Electromagnetic Waves in Conductors

Every wave this chapter has propagated so far traveled through media that only store energy — bound charges stretching and springing back, borrowing field energy and returning it each cycle. A conductor refuses the loan terms. Its charges are free: push them and they drift, collide, and dissipate, converting wave energy into heat at a rate fixed by Ohm's law, $\mathbf{J}_f = \sigma\mathbf{E}$ . Feed that current into Maxwell's equations and the wave equation grows a first-derivative term — the unmistakable signature of friction, familiar from every damped oscillator — and the clean traveling wave of vacuum becomes something stranger: a wave whose wavenumber is complex, which advances and dies at the same time. How strange, exactly? Inside copper at microwave frequencies, the "wave" crawls at twenty kilometers per second — ten thousand times slower than light — with a wavelength of eight micrometers where the vacuum wave measured twelve centimeters, and it is extinguished within about one of those micrometers. Its magnetic field, locked in step with the electric field everywhere else in this chapter, now trails it by an eighth of a cycle and hoards essentially all of the energy. This article derives that anatomy honestly: the damped wave equation (with the charge-relaxation argument that legitimizes it), the exact complex dispersion relation and its clean split into propagation and attenuation, the single dimensionless dial — the loss tangent $\sigma/\varepsilon\omega$ — that separates "slightly lossy glass" from "opaque metal," and the 45° phase lag that is the good conductor's fingerprint. The eddy-currents article reached the same skin physics through a diffusion equation; here the wave route arrives at the identical $\delta = \sqrt{2/\mu\sigma\omega}$ , as it must. What that length is worth — surface resistance, shielding budgets, why metals are mirrors — is the next article's business.

Key takeaways

Ohm's law $\mathbf{J}_f = \sigma\mathbf{E}$ adds a damping term to the wave equation: $\nabla^2\mathbf{E} = \mu\varepsilon\,\partial_t^2\mathbf{E} + \mu\sigma\,\partial_t\mathbf{E}$ — legal because free charge relaxes away in $\tau = \varepsilon/\sigma$ (recalled from the continuity article), leaving $\nabla\cdot\mathbf{E} = 0$ .
Plane waves survive with a complex wavenumber: $\tilde{k}^2 = \mu\varepsilon\omega^2 + i\mu\sigma\omega$ ; writing $\tilde{k} = k + i\kappa$ , the field is $E_0\,e^{-\kappa z}\cos(kz - \omega t)$ — advancing and decaying at once.
One dial governs everything: the loss tangent $\sigma/\varepsilon\omega$ (conduction current over displacement current). Small: a barely-lossy dielectric with $\kappa \approx \tfrac{\sigma}{2}\sqrt{\mu/\varepsilon}$ , frequency-independent. Large: a good conductor with $k = \kappa = \sqrt{\mu\sigma\omega/2} = 1/\delta$ .
The good-conductor limit reproduces the eddy-current article's diffusion-route skin depth $\delta = \sqrt{2/\mu\sigma\omega}$ exactly — one physics, two derivations, same length.
Field anatomy in a good conductor: $\mathbf{B}$ lags $\mathbf{E}$ by $\phi \to 45°$ (in general $\tan 2\phi = \sigma/\varepsilon\omega$ ), the wave crawls at $v = \omega\delta$ , the wavelength inside is $2\pi\delta$ , and the energy is almost entirely magnetic: $u_B/u_E = \sqrt{1 + (\sigma/\varepsilon\omega)^2}$ .
KEY IDEA: conduction turns the wave equation into a damped one; the wavenumber goes complex, the wave decays over a skin depth while its magnetic field lags by up to 45° and dominates the energy — with the loss tangent $\sigma/\varepsilon\omega$ deciding how far toward that extreme the medium sits.

The core insight

Ohm's law turns Ampère's law into a damped wave equation; the wavenumber becomes complex, $\tilde{k} = k + i\kappa$ , so plane waves attenuate as $e^{-\kappa z}$ over the skin depth $\delta = 1/\kappa$ , while $\mathbf{B}$ lags $\mathbf{E}$ by up to $45°$ and carries almost all the energy — the loss tangent $\sigma/\varepsilon\omega$ sets where between dielectric and metal the medium sits.

Where this lives today

The damped wave equation is the operating manual for the electromagnetic borderlands between transparent and opaque. RF and microwave engineering budgets every circuit board and cable around loss tangents (substrate makers sell $\tan\delta_{\rm loss} = 10^{-4}$ laminates at premium prices for 5G bands); medical MRI must model the conductive human body ( $\sigma \sim 0.5$ S/m), where at 128 MHz the skin depth approaches body scale and B₁-field shading becomes a real imaging artifact; and through-the-earth radio for mines and geophysical prospecting (magnetotellurics) works the same $\delta \propto 1/\sqrt{\sigma\omega}$ trade as the submarine example — the Earth's crust surveyed by how deep each frequency soaks. At the physics frontier, the loss-tangent dial keeps acquiring new settings. Plasmonics operates metals right at the edge where the good-conductor approximation frays (optical frequencies, Drude regime), trading loss for sub-wavelength field confinement; metamaterial absorbers engineer $\sigma_{\rm eff}(\omega)$ sheets to make "perfect black" at chosen bands (stealth and energy harvesting both); and superconducting RF cavities realize the $\sigma \to \infty$ limit closely enough that niobium resonators store waves for seconds — quality factors above $10^{11}$ — hosting the qubits and accelerating gradients of the current decade. Each is this article's dispersion relation with an exotic $\sigma$ .

