Why does TEM mode not propagate in single conductor system?

Question

In the Wikipedia article it is said:

"In hollow waveguides (single conductor), TEM waves are not possible, since Maxwell's Equations will give that the electric field must then have zero divergence and zero curl and be equal to zero at boundaries, resulting in a zero field (or, equivalently,$\nabla^2 \Phi=0$ with boundary conditions guaranteeing only the trivial solution)."

My question is why is it required that the electric field be zero at the boundary of a single conductor system. I know that the tangential component of the field must be zero but why does the perpendicular component vanish in this case?

Answer 1

Assuming the hollow waveguide is a perfect conductor, it is an equipotential surface.

Now, imagine that there is an electric field line perpendicular the inner surface. Since there is no charge enclosed by the waveguide, this electric field line must terminate somewhere else on the inner surface (the electric field must have zero curl and so can't form a closed loop).

But that would imply that there is a potential difference between these two points on the inner surface which contradicts that this is an equipotential surface.

Thus, thus the perpendicular component of the field must be zero.