Voltage and current waves in a transmission line

[Txema]

TL;DR Summary

How voltage and current waves in a transmission line fit with TEM waves in the dielectric?

INTRODUCTION
From the boundary conditions of the electromagnetic field in perfect conductors, it is deduced that in a transmission line with a time-varying current, the field vectors E and B in the dielectric lie in planes transverse to the conductors and also that the E field is normal to the conductors while the B field is tangential to them.
Using Maxwell's equations, it is deduced that the fields depend only on the axial coordinate and time: E(z,t), B(z,t), and wave equations are obtained for both fields. These are TEM waves propagating through the dielectric between the conductors.

However, the usual description of Transmission lines is done by voltage and current.
Textbooks are mainly concerned with practical issues. Some authors explain the transition from field waves to v,i waves through the similarity of the E and B fields of the TEM wave in each transverse plane with a situation of static fields. The curl of the electric field in a transverse plane is zero, so its circulation between two points is independent of the path, it means there is a defined voltage, therefore, the Laplace equation is fulfilled in the dielectric. In consequence, the laws that govern static fields can be used. Thus wave equations of v and i are obtained. Some books include a drawing of a coaxial cable in which the electric field of the TEM wave starts in one conductor and ends in the other in a radial fashion.

I can not understand this similarity that allows studying the line by current and voltage waves. It leads me to the following questions.

PREMISE 1
The electric potential is the potential function of a so-called gradient vector field, E = grad v , it is a conservative E field whose scalar sources are the charges, which are the beginning and end of its field lines. Static E field is conservative.
An electromagnetic wave involves radiating E and B fields, of vector origin (magnetic potential A), whose field lines close on themselves. They are fields with non-zero curl, non-conservative.
QUESTION 1
How can the electric field of an electromagnetic wave, with curl nature, be conservative as a static E field is? How can its field lines start and end at the charges instead closing on themselves?

PREMISE 2
In every circuit, the electric current is directed by the surface charge on its conductors, which adjusts the electric field inside the conductors depending on the geometry of the circuit and the resistivity of the conductors. The surface charge creates an electric field not only in the conductors but also in the dielectric.
It is known that acellerated charges create electromagnetic waves.
QUESTION 2.1
Is the voltage that can be measured on a line with a voltmeter the consequence of the electric field produced by the surface charge of both conductors and not the electric field of the wave?
QUESTION 2.2
Are there two electric fields in the dielectric?. A reactive one produced by the surface charge and a radiant one as a result (constructive wave) of the acceleration of the charge carriers in both conductors?

Any help would be appreciated.

 

[jasonRF]

For the first issue, have you tried working out the math to show that the “transverse curl” of the transverse E-field is zero for a TEM wave? A free online book that shows this is
http://eceweb1.rutgers.edu/~orfanidi/ewa/
Look in chapter 9. My books are at work, but I am sure that I have a few that cover this. Foundations for Microwave Engineering by Collin is superb with this kind of stuff. Any edition.

For the other stuff, the surface charges and surface currents on the conductors are self-consistent with the electromagnetic field propagating along the line. Combined they make up the guided wave. I don’t think it is at all helpful (or even makes sense) to try and think of the “field of the wave” and the “field of the charges” as separate things.

Jason

 

[tech99]

Re Question 2.2, there is, in addition to the TEM mode, a longitudinal mode (a single-wire mode) where the lines of the E-field terminate at points along the same conductor. We need a longitudinal field to accelerate the electrons. This mode exists in all transmission lines but is small for close spaced conductors. For a lumped element transmission line, with series L and shunt C, there is a voltage across each inductor, where E field lines connect between the ends of the inductors.

 

[Txema]

For the other stuff, the surface charges and surface currents on the conductors are self-consistent with the electromagnetic field propagating along the line. Combined they make up the guided wave. I don’t think it is at all helpful (or even makes sense) to try and think of the “field of the wave” and the “field of the charges” as separate things.

This comment has been very timely. I don't know for what reason, I had the idea that coulomb and non-Coulomb electric fields lead separate lives, perhaps due to their different sources. I realize that they are both sets of vectors E in space. With this, my objections vanishes.
I will review the books you suggest.
Thank you so much.

 

[Txema]

Re Question 2.2, there is, in addition to the TEM mode, a longitudinal mode (a single-wire mode) where the lines of the E-field terminate at points along the same conductor.

My approach was the simplest, neglecting the other waves compared to the tem wave.
In any case, thanks for responding.