Howie on the TEM wave guided by two conductors.

“ ….elsewhere.  For good but not
perfect conductors the electric field is now no longer purely transverse
but has a small longitudinal component which has opposite sign at the
two conductors.  I also mentioned that this extra small longitudinal E
field component gives rise via Poynting's vector to an energy flow into
each of the conductors compensating for the Joule losses that are going
on there because of the current flow.  The …. “ – A Howie, 19.4.2021, 5.54 or 10.54

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: Ongoing challenges

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Prof. A Howie	19 Apr 2021, 10:54 (8 days ago)
to Malcolm, me, Christopher, Anthony, Brian, HARRY, Steve, Phil, Alex, Forrest, Anthony

Dear Malcolm,

It may be useful to separate your question into two parts.

(a) Do we have a current loop? If we think of a conventional current
loop i.e. charge flowing round a circuit we could break the conducting
circuit at some point.  Current could continue to flow in the conducting
parts of the circuit but charge would build up plus and minus on either
side of the break giving rise to an increasing voltage across the break
i.e, the dE/dt term called displacement current.  Although no charge or
anything else is flowing across the gap the dE/dt term is referred to as
displacement current since it preserves the idea of a current loop. 
Much more important than that of course was that when this extra term
was inserted by Maxwell into Ampere's law the possibility of EM waves
propagating at the velocity of light emerged.

(b) What is driving the current?  In the unbroken parts of the circuit
are actually good conducting (but not perfect i.e, not superconducting)
metals we need an electric field component to drive the current. You
should note that this field component is needed in BOTH the upper and
lower conductor.  As I explained before you will find that this is
analysed in Jackson's book and no doubt elsewhere.  For good but not
perfect conductors the electric field is now no longer purely transverse
but has a small longitudinal component which has opposite sign at the
two conductors.  I also mentioned that this extra small longitudinal E
field component gives rise via Poynting's vector to an energy flow into
each of the conductors compensating for the Joule losses that are going
on there because of the current flow.  The EM field strength in the
guide is then gradually depleted.

I rather fear that repeating this to you is unlikely to have any effect
so it has probably been a waste of my time.  Giving up on this task,
which Ivor prefers to call "omerta" may more accurately be called
"despair".  Nevertheless I live in hope!

Archie Howie.