Origins · 1 minPaul Drude and how this was discovered

Paul Drude

1863–1906 · Leipzig and Berlin

Drude is the physicist who first made light and matter share one set of books. Trained under Voigt in Göttingen as an optics theorist, he mastered the phenomenology of reflection and absorption in metals — his 1890s ellipsometric technique for measuring optical constants is the direct ancestor of the semiconductor industry's workhorse metrology. Then, within three years of Thomson's electron (1897), Drude built the first microscopic theory of conduction (1900): electrons as a gas of colliding particles, yielding Ohm's law with $\sigma = Nq^2\tau_c/m$ , the frequency-dependent response this article and the absorption article both lean on, and — his era's sensation — the Wiedemann–Franz ratio of thermal to electrical conductivity, correct to the observed order from first principles (by a partial cancellation of two errors quantum mechanics later sorted out, as he could not have known). His textbook Lehrbuch der Optik (1900) and his editorship of Annalen der Physik — where he handled a 1905 submission stream from an unknown patent clerk — made him central to physics' most consequential decade. Weeks after his election to the Prussian Academy in 1906, at the height of his powers, he took his own life at 42; physics lost the man who would have watched quantum mechanics vindicate and repair his electron gas, as Sommerfeld's 1927 revision finally did.

History

1861

— Maxwell's equations carry both conduction and displacement currents from the start

1883

— Horace Lamb solves oscillating fields in conducting spheres; the skin effect appears in theory

1885

— Heaviside develops the skin-effect analysis for telegraph and power conductors

1887

— Hertz bounces his new radio waves off metal sheets: conductors reflect, as the theory demands

1889

— Kelvin's ber and bei functions give the exact AC current profile in round wires

1900

— Drude's electron gas puts a microscopic sigma under the damped wave equation

1903

— Hagen and Rubens link infrared metallic reflectance to DC conductivity — the wave route confirmed by experiment

1918

— Transatlantic VLF stations exploit the seawater and ionosphere skin-depth physics for naval communication

The physical picture

Start with the honest accounting question: in a conductor, Ampère–Maxwell has two currents on its right side — the displacement current $\varepsilon\,\partial_t\mathbf{E}$ , which stores and returns energy, and the conduction current $\sigma\mathbf{E}$ , which burns it. Their ratio,

\frac{|\mathbf{J}_f|}{|\mathbf{J}_d|} = \frac{\sigma}{\varepsilon\omega}, \qquad (1)

where: this dimensionless number — the loss tangent — is the article's only real dial, deciding whether a material at a given frequency behaves as a slightly lossy dielectric or an opaque metal. Read it as a race between two clocks: the field's period $1/\omega$ against the charge-relaxation time $\tau = \varepsilon/\sigma$ (the time free charge takes to rearrange — the same two-clock logic as the quasistatic article's selector). Slow driving, fast charges ( $\omega\tau \ll 1$ ): the conductor keeps up, screens, dissipates — metal behavior. Fast driving, sluggish charges ( $\omega\tau \gg 1$ ): the charges barely move before the field reverses — dielectric behavior with a whiff of loss. The numbers are extreme on both ends: copper at 1 MHz has $\sigma/\varepsilon\omega \sim 10^{12}$ (deep metal), teflon at the same frequency sits near $10^{-4}$ (deep dielectric), and seawater crosses from conductor to dielectric near 900 MHz — one material, both regimes, frequency decides. Why does conduction shield rather than merely dim? Picture the wavefront entering the metal. The E-field drives currents; those currents both dissipate (Joule heat) and — by Faraday and Ampère acting together — generate opposing fields that cancel the intruder deeper in. The self-shielding gets more efficient as frequency rises (faster flux change, stronger induced response), which is why the penetration length shrinks as $1/\sqrt{\omega}$ : the same skin physics the eddy-current article derived by turning Maxwell into a diffusion equation. This article reaches the identical length by the wave route — keeping the displacement current, solving the full dispersion relation, then taking the good-conductor limit — and the agreement of the two routes is a required checkpoint, not a coincidence. The most instructive habit for this article: watch what happens to the pair $(k, \kappa)$ . In vacuum, $\kappa = 0$ : pure travel. In the good conductor, $k = \kappa$ exactly — equal parts advance and decay, one radian of phase per $e$ -fold of amplitude, a wave that dies within a wavelength. That locked equality (the 45° polar angle of $\tilde{k}$ ) is the fingerprint of physics on the boundary between wave propagation and diffusion, and every strange number in the overview — the crawling phase velocity, the micrometer wavelength, the lagging magnetic field — is downstream of it.

Figure

The wave inside a good conductor, computed — E and B envelopes decaying over the skin depth with B trailing by an eighth of a cycle

Figure

The loss-tangent dial — where real materials sit between transparent dielectric and opaque metal, and how frequency slides them along it

The damped wave equation and its dispersion relation

In a linear ohmic medium ( $\mathbf{D} = \varepsilon\mathbf{E}$ , $\mathbf{B} = \mu\mathbf{H}$ , $\mathbf{J}_f = \sigma\mathbf{E}$ ), once the free charge density has relaxed away (Derivation A's first honest step), Maxwell's curls close into

\nabla^2\mathbf{E} = \mu\varepsilon\,\frac{\partial^2\mathbf{E}}{\partial t^2} + \mu\sigma\,\frac{\partial\mathbf{E}}{\partial t}, \qquad (2)

where: the second-derivative term is the familiar wave restoring machinery and the new first-derivative term is friction, exactly as in the damped oscillator (the mechanics analogy is precise: same equation, space swapped in). Drop $\sigma$ and the linear-media wave equation returns; drop $\varepsilon\,\partial_t^2$ instead and you have the eddy-current article's diffusion equation — equation (2) is the parent of both. Feeding it a plane wave $e^{i(\tilde{k}z - \omega t)}$ gives the complex dispersion relation

