ELC-05

Conductor skin depth

δ = √(2 ρ /(ω μ)) in a linear conductor.

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ElectromagneticsKelvin skin effect

Governing equation

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

$f \in \left[0.05, 1\times 10^{6}\right]$

where

\rho: Resistivity (Ω·m)
f: Frequency (kHz)
\mu_r: Relative permeability (—)
\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}}

$f \in \left[0.05, 1\times 10^{6}\right]$

$f = 0.05$ $\delta = 9.28$

Along the curve

②

Domain of the variable — blue

f_{\min} = 0.05,\quad f_{\max} = 1\times 10^{6}

$f \in \left[0.05, 1\times 10^{6}\right]$

$f = 0.05$ $f = 0.05$

Along the curve

③

Response curve — green

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

$f \in \left[0.05, 1\times 10^{6}\right]$

$f = 0.05$ $\delta = 9.28$

Along the curve

④

Rate of change — amber

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

$f \in \left[0.05, 1\times 10^{6}\right]$

$f = 6289.4$ $\delta^{\prime} = -7.36\times 10^{-4}$

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-05 — Conductor skin depth) is the form associated with Kelvin skin effect. Working symbols: $\rho$ , $f$ , $\mu_r$ $\rightarrow$ $\delta$ . Maxwell's equations in a good conductor yield an exponentially decaying field with 1/e depth δ, the skin depth.

Purpose

Purpose: compute

\delta

from

\rho

f

\mu_r

in Electrical via

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

δ = √(2 ρ /(ω μ)) in a linear conductor. 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 = 1.700e-8\,\mathrm{Ω·m}

f = 50.000\,\mathrm{kHz}

\mu_r = 1.000\,\mathrm{—}

, the governing relation

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

yields

\delta = 0.2935\,\mathrm{mm}

. Semi-infinite linear metal, μr ≈ 1 unless set. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Resistivity \rho1.7e-8 Ω·mFrequency f50 kHzRelative permeability \mu_r1 —

Outputs

Skin depth \delta0.2935 mm

Reading speed

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Narration of this film

Semi-infinite linear metal, μr ≈ 1 unless set.

Maxwell's equations in a good conductor yield an exponentially decaying field with 1/e depth δ, the skin depth.

Reading speed

Watch on YouTube