Waves in Conductors and Skin Depth

Derivation

Waves in Conductors and Skin Depth

D-173 Home PU-204 Threads waves · matter · fields Depends on Electromagnetic Wave Equation in Vacuum, Structure of Monochromatic Plane Waves

Statement

For a monochromatic plane wave of angular frequency \( \omega \) propagating in a linear, homogeneous, ohmic medium (permittivity \( \varepsilon \), permeability \( \mu \), conductivity \( \sigma \), with \( \vec{J}_f = \sigma \vec{E} \)), Maxwell's equations require the wavevector to be complex, \( \tilde{k} = k + i\kappa \), with \( \tilde{k}^2 = \mu\varepsilon\omega^2 + i\,\mu\sigma\omega \). The fields decay as \( e^{-\kappa z} \) while oscillating as \( e^{i(kz-\omega t)} \). In the good-conductor regime \( \sigma \gg \varepsilon\omega \) this reduces to \( k = \kappa = 1/\delta \) with the skin depth \( \delta = \sqrt{2/(\mu\sigma\omega)} \), and the magnetic field lags the electric field by \( \pi/4 \) with magnitude ratio \( |B_0|/|E_0| = \sqrt{\mu\sigma/\omega} \).

Why it matters

This derivation is the bridge between vacuum electrodynamics and the electrodynamics of real materials. It explains why metals reflect light, why electromagnetic shielding works, why high-frequency currents crowd into a thin surface layer of a wire (raising its effective resistance), and why communicating with submerged submarines forces navies down to extremely low frequencies. One complex number, \( \tilde{k} \), encodes all of it.

Conceptually, it exhibits a smooth interpolation between two archetypes of transport: at \( \sigma \ll \varepsilon\omega \) the fields propagate as weakly damped waves, while at \( \sigma \gg \varepsilon\omega \) the wave equation degenerates into a diffusion equation and the "wave" becomes an overdamped, crawling disturbance. The skin depth is then nothing but the diffusion length of the magnetic field at frequency \( \omega \) — the same mathematics that governs heat penetrating the ground over a daily temperature cycle.

Assumptions

Linear, homogeneous, isotropic medium: \( \vec{D} = \varepsilon\vec{E} \), \( \vec{B} = \mu\vec{H} \) with constant \( \varepsilon, \mu \).If dropped, \( \varepsilon \) and \( \mu \) become tensors or functions of position and the single scalar dispersion relation is replaced by a boundary-value or band-structure problem.

Ohm's law is local and instantaneous: \( \vec{J}_f = \sigma\vec{E} \) with \( \sigma \) real and frequency-independent.If dropped — as it must be when \( \omega \) approaches the electron collision rate \( 1/\tau_c \sim 10^{13}\text{–}10^{14}\,\mathrm{s^{-1}} \) — the Drude form \( \sigma(\omega) = \sigma_0/(1 - i\omega\tau_c) \) applies, the conductor becomes reactive, and above the plasma frequency it turns transparent.

Monochromatic time dependence \( e^{-i\omega t} \); steady state reached.If dropped, one must superpose these solutions; because \( \tilde{k}(\omega) \) is dispersive, pulses distort and spread as they attenuate.

No free volume charge: \( \rho_f = 0 \) inside the conductor.If dropped, an initial charge density decays as \( e^{-t/\tau} \) with \( \tau = \varepsilon/\sigma \); the assumption is self-consistent provided we wait a few relaxation times, which for metals is essentially instantaneous.

Local electrodynamics: skin depth much larger than the electron mean free path, \( \delta \gg \ell \).If dropped (pure metals at cryogenic temperatures, or high frequencies), electrons sample fields over a mean free path and the conduction law becomes nonlocal — the anomalous skin effect, where the effective depth scales as \( \delta_a \propto \omega^{-1/3} \) rather than \( \omega^{-1/2} \).

