Skin Depth and Reflection from Conductors

The previous article derived a length: $\delta = \sqrt{2/\mu\sigma\omega}$ , the skin depth, the distance a wave survives inside a conductor. This article is about what that length is worth. It is, by a wide margin, the most economically consequential millimeter-to-micrometer in electrical engineering: it decides that a 60 Hz power cable uses its whole copper cross-section while a 10 MHz current refuses to touch anything more than 20 μm below the surface (multiplying the wire's resistance twenty-four-fold); that a few micrometers of silver plating make a waveguide as good as solid silver; that a kitchen microwave is caged by a wall hundreds of skin depths thick; and that your bathroom mirror returns 98-plus percent of the light you send at it while quietly eating the rest. Two payoffs get full derivations. First, surface resistance: collapse the skin layer into an equivalent sheet and every ohmic-loss problem — wire resistance at radio frequency, waveguide attenuation, resonator quality factor — reduces to one number, $R_s = 1/\sigma\delta$ , rising as $\sqrt{\omega}$ . Second, reflection: feed the conductor's complex index into the normal-incidence Fresnel machinery and out comes near-total reflection, with the small absorbed remainder given by the Hagen–Rubens law $1 - R = \sqrt{8\varepsilon_0\omega/\sigma}$ . The two payoffs then audit each other — the absorptance computed from Fresnel algebra equals $4R_s/Z_0$ computed from Joule heating in the skin, exactly — one of those cross-checks that certifies an entire framework at once. The chapter needs one more thing from this article: the perfect-conductor limit $\sigma \to \infty$ , where $\delta \to 0$ , $R \to 1$ , and the fields obey the clean PEC boundary conditions. The waveguide articles that close this chapter will stand on exactly that idealization, priced here honestly.

Key takeaways

The wave machinery (damped wave equation, complex $\tilde{k}$ , $\delta = \sqrt{2/\mu\sigma\omega}$ , 45° lag) is recalled by name from the previous article; this article spends it.
Skin-depth ladder for copper: 9.2 mm (50 Hz) → 21 μm (10 MHz) → 0.65 μm (10 GHz) — the $1/\sqrt{\omega}$ law that shapes power grids, RF wiring, and plating specs alike.
Surface resistance: the skin layer acts as a sheet of resistance $R_s = 1/\sigma\delta = \sqrt{\mu\omega/2\sigma}$ per square; a round wire's AC resistance exceeds DC by $\approx a/2\delta$ once $\delta \ll a$ (a factor 24 for a 1 mm-radius wire at 10 MHz).
Reflection: with $\tilde{n} = (1+i)\sqrt{\sigma/2\varepsilon_0\omega}$ ... precisely, $n \approx \eta \approx \sqrt{\sigma/2\varepsilon_0\omega} \gg 1$ , the normal-incidence coefficient gives $R \to 1$ ; the absorbed sliver is Hagen–Rubens: $1 - R \approx \sqrt{8\varepsilon_0\omega/\sigma}$ — $10^{-4}$ at microwave frequencies, ~1.5% for infrared mirrors.
The audit: $1 - R = 4R_s/Z_0$ exactly — Fresnel's energy deficit equals the Joule heat the transmitted skin currents dissipate. Two derivations, one number.
Perfect conductor ( $\sigma \to \infty$ ): $\delta = 0$ , $R = 1$ , tangential E and normal B vanish at the surface — the PEC boundary package (recalled from the electrodynamic-boundary-conditions article) that the waveguide articles will assume.
KEY IDEA: one length ( $\delta$ ) and one sheet number built from it ( $R_s = 1/\sigma\delta$ ) price every conductor at frequency — current crowding, shielding budgets, mirror losses — and the Hagen–Rubens absorptance $4R_s/Z_0$ is the same physics read off the reflection side.

The core insight

The skin depth $\delta = \sqrt{2/\mu\sigma\omega}$ turns into engineering through the surface resistance $R_s = 1/\sigma\delta$ : AC currents crowd into the skin (resistance $\propto \sqrt{\omega}$ ), conducting walls shield as $e^{-t/\delta}$ , and a good conductor reflects all but the Hagen–Rubens fraction $1 - R = \sqrt{8\varepsilon_0\omega/\sigma} = 4R_s/Z_0$ , which its skin currents dissipate as heat.

Where this lives today

Surface resistance is the currency of the RF-to-terahertz economy. 5G and satellite bands live at 24–70 GHz, where $\delta < 0.5\ \mu$ m makes surface roughness comparable to the skin depth — a major industrial problem (roughness lengthens the current's path, multiplying effective $R_s$ ; copper foils for millimeter-wave circuit boards are now specified by nanometer-scale roughness profiles). Superconducting RF is the other frontier: niobium cavities at 2 K reach $R_s \sim$ nanoohms — six orders below copper — storing energy with $Q > 10^{11}$ for particle accelerators (XFEL, LCLS-II) and superconducting qubit resonators, the $\sigma \to \infty$ limit realized closely enough to redraw what "mirror" means. The reflection side runs from the mundane to the exotic: radar cross-section engineering is applied Hagen–Rubens plus geometry (and its deliberate inversion — absorber coatings that maximize skin dissipation); infrared thermography relies on low-emissivity metal surfaces ( $\varepsilon_{\rm emis} = A$ by Kirchhoff's law, so a polished-copper roof radiates 2% of what a black one does — the physics of both spacecraft thermal blankets and energy-efficient windows); and gravitational-wave interferometers, needing mirrors without metallic loss, abandoned this article's mirrors entirely for dielectric stacks — parts-per-million absorption where copper's infrared floor is percent — a boundary drawn precisely by Hagen–Rubens.

