Rectangular waveguides (propagation along z): What is the correct relationship of the axial field components Ez and Hz in TE and TM modes?

Introduction / Context:
Waveguide modes are categorized by which longitudinal field component vanishes. This classification governs boundary conditions and cutoff relations for rectangular guides used in microwave systems.

Given Data / Assumptions:

• Hollow, perfectly conducting rectangular waveguide.
• Propagation is along the z-axis.
• Steady-state sinusoidal fields.

Concept / Approach:

By definition: TE (transverse electric) implies Ez = 0 and Hz generally non-zero; TM (transverse magnetic) implies Hz = 0 and Ez generally non-zero. No true TEM mode exists in a single-conductor hollow waveguide.

Step-by-Step Solution:

Verification / Alternative check:

Mode tables list Ez = 0 for TEmn and Hz = 0 for TMmn. Field probes in lab setups corroborate these properties for common modes like TE10.

Why Other Options Are Wrong:

• Both zero would imply TEM, which is unsupported in a hollow guide.
• Both non-zero violates TE/TM definitions.
• Cutoff-only conditions do not change the definitional property.

Common Pitfalls:

Confusing mode type with polarization; assuming a TEM mode exists in rectangular guides.

Final Answer:

In TE mode Ez = 0 and in TM mode Hz = 0