# Magnetic Field Distribution in TEM (TransverseElectroMagnetic) Modes

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:

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:

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?

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).