Classical, non-superconducting response.If dropped for a superconductor, the London equation replaces Ohm's law and fields are expelled over the temperature-dependent, frequency-independent London penetration depth \( \lambda_L \), not \( \delta(\omega) \).

Derivation

\[ \nabla\cdot\vec{E} = \frac{\rho_f}{\varepsilon}, \qquad \nabla\cdot\vec{B} = 0, \qquad \nabla\times\vec{E} = -\frac{\partial\vec{B}}{\partial t}, \qquad \nabla\times\vec{B} = \mu\sigma\vec{E} + \mu\varepsilon\,\frac{\partial\vec{E}}{\partial t} \]

Maxwell's equations in a linear medium, with the free current eliminated in favour of the field via Ohm's law \( \vec{J}_f = \sigma\vec{E} \). This is the single new physical input relative to the vacuum derivation. C

\[ \frac{\partial\rho_f}{\partial t} = -\nabla\cdot\vec{J}_f = -\sigma\,\nabla\cdot\vec{E} = -\frac{\sigma}{\varepsilon}\,\rho_f \quad\Longrightarrow\quad \rho_f(t) = \rho_f(0)\,e^{-t/\tau}, \quad \tau \equiv \frac{\varepsilon}{\sigma} \]

Charge continuity combined with Ohm's law and Gauss's law. Any free volume charge self-destructs exponentially with relaxation time \( \tau \) (for copper \( \tau \sim 10^{-19}\,\mathrm{s} \), far below any timescale where this macroscopic model is valid). We therefore set \( \rho_f = 0 \), so \( \nabla\cdot\vec{E} = 0 \). B

\[ \nabla\times\left(\nabla\times\vec{E}\right) = -\frac{\partial}{\partial t}\left(\nabla\times\vec{B}\right) = -\mu\sigma\,\frac{\partial\vec{E}}{\partial t} - \mu\varepsilon\,\frac{\partial^2\vec{E}}{\partial t^2} \]

Take the curl of Faraday's law and substitute the Ampère–Maxwell–Ohm equation from Step 1; curl and \( \partial/\partial t \) commute for smooth fields. Same decoupling manoeuvre as in the vacuum wave-equation derivation. B

\[ \nabla\times\left(\nabla\times\vec{E}\right) = \nabla\left(\nabla\cdot\vec{E}\right) - \nabla^2\vec{E} = -\nabla^2\vec{E} \quad\Longrightarrow\quad \nabla^2\vec{E} = \mu\sigma\,\frac{\partial\vec{E}}{\partial t} + \mu\varepsilon\,\frac{\partial^2\vec{E}}{\partial t^2} \]

Vector identity plus \( \nabla\cdot\vec{E} = 0 \) from Step 2. The result is the telegraph equation: a wave equation (second time derivative) with an added diffusive damping term (first time derivative) proportional to \( \sigma \). An identical equation holds for \( \vec{B} \). B

\[ \tilde{\vec{E}}(z,t) = \tilde{E}_0\, e^{i(\tilde{k}z - \omega t)}\,\hat{\mathbf{x}}, \qquad \tilde{k} \in \mathbb{C} \]

Trial monochromatic plane wave propagating along \( \hat{\mathbf{z}} \), polarised along \( \hat{\mathbf{x}} \) (transversality \( \nabla\cdot\vec{E}=0 \) forbids a longitudinal component, exactly as for the vacuum plane wave). We allow \( \tilde{k} \) to be complex because the damping term makes a purely real wavevector inconsistent. C

\[ -\tilde{k}^2 = -\mu\varepsilon\omega^2 - i\,\mu\sigma\omega \quad\Longrightarrow\quad \tilde{k}^2 = \mu\varepsilon\omega^2\left(1 + i\,\frac{\sigma}{\varepsilon\omega}\right) \]

