Home / Math and Physics Files / Physics / Transmission Lines / Notes By Chapter and Appendix / Appendix P eddy currents

Paradox with Appendix P Section 8 field plot figures REVEIWED

DOCX · 27.2 KB

A short Word note dated 5.5.14 from Phil's transmission line and eddy current work. He doubted that the field curves inside the wires should look perfectly straight. He derives Hy at y=0 near the left wire from the two-wire H1 and H2 formulas with step functions, finds a term not linear in x, and checks it with a Maple plot. He concludes the lines only look straight because the linear term dominates, so there is no paradox.

AI-written summary; may contain errors.

Extracted text (machine-read; may contain errors)

Paradox with Appendix P Section 8 field plot figures PhL 5.5.14 I don't think the red lines should be perfectly straight inside the wires in these plots, so I think something is wrong. [ it turns out the just look straight but are not exactly straight, so no paradox ] First, here is the math from "the Surface Currents issue..doc" : H1(r1) = (I/2π)[ θ(r1>a1) (1/r1) + θ(a1>r1) (r1/a12)] H2(r2) = -(I/2π)[ θ(r2>a2) (2/r2) + θ(a2>r2) (r2/a22)] Hx = - (y/r1)H1(r1) - (y/r2) H2(r2) Hy = (x/r1) H1(r1) + ((x-b)/r2) H2(r2) If I look only at Hy we get Hy = (x/r1) H1(r1) + ((x-b)/r2) H2(r2) = (x/r1) [(I/2π)[ θ(r1>a1) (1/r1) + θ(a1>r1) (r1/a12)]] + ((x-b)/r2) [-(I/2π)[ θ(r2>a2) (2/r2) + θ(a2>r2) (r2/a22)]] Now suppose we are close to the center of the left wire and we select using the Heavisides. Then we select the terms I show above as red Hy = (x/r1) [(I/2π)[ θ(a1>r1) (r1/a12)]] + ((x-b)/r2) [-(I/2π)[ θ(r2>a2) (2/r2)]] = (I/2π) [(x/r1) (r1/a12) - ((x-b)/r2) (2/r2)] = (I/2π) [x/a12 - 2 ((x-b)/r22)] r22 = (x-b)2 + y2 Now suppose I look only at y = 0. This then says Hy(x,y=0) = (I/2π) [x/a12 - 2 ((x-b)/(x-b)2)] = (I/2π) [x/a12 - 2/(x-b)] This function is NOT linear in x, but my plots show things as linear in x close to wire 1. This is the paradox. With a1 = 1/2 and b = 2 this says = (I/2π) [4x - 2/(x-3)] But OK, here is a Maple plot of this function: The first term is so dominant