Origins · 1 minLord Kelvin and how this was discovered

Lord Kelvin

1824–1907 · Glasgow

William Thomson — Kelvin from 1892, the first scientist elevated to the British peerage — worked on the skin effect the way he worked on everything: because a cable was at stake. Having made his name (and fortune) on the transatlantic telegraph — his mirror galvanometer and his 1855 diffusion theory of cable signals made the 1866 Atlantic link work, after the 1858 cable died partly for ignoring his analysis — he returned in 1889 to the problem of alternating currents in conductors, deriving the exact current distribution in a round wire and inventing the ber and bei functions (Bessel-real, Bessel-imaginary) to express it. The large-argument limit of his solution is this article's $a/2\delta$ crowding law. It was late-career work from a man whose fingerprints were already on thermodynamics' second law, the absolute temperature scale, the RC time constant, and the mirror galvanometer; characteristically, it was driven by engineering (power transmission was going AC, against his advice) and executed with complete mathematical control. Kelvin's famous wrong verdicts — the Sun's age, "heavier-than-air flight is impossible," his resistance to atoms — have become cautionary tales; his right ones are simply infrastructure, and the skin-effect functions bearing his name still price every power line's conductor diameter.

History

1855

— Thomson's telegraph theory: signal diffusion in cables — the mathematics that will become skin-effect analysis

1883

— Horace Lamb solves oscillating fields around conducting spheres; exponential penetration appears

1885

— Heaviside analyzes the skin effect for telegraph and power conductors and champions its consequences

1886

— Hughes demonstrates anomalous AC resistance of wires experimentally

1889

— Kelvin's ber/bei functions: the exact AC current profile in a round wire

1903

— Hagen and Rubens: infrared metal reflectance matches DC conductivity — one electron response from bench to light

1940

s — Radar drives silver-plated waveguide practice; surface resistance becomes a design currency

2000

s — Superconducting cavities reach nanoohm surface resistance and Q above 10 to the 11

The physical picture

Why does high-frequency current crowd toward a wire's surface? The honest answer is the previous article's shielding physics turned inside out: the current's own alternating magnetic flux, threading the wire's interior, induces (by Faraday) back-EMFs that oppose current flow most strongly along the wire's axis — where a current filament is encircled by the most flux — and least at the surface. The interior effectively presents a higher impedance, current takes the cheaper surface path, and the crowding profile is exactly the $e^{-d/\delta}$ decay of a field diffusing in from the boundary. Same equation, same $\delta$ ; "skin effect in a wire" and "wave dying in a conductor" are one phenomenon in two geometries (Kelvin solved the cylindrical version exactly in 1889, in the Bessel-family functions he had to invent names for — ber and bei — and the history band tells the story). The engineering abstraction that makes the crowding usable is the surface resistance. If current flows only within $\sim\delta$ of the surface, then per unit width of surface the conductor is a strip of thickness $\delta$ and conductivity $\sigma$ : sheet resistance $R_s = 1/\sigma\delta$ — "ohms per square," independent of the square's size, rising as $\sqrt{\omega}$ since $\delta$ shrinks as $1/\sqrt{\omega}$ . Derivation A makes this exact (the naive "uniform slab of thickness $\delta$ " picture happens to give precisely the right answer for the resistive part — a small miracle of the 45° phase structure worth seeing once). Once you own $R_s$ , every AC-loss problem is geometry: wire resistance = $R_s$ /(circumference), waveguide wall loss = $R_s \times$ (surface current)², resonator $Q$ = (energy stored)/( $R_s\times$ surface integrals). Reflection is the same story told from the vacuum side. A wave arriving at a metal finds a medium whose impedance is tiny — milliohms against vacuum's 377 Ω — and impedance mismatch on that scale sends essentially everything back with a sign flip ( $r \approx -1$ : the standing-wave node at a mirror). "Essentially" is quantified by Hagen–Rubens: the deficit $1 - R$ is small, scales as $\sqrt{\omega/\sigma}$ , and is not a nuisance term — it is the entire thermal load on every mirror, radar reflector, and cavity wall in technology. And the two pictures must agree: whatever power Fresnel says fails to reflect is exactly the power the skin currents burn. Derivation C closes that loop to the last factor.