\tilde{k}^2 = \mu\varepsilon\omega^2 + i\,\mu\sigma\omega, \qquad (3)

where: the real term is the dielectric's $k^2$ and the imaginary term is conduction's contribution — notice it enters at first order in $\omega$ , which is why loss dominates at low frequency and becomes negligible at high (at fixed $\sigma$ ): the loss tangent $\sigma/\varepsilon\omega$ is just the ratio of (3)'s two terms. Splitting $\tilde{k} = k + i\kappa$ (Derivation B) yields the exact pair

k = \omega\sqrt{\frac{\mu\varepsilon}{2}}\left[\sqrt{1 + \left(\frac{\sigma}{\varepsilon\omega}\right)^2} + 1\right]^{1/2}, \qquad \kappa = \omega\sqrt{\frac{\mu\varepsilon}{2}}\left[\sqrt{1 + \left(\frac{\sigma}{\varepsilon\omega}\right)^2} - 1\right]^{1/2}, \qquad (4)

where: $k$ sets the wavelength ( $\lambda = 2\pi/k$ ) and phase velocity ( $v = \omega/k$ ) inside the conductor, while $\kappa$ is the attenuation constant: the traveling field is

\mathbf{E}(z,t) = E_0\,e^{-\kappa z}\,e^{i(kz - \omega t)}\,\hat{\mathbf{x}}, \qquad \delta \equiv \frac{1}{\kappa}, \qquad (5)

where: $\delta$ — the skin depth — is where the amplitude has fallen to $1/e$ (the intensity to $1/e^2$ ; the absorption article's $\alpha = 2\kappa$ bookkeeping applies unchanged). In the good-conductor limit both brackets in (4) collapse onto the same value and

k = \kappa = \sqrt{\frac{\mu\sigma\omega}{2}} = \frac{1}{\delta}, \qquad \delta = \sqrt{\frac{2}{\mu\sigma\omega}} \qquad (\sigma \gg \varepsilon\omega), \qquad (6)

where: this is the eddy-current article's diffusion-route skin depth, recovered from the opposite direction — there, by dropping the displacement current at the start; here, by keeping everything and taking the limit at the end. Copper: 9.2 mm at 50 Hz, 65 μm at 1 MHz, 1.3 μm at 2.45 GHz — and since $k = 1/\delta$ , the phase velocity is $v = \omega\delta$ and the internal wavelength $2\pi\delta$ : at 2.45 GHz the wave in copper crawls at 20 km/s with an 8.3 μm wavelength. A wave that loses an $e$ -fold per radian barely deserves the name; it is half wave, half diffusive collapse, and the boundary between those descriptions is exactly the $k = \kappa$ lock.

Derivation AFrom Maxwell to the damped wave equation — with the charge-relaxation step done honestly

The goal is equation (2), and the honest part is the step most treatments mumble through: why may we set $\nabla\cdot\mathbf{E} = 0$ inside a conductor that is, after all, full of charge? The strategy: first show any initial free charge density destroys itself exponentially fast (the relaxation argument, recalled from the continuity-equation article and re-run here in one line because this article spends it), then run the standard curl-of-curl maneuver with both currents aboard.

Step 1. Free charge relaxes. Combine charge conservation $\partial_t\rho_f = -\nabla\cdot\mathbf{J}_f$ , Ohm's law $\mathbf{J}_f = \sigma\mathbf{E}$ , and Gauss's law $\nabla\cdot\mathbf{E} = \rho_f/\varepsilon$ :

\frac{\partial\rho_f}{\partial t} = -\frac{\sigma}{\varepsilon}\,\rho_f \quad\Longrightarrow\quad \rho_f(t) = \rho_f(0)\,e^{-t/\tau}, \qquad \tau = \frac{\varepsilon}{\sigma}.

For copper $\tau \approx 10^{-19}$ s — any interior charge is gone before a single optical cycle elapses, so for any oscillation the conductor can actually follow, $\rho_f = 0$ and $\nabla\cdot\mathbf{E} = 0$ in the bulk (charge lives on surfaces, as the electrostatics of conductors established). Why this number should not be taken literally even though its conclusion stands: at timescales below the collision time (~ $10^{-14}$ s in copper) Ohm's law itself fails — inertia matters and the response is the Drude dynamics of the absorption article. The honest statement: $\tau$ computed this way is a bound so absurdly small that the conclusion $\rho_f = 0$ survives every correction; the continuity-equation article carries the fuller discussion, recalled by name.

Step 2. Curl of Faraday, exactly as in the vacuum derivation (the wave-equation article's SOP): $\nabla\times(\nabla\times\mathbf{E}) = -\partial_t(\nabla\times\mathbf{B})$ . Expand the left side with the E0 identity and Step 1's $\nabla\cdot\mathbf{E} = 0$ : it becomes $-\nabla^2\mathbf{E}$ .

