Coaxial line characteristic impedance — dependence on relative permittivity εr Statement: “The characteristic impedance of a coaxial cable is inversely proportional to εr.” Decide whether this statement is true or false.

Introduction / Context:
Coaxial lines are widely used in RF systems. Their characteristic impedance Z0 depends on geometry and dielectric properties. Knowing how Z0 scales with the relative permittivity εr is essential for cable design and selection.

Given Data / Assumptions:

• Ideal lossless coax with inner radius a and outer radius b.
• Dielectric has relative permittivity εr.

Concept / Approach:
The standard expression for coax is:

Thus Z0 varies inversely with sqrt(εr), not with εr itself. Doubling εr reduces Z0 by a factor of sqrt(2), not by 2. The given statement claims inverse proportionality to εr, which overstates the dependence and is therefore false.

Step-by-Step Reasoning:

Verification / Alternative check:
Handbooks list common 50 Ω and 75 Ω coaxes: replacing polyethylene (εr ≈ 2.25) with foam (lower εr) raises Z0 for the same geometry, consistent with the 1/sqrt(εr) relationship.

Why Other Options Are Wrong:

• True: would imply linear inverse dependence that is not supported by the physical model.

Common Pitfalls:
Confusing velocity factor scaling (also ∝ 1/sqrt(εr)) with a linear inverse rule; ignoring the geometry factor ln(b/a).

Final Answer:
False