Question 1

What is the formula for skin depth?

Accepted Answer

Skin depth is δ = 1/√(πfμ₀μᵣσ) meters, where f is frequency in Hz, μ₀ = 4π×10⁻⁷ H/m is the permeability of free space, μᵣ is relative permeability, and σ is electrical conductivity in S/m. For copper (σ = 5.8×10⁷ S/m, μᵣ = 1) at 50 Hz, δ ≈ 9.3 mm; at 1 MHz, δ ≈ 0.066 mm.

Question 2

How does skin depth change with frequency?

Accepted Answer

Skin depth is inversely proportional to the square root of frequency (δ ∝ 1/√f). Quadrupling the frequency halves the skin depth. For copper: 50 Hz → δ ≈ 9.3 mm, 200 Hz → δ ≈ 4.7 mm, 50 kHz → δ ≈ 0.30 mm. This relationship is used to estimate high-frequency losses in coil conductors.

Question 3

Why is the skin depth in iron much smaller than in copper?

Accepted Answer

Iron has a relative permeability of μᵣ ≈ 200 (copper has μᵣ = 1). Since δ ∝ 1/√(μᵣσ), higher μᵣ reduces skin depth significantly. At 50 Hz, iron (σ = 1×10⁷ S/m, μᵣ = 200) has δ ≈ 1.1 mm — about 1/8 that of copper. This is the primary cause of core losses in transformers and electric motors.

Question 4

What is the relationship between skin depth and AC resistance?

Accepted Answer

For a round conductor of diameter d where d/δ ≫ 1, the AC-to-DC resistance ratio is approximated as R_AC/R_DC ≈ d/(4δ) + 3/4. Current density decays exponentially from the surface: J(d) = J₀ × e^(−d/δ). At a depth of 4.6δ the current density falls to about 1% of its surface value, effectively confining the current to a thin surface layer.

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