Skin Effect

Why current avoids the middle of a wire

When you push DC through a uniform wire, the current spreads evenly across the cross-section: every electron carries an equal share of the load. Switch to AC and the picture changes. As the current oscillates it produces a time-varying magnetic flux inside the conductor itself, and Faraday's law says any changing flux induces an EMF that opposes the change in current. The induced EMF is largest where the enclosed flux is largest, which is on the central axis of the wire. The center fights back hardest, and current density redistributes outward.

The redistribution is not abrupt. Solving Maxwell's equations inside a good conductor gives a current density that decays exponentially as you move inward from the surface:

J(z) = J₀ · e^(−z/δ) · cos(ωt − z/δ)

where z is the depth below the surface, J₀ is the surface current density, and δ is the skin depth. At z = δ the amplitude has fallen to 1/e ≈ 37 %; at z = 5δ it is below 1 %. The decaying wave also picks up a phase shift of one radian per skin depth, so deep in the conductor the small remaining current is out of phase with the surface current.

The skin depth is a property of the metal and the frequency:

δ = √(2 / (ω μ σ)) = √(ρ / (π f μ))

where ω = 2πf is the angular frequency, σ is the conductivity (1/ρ for resistivity ρ), and μ is the magnetic permeability of the conductor. For non-magnetic metals μ ≈ μ₀ = 4π × 10⁻⁷ H/m. For ferromagnetic conductors like iron μ_r is hundreds to thousands, which crashes the skin depth — iron at 60 Hz has δ ≈ 0.5 mm, sixty times less than aluminium of the same conductivity.

Worked example: copper from 60 Hz to 1 GHz

Copper has resistivity ρ_Cu = 1.68 × 10⁻⁸ Ω·m and permeability essentially μ₀. Using δ = √(ρ/(πfμ)):

f       δ_Cu                ratio of cross-section that conducts
                            (for a 5 mm-radius wire)
─────────────────────────────────────────────────────────────
60 Hz   8.5 mm              full cross-section (a < δ)
1 kHz   2.1 mm              ~95 % of area
50 kHz  295 µm              ~22 % (annular shell)
1 MHz   66 µm               ~5.2 %
100 MHz 6.6 µm              ~0.53 %
1 GHz   2.1 µm              ~0.17 %
24 GHz  0.42 µm             ~0.034 %  (a 60-atom-thick shell)

Two conclusions follow. First, at line frequency the skin effect barely matters for thin wires; it only starts to bite when the wire radius is comparable to or larger than δ. Second, at radio frequencies the active shell becomes so thin that conductor surface quality, plating, and oxide layers dominate the losses — and the bulk of the wire is essentially structural.

The AC resistance increase is captured by R_ac/R_dc. For a round wire of radius a in the limit a ≫ δ:

R_ac / R_dc ≈ a / (2δ)

A 12 mm-diameter (a = 6 mm) copper bus bar has R_ac/R_dc ≈ 0.35 at 60 Hz (so the simple limit doesn't yet apply), but rises to about 700 at 1 MHz. That is why an RF inductor wound from solid wire dissipates so much more than its DC resistance would predict.

Where the formula comes from

Start with the conductor equations: Ampère's law ∇×H = J (displacement current is negligible inside a good metal at sub-optical frequencies), Faraday's law ∇×E = −∂B/∂t = −μ ∂H/∂t, and Ohm's law J = σE. Take the curl of Ampère and substitute Faraday:

∇²H = μσ ∂H/∂t

This is a diffusion equation, not a wave equation — energy bleeds into the conductor and is dissipated, it does not propagate freely. Look for a sinusoidal-in-time solution H(z,t) = H₀ e^(jωt) F(z) inside a half-space conductor, and the spatial part satisfies F'' = jωμσ F. The square root of jωμσ has both real and imaginary parts:

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

The decaying solution is therefore F(z) = e^(−z/δ) e^(−jz/δ), an exponentially attenuating wave with the wavelength inside the conductor equal to 2πδ. The skin depth is simultaneously the attenuation length and the phase-shift length — every skin depth you go in, the field both shrinks by a factor of e and rotates by one radian.

The good-conductor approximation is good when the displacement current σ ≫ ωε. For copper that condition fails only above the optical regime (~10¹⁸ Hz), where the ordinary skin-effect formula gives way to the anomalous skin effect and eventually plasmonics.

Skin effect at different frequencies and conductors

Conductor	ρ (×10⁻⁸ Ω·m)	μ_r	δ at 60 Hz	δ at 1 MHz	δ at 1 GHz
Silver	1.59	1	8.3 mm	64 µm	2.0 µm
Copper	1.68	1	8.5 mm	66 µm	2.1 µm
Gold	2.44	1	10.2 mm	79 µm	2.5 µm
Aluminium	2.65	1	10.6 mm	82 µm	2.6 µm
Iron (annealed)	10	~1000	0.65 mm	5.0 µm	0.16 µm
Stainless 304	72	~1	55 mm	425 µm	13 µm
Seawater	~2 × 10⁹	1	~25 m	~250 mm	~8 mm

The seawater row is why ELF radio (76 Hz) is used to wake up nuclear submarines: at 76 Hz the skin depth in seawater is roughly 22 m, so a slow-bit-rate signal can reach a sub a few tens of meters down. Bump to broadcast AM at 1 MHz and the skin depth in the same water collapses to a fraction of a meter — useless for underwater comms.

