Is the magnetic field in an ideal coaxial cable carrying a TEM wave solely defined by the Ampere/Maxwell displacement current?

Question

If I have a plain TEM wave in an ideal coaxial cable, a simple solution to the wave equation assumes a wave in the source free region between the center conductor and the shield. The magnetic field is calculated only from the changing electric field (displacement current) in the Ampere/Maxwell equation.

Is this a complete description of the magnetic field in the dielectric?

I ask because the magnetic field from this changing electric field has non-zero curl in the dielectric. If I now also consider the electron current in the center conductor, occurring from the boundary condition (PEC), then there is a magnetic field with curl only in the conductor (or where the surface current flows in an ideal cable) and a magnetic field with no curl in the dielectric.

Is the non-curling magnetic field in the dielectric (calculated from the current flow) somehow already accounted for in the magnetic field derived as part of the traveling wave? It seems to me that the TEM wave should completely describe the magnetic and electric fields but I'm not clear on how to account for the magnetic field from the current flow.

Answer 1

It turns out that there are vacuum (or linear-dielectric bound-charge) solutions to the Maxwell equations for a coaxial cable (c.f. Griffiths chapter 9), and the associated coupled equations aren't terribly difficult to solve in simple cases such as coaxial cables with circular or rectangular cross sections. However, the solutions typically aren't that different from what you would find if you ignored the magnetic field to solve for the electric field first and then calculated the magnetic field 'induced' by the electric field. The reason for this is essentially that $c$ (the speed of light) is very large by normal standards, even in a typical dielectric, and even for typical electronic signaling applications, and so $\epsilon\mu = 1/c^2$ can be treated as a "small" perturbation.

Solutions with a non-zero $E_z$ component (the component parallel to the wire) are harder to compute as they will tend to induce a nonzero conduction current along the inner and outer layers of the wire. In principle you would need to solve a coupled system of equations involving constitutive relations for the conductors (which may have a surface charge.) If $E_z=0$, as is typical away from the ends of the wire, there is no induced current and the signal propagates purely as an electromagnetic wave.

Answer 2

To answer my own question 😺 ... I think it comes down to the different direction of the curl of the magnetic fields caused by the conduction current density (μJ) and the displacement current from (-$\frac{\partial E}{\partial t}$).

The curl of the magnetic field caused by the conduction current density is in the direction of the current (z direction) but the direction of the curl of the magnetic field from the displacement current is in the radial direction (ρ) of the E field. (The magnetic field lines in both cases are coaxial with the center conductor of course)
(there is a magnetic field with curl in/on the cable but a magnetic field without curl in the dielectric from the conduction current.)

When deriving the integral form of Ampere/Maxwell we use Stokes' theorem and integrate the curl component normal to the surface, over the surface, to get the line integral of the magnetic around the boundary. If this surface is bounded by the Amperian loop coaxial with the center conductor, then the curl of the magnetic field caused by the displacement current (which is radial) has zero contribution normal to the surface and thus plays no part in the calculation. Therefore, although the magnetic field in the dielectric has curl in the radial (ρ) direction, it plays no part the calculation of current in/on the center conductor.

This make intuitive sense when also considering the L/C lumped circuit representation.