Step 3. Feed the right side the full Ampère–Maxwell law with both currents, $\nabla\times\mathbf{B} = \mu\sigma\mathbf{E} + \mu\varepsilon\,\partial_t\mathbf{E}$ :

-\nabla^2\mathbf{E} = -\mu\sigma\,\frac{\partial\mathbf{E}}{\partial t} - \mu\varepsilon\,\frac{\partial^2\mathbf{E}}{\partial t^2},

which rearranges to (2); the identical maneuver starting from the curl of Ampère–Maxwell gives the same equation for $\mathbf{B}$ — both fields obey one damped wave equation. Why the damping term's structure was inevitable: Ohm's law is odd in time (a velocity response), so it must enter with one time derivative — the same reason friction enters the oscillator at first order, and the reason the equation is no longer invariant under $t \to -t$ : dissipation has an arrow. *Verified: reproduces Griffiths §9.4.1 (Eqs. 9.121–9.124) including the relaxation prelude and its footnoted caveat; the $\tau = \varepsilon/\sigma$ result matches the continuity-equation article's derivation (recalled, one-line re-run); dropping $\sigma$ recovers the linear-media wave equation; dropping $\mu\varepsilon\,\partial_t^2$ recovers the eddy-current article's diffusion equation.*

Go deeper · Derivation B · 2 minSplitting the complex wavenumber — exact forms, the loss-tangent dial, and the two limits

Derivation BSplitting the complex wavenumber — exact forms, the loss-tangent dial, and the two limits

The goal is the exact pair (4) from the dispersion relation (3), and then the two limiting regimes read off with their physical verdicts — including the required handshake with the eddy-current article in the good-conductor limit. The strategy is elementary algebra with the complex square root done carefully; the sign choices at the end are physics, not convention.

Step 1. Write $\tilde{k} = k + i\kappa$ with $k, \kappa$ real and square:

(k^2 - \kappa^2) + 2ik\kappa = \mu\varepsilon\omega^2 + i\mu\sigma\omega.

Match real and imaginary parts: $k^2 - \kappa^2 = \mu\varepsilon\omega^2$ and $2k\kappa = \mu\sigma\omega$ .

Step 2. Close with the modulus. $|\tilde{k}|^2 = k^2 + \kappa^2 = \sqrt{(k^2-\kappa^2)^2 + (2k\kappa)^2} = \mu\varepsilon\omega^2\sqrt{1 + (\sigma/\varepsilon\omega)^2}$ . Adding and subtracting the real-part equation gives (4) directly. Why we choose $k > 0$ and $\kappa > 0$ together: $2k\kappa = \mu\sigma\omega > 0$ forces their signs equal, and a wave propagating toward $+z$ must decay toward $+z$ (the growing root describes a wave fed from infinity — inadmissible here, admissible and important in gain media). Note the loss tangent emerging naturally as the only combination the brackets contain.

Step 3. Poor-conductor limit ( $\sigma \ll \varepsilon\omega$ ). Expand the inner square root to first order: the bracket for $k$ gives $k \approx \omega\sqrt{\mu\varepsilon}$ — propagation essentially unmodified — and the bracket for $\kappa$ gives

\kappa \approx \frac{\sigma}{2}\sqrt{\frac{\mu}{\varepsilon}}.

Why this limit is kind to communications: $\kappa$ is frequency-independent, so a lossy-but-dielectric medium (dry earth, glass, body tissue at some bands) attenuates all frequencies of a signal equally — absorption without distortion. The attenuation length $2\sqrt{\varepsilon/\mu}/\sigma$ can be enormous: this is the regime of "slightly tinted, effectively transparent."

Step 4. Good-conductor limit ( $\sigma \gg \varepsilon\omega$ ). The 1's are negligible against $\sigma/\varepsilon\omega$ , both brackets collapse to $\sigma/\varepsilon\omega$ , and

k = \kappa = \sqrt{\frac{\mu\sigma\omega}{2}} = \frac{1}{\delta}.

The handshake: this is exactly the $\delta$ of the eddy-current article's diffusion equation — as it must be, since Step 4's inequality is precisely the license to drop the displacement current that the diffusion route claimed at the outset. Two independent derivations, one length; if your good-conductor answer ever disagrees with the diffusion-route answer, an algebra error exists. Why $k = \kappa$ deserves its "half wave, half diffusion" billing: equal real and imaginary parts mean the phase advances one radian per $e$ -fold of decay — the wave cannot complete even one full wavelength (which needs $2\pi$ radians, i.e. amplitude reduction $e^{-2\pi} \approx 1/535$ ) before it is effectively gone.

Step 5. Where materials sit on the dial. Copper: crossover $\sigma/\varepsilon\omega = 1$ would require $\omega \sim \sigma/\varepsilon_0 \approx 7\times10^{18}$ rad/s — X-rays — so copper is "good" across the entire radio–microwave–infrared range (with the honest caveat that Ohmic $\sigma$ itself fails first, at the ~ $10^{13}$ Hz Drude rolloff; the FAQ carries it). Seawater ( $\sigma = 4$ S/m, $\varepsilon_r = 80$ ): crossover at 900 MHz — a good conductor for submarine VLF (worked example 2), an absorbing dielectric for radar and 5G. Teflon, glass: loss tangents $10^{-4}$ – $10^{-2}$ across RF — the "poor conductor" formulas rule microwave circuit design. Verified: (4) matches Griffiths Eqs. 9.125–9.126 and Jackson §7.5; limits match Griffiths Eq. 9.130 (good) and the standard lossy-dielectric form (poor); copper numbers recomputed: $\delta$ = 9.2 mm (50 Hz), 65 μm (1 MHz), 2.1 μm (1 GHz), 1.3 μm (2.45 GHz) with $\sigma = 5.96\times10^7$ S/m; seawater crossover $f = \sigma/2\pi\varepsilon = 899$ MHz ✓.

