Skin depth scaling: if the depth of penetration is 25 cm at 1 MHz in a given conductor, what is the depth at 4 MHz?

Introduction / Context:The skin depth δ quantifies the exponential decay distance of electromagnetic fields inside a good conductor. It is fundamental in high-frequency power, RF shielding, and transmission line design. This question uses the frequency-scaling property of δ to find a new value at a different frequency.

Given Data / Assumptions:

• At f1 = 1 MHz, skin depth δ1 = 25 cm.
• We seek δ2 at f2 = 4 MHz in the same medium (same σ and μ).
• Good conductor approximation applies so δ ∝ 1/√f.

Concept / Approach:

For good conductors: δ = √(2/(ω μ σ)) ∝ 1/√f when μ and σ are constant. Therefore, δ2/δ1 = √(f1/f2). This provides a quick scaling without recalculating from material constants.

Step-by-Step Solution:

Verification / Alternative check:

Because frequency increased by a factor of 4, the skin depth must halve twice (i.e., reduce by √4 = 2). This matches the computed value.

Why Other Options Are Wrong:

• 6.25 cm: would correspond to a 16× increase in frequency (√16 = 4), not 4×.
• 50 cm and 100 cm: imply that skin depth grows with frequency, which is incorrect.
• 3.125 cm: corresponds to an even larger frequency ratio (64×), not 4×.

Common Pitfalls:

• Using δ ∝ 1/f instead of δ ∝ 1/√f.
• Forgetting to keep units consistent (cm vs. m).

Final Answer:

12.50 cm