Skin Effect in Conductors

Equivalent forms

J(r) = J_0 e^{-(R-r)/\delta}

\omega = 2\pi f

A single length δ collapses the full diffusion of fields into a conductor — it tells you in one number how thin to make a wire before adding copper stops helping.

Symbol	Name	SI unit	Dimension	Range
$δ$	Skin depthoutput Depth at which current density falls to 1/e	m	L	0.000001 – 0.02
$f$	Frequency AC frequency	Hz	T⁻¹	50 – 1000000
$σ$	Conductivity Material conductivity	S/m	M⁻¹·L⁻³·T³·I²	1000000 – 60000000
$µ$	Permeability Magnetic permeability (≈µ₀ for copper)	H/m	M·L·T⁻²·I⁻²	0.00000125 – 0.001

Unit systems

Symbol	SI	CGS	Imperial
$δ$	m	cm	in
$f$	Hz	Hz	Hz
$σ$	S/m	s⁻¹	S/m
$µ$	H/m	-	H/m

Where it holds

Valid for good conductors where

\delta is

much smaller than the conductor radius (high-frequency limit

) and \sigma

≫

\omega \varepsilon

. At very high frequency or in poor conductors the full complex-permittivity treatment is needed.

Dimensional analysis

$\surd (2/(\omega$ µ $\sigma )) = \surd (1 / (s^{-1}\cdot H\cdot m^{-1}\cdot S\cdot m^{-1})) = \sqrt{m^{2}} = m \checkmark$

Discovery

Horace Lamb / Oliver Heaviside · 1885

Lamb analyzed alternating currents in conductors and Heaviside, working out Maxwell's equations for telegraph lines, showed mathematically that high-frequency current crowds toward the surface — explaining why thick conductors were wasteful for telegraphy and radio.

Try this

At high frequency a solid copper bar is mostly dead weight — current avoids the center. Why does AC hug the surface?

Find the skin depth in copper (σ = 5.9×10⁷ S/m) at 50 Hz and at 1 MHz, and explain why power lines and RF wires differ.

Research status: stable

Real-world applications

Litz wire and tubular/hollow conductors for RF
Silver/gold plating of waveguides and connectors
Sizing busbars and overhead lines for AC losses
Induction heating depth control

Common misconceptions

Thicker wire always carries more AC — beyond a few skin depths the extra copper does nothing
Skin effect matters only at radio frequencies — even 60 Hz crowds current in thick busbars
It changes the DC resistance — it raises the effective AC resistance, not the DC value

Experimental verification

Measuring AC resistance of a wire versus frequency shows it rising

as \sqrt{f}

once

\delta

< radius, matching theory. RF engineers routinely verify that silver-plating only a few skin depths thick carries the current as well as solid silver.

Derivation

Inside a good conductor Maxwell's equations reduce to the diffusion equation

\nabla ^{2}J

= µ

\sigma \partial J/\partial t

.

For a field oscillating as

e^{i\omega t}

the solution decays as

e^{-z/\delta }e^{i(\omega t-z/\delta )}

with

\delta = \surd (2/(\omega

µ

\sigma ))

.

Using

\omega = 2\pi f

gives

\delta = 1/\surd (\pi f

µ

\sigma )

.

Limiting cases

f \to 0 (DC)

⟶

\delta \to \infty

Direct current fills the whole cross-section uniformly — no skin effect.

f \to

large⟶

\delta \to 0

At GHz the current rides a micron-thin surface layer; the core is wasted.

ferromagnetic µ large⟶

\delta

smallHigh permeability (iron) shrinks

\delta

sharply — why steel is poor for RF.

What if…

What if you replace copper with iron?

Iron's high µ (hundreds–thousands) shrinks $\delta$ dramatically, so it's a poor RF conductor despite decent $\sigma$ .

What if you hollow out the wire?

You lose almost no current capacity at high f and save weight — the principle behind tubular RF conductors.

What if frequency goes to GHz?

$\delta$ drops $to \sim 1$ µm, so even a thin gold flash plating carries the signal — surface finish dominates loss.

1

Copper at 50 Hz vs 1 MHz

Given ·

σ:: 59000000
µ:: 0.00000125663706212
f:: 50

Find ·

\delta

Steps

At 50 Hz: $\pi f$ µ $\sigma \approx 1.17\times 10^{4}$ , $\delta = 1/\surd \approx 9.3\times 10^{-3} m$
At 1 MHz it scales as $1/\sqrt{f}$ , falling $by \sqrt{20000} \approx 141\times to \sim 66$ µm
Hence power lines use the whole cross-section, but RF wires only need a thin shell

Result ·

\delta (50\,\mathrm{Hz}) = 1/\sqrt{\pi \times 50\times 1.2566\times 10^{-6}\times 5.9\times 10^{7}} \approx 9.3\,\mathrm{mm}

;

\delta (1\,\mathrm{MHz}) \approx 66

µm

Faraday's Law of Induction→Ampère–Maxwell Law→Ohm's Law→