Q1Electromagnetics Waves

A TEM (transverse electromagnetic) wave has both its electric and magnetic fields entirely transverse to the direction of propagation, i.e., Ez = 0 and Hz = 0 simultaneously (z being the direction of propagation). Two-conductor lines such as coaxial cable or parallel-wire lines support TEM because the inner and outer conductors can sustain a genuine static-like transverse E-field pattern (satisfying Laplace's equation in the cross-section) between them, with a return current path on the second conductor.

Maxwell's-equation argument: inside a hollow, single-conductor waveguide (no inner conductor), consider the source-free, time-harmonic Maxwell curl equations for waves varying as e^{−jβz} along the guide:

Decomposing these into transverse and z-components, one obtains the transverse fields Et and Ht in terms of the longitudinal components Ez and Hz and the transverse gradient operator, of the general form:

If we impose the TEM condition Ez = 0 and Hz = 0 directly into these relations, the right-hand side becomes 0/0 (kc² also → 0 for TEM, since TEM requires β = k exactly) — the transverse fields become indeterminate unless kc = 0 is treated as an independent case, requiring Et to satisfy a genuine two-dimensional electrostatic-like boundary value problem, ∇t·Et = 0 and ∇t×Et = 0, inside the guide's cross-section with the guide wall as the only boundary. For a simply-connected, hollow, single conducting boundary (the interior of a rectangular or circular waveguide, with no second, inner conductor), the unique solution to this Laplace-type boundary value problem with Et = 0 (tangential E must vanish on a perfect conductor) on the entire boundary is the trivial solution Et = 0 everywhere inside — a direct consequence of the uniqueness theorem for Laplace's equation with an all-Dirichlet zero boundary on a simply-connected domain. Since Et = 0 forces Ht = 0 as well via the coupled Maxwell relations, the entire field vanishes identically.

Physical interpretation: a genuine TEM field requires two distinct, electrically isolated conductors: one to source the field lines (positive charge/current) and a separate return conductor to terminate them, exactly as in a coaxial line or parallel-wire line. A hollow single-conductor waveguide has only one conducting boundary, so there is no second conductor to close the transverse field lines or to carry a net longitudinal return current; consequently a nontrivial static-like transverse field pattern cannot be supported, and TEM propagation is impossible. Waveguides instead propagate TE modes (Ez = 0, Hz ≠ 0) or TM modes (Ez ≠ 0, Hz = 0), in which the required longitudinal field component supplies exactly the missing degree of freedom that allows a nontrivial, non-static transverse field pattern to exist consistently with the single-conductor boundary and with Maxwell's equations, at the cost of introducing a nonzero cutoff frequency below which the mode cannot propagate — a further fundamental distinction from the dispersion-free, cutoff-free TEM mode of two-conductor lines.