The Skin Effect: Why AC Current Hugs the Surface

A wire that only conducts near its skin

Feed a conductor with direct current and the electrons share the work evenly across the whole cross-section, limited only by the material's resistivity. Feed it alternating current instead, and something stranger happens: the current density is highest right at the surface and fades away toward the centre. This is the skin effect, and at high enough frequency the core of a thick conductor can be carrying almost nothing at all, doing no more electrical work than the plastic insulation around it.

The mechanism is pure electromagnetic induction acting on the conductor itself. A changing current produces a changing magnetic field, both outside the wire and inside it. By Faraday's law that changing internal field induces its own eddy currents in the metal, and by Lenz's law those eddy currents oppose the change that caused them. Near the centre of the conductor this opposition is strongest, effectively cancelling part of the driving current; near the surface it is weaker, so more of the original current survives. The result is a current density profile that piles up at the boundary.

The exponential law and skin depth

Solving Maxwell's equations inside a good conductor under sinusoidal drive turns into a diffusion equation for the electric field and current density, and for a flat conductor surface the solution is remarkably clean: the magnitude of the current density falls off exponentially with depth d below the surface. The characteristic length of that decay is the skin depth, usually written δ.

J(d) = J0 · e^(-d/δ)

δ = √(2ρ / (ω μ)) = √(ρ / (π f μ))

at d = δ   →  J ≈ 37% of the surface value
at d = 3δ  →  J ≈ 5%
at d = 5δ  →  J ≈ 0.7%  (effectively zero — the skin effect is "complete")

Here ρ is the resistivity of the metal, μ is its magnetic permeability and ω = 2πf is the angular frequency of the drive. Every one of those quantities pushes the effect in an intuitive direction: a better conductor (lower ρ) confines current more tightly, a magnetic metal (higher μ, like steel or mild iron) confines it far more tightly than copper does, and higher frequency always shrinks δ.

Numbers: copper from mains to microwave

Copper has a resistivity of about 1.68 × 10⁻⁸ Ω·m and, being non-magnetic, uses the permeability of free space. Plugging those into the formula across the frequency range engineers actually work with shows how fast the effect bites:

   60 Hz  (mains)         δ ≈ 8.5 mm   — a 2 mm wire is barely affected
    1 kHz  (audio)         δ ≈ 2.1 mm
    1 MHz  (medium wave)   δ ≈ 66 μm    — thinner than a human hair
  100 MHz  (FM radio)      δ ≈ 6.6 μm
    1 GHz  (mobile / Wi-Fi) δ ≈ 2.1 μm  — a few hundred atoms deep

At mains frequency an ordinary household cable barely notices the effect. By the time you reach gigahertz frequencies, the conducting layer is thinner than a coat of paint — which is precisely why RF connectors and waveguides are often silver-plated brass rather than solid silver: the plating only needs to be a few skin depths thick to carry essentially all the current.

What it costs: AC resistance, Litz wire, busbars

Once δ is much smaller than the conductor's radius, the effective conducting area shrinks to roughly the perimeter times δ, instead of the full circular cross-section. Resistance is inversely proportional to area, so AC resistance climbs above the DC value, and with it the ohmic losses for a given current. This single fact shapes a surprising amount of electrical engineering.

Litz wire answers it directly: instead of one thick strand, a transformer or induction-heating winding uses dozens of thin, individually insulated strands, twisted and transposed so each strand spends equal time near the surface and near the centre of the bundle. Every strand stays thinner than δ at the operating frequency, so each one conducts through its whole cross-section, and the bundle's total AC resistance stays close to its DC resistance well into the megahertz range. High-voltage transmission lines exploit the same idea the other way around: ACSR cable wraps aluminium strands around a steel core purely for tensile strength, because the skin effect at 50 or 60 Hz already keeps most of the current in the outer aluminium layer, so the steel core barely needs to conduct at all. Busbars in high-current switchgear are shaped as flat plates or hollow tubes rather than solid rods for the same reason — maximising perimeter per unit of copper, since the interior is doing almost no electrical work anyway.

A cousin worth knowing: the proximity effect

Skin effect is a conductor reacting to its own changing field. Put a second current-carrying conductor nearby — the adjacent turns of a transformer winding, or the two conductors of a coaxial cable — and each one's magnetic field also redistributes current in its neighbour. This proximity effect can concentrate current into narrow crescents facing away from (or toward) the neighbouring conductor, and in tightly wound coils it often dominates the total AC loss more than the plain skin effect does. Both effects share the same root cause, electromagnetic induction inside a conductor, and both are why real transformer and inductor design software solves the full field problem rather than trusting the single-wire formula alone.

Frequently asked questions

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