Skin Depth

The formula and the picture

When an AC field hits a conducting surface, the response is not uniform throughout the metal. The field penetrates only a thin shell at the surface; inside, the field decays exponentially. The characteristic decay length is the skin depth:

δ = √(2 / (μ σ ω)) = √(ρ / (π f μ))

where ω = 2πf is the angular frequency, σ is the conductivity (ρ = 1/σ the resistivity), and μ is the magnetic permeability of the conductor. The electric field and current density inside the conductor obey E(z) = E₀ e^(−z/δ) cos(ωt − z/δ), where z is the depth below the surface. Two things to notice: the amplitude decays exponentially with characteristic length δ, and the phase shifts by 1 radian per δ — what little current penetrates deep into the conductor is out of phase with the surface current.

At z = δ the field has fallen to 1/e ≈ 37 % of the surface value. At z = 5δ it is below 1 %. Above 5–10 skin depths the conductor interior is electromagnetically dead — no current, no field. This single fact is what shapes RF and microwave engineering: at high frequency, the "active" conductor is just a thin shell at the surface, and anything thicker than a few δ is just structural mass.

Skin depth across frequency: the canonical copper table

Frequency	δ in Cu	δ in Al	δ in Fe (μ_r ≈ 1000)	Typical context
50/60 Hz	8.5 mm	10.6 mm	0.5 mm	Mains power, transmission lines
1 kHz	2.1 mm	2.6 mm	130 µm	Audio frequencies, motors
50 kHz	295 µm	365 µm	18 µm	SMPS, induction heating, Litz wire band
1 MHz	66 µm	82 µm	4 µm	AM broadcast, RFID
100 MHz	6.6 µm	8.2 µm	0.4 µm	FM/VHF, scanner
1 GHz	2.1 µm	2.6 µm	0.13 µm	Cell, WiFi, GPS
10 GHz	0.66 µm	0.82 µm	0.04 µm	X-band radar, satellite
100 GHz	0.21 µm	0.26 µm	0.013 µm	mmWave 5G, automotive radar

Three patterns: skin depth shrinks as 1/√f (one decade of frequency → 3.2× decrease in δ); aluminium is ~25 % deeper than copper at every frequency because its lower conductivity (σ_Al ≈ 0.63 σ_Cu); iron's high permeability crushes its skin depth by √1000 ≈ 32× versus a non-magnetic conductor of equal σ. The iron column matters for transformers, induction cookware, and magnetic shielding.

Where the formula comes from

Inside a good conductor (ωε ≪ σ — true for copper up to optical frequencies), Maxwell's equations reduce. Ohm's law J = σE eliminates the displacement current; Faraday's law and Ampère's law combine to give a diffusion equation:

∇²H = μσ ∂H/∂t

This is not a wave equation. Inside a conductor, EM fields diffuse — they leak energy into the metal and are dissipated as ohmic heat, rather than propagating freely. Look for a solution H(z, t) = H₀ e^(jωt) F(z) inside a semi-infinite conductor; the spatial part satisfies F''(z) = jωμσ F(z). The square root of jωμσ can be written as

√(jωμσ) = (1 + j) / δ,  where  δ = √(2/(μσω))

The decaying solution is F(z) = e^(−z/δ) e^(−jz/δ): an exponentially attenuating wave with wavelength 2πδ inside the conductor. The simultaneous attenuation and phase shift are the signature of diffusion-type wave propagation.

Worked example: copper at 1 GHz

At 1 GHz, ω = 2π × 10⁹ ≈ 6.28 × 10⁹ rad/s. Copper has σ = 5.96 × 10⁷ S/m and μ ≈ μ₀ = 4π × 10⁻⁷ H/m. Plug in:

δ = √(2 / (4π × 10⁻⁷ × 5.96 × 10⁷ × 6.28 × 10⁹))
  = √(2 / 4.70 × 10¹⁰)
  = √(4.26 × 10⁻¹¹)
  ≈ 2.07 × 10⁻⁶ m  =  2.1 µm

So at 1 GHz the active conducting layer in copper is just over 2 micrometers thick — about 80 atomic layers. The bulk of any copper conductor at this frequency is electromagnetically dead. Cross-check with the 60 Hz value: δ at 60 Hz is √(10⁹/60) ≈ 4100× larger, so 8.5 mm — exactly the textbook number.

The surface resistance follows: R_s = 1/(σδ) = 1/(5.96 × 10⁷ × 2.07 × 10⁻⁶) = 8.1 × 10⁻³ Ω/square at 1 GHz. By the time you reach 24 GHz, R_s has grown to ~0.04 Ω/sq, and a 1 m WR-42 K-band waveguide run dissipates noticeable power per meter of length. Above 100 GHz the only way to stay efficient is to use superconducting Nb at 4 K.

AC resistance and Litz wire

For a round wire of radius a carrying AC, the current distribution is roughly J(r) ∝ I₀(jka)·I₀(jkr)/(a·δ) where I₀ is the modified Bessel function and k = (1+j)/δ. In the high-frequency limit (a ≫ δ) the current confines to an annular shell of thickness δ at the surface. The DC resistance R_dc = ρL/A uses the full cross-section πa²; the AC resistance uses only the active shell of area ≈ 2πaδ:

R_ac / R_dc ≈ a / (2 δ)

A 12 mm diameter copper bus bar at 60 Hz has δ ≈ 8.5 mm, comparable to its radius — skin effect is mild and R_ac/R_dc ≈ 0.35 (full cross-section conducts). At 1 MHz, δ has shrunk to 66 μm and a/(2δ) ≈ 90 — the AC resistance is 90× higher than DC. At 1 MHz a 12 mm conductor wastes 99 % of any DC-rated I²R headroom.

