Skin Depth Calculator

Enter frequency, resistivity, and relative permeability to find the electromagnetic skin depth (δ) of a conductor, plus how deep the induced AC current actually penetrates.

Quick Facts

Skin depth formula
δ = √(ρ / (πfμ₀μr))
Depth where AC current density falls to 1/e (≈36.8%) of its surface value.
Current distribution
≈86% within 2δ, ≈95% within 3δ
Current decays exponentially with depth, J(x) = J₀e^(−x/δ).
Permeability of free space
μ₀ = 4π × 10⁻⁷ H/m
Magnetic materials (μr ≫ 1) have far smaller skin depth than resistivity alone suggests.

Your Results

Calculated
Skin Depth (δ)
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δ = √(ρ / (πfμ₀μr))
Skin Depth (mils)
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1 mil = 0.0254 mm
95% Current Depth (3δ)
-
Depth containing ≈95% of induced current
Angular Frequency (ω)
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ω = 2πf, used in the formula

Ready

Enter frequency, resistivity, and permeability, then press Calculate.

Formula and Method for Skin Depth

At DC, current is spread evenly across a conductor's cross-section. At AC, self-induced eddy currents oppose current flow in the interior of the conductor, concentrating it near the surface — the skin effect. The skin depth δ is the depth below the surface at which the current density has fallen to 1/e (≈36.8%) of its value at the surface, given by the standard formula δ = √(ρ / (πfμ₀μr)), where ρ is resistivity, f is frequency, μ₀ = 4π×10⁻⁷ H/m, and μr is the material's relative permeability.

How the calculation works

Enter the operating frequency and its unit, then either pick a material (which fills in typical resistivity and relative permeability) or enter custom values. The calculator converts resistivity from μΩ·cm to Ω·m and frequency to Hz, computes μ = μ₀μr, and evaluates δ = √(ρ / (πfμ)). It also reports δ in mils, the depth containing roughly 95% of the induced current (3δ, since current decays as J(x) = J₀e^(−x/δ)), and the angular frequency ω = 2πf used inside the formula.

Common mistakes

  • Ignoring relative permeability: for magnetic materials like steel, iron, or nickel, μr can be 100-2000+, which shrinks skin depth far more than resistivity alone would suggest — leaving μr at 1 for a magnetic material gives a badly wrong answer.
  • Mixing resistivity units: this calculator expects resistivity in μΩ·cm (copper ≈ 1.68); entering a raw Ω·m value like 0.0000000168 without converting will overstate the skin depth by a factor of 10⁸.
  • Treating δ as a hard cutoff: current does not stop at depth δ — it decays exponentially, so meaningful current still flows several skin depths into the conductor.

Real-world applications

  • Power engineering: at 50/60 Hz, copper's skin depth is roughly 9/8.5 mm, which is why large busbars and cables are stranded, hollow, or Litz-wired to keep AC resistance close to DC resistance.
  • RF and antenna design: at MHz-GHz frequencies, copper's skin depth shrinks to micrometers, so conduction happens almost entirely in a thin outer layer — surface plating quality matters more than bulk conductor material.
  • Induction heating: heating efficiency depends on concentrating eddy currents in a thin skin layer, so engineers choose frequency specifically to target a skin depth close to the desired case-hardening depth.
  • Electromagnetic shielding: required shield thickness for a given attenuation is often specified as a multiple of skin depth (commonly 3-5δ) at the frequency being shielded against.

Frequently Asked Questions

What is skin depth?
Skin depth (δ) is the depth below a conductor's surface at which the AC current density has fallen to about 1/e (≈36.8%) of its value at the surface. It results from the skin effect, in which self-induced eddy currents oppose current flow in the interior of a conductor at higher frequencies, effectively confining current to a thin outer layer.
How does frequency affect skin depth?
Skin depth is inversely proportional to the square root of frequency: δ = √(ρ / (πfμ)). Quadrupling the frequency halves the skin depth, which is why skin depth is measured in millimeters at 50/60 Hz power frequencies but shrinks to micrometers at radio frequencies.
Why is skin depth so much smaller in steel or iron than in copper?
Skin depth depends on both resistivity and relative permeability (δ is proportional to 1/√μr). Ferromagnetic materials like iron, steel, and nickel have relative permeabilities from the hundreds to the thousands, so even though their resistivity is only a few times higher than copper's, their skin depth can be 10-40 times smaller.
How much of the current actually flows within one skin depth?
Because current density decays exponentially with depth, J(x) = J₀ × e^(−x/δ), about 63% of the total current flows within the first skin depth, about 86% within two skin depths, and about 95% within three skin depths.

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