Skin depth δ in a good conductor varies with frequency as δ ∝ 1 / √f. Compared to 500 MHz, what is the skin depth at 1000 MHz (expressed as a ratio δ_1000MHz / δ_500MHz)?

Introduction / Context:
Skin effect confines AC current to a thin layer near a conductor’s surface at high frequencies, increasing effective resistance and loss in RF components, transmission lines, and transformers. Quantifying how skin depth changes with frequency helps in conductor sizing and plating decisions.

Given Data / Assumptions:

• Skin depth δ ∝ 1 / √f for a given material (constant conductivity and permeability).
• Two frequencies: f1 = 500 MHz, f2 = 1000 MHz.
• Material properties are unchanged and temperature effects are neglected.

Concept / Approach:

Since δ ∝ f^−1/2, the ratio of skin depths at two frequencies is δ2 / δ1 = √(f1 / f2). Doubling the frequency halves the argument inside the square root, reducing δ by a factor of √2 ≈ 1.414.

Step-by-Step Solution:

Verification / Alternative check:

For copper at room temperature, δ ≈ 66 / √f(MHz) micrometers is a common rule of thumb, confirming that δ at 1000 MHz is approximately 66/√1000 ≈ 2.09 μm and at 500 MHz is ≈ 2.95 μm; their ratio ≈ 0.71.

Why Other Options Are Wrong:

• 2.0 and 12.0: Imply larger skin depth at higher frequency, opposite of skin effect.
• 0.5 or 0.25: Would correspond to 4x or 16x frequency, not 2x.

Common Pitfalls:

Using linear instead of square-root scaling; forgetting that increased frequency increases surface resistance and insertion loss; ignoring changes in permeability for magnetic materials.

Final Answer:

0.707