Litz wire is the engineering response: a bundle of many enamel-insulated strands, each thinner than δ at the operating frequency, twisted so each strand spends equal time in every position within the bundle. The strands' parallel resistances combine like DC; skin effect is defeated up to ~1 MHz. Modern SMPS designs at 100–500 kHz universally use Litz for high-Q inductors and transformer windings.

Where skin depth shows up in engineering

• HF antennas as hollow tubes. A 28 MHz amateur radio yagi at 15 dBi uses 1.25-inch aluminium tubing throughout — wall thickness 1.65 mm, conductor surface area equivalent to a solid rod of the same diameter. δ at 28 MHz is ~12 μm; the bulk material is purely structural.
• Microwave waveguides electroplated. WR-90 X-band waveguide walls are brass with 0.1–10 μm of silver or gold plating. δ_silver at 10 GHz is 0.62 μm, so a 5 μm plating gives 8 skin depths — plenty to confine the field, while the brass below provides mechanical strength.
• Induction cooktops. A 50 kHz coil under a steel pot drives eddy currents with δ_steel ≈ 0.4 mm. All the absorbed power deposits in a ~2 mm surface layer, heating food rapidly. Aluminium and glass pots don't work because they have either zero permeability (too-deep δ_Al ≈ 365 μm wastes coupling) or zero conductivity.
• Litz wire in SMPS transformers. Modern 100 kHz switching power supplies use Litz wire with ~50 strands of 0.1 mm diameter, each well below δ_Cu at 100 kHz (≈ 200 μm). This keeps R_ac/R_dc near 1.5 even with 20-layer windings, where solid wire would have R_ac/R_dc > 50.
• Submarine ELF communications. Seawater conductivity σ ≈ 4 S/m gives δ at 76 Hz (Project Sanguine) ≈ 28 m, large enough to reach a submarine 50 m below surface. At 10 kHz δ drops to 2.5 m — useless. The low-bit-rate cost of ELF (a few baud) is the unavoidable price of working with sea water as the propagation medium.

Variants and extensions

• Anomalous skin effect. When the mean free path λ_mfp of conduction electrons exceeds δ (very pure metals at low temperature, or very high frequencies), the local-Ohm's-law assumption fails. The local J = σE is replaced by a non-local response, and δ becomes weakly frequency-dependent, saturating at a residual value set by Fermi-surface geometry.
• London penetration depth. The superconducting analog of skin depth, set by Cooper-pair density rather than σ. For Nb, λ_L ≈ 50 nm; for Pb, λ_L ≈ 39 nm. RF surface resistance of superconducting cavities at GHz is residual but not zero, dominated by residual normal-state quasiparticles below T_c/2.
• Plasmonic skin depth. At optical and near-infrared frequencies, conductor response is dominated by collective electron plasma oscillations. The penetration depth becomes δ_p = c/ω_p where ω_p is the plasma frequency — for silver, δ_p ≈ 22 nm, the basis of surface plasmon polariton waveguides.
• Magnetic skin depth. An ELF magnetic field passing through a metallic shield decays by the same δ that defines the AC current penetration. Mu-metal shields work partly because their high permeability crashes δ, forcing flux to flow in a thin surface layer that closes the magnetic circuit instead of leaking through.
• Thermal skin depth. The same √(2/ω·) form appears in heat diffusion: the seasonal thermal skin depth in soil is ~2 m, the diurnal ~10 cm. Wine cellars and root cellars work because the annual temperature wave penetrates only that far.

Common pitfalls

• Using μ₀ for iron. The formula's μ is the conductor's permeability — μ_r × μ₀. Iron's μ_r ≈ 1000 at low field means iron's skin depth is √1000 ≈ 32× smaller than copper's at the same frequency. Plugging μ₀ for steel mains busbars gives the wrong answer by a factor of 32.
• Forgetting μ_r is nonlinear. Ferromagnetic materials lose their high permeability at high B (~1 T saturation in soft iron), so skin depth grows abruptly when current spikes drive the core into saturation. Important for short-circuit fault analysis on transformers.
• Conflating skin depth with conductor diameter. R_ac/R_dc = a/(2δ) is the high-frequency limit, valid when a ≫ δ. For a ≈ δ or smaller, the full Bessel-function solution must be used. A 1 mm wire at 60 Hz still carries current essentially uniformly because a < δ.
• Plating thinner than a few skin depths. Silver-plating an RF connector with only one δ thickness still leaves measurable conduction in the underlying brass substrate. Industry standards specify ≥5δ at the highest operating frequency, so that the substrate contribution to surface resistance is below 1 %.
• Ignoring proximity effect. Skin depth captures self-induction within a single conductor. Two adjacent wires force current toward (or away from) their facing surfaces by mutual flux — proximity effect — with the same √f scaling. RF inductors in solid wire have R_ac/R_dc 5–10× higher than the skin-effect estimate suggests because of this. Litz wire fixes both.