wire). Fig B.8 Az wanders down as ~ -ln(r) for large r; Bθ jumps down at r = a while Hθ is continuous there.