In a rectangular waveguide, which listed mode corresponds to the trivial (non-propagating) solution where all field components are zero?

Introduction / Context:
Waveguide modes are labeled TEmn or TMmn. Not every index pair represents a physical propagating solution. Some index choices lead to a trivial zero-field solution, which is excluded from the set of allowed modes.

Given Data / Assumptions:

• Hollow rectangular waveguide with perfectly conducting walls (idealized).
• Mode families: TE and TM with integer indices m, n ≥ 0.
• Boundary conditions require tangential electric field to vanish at walls.

Concept / Approach:

For TM modes, both tangential electric field components must satisfy standing-wave conditions along the cross-section. If m = n = 0 (TM00), the field solution reduces to zero everywhere to satisfy boundary conditions—hence it is a non-existent (trivial) mode. By contrast, TE10 is the dominant physical mode.

Step-by-Step Solution:

Verification / Alternative check:

Standard mode tables omit TM00 for rectangular waveguides; laboratory measurements never observe TM00 propagation.

Why Other Options Are Wrong:

• TM11, TM01, TM10, TE10: All describe legitimate non-trivial modes with finite cutoff (except TM00).

Common Pitfalls:

Assuming every index pair yields a mode; confusing the TE/TM naming with TEM, which does not propagate in hollow rectangular guides.

Final Answer:

TM00