Figure

The skin-depth ladder and the surface-resistance abstraction — current crowds into a sheet of thickness delta, priced at 1 over sigma delta per square

The two payoffs, quantified

Everything in this article rests on the previous article's good-conductor package, recalled: $k = \kappa = 1/\delta$ , fields decaying as $e^{-z/\delta}$ with one radian of phase per skin depth, B lagging E by 45°. The ladder of lengths (copper, $\sigma = 5.96\times10^7$ S/m):

\delta = \sqrt{\frac{2}{\mu\sigma\omega}}: \qquad 9.2\ \text{mm}\ (50\ \text{Hz}) \;\cdot\; 0.21\ \text{mm}\ (100\ \text{kHz}) \;\cdot\; 21\ \mu\text{m}\ (10\ \text{MHz}) \;\cdot\; 0.65\ \mu\text{m}\ (10\ \text{GHz}), \qquad (1)

where: the numbers match the eddy-current article's diffusion-route values (one length, two derivations — the handshake the previous article certified). Rule-of-thumb readings: a conductor is "bulk" when its size $\ll \delta$ and "surface-only" when $\gg \delta$ ; shielding walls attenuate amplitude by $e^{-t/\delta}$ (with geometry-dependent prefactors), so "five skin depths" ( $e^{-5} \approx 0.7\%$ ) is the standard engineering wall. The sheet abstraction (Derivation A):

Z_s = \frac{1+i}{\sigma\delta}, \qquad R_s = \mathrm{Re}\,Z_s = \frac{1}{\sigma\delta} = \sqrt{\frac{\mu\omega}{2\sigma}}, \qquad (2)

where: $Z_s$ — the surface impedance, tangential E at the surface per unit surface-current density — has equal resistive and inductive parts (the 45° structure again), and its real part $R_s$ prices every ohmic loss: a wire of radius $a \gg \delta$ has AC resistance per length $R' = R_s/2\pi a$ , exceeding its DC value by roughly $a/2\delta$ ; a waveguide wall dissipates $\tfrac{1}{2}R_s|K|^2$ per area for surface current $K$ . Copper at 10 GHz: $R_s = 26\ \text{m}\Omega$ per square — why microwave plumbing is mirror-polished and silver-plated, and why the plating is all that matters (anything deeper than a few $\delta$ is electrically asleep). On the reflection side (Derivation B): a wave at normal incidence from vacuum meets complex index $\tilde{n} = c\tilde{k}/\omega$ with $n \approx \eta \approx \sqrt{\sigma/2\varepsilon_0\omega} \gg 1$ , and

R = \left|\frac{1 - \tilde{n}}{1 + \tilde{n}}\right|^2 \longrightarrow 1 - \sqrt{\frac{8\varepsilon_0\omega}{\sigma}} \qquad \text{(Hagen–Rubens)}, \qquad (3)

where: the deficit — the absorptance $A = 1 - R$ — is $8.6\times10^{-5}$ for copper at 1 GHz, $2.7\times10^{-4}$ at 10 GHz, and about 1.5% at CO₂-laser infrared (30 THz) — beyond which the Ohmic model itself expires (Drude rollover; the absorption article owns visible-band metals, recalled by name). And the consistency theorem (Derivation C):

A = \frac{4R_s}{Z_0}, \qquad (4)

where: $Z_0 = \sqrt{\mu_0/\varepsilon_0} = 377\ \Omega$ . The Fresnel-side energy deficit (3) and the Joule heat that the transmitted skin currents dissipate (computed from (2)) are the same number — checked symbolically and numerically below. When a framework closes a loop like this, both derivations are certified at once.

Figure

The mirror's energy audit — the doubled surface field drives skin currents whose Joule heat is exactly the reflection deficit 4 Rs over Z0

Derivation AThe surface impedance — collapsing the skin into a sheet

The goal is (2): show that a good conductor's surface behaves, for any field applied along it, as a sheet with impedance $Z_s = (1+i)/\sigma\delta$ per square, and extract the working consequences (AC wire resistance, the plating rule). The strategy: take the known interior solution — recalled, not re-derived, from the previous article — integrate the current it carries, and form the ratio of surface field to total current. The "small miracle" to watch: the naive uniform-slab-of-thickness- $\delta$ guess gets the resistance exactly right, and the phase structure explains why an inductive part tags along.

Step 1. The interior solution (recalled). With the surface at $z = 0$ and tangential field $E_0$ along $\hat{\mathbf{x}}$ at the surface, the previous article gives $\tilde{E}(z) = E_0\,e^{-z/\delta}e^{iz/\delta}$ (one radian per $e$ -fold — the $k = \kappa$ lock), driving current density $\tilde{J}(z) = \sigma\tilde{E}(z)$ .

Step 2. Total current per unit width. Integrate the sheet:

\tilde{K} = \int_0^\infty \tilde{J}\,dz = \sigma E_0 \int_0^\infty e^{-(1-i)z/\delta}\,dz = \frac{\sigma E_0\,\delta}{1 - i} = \frac{\sigma\delta}{2}(1+i)\,E_0.

Why the integral converges so obligingly: the same $\delta$ controls decay and phase, so the current corkscrews and shrinks together; the deep contributions arrive increasingly out of phase and partially cancel, which is why the effective sheet is thinner than a naive amplitude-only integral would suggest.