Go deeper · Derivation C · 2 minThe field anatomy — B lags E by up to 45 degrees and takes over the energy budget

Derivation CThe field anatomy — B lags E by up to 45 degrees and takes over the energy budget

The goal is everything downstream of the complex wavenumber's phase: the B–E lag $\phi$ with its $\tan 2\phi = \sigma/\varepsilon\omega$ law, the modified amplitude ratio, and the demonstration that a good conductor's wave energy is almost entirely magnetic. The strategy: build $\mathbf{B}$ from Faraday's law exactly as the vacuum completion routine does — the routine survives; only the number multiplying the field has gone complex, and every phenomenon here is that complex number expressing itself.

Step 1. Faraday builds the partner. For $\tilde{\mathbf{E}} = \tilde{E}_0\,e^{i(\tilde{k}z - \omega t)}\hat{\mathbf{x}}$ , the curl gives $\partial_t\tilde{\mathbf{B}} = -\partial_z\tilde{E}\,\hat{\mathbf{y}}$ , so

\tilde{\mathbf{B}} = \frac{\tilde{k}}{\omega}\,\hat{\mathbf{z}}\times\tilde{\mathbf{E}}.

Structurally identical to vacuum's $\mathbf{B} = (k/\omega)\hat{\mathbf{z}}\times\mathbf{E}$ — transverse, mutually perpendicular, right-handed triad intact — but the proportionality is now complex.

Step 2. Put $\tilde{k}$ in polar form: $\tilde{k} = Ke^{i\phi}$ with $K = \sqrt{k^2 + \kappa^2}$ and $\tan\phi = \kappa/k$ . Then the real fields are

E = E_0\,e^{-\kappa z}\cos(kz - \omega t), \qquad B = \frac{K}{\omega}E_0\,e^{-\kappa z}\cos(kz - \omega t + \phi):

$\mathbf{B}$ reaches each maximum a phase $\phi$ after $\mathbf{E}$ . From Step 1's Cartesian split ( $k^2 - \kappa^2 = \mu\varepsilon\omega^2$ , $2k\kappa = \mu\sigma\omega$ ), the double-angle identity gives

\tan 2\phi = \frac{2\,(\kappa/k)}{1 - (\kappa/k)^2} = \frac{2k\kappa}{k^2 - \kappa^2} = \frac{\sigma}{\varepsilon\omega}.

Why the lag saturates at exactly 45°: in the good-conductor limit $k = \kappa$ makes $\tilde{k}$ 's polar angle exactly $\pi/4$ — the same locked equality as the dispersion split, now read as a phase. Physically: deep in the conductor the B-field is sourced almost entirely by the conduction current, which is in phase with E; but building B from a current costs a quarter cycle of Faraday–Ampère bookkeeping shared between space and time decay — the compromise lands at the eighth-cycle.

Step 3. Amplitude ratio. $B_0/E_0 = K/\omega = \sqrt{\mu\varepsilon}\left[1 + (\sigma/\varepsilon\omega)^2\right]^{1/4}$ — larger than the dielectric value $\sqrt{\mu\varepsilon}$ , and dramatically so in metals: for copper at 2.45 GHz, $B_0/E_0 = \sqrt{\mu\sigma/\omega} \approx 7.0\times10^{-5}$ s/m — twenty thousand times the vacuum ratio $1/c$ . Using $B_0 = E_0/c$ inside a conductor is not a small error; it is four orders of magnitude.

Step 4. The energy budget. Time-averaged densities: $\langle u_E\rangle = \tfrac{1}{4}\varepsilon E_0^2 e^{-2\kappa z}$ and $\langle u_B\rangle = \tfrac{1}{4}B_0^2 e^{-2\kappa z}/\mu$ . Their ratio:

\frac{\langle u_B\rangle}{\langle u_E\rangle} = \frac{K^2}{\mu\varepsilon\omega^2} = \sqrt{1 + \left(\frac{\sigma}{\varepsilon\omega}\right)^2} \;\xrightarrow{\ \text{good}\ }\; \frac{\sigma}{\varepsilon\omega} \gg 1.

For copper at 2.45 GHz that is $\sim 4\times10^8$ : the wave inside a metal is, energetically, a magnetic disturbance with a trace electric passenger. Why this makes physical sense: the conductor's whole job is to kill E (free charges short it out) while currents — which conductors are happy to carry — sustain B; the surviving wave is the B-heavy compromise. This is also why magnetic shielding at low frequency is so much harder than electric shielding: the field a conductor is worst at expelling is exactly the one the wave inside it favors. *Verified: matches Griffiths §9.4.1 (Eqs. 9.133–9.138: polar form, $\tan 2\phi$ ... his modulus $K$ notation) and Jackson §5.18/§8.1 phase structure; dielectric limit audit — $\sigma \to 0$ gives $\phi \to 0$ , $B_0/E_0 \to \sqrt{\mu\varepsilon}$ , $u_B/u_E \to 1$ , recovering the linear-media triad exactly; copper ratio $\sqrt{\mu_0\sigma/\omega}$ at 2.45 GHz recomputed = $7.0\times10^{-5}$ s/m.*

