Assertion (A): In a rectangular waveguide operating in TE10 mode, the cutoff frequency is f_c = c / (2a), where a is the broad-wall dimension and c is the speed of light. Reason (R): TE10 is the dominant mode in a rectangular waveguide because it has the lowest cutoff frequency among all allowed modes.
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ABoth A and R are correct and R is correct explanation of A
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BBoth A and R are correct but R is not correct explanation of A
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CA is correct but R is wrong
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DA is wrong but R is correct
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EBoth A and R are wrong
Answer
Correct Answer: Both A and R are correct but R is not correct explanation of A
Explanation
Introduction / Context:A rectangular waveguide supports discrete TE and TM modes. Each mode has a cutoff frequency determined by the waveguide cross-section. Understanding the dominant mode and the exact cutoff formula for TE10 is foundational for single-mode operation, bandwidth planning, and loss estimation.
Given Data / Assumptions:
- Rectangular waveguide with dimensions a (broad wall) and b (narrow wall).
- Electromagnetic wave in TE10 mode.
- Free space inside (for simplicity); speed of light is c.
Concept / Approach:
For a rectangular guide, the cutoff frequency for TEmn is f_c = (c/2) * sqrt((m/a)^2 + (n/b)^2). For TE10 (m = 1, n = 0) this simplifies to f_c = c/(2a). The dominant mode is the one with the lowest cutoff frequency; in standard hollow rectangular guides, that is TE10 because TE00 does not exist and other allowed modes have higher indices.
Step-by-Step Solution:
1) Start from f_c(TEmn) = (c/2) * sqrt((m/a)^2 + (n/b)^2).2) Substitute m=1, n=0 ⇒ f_c(TE10) = c/(2a).3) Compare cutoffs: TE10 < TE20 (m=2), TE01 (n=1), etc., so TE10 is dominant.4) Note that being dominant explains “lowest cutoff,” not the exact algebraic formula for f_c.Verification / Alternative check:
Waveguide handbooks and measurements show TE10 propagates first as frequency increases; the first higher modes (TE20 or TE01) appear only at higher frequencies, confirming the ordering implied by the formula.
Why Other Options Are Wrong:
- Both correct and explanatory: Dominance explains which mode cuts on first, but not the specific equation f_c = c/(2a).
- A correct, R wrong: R is true—TE10 is the dominant mode.
- A wrong, R correct / Both wrong: Conflicts with standard theory.
Common Pitfalls:
Confusing “dominant” with “fundamental TEM” (TEM does not exist in hollow rectangular guides); forgetting that b does not appear for TE10 because n = 0.
Final Answer:
Both A and R are correct but R is not correct explanation of A