Rectangular resonant cavity — mode existence rules Which of the following modes cannot exist in a rectangular resonant cavity?

Introduction / Context:
Electromagnetic modes in a rectangular cavity are classified as TEmnp or TMmnp, where m, n, p count half-wave variations along the a, b, and d dimensions. Existence rules differ for TE and TM families because of boundary conditions on tangential fields at conducting walls.

Given Data / Assumptions:

• Perfectly conducting rectangular cavity with dimensions a, b, d.
• Mode indices m, n, p are nonnegative integers.

Concept / Approach:
For TM modes in a cavity, the axial electric field component must be nonzero and satisfy boundary conditions on all six walls. This enforces that all three indices m, n, and p must be nonzero (≥ 1). If any of m, n, or p is zero, the required field configuration cannot satisfy the boundary conditions for TM in a closed cavity. In contrast, TE modes may have one index equal to zero, provided not all are zero simultaneously.

Step-by-Step Solution:

Verification / Alternative check:
Standard cavity-mode tables list TMmnp only with m, n, p ≥ 1, while TEmnp may include zeros. This corroborates the exclusion of TM110.

Why Other Options Are Wrong:

• TE110 and TE011: valid TE modes with appropriate boundary conditions.
• TM111: all indices nonzero → valid TM mode.

Common Pitfalls:
Assuming waveguide rules carry over directly; in a closed cavity, TM requires three standing-wave variations.

Final Answer:
TM110