Substitute the ansatz: \( \nabla^2 \to (i\tilde{k})^2 \), \( \partial_t \to -i\omega \). This is the dispersion relation; the dimensionless loss parameter \( \sigma/(\varepsilon\omega) \) — the ratio of conduction to displacement current — controls everything that follows. A

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

Write \( \tilde{k}^2 = k^2 - \kappa^2 + 2ik\kappa \), equate real and imaginary parts to Step 6 ( \( k^2-\kappa^2 = \mu\varepsilon\omega^2 \), \( 2k\kappa = \mu\sigma\omega \) ), and solve the resulting quadratic in \( k^2 \), keeping the root with \( k,\kappa > 0 \) so the wave propagates and decays in \( +z \). These are exact for any \( \sigma \). A

\[ \sigma \gg \varepsilon\omega: \qquad \tilde{k}^2 \approx i\,\mu\sigma\omega \quad\Longrightarrow\quad \tilde{k} \approx \sqrt{\mu\sigma\omega}\;e^{i\pi/4} = (1+i)\sqrt{\frac{\mu\sigma\omega}{2}} \equiv \frac{1+i}{\delta}, \qquad \delta = \sqrt{\frac{2}{\mu\sigma\omega}} \]

Good-conductor limit: the displacement current is negligible against the conduction current, so the \( 1 \) in Step 6 is dropped and \( \sqrt{i} = e^{i\pi/4} \) is taken with positive real part. Real and imaginary parts become equal, defining the skin depth \( \delta \) as the common \( e \)-folding and radian length. C

\[ \tilde{\vec{B}} = \frac{\tilde{k}}{\omega}\,\hat{\mathbf{z}}\times\tilde{\vec{E}} \quad\Longrightarrow\quad \frac{|B_0|}{|E_0|} = \frac{|\tilde{k}|}{\omega} \xrightarrow{\;\sigma\gg\varepsilon\omega\;} \sqrt{\frac{\mu\sigma}{\omega}}, \qquad \arg\tilde{k} = \frac{\pi}{4} \]

Faraday's law applied to the ansatz fixes the magnetic field: still transverse and perpendicular to \( \vec{E} \), but multiplied by the complex number \( \tilde{k}/\omega \). Its modulus gives the magnitude ratio; its argument \( \pi/4 \) means \( \vec{B} \) reaches its maximum one-eighth of a period after \( \vec{E} \) — the fields are no longer in phase. B

Result

\[ \tilde{k}^2 = \mu\varepsilon\omega^2 + i\,\mu\sigma\omega, \qquad \vec{E} = E_0\, e^{-z/\delta} \cos\!\left(\frac{z}{\delta} - \omega t\right)\hat{\mathbf{x}}, \qquad \delta = \sqrt{\frac{2}{\mu\sigma\omega}} \;\;(\sigma \gg \varepsilon\omega) \]

Reading. Inside a conductor an electromagnetic wave is strangled as it propagates: its amplitude falls by a factor \( e \) every skin depth \( \delta \), and in a good conductor the wavelength inside the metal is only \( 2\pi\delta \) — the wave barely completes one oscillation before it is extinguished. Higher frequency, higher conductivity, or higher permeability all thin the skin, each under a square root. The magnetic field is enormously enhanced relative to \( E \) (by \( \sqrt{\sigma/\varepsilon\omega} \) compared with a vacuum wave) and lags it by \( 45^\circ \): energy in a conductor is stored overwhelmingly in the magnetic field.

Units check. \( [\mu\sigma\omega] = \dfrac{\Omega\,\mathrm{s}}{\mathrm{m}}\cdot\dfrac{1}{\Omega\,\mathrm{m}}\cdot\dfrac{1}{\mathrm{s}} = \mathrm{m^{-2}} \), so \( \delta = \sqrt{2/(\mu\sigma\omega)} \) is a length, and \( \tilde{k} \) an inverse length, as required. In the dispersion relation, \( [\mu\varepsilon\omega^2] = (1/c_{\text{medium}}^2)\,\omega^2 = \mathrm{m^{-2}} \) matches \( [\mu\sigma\omega] = \mathrm{m^{-2}} \): both terms are inverse lengths squared.

