ELC-23

Skin depth

δ = √(2ρ / ωμ) = √(ρ / π f μ). AC current hugs the surface.

Reading speed

FieldsKelvin

Governing equation

\delta=\sqrt{\dfrac{2\rho}{\omega\mu}}=\sqrt{\dfrac{\rho}{\pi f\mu}}

$f \in \left[1, 1\times 10^{6}\right]$

where

\rho: Resistivity (nΩ·m)
f: Frequency (Hz)
\mu_r: Relative μ (—)
\delta: Skin depth (mm)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

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①

Governing law — red

\delta=\sqrt{\dfrac{2\rho}{\omega\mu}}=\sqrt{\dfrac{\rho}{\pi f\mu}}

$f \in \left[1, 1\times 10^{6}\right]$

$f = 1$ $\delta = 65.62$

Along the curve

②

Domain of the variable — blue

f_{\min} = 1,\quad f_{\max} = 1\times 10^{6}

$f \in \left[1, 1\times 10^{6}\right]$

$f = 1$ $f = 1$

Along the curve

③

Response curve — green

\delta=\sqrt{\dfrac{2\rho}{\omega\mu}}=\sqrt{\dfrac{\rho}{\pi f\mu}}

$f \in \left[1, 1\times 10^{6}\right]$

$f = 1$ $\delta = 65.62$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}\delta}{\mathrm{d}f}

$f \in \left[1, 1\times 10^{6}\right]$

$f = 6290.3$ $\delta^{\prime} = -0.005$

Along the curve

Lecture brief

Historical brief

Ohm (1827), Kirchhoff (1845) and Maxwell’s circuit reduction still run every board: RLC transients, transformers, skin effect and three-phase power. The sheets are those network laws, not a SPICE deck. This sheet (ELC-23 — Skin depth) is the form associated with Kelvin. Working symbols: $\rho$ , $f$ , $\mu_r$ $\rightarrow$ $\delta$ . Eddy currents cancel the field in the bulk, leaving an exponential skin of thickness δ.

Purpose

Purpose: compute

\delta

from

\rho

f

\mu_r

in Electrical via

\delta=\sqrt{\dfrac{2\rho}{\omega\mu}}=\sqrt{\dfrac{\rho}{\pi f\mu}}

δ = √(2ρ / ωμ) = √(ρ / π f μ). AC current hugs the surface. Use it when a real electrical question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

\rho = 17.000\,\mathrm{nΩ·m}

f = 50.000\,\mathrm{Hz}

\mu_r = 1.000\,\mathrm{—}

, the governing relation

\delta=\sqrt{\dfrac{2\rho}{\omega\mu}}=\sqrt{\dfrac{\rho}{\pi f\mu}}

yields

\delta = 9.2802\,\mathrm{mm}

. A round wire, a current annulus, a δ ring. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Resistivity \rho17 nΩ·mFrequency f50 HzRelative μ \mu_r1 —

Outputs

Skin depth \delta9.2802 mm

Reading speed

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Narration of this film

A round wire, a current annulus, a δ ring.

Eddy currents cancel the field in the bulk, leaving an exponential skin of thickness δ.

Reading speed

Watch on YouTube