Why is del(E_t) zero in TEM mode of transmission lines?

[Adit112]

I was going through MIT opencourseware https://ocw.mit.edu/courses/electri...s-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? I can always construct a surface in the transverse plane to include the transmission line which has charge and so the del.Et should be equal to the surface charge density isn't?

 

[NFuller]

The divergence of E is zero in between the conductors because there is no charge in that space. The surface charge density is only used for boundary conditions when solving for the fields.