Frequency dependence of resistance in a metallic wire (skin effect) As operating frequency f increases from low frequency into the radio-frequency range, how does the effective AC resistance of a metallic wire change?

Introduction / Context:In AC conditions, current distribution in conductors becomes non-uniform with frequency. The skin effect forces current toward the surface of the conductor, effectively reducing the cross-sectional area available for conduction and increasing the AC resistance compared with DC resistance. This behavior is fundamental in RF design and power transmission at high frequencies.

Given Data / Assumptions:

• Good conductor (e.g., copper) with standard conductivity.
• Frequency f varied from low to high without resonance structures.
• Straight uniform wire to isolate skin effect.

Concept / Approach:

The skin depth δ = √(2 / (ω μ σ)) decreases with increasing angular frequency ω = 2π f. As δ shrinks, current crowds near the surface, reducing effective conducting area and raising AC resistance RAC. Typically, RAC ∝ 1/δ ∝ √f for round wires at sufficiently high frequencies.

Step-by-Step Solution:

Verification / Alternative check:

Practical measurements show wire resistance and attenuation per unit length rising with frequency; Litz wire mitigates this by distributing strands to equalize current distribution.

Why Other Options Are Wrong:

(b) contradicts skin effect; (c) ignores frequency dependence; (d) suggests a decrease at high f not observed in simple conductors; (e) resonance requires specific reactive networks, not a straight uniform wire.

Common Pitfalls:

Confusing skin effect with proximity effect (current crowding due to nearby conductors); overlooking temperature rise which further increases resistance.

Final Answer:

It increases as f increases (due to skin effect)