old sections D_4,D_5,D_6 ARCHIVE

Old Sections D.4, D.5 and D.6 (retired on 9/21/14) 9.21.14 D.4 Computation of the B fields in the round wire The B field components may be computed from the Maxwell curl E equation (1.1.2) - ∂tB = curl E . Maxwell curl E equation (1.1.2) (D.4.1) In cylindrical coordinates one has from (D.1.14), curl E = [ r-1∂θEz - ∂zEθ] + [∂zEr - ∂rEz] + [ r-1∂r(rEθ) - r-1∂θEr ] (D.4.2) where the fields are of the traveling wave form shown in (D.1.1) which we assume also for the B field. Thus, combining (D.1.1) with (D.1.3a), one has E(r,θz,t) = ej(ωt-kz) E(r,θ) = ej(ωt-kz) !Syntax Error, I E(r,m) ejmθ (D.4.3) B(r,θz,t) = ej(ωt-kz) B(r,θ) = ej(ωt-kz) !Syntax Error, I B(r,m) ejmθ . (D.4.4) Inserting the three cylindrical components of the E expansion (D.4.3) into (D.4.2), one finds that these replacements may be made, ∂t → +jω ∂z → -jk ∂θ → +jm . (D.4.5) Similarly, inserting the B expansion (D.4.4) into -∂tB one may replace ∂t→ +jω. After doing this, both sides of (D.4.1) are expansions having the general form of (D.4.3) and one may then equate terms in the m sum [completeness of the ejmθ on (-π.π)] to find that -jωB(r,m) = [ r-1jmEz +jkEθ] + [-jkEr - ∂rEz] + [ r-1∂r(rEθ) - r-1jmEr ] (D.4.6) and this then gives the three components of the B field Br(r,m) = (j/ω) [curl E]r = (j/ω) [r-1jmEz +jkEθ] Bθ(r,m) = (j/ω) [curl E]θ = (j/ω)[-jkEr - ∂rEz] Bz(r,m) = (j/ω) [curl E]z = (j/ω) [r-1∂r(rEθ) - r-1jmEr] . (D.4.7) It is now a mechanical task to insert our E field components, and such tasks are grist for Maple's mill. We use the E component forms summary box (D.2.21) which have the am and Km constants not yet specified. The alias line "unaliases" I, sets j = in place of the default I, and allows simple reference to the Bessel functions of interest. Diff(Ez,r) represents ∂rEz, but in an "inert" form which is not executed until later after Ez has been specified. The resulting B field expressions are somewhat ugly but can be cleaned up using a few more Maple manipulations. Having seen the results, we extract certain factors as shown in the following commands, which we then translate back into our normal notation, (ω/β')Bz(r,m) = ( + )Jm(x) (ω/jβ')Br(r,m) = + ( m - am) Jm(x) + ( + ) Jm+1(x) (ω/β')Bθ(r,m) = - ( m - am) Jm(x) + ( + ) Jm+1(x) . (D.4.8) The last two equations contain the same factor which can be written as (recall x = rβ') ( m - am) = ( m - am) = ( m - am)(1/x) The three equations for the exact B field components in the round wire are then shown in the summary box below which includes the earlier E field results as well: Summary of E and B fields inside a round wire (D.4.9) Ez(r,m) = - j (β'/k) Jm(x) x = β'r β'2 = β2 - k2 Er(r,m) = am x-1 Jm(x) + Jm+1(x) . jEθ(r,m) = - am x-1 Jm(x) + ( + ) Jm+1(x) (D.2.21) Bz(r,m) = (β'/ω) ( + )Jm(x) Br(r,m) = j(β'/ω){ + ( m - am) x-1Jm(x) + ( + ) Jm+1(x) } Bθ(r,m) = (β'/ω){ - ( m - am) x-1Jm(x) + ( + ) Jm+1(x) } , (D.4.8) where the constants are given in (D.2.28), which we rewrite using (D.2.32), am = (j/4) (ak) ηm I Rdc * 2m = (j/4) (ak) ηm I Rdc * [ – ] (+ ) = (j/4) (ak) ηm I Rdc * [ + ] . The three constant quantities at the end of the above summary box are roughly the same size in terms of scale. Using this fact, and the fact that for a low-loss line |β'| >> |k| = |βd| (so β = ≈ β') we can simplify (D.4.9) to read, Bz(r,m) = (β/ω) ( + )Jm(x) β' ≈ β, x = βr Br(r,m) = j(β/ω){ + ( m ) x-1Jm(x) } // m ≠ 0 Bθ(r,m) = (β/ω){ - ( m ) x-1Jm(x) + ( ) Jm+1(x) } . (D.4.10) The last line of (D.4.10) can be further simplified, Bθ(r,m) = (β/ω){ - ( m ) x-1Jm(x) + ( ) Jm+1(x) } = (β/ω) { - 2mx-1Jm(x) + 2Jm+1(x) } = (β/ω) { - Jm+1(x) - Jm-1(x) + 2Jm+1(x) } // Spiegel 