Hi all,
the following is not a proof, however, it shows what was asked in the question:
Consider a rectangular/circular waveguide (no inner conductor), cartesian coordinate system.
TEM mode is assumed and direction of propagation is the z-axis.
a. Maxwell: rot x H = jwεE
b. Choosing a surface cut in the transverse plane of the waveguide and integrating both sides of the above:
∫(rot x H)•ds = ∫(jwεE)•ds
c. Using Stokes' theorem (forming Ampere's law):
∫(H)•dl = ∫(jwεE)•ds
d. Following the TEM mode assumption: Ez=Hz=0, while Ex,Ey,Hx,Hy≠0.
ds=ds•1z
dl=a function of x and y unit vector.
e. Solving c using d zeros the right side whereas the left side ≠ 0.
An obvious contradiction!
Hence, TEM mode cannot exist without an inner conductor that satisfies Ampere's law.
Regards,
P.