Limiting cases

\( \sigma \to 0 \): \( \tilde{k} \to \omega\sqrt{\mu\varepsilon} \), real — the undamped plane wave in a dielectric is recovered, with \( E \) and \( B \) back in phase.
Poor conductor \( \sigma \ll \varepsilon\omega \): expanding Step 7, \( \kappa \approx \frac{\sigma}{2}\sqrt{\mu/\varepsilon} \) — attenuation independent of frequency, while \( k \approx \omega\sqrt{\mu\varepsilon} \) is barely modified.
Good conductor \( \sigma \gg \varepsilon\omega \): \( k = \kappa = 1/\delta \), wavelength \( \lambda = 2\pi\delta \), phase velocity \( v_p = \omega\delta \ll c \), and \( \arg(\tilde{B}_0/\tilde{E}_0) = \pi/4 \).
\( \omega \to 0 \): \( \delta \to \infty \) — static fields fully penetrate; the telegraph equation degenerates to the magnetic diffusion equation \( \nabla^2\vec{B} = \mu\sigma\,\partial_t\vec{B} \).
\( \sigma \to \infty \): \( \delta \to 0 \) — the ideal-conductor boundary condition emerges: fields expelled, currents confined to a true surface sheet.

Breaks when

Frequencies approaching the collision rate (infrared and above, \( \omega\tau_c \gtrsim 1 \)): the static Ohm's law fails; with the Drude form \( \sigma(\omega) = \sigma_0/(1 - i\omega\tau_c) \) the response becomes inductive, and above the plasma frequency \( \omega_p \) (ultraviolet for most metals) the metal transmits rather than attenuates — alkali metals really do become transparent in the UV.
Anomalous skin effect: in pure metals at low temperature the mean free path \( \ell \) exceeds \( \delta \); electrons then feel a field that changes over one free flight, conduction becomes nonlocal, and the measured surface resistance departs from this theory (Pippard's regime, \( \delta_{\text{eff}} \propto \omega^{-1/3} \)).
Superconductors: Ohm's law is replaced by the London equation; screening occurs over the London depth \( \lambda_L \), which is frequency-independent and survives at \( \omega = 0 \) — qualitatively different physics from the skin effect.
Strongly inhomogeneous or magnetically nonlinear media (ferromagnetic steel near saturation, plasmas with density gradients): \( \mu \) or \( \sigma \) become field- or position-dependent and the single complex exponential is no longer a solution.

Failure modes

Amplitude–power confusion: fields fall as \( e^{-z/\delta} \) but power and intensity fall as \( e^{-2z/\delta} \); quoting "one skin depth = 8.7 dB" (a power statement) as a field ratio, or vice versa.
Forgetting the square root: writing \( \delta \propto 1/f \) instead of \( \delta \propto 1/\sqrt{f} \) — a factor-of-\(\sqrt{10}\) error per decade of frequency.
Regime abuse: applying \( \delta = \sqrt{2/\mu\sigma\omega} \) to media that are not good conductors at the frequency in question (dry soil at GHz, semiconductors, fresh water at HF) instead of the exact \( \kappa \) of Step 7. Always check \( \sigma/(\varepsilon\omega) \) first.
Inserting a metal's "dielectric constant": for \( \sigma \gg \varepsilon\omega \) the result is independent of \( \varepsilon \); hunting for \( \varepsilon_r \) of copper signals a misreading of which term survived.
Assuming \( E \) and \( B \) in phase: carrying the vacuum relation \( B = E/c \) (in phase, ratio \( 1/c \)) into a conductor, missing both the \( 45^\circ \) lag and the enormous magnitude enhancement.
Using relative instead of absolute \( \mu \): for ferromagnetic conductors (steel, mu-metal) forgetting \( \mu = \mu_r\mu_0 \) with \( \mu_r \sim 10^2\text{–}10^5 \), which shrinks \( \delta \) dramatically.