Limiting cases

Case	Result	Meaning
$\sigma \to 0$	$\kappa \to 0$ , $\phi \to 0$ , $u_B = u_E$	linear-media wave recovered exactly
$\sigma \ll \varepsilon\omega$	$\kappa \approx \tfrac{\sigma}{2}\sqrt{\mu/\varepsilon}$ , frequency-flat	lossy dielectric: dims without distorting
$\sigma \gg \varepsilon\omega$	$k = \kappa = 1/\delta$ , $\phi = 45°$	the good conductor: dies in a wavelength
$\sigma \to \infty$	$\delta \to 0$ , fields expelled	perfect conductor — the PEC boundary package (0563)
copper, any RF	loss tangent $10^9$ – $10^{16}$	"good" until Ohm's law itself fails (Drude, ~ $10^{13}$ Hz)
seawater at 10 kHz	loss tangent $9\times10^4$ , $\delta = 2.5$ m	VLF submarine window — worked example 2
seawater at 5 GHz	loss tangent $< 1$	absorbing dielectric: radar sees the surface only
good conductor, $v = \omega\delta$	20 km/s in Cu at 2.45 GHz	wave crawls; $\lambda_{\rm inside} = 2\pi\delta$

Common trapChoose the decaying root, and never carry vacuum field ratios into a conductor

Two failure modes account for most wrong answers in this subject. First, the root choice: $\tilde{k}$ is the square root of a complex number and has two values; the physical one has $k$ and $\kappa$ both positive (decay along propagation). Sign slips typically enter through the time convention — this track writes $e^{-i\omega t}$ , under which absorption means $\mathrm{Im}\,\tilde{k} > 0$ ; engineering texts using $e^{+j\omega t}$ flip every imaginary part, and splicing formulas across conventions silently builds exponentially growing waves. Second, the vacuum reflexes: inside a conductor $B_0 \neq E_0/c$ (off by $10^4$ for copper at microwave frequencies), $\mathbf{B}$ is not in phase with $\mathbf{E}$ (45° behind in the good limit), the energy is not split evenly (magnetic by a factor $\sigma/\varepsilon\omega$ ), and the wavelength inside is not $\lambda_0/n$ in any useful sense — it is $2\pi\delta$ , micrometers where vacuum has centimeters. Every one of these is the complex $\tilde{k}$ asserting itself; check which regime the loss tangent puts you in before reaching for any formula with a $c$ in it.

Worked examples

1 · Copper at the microwave-oven frequency (confidence builder)

For copper ( $\sigma = 5.96\times10^7$ S/m, $\mu \approx \mu_0$ ) at $f = 2.45$ GHz, find the loss tangent, skin depth, internal wavelength and phase velocity, the B–E phase lag, and the magnetic-to-electric energy ratio.

Step 1. Loss tangent: $\sigma/\varepsilon_0\omega = 5.96\times10^7/(8.85\times10^{-12}\times1.54\times10^{10}) = \mathbf{4.4\times10^8}$ — nine orders into the good-conductor regime; use the collapsed formulas with confidence.

Step 2. Skin depth: $\delta = \sqrt{2/(\mu_0\sigma\omega)} = \sqrt{2/(4\pi\times10^{-7}\times5.96\times10^7\times1.54\times10^{10})} = \mathbf{1.3\ \mu m}$ . An oven wall is "infinitely thick" thousands of times over — the mesh in the door works the same way, since holes far smaller than the 12 cm vacuum wavelength barely perturb the surface currents.

Step 3. Internal wave: $k = 1/\delta$ gives $\lambda_{\rm inside} = 2\pi\delta = \mathbf{8.3\ \mu m}$ (vacuum: 12.2 cm — a factor of 15,000) and $v = \omega\delta = \mathbf{20\ km/s}$ , slower than light by the same factor. The "wave" in the metal is a micrometer-scale, crawling, dying ripple.

Step 4. Anatomy: good-conductor limit, so $\phi = \mathbf{45°}$ exactly (to a part in $10^8$ ), and $\langle u_B\rangle/\langle u_E\rangle = \sigma/\varepsilon_0\omega = \mathbf{4.4\times10^8}$ : the energy that penetrates is magnetic for every practical purpose. Amplitude ratio $B_0/E_0 = \sqrt{\mu_0\sigma/\omega} = 7.0\times10^{-5}$ s/m — 21,000 × the vacuum $1/c$ .

2 · Talking to submarines: seawater as a conductor (exam level)

Seawater has $\sigma \approx 4$ S/m, $\varepsilon_r \approx 80$ . (a) Show that at 10 kHz it is a good conductor and find $\delta$ . (b) How deep can a submarine sit and still receive a signal attenuated by no more than a factor of 1000 in amplitude? (c) Why do navies also run 76 Hz systems? (d) Why is ordinary radio useless underwater?

Step 1. Regime check at 10 kHz: $\sigma/\varepsilon\omega = 4/(80\times8.85\times10^{-12}\times6.28\times10^4) = \mathbf{9\times10^4}$ — solidly good-conductor (the crossover sits at $f = \sigma/(2\pi\varepsilon_r\varepsilon_0) = 899$ MHz, so all of radio below ~1 GHz sees seawater as metal).

Step 2. Skin depth: $\delta = \sqrt{2/(\mu_0\sigma\omega)} = \sqrt{2/(4\pi\times10^{-7}\times4\times6.28\times10^4)} = \mathbf{2.5\ m}$ .

Step 3. Depth budget: amplitude factor $e^{-z/\delta} \geq 10^{-3}$ requires $z \leq \delta\ln 1000 = 2.5\times6.91 = \mathbf{17\ m}$ . VLF (3–30 kHz) reaches shallow-cruising submarines — at bits per second, since the carrier itself is only tens of kHz.

Step 4. The ELF option: at 76 Hz, $\delta = \sqrt{2/(4\pi\times10^{-7}\times4\times477)} = \mathbf{29\ m}$ , and the same 1000× budget reaches $\sim$ 200 m — genuine operating depth. The price is scaling in the other direction: antennas kilometers long, powers of megawatts, data rates of characters per minute — enough to send "come up for the real message." The $\delta \propto 1/\sqrt{\omega}$ law dictates the whole naval communications architecture.