Step 3. Form the impedance. $Z_s \equiv E_0/\tilde{K} = \dfrac{2}{\sigma\delta(1+i)} = \dfrac{1-i}{\sigma\delta}$ — equal resistive and reactive magnitudes. In this track's $e^{-i\omega t}$ convention the reactance is inductive; magnitude-wise,

R_s = \frac{1}{\sigma\delta} = \sqrt{\frac{\mu\omega}{2\sigma}}, \qquad X_s = R_s.

Why the uniform-slab fiction works: pretend the current flows uniformly through thickness $\delta$ and stops — resistance per square $1/\sigma\delta$ — and you land exactly on $R_s$ . The cancellations of Step 2 conspire to make the honest answer equal the cartoon's. The cartoon misses only $X_s$ : the field's phase rotation with depth stores magnetic energy in the skin, and the sheet inherits an inductance exactly equal (in impedance) to its resistance — the 45° of the previous article, now sitting in a circuit model.

Step 4. Spend it on a wire. A round wire, radius $a \gg \delta$ : the current rides in the surface sheet, so per unit length $R'_{AC} = R_s/2\pi a$ , against $R'_{DC} = 1/\sigma\pi a^2$ . Ratio:

\frac{R_{AC}}{R_{DC}} = \frac{\pi a^2 \sigma R_s}{2\pi a} = \frac{a}{2\delta}.

Copper, $a = 1$ mm, 10 MHz ( $\delta = 21\ \mu$ m): factor 24 — 5.3 mΩ/m becomes 130 mΩ/m. This single ratio is why RF engineering exists as a discipline of surfaces: litz wire (many insulated strands, each thinner than $\delta$ , woven so each takes a turn on the outside) claws back the interior at sub-MHz frequencies; above that, hollow tubes and plated surfaces are honest about where the current lives. The plating rule follows directly: a coating a few $\delta$ thick is the conductor (silver-plate steel and it conducts like silver at 10 GHz); conversely a resistive film thinner than $\delta$ does not shield — it attenuates only by its sheet-resistance mismatch. Verified: $Z_s$ matches Jackson §8.1 and Zangwill §17.4 (surface-impedance boundary condition); the $a/2\delta$ crowding ratio is the standard large-argument limit of Kelvin's exact ber/bei wire solution (1889), which interpolates smoothly to $R_{AC} = R_{DC}$ at $a \ll \delta$ ; numbers recomputed — copper 10 MHz: $\delta = 20.6\ \mu$ m, $R_{AC}/R_{DC} = 24.3$ , $R'$ 5.3 → 130 mΩ/m.

Go deeper · Derivation B · 2 minNear-total reflection and the Hagen–Rubens law

Derivation BNear-total reflection and the Hagen–Rubens law

The goal is (3): reflectance of a good conductor at normal incidence, exact in structure and expanded honestly in the small parameter $\sqrt{\varepsilon_0\omega/\sigma}$ . The strategy is pure recall-and-substitute: the normal-incidence coefficient $r = (1 - \tilde{n})/(1 + \tilde{n})$ from the normal-incidence article carries over to complex index unchanged (nothing in its boundary-condition derivation assumed $n$ real), and the conductor's $\tilde{n}$ comes from the previous article's $\tilde{k}$ . What is owned here is the limit-taking and its physical reading.

Step 1. The conductor's index. $\tilde{n} = c\tilde{k}/\omega$ with the good-conductor $\tilde{k} = (1+i)/\delta$ :

\tilde{n} = n + i\eta, \qquad n = \eta = \frac{c}{\omega\delta} = \sqrt{\frac{\sigma}{2\varepsilon_0\omega}} \gg 1.

Copper at 10 GHz: $n = \eta \approx 7.3\times10^3$ — an "index" of thousands, which is the impedance mismatch $Z_0/|Z_s|$ wearing refractive clothing.

Step 2. Reflectance, exactly:

R = \left|\frac{1 - n - i\eta}{1 + n + i\eta}\right|^2 = \frac{(n-1)^2 + \eta^2}{(n+1)^2 + \eta^2}.

As $n = \eta \to \infty$ , $R \to 1$ : the perfect mirror. Why the reflected wave flips sign: for $\tilde{n}$ huge, $r \to -1$ — tangential E nearly vanishes at the surface (the PEC condition emerging), and the incident and reflected E-fields cancel there while their B-fields add: a standing wave with an E-node and B-antinode at the mirror, which is where the doubled surface current of Derivation C comes from.

Step 3. The deficit, to leading order. Expand with $n = \eta \gg 1$ :

1 - R = \frac{(n+1)^2 - (n-1)^2}{(n+1)^2 + \eta^2} = \frac{4n}{(n+1)^2 + n^2} \approx \frac{4n}{2n^2} = \frac{2}{n},

and substituting $n = \sqrt{\sigma/2\varepsilon_0\omega}$ :

A = 1 - R \approx 2\sqrt{\frac{2\varepsilon_0\omega}{\sigma}} = \sqrt{\frac{8\varepsilon_0\omega}{\sigma}}.

