As I understand, in an ideal TEM transmission line we can set up the telegrapher equations and solve to show that the line propagates voltage and current waves.
My confusion arises when we recognize that the boundary conditions we have enforced for this ideal situation imply the electric field in the system is always perpendicular to the surfaces of the conductors, and the magnetic field is always parallel to the surfaces but perpendicular to the direction of propagation (direction of the line).
If there is AC current in the line, then locally charges are being accelerated in the direction of propagation.
If charges are being accelerated, locally, along the line in one direction or the other, then doesn't there need to be, locally, either some electric field that is parallel to the direction of propagation, or (more weirdly) some velocity of charge carriers in the line that is perpendicular both to the magnetic field and the direction of propagation?
Furthermore, doesn't the mere fact that we have modulated charge density along the line (before or because we have modulated current density) imply that there must be electric field pointing between the peaks and the valleys of this modulated charge density, and hence along surface of the line, not perpendicular to it?
I believe that in situations with nonideal conductors electric field along the direction of propagation is not forbidden roughly up to the skin depth of the conductor, but as I understand it this is not the case for the ideal situation of lossless conductors, which is the system confusing me right now.
Am I wrong about the assumption that there is no electric field along the conductor? Is there some sense in which the lumped element approximation of the transmission line buries this fact? Can, in classical electrodynamics, charges be accelerated without any local electric or magnetic fields (something that would be surprising to me and indicate I really have misunderstood something)?
Fundamentally my question is 'where, explicitly, is the force coming from that is accelerating the charges along the line?' I don't know if this is possible, but I would appreciate the answer being in terms of electric and magnetic fields, not scalar or vector potentials.
As always, thanks for your attention.