Discussion

The deepest way to read the telegraph equation of Step 4 is as a competition between two transport archetypes. When the second time derivative dominates ( \( \varepsilon\omega \gg \sigma \) ), the equation is hyperbolic and disturbances propagate; when the first derivative dominates, it is parabolic and disturbances diffuse. The good-conductor skin effect is diffusion in disguise: dropping the displacement term entirely gives \( \nabla^2\vec{B} = \mu\sigma\,\partial_t\vec{B} \), a diffusion equation with diffusivity \( D = 1/(\mu\sigma) \), and the skin depth is exactly the diffusion length \( \delta = \sqrt{2D/\omega} \) over one radian of oscillation. This is the same mathematics as the annual temperature wave penetrating the soil — including the \( 45^\circ \) phase lag between "flux" and "gradient" quantities characteristic of oscillatory diffusion.

The result also underlies the optics of metals. At a vacuum–conductor interface, the impedance mismatch implied by \( |B_0|/|E_0| = \sqrt{\mu\sigma/\omega} \) is so severe that almost all incident amplitude is reflected — which is why good conductors are shiny and why a thin metal film shields electromagnetically far better than its thickness in skin depths would naively suggest (reflection loss adds to absorption loss). In circuit language, the surface of a good conductor presents a surface impedance \( Z_s = (1+i)/(\sigma\delta) \), whose real part \( R_s = 1/(\sigma\delta) \) sets the wall losses of waveguides and cavities and the AC resistance of conductors: a wire carrying current at frequency \( f \) behaves as if only an annulus of thickness \( \sim\delta \) were conducting, so \( R_{\text{AC}}/R_{\text{DC}} \approx a/(2\delta) \) for a wire of radius \( a \gg \delta \). This is why high-frequency conductors are hollow, stranded (Litz wire), or silver-plated, and why induction hobs deposit their power precisely in the base of a steel pan ( \( \mu_r \) large, \( \delta \) tiny).

At a more rigorous level, the dispersion relation is the \( \omega\tau_c \ll 1 \) limit of a single unified description. Writing the conductor as a dielectric with complex permittivity \( \tilde{\varepsilon}(\omega) = \varepsilon + i\sigma/\omega \), then inserting the Drude conductivity \( \sigma(\omega) = \sigma_0/(1-i\omega\tau_c) \), yields \( \tilde{\varepsilon}(\omega)/\varepsilon_0 \approx 1 - \omega_p^2/(\omega^2 + i\omega/\tau_c) \) with \( \omega_p^2 = ne^2/(m\varepsilon_0) \). Three regimes follow: the Hagen–Rubens (skin-effect) regime \( \omega \ll 1/\tau_c \) derived here; the relaxation regime \( 1/\tau_c \ll \omega \ll \omega_p \), where the field decays over the collisionless penetration depth \( c/\omega_p \) — essentially the London-like inertial screening of free electrons — with little dissipation; and transparency for \( \omega > \omega_p \). The classical skin depth formula is thus a low-frequency asymptote of the metal's full dielectric function, and its breakdown frequencies are not arbitrary: they are \( 1/\tau_c \) and \( \omega_p \), both measurable in optical data.

Common misconceptions. The skin depth is not a wall: fields at depth \( 2\delta \) or \( 3\delta \) are attenuated, not zero, and shielding calculations must count how many \( \delta \) thick the barrier is. The wave in the conductor is not "absorbed at the surface" — Joule dissipation \( \frac{1}{2}\sigma|E|^2 \) is distributed through the skin layer. And \( \delta \) is not the London penetration depth, nor the Debye length; all three describe field screening but arise from dissipative, inertial, and thermodynamic physics respectively.