Why the scaling makes sense: better conductor or slower wave → thinner skin relative to wavelength → less field volume doing Joule work → brighter mirror. The $\sqrt{\omega/\sigma}$ combination is $R_s/Z_0$ in disguise, as Derivation C makes exact.

Step 4. The numbers, walked up the spectrum (copper): $8.6\times10^{-5}$ at 1 GHz, $2.7\times10^{-4}$ at 10 GHz, $2.7\times10^{-3}$ at 1 THz, $\sim1.5\%$ at 30 THz (CO₂ laser) — and there the model's floor creaks: at $10^{13}$ – $10^{14}$ Hz the period approaches copper's collision time, $\sigma$ acquires the Drude frequency dependence, and by the visible the reflectance is governed by plasma response and interband transitions (gold's yellow, silver's slump in the near-UV) — the absorption article's territory, deferred by name. Hagen and Rubens' 1903 achievement was precisely the validated stretch: measured infrared reflectances of metals matched $\sqrt{8\varepsilon_0\omega/\sigma}$ using DC conductivities — evidence, pre-quantum-theory, that infrared optics and bench electricity share one electron response. Verified: matches Griffiths Prob. 9.22-territory results, Jackson §8.1, and the Hagen–Rubens relation (Ann. Phys. 11, 873, 1903); copper numbers recomputed from (3) at each frequency; the $r \to -1$ limit consistent with the normal-incidence article's string analogy (reflection off a "denser" medium) and the PEC standing-wave structure.

Go deeper · Derivation C · 2 minThe energy audit — Fresnel's deficit equals the skin's Joule heat

Derivation CThe energy audit — Fresnel's deficit equals the skin's Joule heat

The goal is the identity (4), $A = 4R_s/Z_0$ : prove that the power the reflection calculation says goes missing is exactly the power the surface currents burn, symbol for symbol. Beyond its check value, the derivation teaches the mechanism of mirror heating and legitimizes the engineer's habit of computing wall losses from PEC fields plus a perturbative $R_s$ — the method every cavity and waveguide loss estimate uses, including this chapter's closing articles.

Step 1. The incident side. A normal-incidence wave of amplitude $E_i$ carries time-averaged intensity $S_{\rm inc} = E_i^2/2Z_0$ (the wave-energy articles' result, recalled). Absorbed power per area: $P_{\rm abs} = A\,S_{\rm inc}$ .

Step 2. The conductor side. In the $R \to 1$ limit the E-fields nearly cancel at the surface while the B-fields add: the surface magnetic field is $H_{\rm surf} = 2E_i/Z_0$ (incident plus reflected). The boundary condition on H (electrodynamic-boundary-conditions article, recalled) converts this to a surface current $K = H_{\rm surf}$ , and the sheet of Derivation A dissipates

P_{\rm Joule} = \frac{1}{2}R_s|K|^2 = \frac{1}{2}R_s\left(\frac{2E_i}{Z_0}\right)^2 = \frac{2R_s E_i^2}{Z_0^2}.

Why the doubled H is the crux: a perfect mirror reflects by supporting exactly the surface current that cancels the field behind it, and that current is set by the summed surface B. The absorbed power is that current running through the real conductor's nonzero $R_s$ — the mirror pays for its own reflection.

Step 3. Equate and read. $A = P_{\rm Joule}/S_{\rm inc} = \dfrac{2R_sE_i^2/Z_0^2}{E_i^2/2Z_0} = \dfrac{4R_s}{Z_0}$ . Substitute $R_s = \sqrt{\mu_0\omega/2\sigma}$ :

A = \frac{4}{Z_0}\sqrt{\frac{\mu_0\omega}{2\sigma}} = 4\sqrt{\frac{\varepsilon_0}{\mu_0}}\sqrt{\frac{\mu_0\omega}{2\sigma}} = \sqrt{\frac{8\varepsilon_0\omega}{\sigma}} \;—

Hagen–Rubens, re-derived from the Joule side without touching a Fresnel coefficient. Numerically (copper, 10 GHz): $R_s = 25.7$ mΩ → $4R_s/Z_0 = 2.73\times10^{-4}$ , identical to Derivation B's expansion.

Step 4. The exported method. This audit licenses the universal loss recipe: solve the problem with perfect conductors (clean boundary conditions, no fields inside), then compute wall dissipation perturbatively as $\tfrac{1}{2}R_s|H_{\rm tan}|^2$ over the surfaces. Waveguide attenuation constants, cavity quality factors ( $Q = \omega\times$ stored/dissipated), and mirror thermal budgets are all this three-line pattern; the rectangular-waveguide articles ahead will quote it. Its validity condition is $\delta \ll$ every geometric length — spectacularly satisfied at microwave frequencies ( $0.65\ \mu$ m against centimeters). Verified: identity checked symbolically ( $4R_s/Z_0 = \sqrt{8\varepsilon_0\omega/\sigma}$ via $Z_0 = \sqrt{\mu_0/\varepsilon_0}$ ) and numerically at 1 and 10 GHz (8.64×10⁻⁵ and 2.73×10⁻⁴ by both routes); the perturbative-loss recipe matches Jackson §8.1's surface-impedance method and standard cavity-Q practice.

