TEM mode in a waveguide From Jackson: "The TEM mode cannot exist inside a single hollow, cylindrical conductor of infinite conducivity. The surface is an equipotential; the electric field therefore vanishes inside". I didn't understand this sentence. If the surface is an equipotential, and therefore the electric field inside vanishes, why only the TEM mode is prohibited? Why not the the TM and TE modes too? 
 A: In a transverse electromagnetic EM wave, the electric field is perpendicular to the magnetic field, and both are perpendicular to the axis of the cylindrical conductor.
The surface of the a hollow, cylindrical conductor of infinite conductivity is equipotential, meaning that the electric potential is the same everywhere on the surface. This means that the electric field inside the conductor must be zero, because there is no potential difference between any two points inside the conductor.
Because the electric field is a fundamental component of the TEM wave, it cannot exist inside the conductor.
The Transverse Magnetic (TM) and Transverse Electric (TE) modes, however, do not have this restriction. In a TE wave, the electric field is transverse to the direction of propagation and the magnetic field is parallel to the direction of propagation. The magnetic field does not need to be zero inside the conductor for the wave to exist. Same is  true for TM wave. So, both modes exist in this case.
A: A TEM mode is such that the fields are both lamellar (curl=0) and solenoidal (div=0), hence it can be derived from a single scalar potential function that satisfies the Laplace equation. As such it must be identically zero (or constant) within a closed singly connected domain on whose boundary the function is 0 (or constant). In plain English this means you need at least two pieces of unconnected metal at different potentials (voltages) to define a static field within a finite region.
