in many books (Electromagnetics, M. Notarov; Foundations for Microwave Engineering, Collin; Microwave Engineering, Pozar etc) I have read that in a certain waveguide or transmission line, the transversal electric field and the magnetic field have exactly the same spatial distribution that they have in electrostatic and magnetostatic problem.

From a math point of view, this is due to the fact that, under TEM hypothesis (Ez = 0, Hz = 0, with z = longitudinal axis of the line/waveguide), Maxwell's equations lead to the result that both H and E are solenoidal (divergence equal to 0) and irrotational (curl equal to 0).

Here you see one of the references I quoted before:

enter image description here

This text expresses what I have told before by using the Laplace equation, which is obtained by combining the equations:

  • $\nabla\cdot E= 0$
  • $\nabla\times E = 0$

and, for H:

  • $\nabla\cdot H= 0$
  • $\nabla\times H = 0$

Now, let's consider for instance a coaxial cable, whose TEM distribution is shown in the following picture:

enter image description here

I have some difficulties on understanding the meaning of the previous equations and their correspondence with field lines:

1) About E: previous equations say that it does not diverge and does not rotate. If you look at the field lines, it is ok that it does not rotate. But it diverges (it is radial). Why?

2) About H: previous equations say that it does not diverge and does not rotate. If you look at the field lines, it is ok that it does not diverge, but it rotates. Why?


1 Answer 1


Let’s avoid getting confused by manipulations of Maxwell’s Equations and return to the Equations themselves (electrostatic + in free space):

Gauss’ Law: $\nabla\cdot E=\frac{\rho}{\epsilon_0}$

Ampere’s Law: $\nabla\times H = J$

So to me, the reason the fields are the way they are is straightforward. $E$ is diverging from a charge density $\rho$ on the center conductor (and diverging into a negative charge density in the shield). $H$ is rotating around a current density $J$ into the page in the center conductor (and an opposite current in the shield prevents magnetic field from extending farther out).

  • $\begingroup$ With this kind of analysis, I understand it. Regarding what I have written in my question, some books add also that E and H are both solenoidal and irrotational because in the dielectric region there are no charge and no current. So if for instance we analyze the electric field in the dielectric region, we will conclude that it does not diverges, but in reality it does. $\endgroup$
    – Kinka-Byo
    Commented Feb 29, 2020 at 8:32
  • $\begingroup$ @Kinka-Byo well it’s solenoidal and irrotational where there are no sources. But, for instance, electric field diverges from a single point charge. I think you’re getting confused by the differential forms of these equations, which you might (mistakenly) read that a charge has to be present everywhere there’s any E-field divergence. So try the integral forms. With these, e.g., you take a surface integral around a charge and find that the field is penetrating through the surface, thus explaining the coaxial waveguide. $\endgroup$
    – Gilbert
    Commented Feb 29, 2020 at 15:05

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