Skin Depth Calculator — The AC Surface

Alternating current does not use the whole wire — it crowds toward the surface, and the higher the frequency, the thinner the skin it lives in. Enter the frequency and the conductor’s resistivity and permeability, and this calculator returns the skin depth δ: the depth at which the current density has fallen to 1/e of its surface value.

Frequency in question (Hz) — mains 60, AM radio 1,000,000, Wi-Fi 2,400,000,000

Resistivity ρ (Ω·m) — copper 1.68×10⁻⁸, aluminium 2.82×10⁻⁸, silver 1.59×10⁻⁸

Relative permeability μr — copper and aluminium are 1; iron runs into the hundreds (and varies)

The Current’s Shallow Habits

Maxwell’s equations in a conductor make the fields diffuse inward with attenuation: the changing magnetic field of the current itself induces eddy currents that oppose the flow in the interior, exiling it to a surface layer of characteristic depth δ = √(2ρ ÷ ωμ). Three trends to internalise: δ falls with the square root of frequency (quadruple the frequency, halve the skin), grows with the square root of resistivity (worse conductors let the field deeper, having weaker eddies to resist it — the formula’s counterintuitive mercy), and shrinks with permeability, which is why steel’s skin at radio frequencies is a micron-scale varnish while copper’s is a paint coat. The consequences structure entire industries: hollow waveguides, plated connectors, foil inductors, the induction hob that heats only a pan’s skin-deep layer, and the security tag deactivator at the shop door. The calculator puts δ in mm or µm as the scale demands, with the exponential’s census (at 5δ, under 1% remains) in its note.

Worked Example: Copper at Mains and at Radio

The defaults — copper, 60 Hz — return δ = 8.42 mm: at mains frequency the skin is generous, whole house-wiring cross-sections participate, and DC intuition survives nearly intact (though transmission engineers already derate fat conductors for it). Now change only the frequency to 1 MHz: the calculator returns 65 µm — a human hair’s width carries an AM-radio-frequency current, and a solid rod of copper at that frequency is mostly ballast inside, its AC resistance set by the thin annulus doing the work (the resistance rises as √f once the skin is thin). At Wi-Fi’s 2.4 GHz, δ ≈ 1.3 µm, which is why microwave plumbing is silver-plated microns-deep and why the tracks on a phone’s board are finished like jewellery. Three frequencies, three industries, one square root. Keep the anchor figures in memory (8.4 mm at the mains, 66 µm at a megahertz, about 2 µm at a gigahertz, all for copper) and every conductor you meet afterwards can be sanity-checked against them by square roots alone: the reader who internalises those three points owns the curve, and the calculator becomes a confirmation rather than an oracle — which is the proper fate of every tool on this hub.

Litz Wire and Other Countermeasures

Engineering fights the exile with geometry: litz wire braids hundreds of individually-insulated strands, each thinner than the skin depth at the working frequency and transposed so every strand takes its turn at the bundle’s surface and interior — the whole cross-section works again, AC resistance falls, and switch-mode power supplies and induction cookers breathe easier up to a megahertz or two. Beyond that, surface finish itself becomes the resistor (roughness comparable to δ adds loss, a real effect in millimetre-wave design) and conductors become plated tubes and cavities. The calculator’s δ is the specification every one of these countermeasures is dimensioned against: strand gauges chosen under it, plating thicknesses chosen over it.

The Induction Hob’s Party Trick

An induction hob drives its coil at some 25 kHz, where this page gives copper a skin near 0.42 mm and — through a pan base’s much higher resistivity and permeability (cast iron’s μr in the hundreds) — the pan’s own skin shrinks to tens of micrometres: a foil-thin layer of the pan’s bottom absorbs the entire field and heats, the glass top above it staying cool because glass is no conductor at all. The trick that feels like magic on the showroom floor is the square root evaluated twice, once for the coil’s litz wire (see below) and once for the pan. The calculator prices both sides of the glass with the same three inputs.

Geophysics: Skin Depth as a Prospecting Tool

Scale the formula to the planet and it becomes an exploration method: magnetotellurics broadcasts nothing at all but listens to natural electromagnetic fluctuations, whose penetration into the Earth follows this page’s square root with rock resistivities — millihertz fields skin hundreds of kilometres deep, audio frequencies mere metres — so a frequency sweep sounds the crust’s resistivity in layers, hunting ore bodies, geothermal fields and groundwater. The conductor here is a continent and δ is measured in kilometres, but the mathematics is the induction hob’s verbatim. Few formulas on this hub span nine orders of magnitude of application without changing a symbol; the skin depth is one, and the calculator’s inputs (a resistivity, a frequency, a permeability) are exactly a geophysicist’s.

