Rectangular resonant cavity — mode existence rules Which of the following modes cannot exist in a rectangular resonant cavity?
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ATE110
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BTE011
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CTM110
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DTM111
Answer
Correct Answer: TM110
Explanation
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:
Check TM110: p = 0 → violates TM requirement that m, n, p ≥ 1 → cannot exist.Check TE110: allowed; TE permits a zero or more in indices as long as fields satisfy boundary conditions.Check TE011: allowed for the same reason (TE with m = 0 is permissible).Check TM111: allowed; all indices nonzero.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