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Sisyphus paradox for 11_27.13 REVIEWED
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A short working note by Phil, dated 11.27.13, in his transmission lines material on the round wire. He reviews the logic of Section 2.5: the dielectric wave equation, the ansatz of negligible z dependence, and the resulting Hankel function E field outside the wire. He then argues that a long-wavelength wave with k = ω/v = βd makes the transverse equation lose its βd² term, contradicting the ansatz, so Section 2.5 may need rewriting.
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Sisyphus Paradox of the Day 11_27_13 PhL 11.27.13
See History section in "Finding a Meaning" for what is going on here.
1. In my new "section 2.5" I have assumed the following logic thread:
(a) First, we know from Maxwell that (dielectric does not conduct we shall say)
(2 + βd2)E(x,y,z,ω) = 0 in any region of dielectric where βd = ω
This is "the wave equation" for the E field in a region of space absent any current and any charge. It appears in my Chapter 1 as (1.2.1). It applies therefore to the region "outside" my round wire.
(b) Then without any justification, I made the ansatz that the E field has "weak z dependence" so that we can basically ignore ∂z E(x,t,z,ω) . In that case, the above becomes
(22D + βd2)E(x,y,z0,ω) = 0 region 1
where I just pick z0 to represent the z value, since E is basically assumed constant in z.
(c) With this assumption, and assuming E = Ez, I write Section 2.5 and I obtain various results solving the ODE which is (22D + βd2)Ez(r) = 0. For example, I find that ( 1 = dielectric, β1= βd )
E(r) = ~ constant * H0(1)(β1r) r > a
I was very pleased with myself for finding this sensible result for the E field outside the wire, although I did note at the time that it seemed to decay with r "rather slowly" which seemed odd.
Now here is the paradoxical issue:
In what physical situation would the above ansatz apply?
I find that I am unable to construct any such situation for the round wire.
Failed Situation #1.
At first I thought I could construct some kind of "long wavelength" situation for the wire where
E(x,y,z0,ω) = e-jkz E0(x,y,ω)
where somehow k is very small so we can ignore it. When this form is put into the wave equation above, the result is
(22D + βd2 - k2 )E0(x,y,ω) = 0 region 1
Then I just have to assume that k << βd and then I have my ansatz scenario exactly! So I then said the physical situation was just "some long wavelength wave" going down the wire.
However, waves do not go faster than c, and generally even in a transmission line only go at roughly velocity c. They go at the wave velocity in the dielectric medium I am pretty sure. There is then a relationship between k and ω which is v = ω/k. One cannot arbitrarily set ω and k at the same time. In fact, what you have is
k = ω/v = ω = βd
This says that k are βd are the same thing! Thus, the correct wave equation outside the dielectric must be this:
(22D + 0 )E0(x,y,ω) = 0 region 1
This equation is in complete conflict with the form I assumed above in region 1
(22D + βd2)E(x,y,z0,ω) = 0 region 1
and therefore my section 2.5 is completely wrong, or I have done something else wrong somewhere.
This then is my Paradox of the Day. I am not yet sure what implication this has for my Chapter 2 and the interior wire work I did there.
Idea: Maybe try to rewrite Section 2.5 in light of the new equation.