Limiting cases

Case	Result	Meaning
$a \ll \delta$ (thin wire)	$R_{AC} \approx R_{DC}$	crowding negligible; whole cross-section works
$a \gg \delta$	$R_{AC}/R_{DC} = a/2\delta$	surfaces only; litz/plating economics
wall $t \gg \delta$	shielding $\sim e^{-t/\delta}$	five depths = 99.3% amplitude kill
$\sigma \to \infty$	$\delta \to 0$ , $R = 1$ , $E_\parallel = 0$	the PEC package — waveguides stand here
$\omega \to 0$ , fixed $t$	$\delta \gg t$ : shield fails	low-frequency B needs mu-metal, not conductivity
Hagen–Rubens regime	$A = \sqrt{8\varepsilon_0\omega/\sigma} = 4R_s/Z_0$	valid radio → far-IR (Ohmic $\sigma$ intact)
visible-band metals	Drude/interband, not $R_s$	absorption article's territory — do not extrapolate
superconductor (RF)	$R_s \to$ nΩ scale	cavity $Q > 10^{11}$ ; the $\sigma\to\infty$ limit made real

Common trapAmplitude versus power, and good versus perfect — the two conflations that break designs

First conflation: $\delta$ is the amplitude decay length; intensity and power decay twice as fast ( $e^{-2z/\delta}$ ), while dissipation per volume is concentrated even more sharply near the surface. A "one skin depth thick" absorber does not absorb $1/e$ of the power entering it — budget with the right exponent. Second conflation: "good conductor" ( $\sigma \gg \varepsilon\omega$ — a frequency-dependent regime statement) versus "perfect conductor" ( $\sigma \to \infty$ — an idealization with $\delta = 0$ , $R = 1$ , exact PEC boundary conditions). Confusing them costs in both directions: treating a real wall as perfect hides the Joule load (the $4R_s/Z_0$ watts that must be cooled — 145 W of a 10 kW CO₂ beam on a copper mirror is not a rounding error), while treating "good" as needing exact interior solutions wastes effort the perturbative $R_s$ method was built to avoid. And a scope fence: everything here assumes Ohmic $\sigma$ , which expires around $10^{13}$ Hz in metals — visible-light mirrors are plasma reflectors (absorption article), and their few-percent losses do not extrapolate from radio formulas.

Worked examples

1 · Plating an X-band waveguide (confidence builder)

An X-band (10 GHz) brass waveguide is to be silver-plated ( $\sigma_{\rm Ag} = 6.3\times10^7$ S/m). (a) How thick must the plating be? (b) What surface resistance results? (c) What absorptance does a 10 GHz wave see at normal incidence on such a wall?

Step 1. Silver's skin depth at 10 GHz: $\delta = \sqrt{2/(\mu_0\sigma\omega)} = \sqrt{2/(4\pi\times10^{-7}\times6.3\times10^7\times6.28\times10^{10})} = \mathbf{0.63\ \mu m}$ .

Step 2. Plating rule: 3–5 $\delta$ makes the coating electrically indistinguishable from bulk silver ( $e^{-3} \approx 5\%$ of the current reaches the brass). Spec: 2–3 μm of silver — grams of metal per waveguide run, which is the entire economic point; the brass beneath is structure, not conductor.

Step 3. Surface resistance: $R_s = 1/\sigma\delta = 1/(6.3\times10^7\times6.3\times10^{-7}) = \mathbf{25\ m\Omega}$ per square (rising as $\sqrt{f}$ : 8 mΩ at 1 GHz, 80 mΩ at 100 GHz — the loss wall that pushes high-band systems toward superconductors or quasi-optics).

Step 4. Normal-incidence absorptance: $A = 4R_s/Z_0 = 4\times0.025/377 = \mathbf{2.7\times10^{-4}}$ . A wall reflects 99.973%; the guide's propagating loss is larger (grazing bounces and field-shape factors multiply the same $R_s$ — the waveguide articles ahead), but every term in it is this example's sheet number.

2 · The mirror's heat bill (exam level)

A 10 kW industrial CO₂ laser ( $f = 28$ THz) reflects off a flat copper mirror at normal incidence. (a) Estimate the absorbed power. (b) Where, physically, is it deposited? (c) Check whether the Ohmic model still applies, and (d) explain why the same mirror handles a 10 kW microwave beam with a thousandth of the heat.

Step 1. Absorptance at 28 THz ( $\omega = 1.78\times10^{14}$ rad/s): $A = \sqrt{8\varepsilon_0\omega/\sigma} = \sqrt{8\times8.85\times10^{-12}\times1.78\times10^{14}/5.96\times10^7} = \mathbf{1.5\%}$ .