Related Tools

More from the Physics Calculators hub: the Drift Velocity Calculator — how slowly the charge carriers actually move, skin or no skin — and the Wheatstone Bridge Calculator, for measuring the resistances this effect complicates.

Frequently Asked Questions

Does skin effect apply to DC?

No — at DC (ω = 0) the formula’s δ diverges to infinity, which is the mathematics saying the current uses the whole conductor uniformly, as Ohm’s-law intuition expects. The effect grows in continuously with frequency; there is no threshold, only the gradual exile the square root describes. Practical engineers start caring when the conductor’s radius becomes comparable to δ — for household copper at 60 Hz that is centimetre-scale busbar territory, for RF it is everything. The calculator spans the range without a mode switch because the formula never has one.

Why does a worse conductor have a deeper skin?

Because the exiling agent is the induced eddy current, and a high-resistivity material supports feebler eddies: the fields penetrate further before the opposition builds. Hence nichrome’s skin is deeper than copper’s at the same frequency, and seawater (ρ ≈ 0.23 Ω·m, a wretched conductor) lets radio-frequency fields reach metres — except that the attenuation over those metres is still severe, which is why submarines trail very-low-frequency antennas (at 76 Hz, δ in seawater is tens of metres) for their one-way bells. Depth of penetration and goodness of transmission part company in lossy media; the formula prices the first, and this FAQ warns against inferring the second.

How is skin depth measured or verified?

Indirectly, through AC resistance: a wire’s resistance rises as the current’s annulus thins — R_ac ÷ R_dc following √f once δ is well under the radius — and precision impedance analysers map the transition beautifully against the formula (the proximity effect between neighbouring conductors complicates the clean case, as every transformer designer knows). Direct field probes inside solid metal are impractical at these scales; the calculator’s δ is verified the way most field quantities are, by its consequences matching measurement to the third digit across a century of radio engineering.

What about the skin depth in superconductors?

A different mechanism with its own length: below the critical temperature, magnetic fields penetrate a superconductor only the London penetration depth (tens of nanometres), set by the superfluid density rather than by resistivity and frequency at all. The analogy instructs — both are surface-exile phenomena — and the difference instructs more: the classical δ of this page is dissipative diffusion, the London depth is dissipationless screening. The calculator’s jurisdiction is the normal conductor stated in its labels, into which the superconducting case would import a foreign formula.

Does the effect matter in lightning protection?

Lightning is a fast transient — its spectrum reaches megahertz — so its current rides the down-conductor’s skin, and the conductor’s inductance (set by geometry, aggravated by bends) matters more than its bulk resistance: a fat copper strap and a hollow tube of the same surface perform alike for the stroke. This is why lightning conductors are broad straps routed straight, and why a coiled spare length in the run is forbidden — the calculator’s frequency fields, run at a megahertz, show the skin the stroke actually inhabits.

How thick should RF shielding be?

Five skin depths is the engineer’s rule for “opaque”: at 5δ the field is down to e⁻⁵, under 1% in amplitude (and 0.005% in power terms expressed as shielding effectiveness of about 43 dB from absorption alone; the reflection at the surface adds far more in practice). At 1 GHz in copper, δ ≈ 2.1 µm, so a 10 µm foil — household foil is several times that — is a serious shield, which is why a biscuit tin makes a respectable Faraday enclosure for a phone and why shielded-room walls need no great thickness, only unbroken continuity at the seams. The calculator turns any shielding question into its three-input form.

Who worked out the skin effect?

The mathematics descends from Maxwell’s equations through Horace Lamb and Oliver Heaviside in the 1880s, with Kelvin contributing the conductor-diffusion analysis — Heaviside, characteristically, had the operational picture before the profession wanted it. Its engineering canonisation came with radio: by the 1920s the δ formula was dimensioning the young industry’s coils, and the “skin effect” entry in every electrical handbook since is this square root. The calculator returns Heaviside’s diffusion depth in modern dress — SI units and a pink button, the physics unchanged.