Rectangular waveguide TE10 — maximum power handling expression Given Z_TE as the wave impedance for TE waves, E_d as the maximum dielectric strength of the insulating medium, and a (broad wall width) and b (narrow wall height) as the guide dimensions, choose the correct expression for the maximum power handling capability P_max in TE10.

Introduction / Context:
Waveguide power handling is limited by the maximum electric field before breakdown. For a rectangular waveguide operating in the dominant TE10 mode, there is a known relationship between the peak electric field amplitude and the time-average power flow. This item asks you to identify the correct formula using Z_TE and breakdown strength E_d.

Given Data / Assumptions:

• Mode: TE10 in a rectangular waveguide.
• Dimensions: a (broad wall), b (narrow wall).
• Z_TE: wave impedance for the TE mode at the operating frequency.
• E_d: maximum allowable electric field (dielectric strength) of the filling medium.
• Lossless, steady-state sinusoidal fields.

Concept / Approach:
For TE10, the transverse electric field varies as cos(pi x/a) across the broad wall, with a maximum field magnitude E_max. The average power flow P in a lossless guide can be written in terms of E_max as P = (a * b / 4) * (E_max^2 / Z_TE). Setting E_max = E_d at the threshold gives the maximum permissible power P_max. The 1/4 factor arises from spatial averaging of the field distribution across the guide cross-section.

Step-by-Step Solution:

Verification / Alternative check:
Dimensional analysis: E^2 / Z_TE has dimensions of power density; multiplying by area ab yields power. The quarter factor is consistent with integrating cos^2(pi x/a) over the width.

Why Other Options Are Wrong:

• (ab/2)(E_d^2Z_TE): incorrectly places Z_TE in numerator and has the wrong factor.
• (ab)(E_d/Z_TE): linear in E_d rather than quadratic; dimensionally inconsistent.
• (a/b)*(E_d^2/Z_TE): incorrect geometric dependence; power scales with cross-sectional area, not aspect ratio alone.

Common Pitfalls:
Forgetting the spatial averaging factor; mixing up Z_TE in numerator vs denominator; ignoring mode dependence of field distribution.

Final Answer:
P_max = (a * b / 4) * (E_d^2 / Z_TE)