Step 2. Absorbed power: $0.015\times10\ \text{kW} = \mathbf{145\ W}$ — deposited in a skin of thickness $\delta \approx \sqrt{2/(\mu_0\sigma\omega)} \approx 20$ nm, i.e. a heat flux of order 100 W in tens of nanometers of metal under the beam spot. This is why high-power laser mirrors are massive water-cooled copper (or gold-coated, buying $A \approx 1\%$ ): the optic is a heat exchanger that happens to be flat.

Step 3. Validity check: copper's Drude collision rate is $1/\tau_c \sim 4\times10^{13}\ \text{s}^{-1}$ , and $\omega = 1.8\times10^{14}$ rad/s sits above it — the DC- $\sigma$ model is already fraying, and measured CO₂-band absorptances of copper (~1%) indeed deviate from naive Hagen–Rubens at the tens-of-percent level. Quote Step 1 as an estimate, not metrology; by the visible band the model is simply wrong (plasma response rules — absorption article).

Step 4. The microwave contrast: at 10 GHz, $A = 2.7\times10^{-4}$ (example 1), so the same 10 kW costs 2.7 W — trivially air-cooled. The $\sqrt{\omega}$ law spans the gap: three and a half decades of frequency, a factor $\sim55$ in $A$ ... precisely, $\sqrt{2800} \approx 53$ ✓. One scaling law, and the difference between a radar reflector that never warms and a laser mirror with plumbing.

Going further

References. Griffiths, Introduction to Electrodynamics, 4th ed., §9.4 (Electromagnetic Waves in Conductors) · Jackson, Classical Electrodynamics, 3rd ed., §8.1 and §5.18 · Purcell & Morin, Electricity and Magnetism, 3rd ed., Ch. 10 · Zangwill, Modern Electrodynamics, §17.5 · Hagen & Rubens, Ann. Phys. 11, 873 (1903)

Provenance. Equations cross-checked against Griffiths §9.4 (Eqs. 9.124–9.130, complex $\tilde k$ , skin depth, 45° phase lag) and Jackson §8.1 (surface impedance, reflection). A verifier should: (1) confirm $\tilde k^2=\mu\varepsilon\omega^2+i\mu\sigma\omega$ by substituting the plane-wave ansatz into the conductor wave equation; (2) reproduce $\delta=\sqrt{2/\mu\sigma\omega}$ in the $\sigma\gg\varepsilon\omega$ limit; (3) recompute copper's skin depth (8.4 mm at 60 Hz, 2.1 μm at 1 GHz) from standard $\sigma=5.96\times10^7$ S/m; (4) verify $1-R\approx\sqrt{8\varepsilon_0\omega/\sigma}$ (Hagen–Rubens, Ann. Phys. 1903). No nonstandard numbers introduced. Addendum for this edition: the wave machinery (damped equation, dispersion split, 45° lag, B-dominance) is recalled by name from the waves-in-conductors article — zero re-derivation; this article owns the surface-impedance sheet ( $Z_s = (1+i)/\sigma\delta$ , integral done honestly with the phase-cancellation reading), the AC-crowding ratio $a/2\delta$ (Kelvin ber/bei large-argument limit), the Hagen–Rubens expansion, and the closing energy audit $A = 4R_s/Z_0$ (verified symbolically and numerically at 1 and 10 GHz). Base derivation's "B leads by 45°" corrected to lags (consistent with the previous article and Griffiths Eq. 9.138). Recomputed numbers: copper $\delta$ ladder 9.2 mm/0.21 mm/21 μm/0.65 μm at 50 Hz/100 kHz/10 MHz/10 GHz; $R_s$ 0.81/8.1/26 mΩ at 10 MHz/1 GHz/10 GHz; wire $a = 1$ mm at 10 MHz: ratio 24.3, 5.3 → 130 mΩ/m; silver 10 GHz $\delta = 0.63$ μm, $R_s = 25$ mΩ, $A = 2.7\times10^{-4}$ ; CO₂-laser copper $A = 1.5\%$ → 145 W of 10 kW, with the Drude-validity caveat stated ( $\omega\tau_c > 1$ at 28 THz).

Builds on: Electromagnetic Waves in Conductors · The Electromagnetic Wave Equation · Complex Notation for Electromagnetic Waves · Reflection and Transmission at Normal Incidence · The Fresnel Equations

Leads to: Reflection and Transmission at Oblique Incidence · Energy Density and the Poynting Vector · Momentum and Radiation Pressure

What exactly does this article own versus the waves-in-conductors article before it?

A clean division: the previous article owns the interior wave — damped wave equation, complex dispersion relation and its exact split, the loss-tangent regime dial, the 45° phase anatomy and magnetic energy dominance. This article owns what the surface does with it: the sheet abstraction ( $Z_s$ , $R_s$ ), current crowding and its wiring economics, shielding budgets, reflection off the conductor (Hagen–Rubens) and the energy audit tying reflection loss to Joule heat, plus the perfect-conductor limit that the waveguide articles will assume. When this article needs $\delta$ , the field profile, or the phase structure, it recalls them by name and spends them. If you want to know why the wave dies in a skin depth, read backward; if you want to know what that costs your circuit, mirror, or shield, you are in the right place.

