Microwave coaxial line (inner radius = a, inner radius of outer conductor = b): the approximate cutoff wavelength for the first TE mode (TE11) is closest to

Introduction:
Although coax supports a TEM mode without cutoff, higher-order modes (TE/TM) have finite cutoff frequencies. Knowing the TE11 cutoff helps set the upper frequency limit for single-mode coaxial operation.

Given Data / Assumptions:

• Coax geometry: inner conductor radius a, inner radius of outer conductor b.
• We consider the first higher-order TE mode (TE11).
• Air dielectric; standard boundary conditions.

Concept / Approach:

Analytical solutions give an approximate TE11 cutoff wavelength λc ≈ π(a + b), which is a widely used design rule. Operation should remain well below this cutoff frequency (i.e., at wavelengths ≪ λc) to avoid multimoding.

Step-by-Step Solution:

Verification / Alternative check:

More exact expressions involve solving characteristic equations with Bessel functions; the π(a + b) rule closely tracks the true cutoff and is sufficient for design checks.

Why Other Options Are Wrong:

• a + b, πa, πb: ignore the dependence on both radii with the correct coefficient.
• 2π(b − a): relates to circumference gap, not TE11 cutoff.

Common Pitfalls:

Confusing TEM (no cutoff) with TE11 (finite cutoff); the upper frequency limit for single-mode coax is set by TE11.

Final Answer:

π(a + b)