Where the skin effect shows up

• 60 Hz transmission lines. Long-distance power lines use ACSR (aluminum conductor steel-reinforced) cable in which a tensile steel core is sheathed by aluminum strands. At 60 Hz δ_Al ≈ 11 mm, comparable to typical conductor radii of 10–25 mm, so most current still uses the cross-section but the core gives diminishing returns. Stranding also reduces proximity-effect losses between phase conductors.
• Litz wire in switching power supplies. Litz cable is 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. It defeats both skin and proximity losses up to ~1 MHz and is the standard winding for high-Q inductors and transformers in 100 kHz–500 kHz SMPS designs.
• Microwave waveguides and cavities. WR-90 X-band rectangular guide has internal walls plated with thin silver. The current rides in a layer ~2 µm deep so the silver only needs to be a few skin depths thick — yet the surface roughness becomes a dominant loss term, motivating electropolishing of accelerator cavities and superconducting RF for the highest Q-factors.
• Magnetic shielding and induction heating. Mu-metal shields work partly because the small skin depth in high-μ alloys forces stray flux to flow in a thin surface layer that closes the magnetic circuit. Induction heaters do the inverse: a 50 kHz field penetrates only ~2 mm into a steel pot, depositing all the heat in a thin shell that warms the food above quickly.
• Geophysical exploration and ELF submarine comms. Magnetotelluric surveys exploit the fact that lower frequencies have larger skin depths in earth and so probe deeper formations; the FAA's ELF "Project Sanguine" used 76 Hz signals partly because seawater's skin depth at that frequency is large enough for submerged reception.

Proximity effect, the skin's noisy cousin

Skin effect is what a conductor does to itself. Proximity effect is what neighbours do to it. When two parallel wires carry AC, each one's magnetic field induces eddy currents in the other, redistributing current toward (or away from) the facing surface depending on whether the two carry currents in the same or opposite direction. The result is the same kind of crowding, but driven externally.

In a transformer winding the proximity effect can be much worse than the skin effect. A solenoid with twenty layers of solid wire forces the inner layers to carry their currents in narrow sheets right against the layer-to-layer boundary, and the AC resistance of the inner layers can be 10–50× the DC resistance. Litz wire fixes both at once by ensuring that each sub-strand spends equal time in regions of high and low field, so its average eddy contribution cancels.

The two effects share the same physics — induced eddy currents from time-varying flux — and the same scaling: both ∝ √f. Designers usually treat them with one combined factor, the AC resistance ratio R_ac/R_dc, computed by Dowell's equations or finite-element simulation.

Surface resistance and the cost of being shiny

At RF and microwave frequencies the skin depth is so small that the relevant figure of merit is not bulk conductivity but surface resistance R_s = 1/(σδ) = √(πfμ/σ). Power dissipation per unit area on a conductor surface is ½ R_s |H_||²|, where H_|| is the tangential magnetic field. R_s grows as √f, so doubling the frequency raises losses by 41 %.

Material         R_s at 10 GHz
─────────────────────────────────
Silver           0.026 Ω/sq
Copper           0.027 Ω/sq
Gold             0.032 Ω/sq
Aluminum         0.034 Ω/sq
Brass            0.054 Ω/sq
Stainless 304    0.16 Ω/sq
Niobium @ 4 K (SC) ~10⁻⁵ Ω/sq

This is why expensive accelerator cavities are coated in superconducting niobium and cooled to a few kelvin: the surface resistance falls by four orders of magnitude, and a cavity that would cost megawatts of klystron power as copper can be driven by tens of watts when superconducting.

Variants and extensions

• Anomalous skin effect. When the mean free path of conduction electrons becomes longer than δ (very pure metals at very low temperatures, or very high frequencies), the local Ohm's-law assumption breaks down. The classical formula overestimates conductivity; the surface resistance saturates at a residual value that depends on Fermi-surface geometry rather than bulk conductivity.
• Litz wire. Deliberately stranded, individually insulated, periodically transposed bundles that defeat skin and proximity effects up to ~1 MHz. The strand diameter must be smaller than δ; the number of strands sets the bundle's overall current rating.
• Proximity effect. Externally driven eddy redistribution from neighbouring AC currents. Dowell's equations parameterize transformer-winding losses combining skin and proximity terms.
• Plasmonic skin depth. At optical and near-infrared frequencies the conductor's response is dominated by free-electron plasma oscillations. The penetration depth becomes the plasma skin depth δ_p = c/ω_p ≈ 22 nm in silver — the basis of surface-plasmon-polariton waveguides and metasurfaces.
• Skin effect in superconductors. A superconductor expels magnetic flux up to its London penetration depth λ_L (~50 nm in Nb), which plays the role of δ but is set by the Cooper-pair density, not σ. RF surface resistance of Nb cavities is residual but not zero.

Common pitfalls

• Forgetting magnetic permeability for ferrous conductors. Iron's μ_r ≈ 1000 at low fields, so its skin depth is √1000 ≈ 32× shallower than copper at the same frequency. Engineers who plug μ₀ into the formula for steel mains busbars get the wrong answer by a factor of more than thirty.
• Confusing skin depth with conductor radius. The simple R_ac/R_dc = a/(2δ) holds only when a ≫ δ; for a comparable to δ 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 a skin only one δ thick leaves residual conduction in the underlying brass. Plating standards typically specify ≥ 5δ at the highest operating frequency to attenuate the contribution from the worse-conducting substrate to <1 %.
• Treating the skin effect as resistance only. The skin effect also adds inductive reactance because of the in-phase / quadrature decomposition. Both R_ac and the internal inductance per unit length scale as √f. RF designers using a pure-resistance approximation at high frequency miss the phase shift.
• Ignoring proximity effect in tightly wound coils. Two wires next to each other can have AC resistance 5–10× higher than each in isolation. Solving the skin-effect formula for a single isolated wire and assuming the bundle multiplies it linearly understates losses by an order of magnitude.