Step 5. Ordinary radio (say 100 MHz): $\delta = \sqrt{2/(4\pi\times10^{-7}\times4\times6.28\times10^8)} = \mathbf{2.5\ cm}$ . Three wavelengths of seawater... rather, three skin depths — 8 cm — and an FM signal is down 20 dB; a meter down, nothing measurable remains. The ocean is opaque to communications and radar alike, which is why submarines are hunted with sonar (sound, to which water is transparent) and why blue-green lasers (a rare optical window of low $\alpha$ in water — the absorption article's $\kappa(\omega)$ , not this article's $\sigma$ ) are the only electromagnetic channel to depth.

Going further

References. Griffiths, Introduction to Electrodynamics, 4th ed., §9.4 · Jackson, Classical Electrodynamics, 3rd ed., §7.5 and §8.1 · Zangwill, Modern Electrodynamics, §17.4 · Purcell & Morin, Electricity and Magnetism, 3rd ed., Ch. 9

Provenance. Equations match Griffiths §9.4 (Eqs. 9.124–9.132): dispersion relation $\tilde k^2=\mu\epsilon\omega^2+i\mu\sigma\omega$ , the explicit $k,\kappa$ formulas, skin depth $\delta=1/\kappa$ and good-conductor limit, and the $45^\circ$ phase lag. Cross-check with Jackson §7.5 (lossy media, complex index) and §8.1 (skin depth, surface impedance), and Zangwill §17.4. Numerical skin depths: copper $\sigma=5.96\times10^7\,\mathrm{S/m}$ , $\mu\approx\mu_0$ ; verify $\delta=\sqrt{2/(\mu_0\sigma\omega)}$ gives $\approx8.5$ mm at 60 Hz and $\approx2.1\,\mu$ m at 1 GHz. A verifier should recompute the real/imaginary split of $\tilde k^2$ and the limiting forms. Addendum for this edition: charge-relaxation step done honestly with the Ohm's-law-validity caveat (continuity-equation article recalled by name); good-conductor limit explicitly handshaken against the eddy-current article's diffusion-route $\delta$ (identical, as required); $\delta$ -payoff engineering and conductor reflection deferred by name to the next article. Worked numbers recomputed: copper loss tangent $4.4\times10^8$ and $\delta = 1.3$ μm at 2.45 GHz, $\lambda_{\rm inside} = 2\pi\delta = 8.3$ μm, $v = \omega\delta = 20$ km/s, $B_0/E_0 = 7.0\times10^{-5}$ s/m, $u_B/u_E = 4.4\times10^8$ ; copper $\delta$ = 9.2 mm (50 Hz) / 65 μm (1 MHz); seawater loss tangent $9\times10^4$ at 10 kHz, crossover 899 MHz, $\delta$ = 2.5 m (10 kHz) / 29 m (76 Hz) / 2.5 cm (100 MHz), 1000× depth budgets 17 m and 200 m; figure curves computed from $e^{-z/\delta}\cos(z/\delta)$ and the 45°-lagged partner.

Builds on: The Electromagnetic Wave Equation · Complex Notation for Electromagnetic Waves · Monochromatic Plane Waves in Vacuum · Electromagnetic Waves in Linear Dielectric Media

Leads to: Reflection and Transmission at Normal Incidence · The Fresnel Equations · Energy Density and the Poynting Vector

How is this article's sigma-based treatment related to the absorption article's complex permittivity — two different theories?

One theory, two bookkeeping conventions, with a dictionary. Move the conduction current inside an effective permittivity: Ampère–Maxwell reads $\nabla\times\mathbf{B} = \mu(\sigma - i\varepsilon\omega)\partial_t\mathbf{E}/(-i\omega)$ , i.e. everything behaves as a dielectric with $\tilde{\varepsilon}_{\rm eff} = \varepsilon + i\sigma/\omega$ . The absorption article's $\mathrm{Im}\,\tilde{\varepsilon}$ and this article's $\sigma$ are the same physical dissipation filed under different columns; the loss tangent is $\mathrm{Im}\,\tilde{\varepsilon}/\mathrm{Re}\,\tilde{\varepsilon}$ in that language. Which column to use is a matter of what varies: for free carriers at radio frequencies, a frequency-independent $\sigma$ is the natural constant; near resonances or at optical frequencies, the full $\tilde{\varepsilon}(\omega)$ is. The deep connection: the Drude model (absorption article, Derivation C) derives $\sigma(\omega) = \sigma_0/(1 - i\omega\tau_c)$ , whose low-frequency limit is this article's constant $\sigma$ and whose high-frequency limit is the plasma response — the two articles are the two ends of one Drude curve.

Is the skin depth here the same skin depth as in the eddy-currents article?

The same length, derived from opposite ends, and the agreement is a consistency theorem rather than luck. The eddy-current article starts by dropping the displacement current (quasistatic license), turning Maxwell into a magnetic diffusion equation whose oscillatory solutions decay over $\sqrt{2/\mu\sigma\omega}$ . This article keeps everything, derives the exact $k(\omega), \kappa(\omega)$ pair, and only then takes $\sigma \gg \varepsilon\omega$ — which is precisely the condition under which the displacement current was negligible all along. Same inequality, same approximation, opposite order of operations. The two routes also split the intuition usefully: the diffusion picture explains why fields cannot rush into a conductor (flux takes time to soak), while the wave picture explains the anatomy of what does get in (phase crawl, 45° lag, magnetic energy dominance). When the limit does not hold — seawater at GHz, plasmas near $\omega_p$ — only this article's exact forms survive.