Worked examples

Example 1 — mains-frequency skin depth in copper. Copper has \( \sigma = 5.96\times 10^{7}\,\mathrm{S\,m^{-1}} \), \( \mu \approx \mu_0 = 4\pi\times10^{-7}\,\mathrm{H\,m^{-1}} \). Find \( \delta \) at \( f = 60\,\mathrm{Hz} \).

\[ \frac{\sigma}{\varepsilon_0\omega} = \frac{5.96\times10^{7}}{(8.85\times10^{-12})(2\pi\cdot 60)} \approx 1.8\times10^{16} \gg 1 \]

Verify the good-conductor condition before touching the formula: conduction current exceeds displacement current by sixteen orders of magnitude, so Step 8 applies with overwhelming accuracy. C

\[ \delta = \sqrt{\frac{2}{\mu_0\sigma\omega}} = \sqrt{\frac{2}{(4\pi\times10^{-7})(5.96\times10^{7})(2\pi\cdot 60)}} \]

Symbols first, then numbers: \( \mu_0\sigma\omega = (1.257\times10^{-6})(5.96\times10^{7})(377) = 2.82\times10^{4}\,\mathrm{m^{-2}} \). A

\[ \delta = \sqrt{\frac{2}{2.82\times10^{4}\,\mathrm{m^{-2}}}} = 8.4\times10^{-3}\,\mathrm{m} \]

Arithmetic; the answer carries units of metres as required. A

\[ \delta_{\mathrm{Cu}}(60\,\mathrm{Hz}) \approx 8.4\,\mathrm{mm} \]

Reading. Mains current fills conductors up to about a centimetre thick almost uniformly — but in the massive multi-centimetre busbars of power stations, the core is already dead weight. At \( 10\,\mathrm{MHz} \) the same formula gives \( \delta \approx 21\,\mathrm{\mu m} \): radio-frequency current is a pure surface phenomenon, which is why RF conductors are silver-plated tubes.

Units check. \( \mathrm{H\,m^{-1}\cdot S\,m^{-1}\cdot s^{-1}} = \mathrm{m^{-2}} \), square root of the reciprocal gives \( \mathrm{m} \).

Example 2 — radio waves in seawater. Seawater has \( \sigma \approx 4\,\mathrm{S\,m^{-1}} \), \( \varepsilon \approx 81\,\varepsilon_0 \), \( \mu \approx \mu_0 \). Find the skin depth at \( f = 10\,\mathrm{kHz} \) and the depth at which the field amplitude has fallen to \( 1\% \).

\[ \frac{\sigma}{\varepsilon\omega} = \frac{4}{(81)(8.85\times10^{-12})(2\pi\times10^{4})} \approx 8.9\times10^{4} \gg 1 \]

Despite being a mediocre conductor by metallic standards, at \( 10\,\mathrm{kHz} \) seawater is still deep in the good-conductor regime — the criterion is frequency-dependent, not a property of the material alone. C

\[ \delta = \sqrt{\frac{2}{\mu_0\sigma\omega}} = \sqrt{\frac{2}{(1.257\times10^{-6})(4)(6.28\times10^{4})}} = \sqrt{\frac{2}{0.316\,\mathrm{m^{-2}}}} = 2.5\,\mathrm{m} \]

Direct substitution into the good-conductor formula. A

\[ e^{-z/\delta} = 10^{-2} \quad\Longrightarrow\quad z = \delta\,\ln 100 = (2.5\,\mathrm{m})(4.61) \approx 12\,\mathrm{m} \]

Invert the exponential decay law for the amplitude ratio; \( 1\% \) amplitude corresponds to \( 40\,\mathrm{dB} \) of field attenuation. A

\[ \delta_{\text{sea}}(10\,\mathrm{kHz}) \approx 2.5\,\mathrm{m}, \qquad z_{1\%} \approx 12\,\mathrm{m} \]

Reading. A \( 10\,\mathrm{kHz} \) signal is effectively gone a dozen metres below the surface. Since \( \delta \propto 1/\sqrt{f} \), reaching a submarine at depth of order \( 100\,\mathrm{m} \) forces frequencies down to tens of hertz — the ELF band, with its kilometre-scale antennas and data rates of characters per minute. The ocean is an electromagnetic shield by the same physics that makes a copper box one.

