How this instrument works
Skin depth is the distance below a conductor's surface at which current density has fallen to 1/e, about 37 percent, of its value right at the surface. It exists because a changing current produces a changing magnetic field inside the conductor, and that changing field induces its own eddy currents in the interior that oppose the original flow — Lenz's law working against you from within the metal. The faster the current alternates, the stronger that internal opposition, and the more the real current is pushed outward toward the surface.
The formula δ = √(ρ ⁄ (π f μ)) falls straight out of solving Maxwell's equations inside a good conductor, where the conduction current so completely dominates the displacement current that the electric field obeys a diffusion equation rather than a wave equation. Resistivity ρ sits on top because a poorer conductor supports weaker eddy currents for the same changing field, so it resists the field's penetration less and lets current reach deeper before being cancelled. Frequency f and permeability μ sit underneath because both strengthen that internal cancellation — raise either one and the skin shrinks, and because the relationship is a square root, quadrupling the frequency only halves the depth.
The formula is the good-conductor limit, not a universal law. Push frequency down toward zero and δ grows without bound, correctly recovering the DC case where current fills the whole cross-section and plain Ohm's law applies. Push a material's permeability up instead — steel and iron run hundreds of times higher than copper's — and the same frequency yields a far shallower skin, which is exactly why induction furnaces couple so efficiently into ferromagnetic stock and so weakly into aluminum or brass.
- Enter Resistivity in Ω·m. Annealed copper at 20°C is about 1.68 × 10⁻⁸ Ω·m; aluminum runs closer to 2.65 × 10⁻⁸.
- Enter Frequency in Hz — 60 for US mains power, 50 for most of the rest of the world, or into the kilohertz and megahertz range for induction heating or RF work.
- Enter Absolute permeability in H/m. For non-magnetic metals like copper or aluminum, use μ0 = 1.25663706212 × 10⁻⁶ H/m. For steel or iron, multiply μ0 by the material's relative permeability first.
- Read Skin depth in metres — the depth at which current density has dropped to about 37 percent of its surface value, not the depth where the flow stops.
Worked example — copper busbar at 60 Hz mains frequency
Take a copper busbar carrying ordinary 60 Hz mains current. Copper's resistivity is ρ = 1.68 × 10⁻⁸ Ω·m and, being non-magnetic, its absolute permeability equals μ0 = 1.25663706212 × 10⁻⁶ H/m. Feeding both into δ = √(ρ ⁄ (π f μ)) with f = 60 Hz gives δ = √(1.68 × 10⁻⁸ ⁄ (π × 60 × 1.25663706212 × 10⁻⁶)) = 0.00842168798466 m, about 8.42 mm.
Eight millimetres is generous next to a typical busbar's thickness of a centimetre or two, which is exactly why skin effect is usually ignored at power-line frequency — conduction still fills nearly the whole bar. Push the same copper to 60 kHz, a common induction-heating frequency, and depth falls by the square root of that thousand-fold frequency jump to roughly 0.27 mm, confining the flow to a shell you could barely resolve by eye.
Questions
Does AC current really flow only on the surface of a conductor?
No — skin depth is not a hard boundary where the flow of charge stops. Current density decays exponentially with depth, and δ marks where it has fallen to 1/e, about 37 percent, of the surface value. Roughly 87 percent of the total flow sits within the outer two skin depths and 95 percent within three, but some charge still moves at every depth.
Why does skin depth shrink as frequency increases?
A changing magnetic field inside the conductor induces eddy currents that oppose the original flow in the interior. The faster it alternates, the faster that field changes and the stronger the opposition, so charge is pushed toward the surface. Because δ scales with the inverse square root of f, quadrupling frequency only halves the skin depth, not quarters it.
Why does higher resistivity increase skin depth instead of shrinking it?
A more resistive material supports weaker induced eddy currents for the same changing field, so it opposes and cancels that field less effectively, letting current penetrate deeper before it fades. That is why ρ sits in the numerator: brass or nichrome, both poorer conductors than copper, show noticeably deeper skin depths at the same frequency.
How much does permeability change the answer for steel versus copper?
Steel's relative permeability can run from a few hundred to several thousand, against copper's 1. Since μ sits under the square root in the denominator, that difference alone shrinks steel's skin depth by a factor of roughly 10 to 60 compared with an equally resistive non-magnetic metal at the same frequency — a large part of why induction furnaces heat steel stock so efficiently.
What permeability value should I enter for copper, aluminum, or gold?
Use μ0, the permeability of free space: 1.25663706212 × 10⁻⁶ H/m, the CODATA value. These metals have relative permeability close enough to 1 that μ0 alone is accurate. Only for a ferromagnetic conductor do you need to multiply μ0 by the material's relative permeability.
Does this formula still hold at radio and microwave frequencies?
Yes, for good conductors — metals in which conduction current dominates displacement current, true for copper from mains frequency well up into the infrared. The same √(ρ ⁄ (π f μ)) relation sets the minimum copper thickness on RF circuit boards and the wall thickness needed inside a waveguide or coaxial shield.