Assertion–Reason (Transmission/Guided Waves): The phase velocity of electromagnetic waves on overhead lines and on coaxial cables is the same. Reason: Free space has an intrinsic impedance of 377 Ω.

Introduction:Assertion–Reason questions test conceptual linkage. Here, we compare phase velocity on different transmission media and evaluate a statement about the free-space intrinsic impedance to see whether it explains the assertion.

Given Data / Assumptions:

• Overhead lines are largely surrounded by air (effective relative permittivity εr ≈ 1), so v ≈ c / sqrt(εeff) ≈ c.
• Coaxial cables use a dielectric (e.g., polyethylene with εr ≈ 2.25 to 2.3), so v ≈ c / sqrt(εr) < c.
• Free space intrinsic impedance η0 ≈ 377 Ω is a material property defined by sqrt(μ0 / ε0).

Concept / Approach:

Phase velocity on a transmission line is determined by effective permittivity and permeability of the medium guiding the fields: v = 1 / sqrt(L′C′) = c / sqrt(εeff μeff). Overhead lines mainly see air, while coax fields reside within a dielectric. Therefore, velocities are generally not equal. The reason statement about η0 = 377 Ω is true but unrelated to whether two guided structures have the same phase velocity.

Step-by-Step Solution:

Verification / Alternative check:

Typical coaxial velocity factor ≈ 0.66 for solid polyethylene and ≈ 0.8–0.9 for foam dielectrics; overhead line velocity factor ≈ 1.

Why Other Options Are Wrong:

• Options a, b, c: all assume A is correct, which it is not.
• Option e: R is true; thus “both wrong” is incorrect.

Common Pitfalls:

Confusing impedance with velocity: η and v involve ε and μ differently; equal impedances or a known η0 value do not imply equal phase velocities in dielectrics.

Final Answer:

A is wrong but R is correct