# Divergence of Et in TEM propagation in transmission lines

I was going through MIT opencourseware https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-013-electromagnetics-and-applications-spring-2009/readings/MIT6_013S09_chap07.pdf and I didnt understand exactly why in equation 7.1.37 and 7.1.38 in page 192 the divergence of transverse electric field is taken as zero ? The only reason this would be true is if the region is source free but we clearly see that this field does setup a surface charge density on the surface of the transmission line so intuitively and mathematically why would the divergence of Et be zero?

For example, according to 7.1.37 $$\text{div}\textbf{H}=\text{div}_T\textbf{H}_T+\frac{\partial}{\partial z}\hat {\textbf{z}} \cdot \textbf{H}_T$$ but you know that there are no magnetic space charges $\text{div}\textbf{H}=0$ and by definition $\hat {\textbf{z}} \cdot \textbf{H}_T$, therefore you must also have $$\text{div}_T\textbf{H}_T=0$$; same consideration for the $\textbf{E}$ field.

The induced surface charges or currents play no role in the spatial divergence of the fields but they are important to define the boundary conditions: the E-field is perpendicular and $\propto \sigma$ and H-field is tangential with the ideal metal surface and $\propto \K$.

Here's my understanding. This is an excellent question, which i think should attract a lot of discussion.

The question is about TEM wave between two conductors, which are not necessarily planar, but having a cross-section that is independent of z.

//In planar (parallel plate tx line) case, we have followed these steps:

-set $$\\mathbf{E}\$$ and $$\\mathbf{H}\$$, such that the general boundary condition requirement ($$\\mathbf{E}\$$ perp, $$\\mathbf{H}\$$ parallel, to metal) is satisfied. Our assumption that E and H field are TEM ($$\\mathbf{E}=\mathbf{E}_T, \mathbf{H}=\mathbf{H}_T\$$), as well as, uniform plane wave ($$\\mathbf{E}=\mathbf{E}_T(z,t)=\hat{x}E_0cos(wt-kz)\$$ and $$\\mathbf{H}=\mathbf{H}_T(z,t)\$$, in eq.7.1.1) satisfy these constraints.

-find $$\\mathbf{E}\$$ and $$\\mathbf{H}\$$ wave equation, as they travel in the medium between the planes, which is indeed a source free region, so $$\J=0=Q\$$, and $$\\mathbf{\nabla}.\mathbf{E}=0\$$. Verify: $$\E=\hat{x}E_0cos(wt-kz) \implies \mathbf{\nabla}.\mathbf{E}=\frac{\partial E_x}{\partial x}+0+0=0\$$. For TEM, $$\\mathbf{\nabla}.\mathbf{E}=\mathbf{\nabla _T}.\mathbf{E_T}\$$, so $$\\mathbf{\nabla _T}.\mathbf{E_T}=0\$$ follows.

-The $$\\mathbf{E}\$$ and $$\\mathbf{H}\$$ so obtained, must be supported by surface charges and currents at the metal surface, so we use the (integral form of) $$\\mathbf{\nabla}.\mathbf{E}=\pm Q/\epsilon_0\$$ at $$\x=0\$$, and $$\x=d\$$, to find $$\Q\$$ in terms of $$\\mathbf{E}\$$ [then find $$\V\$$ in terms of $$\\mathbf{E}\$$, thus find Capacitance $$\C\$$].

//Similar logic and steps for non-planar tx line case:

-set $$\\mathbf{E}\$$ and $$\\mathbf{H}\$$, such that the general boundary condition requirement ($$\\mathbf{E}\$$ perp, $$\\mathbf{H}\$$ parallel, to metal) is satisfied. But now the two surfaces are arbitrary, so we can no longer impose plane wave form, but only TEM form: $$\\mathbf{E}=\mathbf{E}_T(x,y,z,t)\$$ and $$\\mathbf{H}=\mathbf{H}_T(x,y,z,t)\$$.

-find $$\\mathbf{E}\$$ and $$\\mathbf{H}\$$ wave equation, as they travel in the medium between the planes, which is indeed a source free region, so $$\J=0=Q\$$, and $$\\mathbf{\nabla}.\mathbf{E}=0\$$. For TEM, $$\\mathbf{\nabla}.\mathbf{E}=\mathbf{\nabla _T}.\mathbf{E_T}\$$, so $$\\mathbf{\nabla _T}.\mathbf{E_T}=0\$$ follows.

-The $$\\mathbf{E}\$$ and $$\\mathbf{H}\$$ so obtained, must be supported by surface charges and currents at the metal surface, so we use the (integral form of) $$\\mathbf{\nabla}.\mathbf{E}=\pm Q/\epsilon_0\$$ on a Gaussian surface covering one of the conductors, to find $$\Q\$$ in terms of $$\\mathbf{E}\$$ [in eq 7.1.39, then find $$\V\$$ in terms of $$\\mathbf{E}\$$, thus find Capacitance $$\C\$$].