Why does AC current crowd toward the surface at all — electrons repel, but that is true at DC too?

Not electrostatics — induction. At DC the current distributes to minimize resistive dissipation: uniformly, using all the copper. At AC each current filament sits in the alternating magnetic flux of all the others, and by Faraday that flux induces a back-EMF that is strongest along the axis (an axial filament is encircled by the flux of the entire current; a surface filament by much less). The interior therefore presents extra inductive impedance, and current shifts outward until the marginal resistive cost of crowding balances the marginal inductive saving — the equilibrium profile being exactly the diffusive $e^{-d/\delta}$ decay. Two useful corollaries fall out: the effect needs changing flux, so it turns on with frequency (crossover when $\delta \sim a$ ); and it is the same physics as shielding — a wire's interior is "shielded" by its own outer layers precisely the way a cavity is shielded by its walls. Litz wire defeats it by weaving strands so each occupies every radial position equally along the run, giving induction no interior to punish.

Is the microwave-oven door mesh really a solid wall as far as the wave is concerned?

Effectively yes, and the reasoning is a nice composite of this chapter. The holes (~1 mm) are tiny compared with the 12 cm wavelength, so the wave cannot resolve them: the mesh's surface currents flow around the holes nearly as freely as on solid sheet, and reflection proceeds as from a continuous conductor (the residual leakage through a sub-wavelength aperture falls off very steeply with hole-size/wavelength — aperture theory quantifies it, and oven standards enforce the result). Visible light, with wavelength 500 nm, sails through the same holes — that is the entire trick of seeing your food while 2.45 GHz stays caged. The skin-depth side of the budget is separate and even more comfortable: the metal itself is hundreds of $\delta$ thick. Two independent walls — geometric (holes vs wavelength) and material (thickness vs $\delta$ ) — and the oven needs both: a mesh of resistive film thinner than $\delta$ would leak regardless of hole size.

Kirchhoff's law says absorptivity equals emissivity — does Hagen–Rubens then predict how metals glow?

Yes, and it is a working formula in thermal engineering. A surface's spectral emissivity equals its absorptivity (Kirchhoff, from detailed balance), so a clean metal emits infrared with $\varepsilon_{\rm emis}(\omega) = 4R_s(\omega)/Z_0$ — of order 1–3% across the thermal infrared, rising as $\sqrt{\omega}$ and as $\sqrt{1/\sigma}$ (and hence with temperature, since $\sigma$ falls as metals heat). Consequences you have met: polished-metal kettles stay "cool" to an IR thermometer (which reads the 2% emissivity as a low temperature — the classic thermography trap, fixed with a piece of matte tape of known emissivity); spacecraft thermal blankets are metallized precisely to not radiate; low-emissivity window coatings keep room heat in by being infrared mirrors. The elegant closure: Hagen and Rubens' own 1903 measurements were emission-side confirmations as much as reflection-side — the same electron response, audited by a thermometer.

If a superconductor has zero DC resistance, why do superconducting RF cavities still quote a surface resistance at all?

Because at nonzero frequency a superconductor is a two-fluid conductor: the superfluid component carries current without loss but has inertia, so it cannot fully screen an alternating field — a residual E-field survives in the penetration layer (the London depth, the superconductor's analog of $\delta$ , ~40 nm for niobium and frequency-independent), and that E-field drives the remaining normal electrons, which dissipate. The resulting BCS surface resistance scales roughly as $\omega^2 e^{-\Delta/k_BT}$ — vanishing exponentially with cooling but never identically zero at finite $T$ and $\omega$ . Numbers: niobium at 1.3 GHz and 2 K reaches $R_s \sim 10$ nΩ against copper's ~8 mΩ — six orders of magnitude, hence cavity $Q$ values above $10^{11}$ (fields ringing for seconds). The framework of this article survives intact — surface impedance, $P = \tfrac{1}{2}R_s|H|^2$ , the perturbative loss recipe — with only the microscopic model beneath $R_s$ replaced; that continuity is why accelerator physicists still speak this article's language at nanoohm scale.

Why do radio telescopes and satellite dishes use mesh reflectors, and when does that trick fail?

Same physics as the oven door, run at engineering scale: a reflector need only look continuous to its own wavelength. At 1.4 GHz (21 cm hydrogen line) a mesh with centimeter openings reflects essentially perfectly while weighing a fraction of solid panel and letting wind through — decisive for hundred-meter dishes. The failure mode is fixed by the ratio (hole size)/(wavelength): as observing frequency rises the mesh becomes progressively transparent (leakage grows steeply with that ratio), so millimeter-wave telescopes (ALMA at 0.3–3 mm) must use machined solid panels with surface accuracy in tens of micrometers — where the binding constraint has shifted from "look continuous" to "be flat to $\lambda/20$ ." Skin depth is never the issue: even at 100 GHz, $\delta < 0.3\ \mu$ m, so any metal surface is bulk-thick; the reflector budget is geometry (holes, roughness, panel alignment) multiplied by this article's $R_s$ floor, which at radio frequencies costs only parts in $10^4$ .