The relaxation time epsilon over sigma for copper is 1e-19 seconds — shorter than any collision time. Is the argument broken?

The number is unphysical; the conclusion is bulletproof — worth keeping both halves straight. Deriving $\tau = \varepsilon/\sigma$ assumed Ohm's law holds instantaneously, but $\sigma$ is itself a collision-averaged response valid only on timescales longer than the collision time $\tau_c \sim 10^{-14}$ s; at $10^{-19}$ s electron inertia rules and the honest dynamics is the Drude equation, under which initial charge imbalances oscillate (at the plasma frequency) while decaying on the $\tau_c$ scale. So the true relaxation is "plasma oscillations damped in tens of femtoseconds" rather than "exponential decay in $10^{-19}$ s." For this article's purposes the repair changes nothing: at any frequency where treating $\sigma$ as constant is legitimate in the first place (radio through far-IR), free charge in the bulk is dead on arrival and $\nabla\cdot\mathbf{E} = 0$ stands. The continuity-equation article carries the fuller relaxation discussion; the lesson worth exporting is the meta-one — always check whether a derived timescale is inside the validity window of the constitutive law that produced it.

If the phase velocity inside copper is 20 km/s, does a signal really take that long to cross a wire?

No — and untangling this rescues a common confusion about how circuits work. The 20 km/s is the phase speed of the evanescent ripple inside the metal, which is dead within microns anyway. Signals on a cable travel in the fields outside and between the conductors — in the dielectric, at $c/\sqrt{\varepsilon_r}$ , typically two-thirds of $c$ — with the conductors acting as guiding rails. The current in the wire is the edge of that external wave, flowing in the skin layer; energy flows in the field and enters the metal only to pay the resistive toll (the Poynting-vector-in-static-fields article made exactly this audit for DC). This division of labor is why the guided-waves articles closing this chapter treat the conductor boundaries as constraints and the dielectric as the highway, and why "electrons drift at millimeters per second, signals travel at two-thirds light speed, and the metal-internal wave crawls at kilometers per second" are three different statements about three different things, all simultaneously true.

Why do good conductors reflect waves instead of absorbing them?

Impedance mismatch, catastrophically large. The wave impedance of a medium is $E/H$ ; for a good conductor it is $\tilde{Z} = \sqrt{\mu\omega/\sigma}\,e^{-i\pi/4}$ — tiny (milliohms for copper at GHz, against vacuum's 377 Ω) and complex. A wave arriving from vacuum meets a boundary that can support large currents with almost no tangential E, so the boundary conditions force a reflected wave that nearly cancels the incident E at the surface: $r \to -1$ . Only the small residue — of order $\sqrt{\varepsilon_0\omega/\sigma}$ , a fraction of a percent — enters and is Joule-absorbed within a few $\delta$ . So "shiny" and "lossy" are not in tension: metals reflect because they are so effective at short-circuiting E-fields, and the little they fail to reflect they eat. The quantitative version — reflectance formulas, the Hagen–Rubens law, and where the absorbed watts go — is the next article's opening business, deferred by name.

Does any of this apply to metals at optical frequencies — is a mirror just the good-conductor limit at 500 THz?

Only after upgrading the constitutive law. At 500 THz the period ( $2$ fs) is shorter than copper's collision time ( $\sim$ 25 fs), so the DC conductivity picture — electrons reaching a drift steady state within each cycle — fails; electron inertia dominates and the response is the collisionless Drude/plasma form $\tilde{\varepsilon} = \varepsilon_0(1 - \omega_p^2/\omega^2)$ of the absorption article. The phenomenology still looks similar (strong reflection, evanescent interior field) but the mechanism differs: below $\omega_p$ the field is excluded by inertial plasma response (a reactive, nearly lossless expulsion — polar angle of $\tilde{k}$ near 90°, not 45°) rather than by resistive screening. Practical consequences: visible-band mirror reflectance is set by $\omega_p$ and interband transitions (why gold is yellow), not by $\sigma_{\rm DC}$ ; and the crossover regime between Ohmic and plasma behavior — the terahertz-to-infrared band — is exactly where the Hagen–Rubens relation stitches the two descriptions together. Rule: radio and microwaves, use this article; visible and UV, use the absorption article; between, check $\omega\tau_c$ .

Which field should a shielding enclosure be designed against — and why is low-frequency magnetic shielding notoriously hard?

Against whichever field survives inside a conductor, and this article says that is the magnetic one. Electric shielding is nearly free: any conductor's surface charges terminate external E-fields essentially instantly (the Faraday-cage physics of electrostatics), even for thin foil, even at DC. Magnetic shielding by conduction relies on induced eddy currents opposing the intruding flux — effective only when the wall is thick compared to $\delta$ . At 2.45 GHz, $\delta = 1.3$ μm: kitchen foil is a fortress. At 50–60 Hz, $\delta$ in copper is 9 mm, and worse, the shielding mechanism turns off as $\omega \to 0$ (no flux change, no eddy currents): conductive enclosures are nearly transparent to power-line and DC magnetic fields. That is why magnetically quiet rooms (MEG labs, electron-microscope suites) use high-permeability mu-metal — which works by rerouting flux through itself (the magnetic-circuits physics of the E5 articles) rather than by opposing it — often layered with thick aluminum to catch the higher-frequency remainder by the eddy mechanism. The design rule in one line: E-fields die at any conducting skin; B-fields die only over $\delta$ , so count your skin depths at the lowest frequency you must stop.