Short note: Skin depth

When a plane wave enters a conducting medium, its fields decay as e^{−αz} and the induced currents flow mainly near the surface. Skin depth (depth of penetration) δ is the distance over which the amplitude falls to 1/e (about 37 %) of its surface value:

δ = 1/α

For a good conductor (σ ≫ ωε), α = β = √(πfμσ), so

δ = 1/√(πfμσ) = √(2/(ωμσ))

Points:

• δ decreases as frequency, conductivity or permeability increases. At high frequency the current is confined to a thin skin; this is the skin effect, which raises the AC resistance of a wire.
• Example: copper (σ = 5.8×10⁷ S/m) at 1 MHz has δ ≈ 66 μm; at 60 Hz it is about 8.5 mm.
• Practical uses: shielding (a few skin depths of metal block fields), surface hardening, plating conductors with silver, hollow conductors/waveguides for microwaves.
• Wave velocity in a good conductor is v = ω/β = ωδ, and the wave is attenuated by e^{−1} per skin depth (8.686 dB per δ).