Skin Depth Calculator

Calculate electromagnetic skin depth and attenuation of AC current in conductors.

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AC Fields Penetrate Conductors Only a Finite Distance

Skin depth is the characteristic distance over which an alternating electromagnetic field decays inside a good conductor. For a good conductor, δ=√(2/(ωμσ))=√(1/(πfμσ)). Current density and field amplitude fall approximately as e−x/δ, so higher frequency, greater permeability, or greater conductivity produces a smaller penetration depth.

The skin effect pushes AC current toward the conductor surface and increases effective resistance compared with DC. At one skin depth the field amplitude is about 36.8% of its surface value; power density falls even faster because it depends on amplitude squared. Very thin conductors, low frequencies, or poor conductors may require more general electromagnetic treatment.

δ=√(1/(πfμσ))
SymbolMeaningWhy it appears / units
δSkin depthm; amplitude e-folding distance.
fFrequencyHz; increasing f reduces δ.
μMagnetic permeabilityH/m; often μ=μ0μr.
σElectrical conductivityS/m; higher conductivity reduces penetration depth.

If conductor thickness is many skin depths, most AC current is concentrated near the surface. This motivates litz wire at some frequencies and surface-plated conductors or hollow waveguide structures in high-frequency systems.

Skin depth follows inverse square-root scaling. Multiplying frequency by four should halve δ when μ and σ stay fixed. The same check applies to conductivity and permeability. Large discrepancies often come from confusing relative permeability with absolute μ or from using resistivity where conductivity is required.

Worked Examples

Example 1: Copper at 50Hz (mains)
δ=1/√(π×50×1×4π×10⁻⁷×5.8×10⁷)
Result: 9.4 mm — most wire is within 1 skin depth
50Hz current uses full wire cross-section
Example 2: Copper at 1MHz
δ scales as 1/√f
Result: 0.066 mm — only outer 66μm carries current!
RF cables use hollow conductors or stranded wire
Example 3: Frequency scaling
frequency increases from 1kHz to 100kHz
Result: skin depth becomes 1/10 as large
Because δ is proportional to 1/√f, a 100× frequency increase gives a 10× decrease.
Example 4: Two skin depths
x=2δ
Result: amplitude=e−2≈13.5% of surface value
Skin depth is an exponential decay length, not a hard boundary.

Common Mistakes

⚠️
Treating skin depth as the depth where current becomes zero

Fields decay exponentially and never abruptly vanish at δ.

⚠️
Using relative permeability without multiplying by μ₀

The formula requires absolute permeability in H/m unless a specially normalized form is used.

⚠️
Assuming DC has a finite skin depth from this approximation

As frequency approaches zero the good-conductor AC skin-depth expression tends toward infinity, consistent with more uniform DC current distribution.

Frequently Asked Questions

Why does skin effect matter?
AC current crowds toward conductor surface (skin effect). At high frequency, effective resistance increases as R_ac = R_dc × (wire_radius/δ). RF engineers use hollow tubes or Litz wire (many thin insulated strands) to overcome this.
Skin depth in seawater?
At 1 Hz: δ≈250m. At 10 kHz: δ≈8m. Submarines use ELF (3-30 Hz) for communication — very long wavelengths penetrate seawater. Higher frequencies are quickly attenuated.
Why does AC resistance rise because of skin effect?
Current is crowded into a smaller effective cross-sectional area near the surface. Less conducting area means greater resistance than the DC value for the same conductor.
What fraction remains at one skin depth?
Field or current-density amplitude is e⁻¹, about 36.8% of its surface value. Power-related quantities proportional to amplitude squared are about 13.5%.
Why does higher conductivity reduce skin depth?
A highly conducting material supports stronger induced currents that oppose field penetration, confining alternating fields closer to the surface.
When is the simple skin-depth formula valid?
It is a good approximation for good conductors when conduction current dominates displacement current and material parameters can be treated as approximately uniform at the frequency of interest.
Why does skin depth decrease as frequency increases?
For a good conductor, δ≈√(2/(ωμσ)). Higher frequency, permeability, or conductivity increases attenuation per unit depth, so the field penetrates less deeply. Skin depth is the distance over which field amplitude falls by a factor e, not the depth at which current abruptly becomes zero.

Formula Explorer connections

Interpretation: This relationship connects magnetic fields, moving charge, flux, induction or electromagnetic material response. Assumption: Specify field direction and sign convention. Uniform fields, linear materials, negligible edge effects or sinusoidal steady state may be assumed.

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