24.17 identity = (β/ω) { Jm+1(x) - Jm-1(x) } . (D.4.11) Therefore, in the limit |β'| >> |k| = |βd| equations (D.4.10) become Bz(r,m) = (β/ω) ( + )Jm(x) β' ≈ β Br(r,m) = j(β/ω) { m x-1Jm(x) } // m ≠ 0 Bθ(r,m) = (β/ω) [ Jm+1(x) - Jm-1(x)] . (D.4.12) For m>0, |Br| and |Bθ| are larger than |Bz| by the large factor | β/βd|. For m = 0, Br ≈ 0 and |Bθ| >> |Bz|. It is this large Bθ field which appears in Chapter 2 as Bθ. D.5 Verification that the E and B fields satisfy the Maxwell equations The Maple program discussed above goes on to verify that the exact E and B fields obtained above for the round wire in fact satisfy Maxwell's equations. Since the B equations were obtained from the curl E Maxwell equation, this one is not verified. The other three Maxwell equations are projected into their partial wave versions analogous to (D.4.6) above : div B(r,m) = r-1∂r(rBr) + r-1∂θBθ + ∂zBz = r-1∂r(rBr) +r-1jmBθ -jk Bz (D.5.1) div E(r,m) = r-1∂r(rEr) + r-1∂θEθ + ∂zEz = r-1∂r(rEr) + r-1jmEθ - jkEz (D.5.2) curl B(r,m) = [ r-1∂θBz - ∂zBθ] + [∂zBr - ∂rBz] + [ r-1∂r(rBθ) - r-1∂θBr ] = [ r-1jmBz + jkBθ] + [-jkBr - ∂rBz] + [ r-1∂r(rBθ) - r-1jmBr ] . (D.5.3) Inside the round wire we expect to find div B = 0 // since B = (1/jω) curl E so div B = (1/jω) div curl E = 0 div E = 0 // no free charge curl B = μ J + μ jωεE = μ(σ + jωε) E = μ(jω)( ε - jσ/ω) E = jω μξ E = j (β2/ω) E . // see (1.5.1c) Thus, for the divergence equations we just compute the divergence as shown and see if it comes out zero, while for the curl B equation we verify that curl B - j(β2/ω) E = 0 (D.5.4) for each component. The fact (D.2.2) that β'2 = β2 - k2 is also used. Here then is the Maple code which does the verification of the three Maxwell equations: In Maple % refers to the last quantity computed. Prior to each simplify(%) statement we find a huge mess for the expression at hand, but simplify then shows it is really zero. As an example, here is the execution of the verification that [curl B]z - j(β2/ω) Ez = 0 : No approximations were made in the E fields, the B fields, or in these Maxwell verifications. D.6 The exact E and B fields for the m=0 partial wave The m=0 partial wave is all there is for an axially symmetric problem like that considered in Chapter 2, where the round wire is imagined in isolation, but is operationally the central conductor of a coaxial cable with a very distant return cylinder (outer shield). Here is the reduction of box (D.4.9) for m = 0, making use of the m=0 coefficients noted in box (D.2.28), namely, m = 0 a0 = 0 ( + ) = 0 = (jω/σ) N0 Summary of E and B fields inside a round wire ( m = 0 only ) (D.6.1) Ez(r,0) = - j (β'/k) J0(x) // large x = β'r β'2 = β2 - k2 Er(r,0) = J1(x) . // small = (j/2) (ak) I Rdc jEθ(r,0) = 0 Rdc = Bz(r,0) = 0 Br(r,0) = 0 Bθ(r,0) = (β'/ω) ( + ) J1(x) // large ~ β' (β'/k) No approximations have been made in these results, but a very good approximation for a low loss line is that |β| >> |k| = |βd| which means β' ≈ β, as discussed below equation (D.2.2). With this approximation, we have commented in the above box on the size of the various field components. The dominant components are Ez(r,0) = - j (β/k) J0(x) = - j (β/k) (j/2) (ak) I Rdc = (1/2) β a I Rdc = (ω/β) (1/2) (aβ2/ω) I Rdc Bθ(r,0) = (β/ω) ( ) (j/2) (ak) I Rdc = (j/2) (aβ2/ω) I Rdc . As shown in (2.2.3) we can write β2/ω ≈ - jμσ so that (1/2) (aβ2/ω) I Rdc = (1/2) a (- jμσ) I = - j and then the dominant components above become Ez(r,0) = (ω/β) [ - j ] = -j (ω/β) Bθ(r,0) = j [- j ] = . (D.6.2) These results are in agreement with E(r) and B(r) shown in summary box (2.2.30) from the Chapter 2 calculation where we assumed E = E(r) and B = B(r)