Units check. \( \mu_0\sigma\omega \) in \( \mathrm{m^{-2}} \) as before; \( z = \delta\ln(\text{ratio}) \) is a length times a pure number.

Problems

Silver ( \( \sigma = 6.30\times10^{7}\,\mathrm{S\,m^{-1}} \), \( \mu \approx \mu_0 \) ) is used to plate microwave components. Compute the skin depth at \( 1\,\mathrm{GHz} \) and comment on how thick the plating needs to be.
Solution
\( \omega = 2\pi\times10^{9} = 6.28\times10^{9}\,\mathrm{s^{-1}} \). Then \( \mu_0\sigma\omega = (1.257\times10^{-6})(6.30\times10^{7})(6.28\times10^{9}) = 4.97\times10^{11}\,\mathrm{m^{-2}} \), so \( \delta = \sqrt{2/4.97\times10^{11}} = 2.0\times10^{-6}\,\mathrm{m} = 2.0\,\mathrm{\mu m} \). A plating of \( 5\delta \approx 10\,\mathrm{\mu m} \) carries \( \gt 99\% \) of the current ( \( 1 - e^{-5} \approx 0.993 \) ), so a few microns of silver make the component electrically indistinguishable from solid silver at this frequency — the substrate metal is irrelevant.
Find the frequency at which the conduction and displacement currents in copper would be equal ( \( \sigma = \varepsilon_0\omega \), taking \( \varepsilon \approx \varepsilon_0 \) ), and explain why the good-conductor approximation nevertheless fails long before this frequency is reached.
Solution
\( \omega = \sigma/\varepsilon_0 = (5.96\times10^{7})/(8.85\times10^{-12}) = 6.7\times10^{18}\,\mathrm{rad\,s^{-1}} \), i.e. \( f = \omega/2\pi \approx 1.1\times10^{18}\,\mathrm{Hz} \) — soft X-rays. But the static Ohm's law already fails when \( \omega\tau_c \sim 1 \) with the collision time \( \tau_c \sim 2.5\times10^{-14}\,\mathrm{s} \) for copper, i.e. around \( \omega \sim 4\times10^{13}\,\mathrm{rad\,s^{-1}} \) (mid-infrared) — five orders of magnitude lower. Above that, \( \sigma \) must be replaced by the complex Drude \( \sigma(\omega) \), and above the plasma frequency ( \( \hbar\omega_p \approx 9\,\mathrm{eV} \) for Cu's free electrons) the metal ceases to attenuate at all. The naive crossover frequency is never physically reached within the model's domain of validity.
For copper at \( 60\,\mathrm{Hz} \) (use \( \delta = 8.4\,\mathrm{mm} \) from Example 1), compute the wavelength and phase velocity of the electromagnetic wave inside the metal, and compare with the vacuum values.
Solution
In a good conductor \( k = 1/\delta = 119\,\mathrm{m^{-1}} \), so \( \lambda = 2\pi\delta = 2\pi(8.4\times10^{-3}) = 5.3\times10^{-2}\,\mathrm{m} = 5.3\,\mathrm{cm} \), versus the vacuum wavelength \( c/f = 5000\,\mathrm{km} \) — a compression by a factor of \( \sim10^{8} \). The phase velocity is \( v_p = \omega/k = \omega\delta = (377\,\mathrm{s^{-1}})(8.4\times10^{-3}\,\mathrm{m}) = 3.2\,\mathrm{m\,s^{-1}} \): walking pace. The disturbance inside the conductor is not a propagating wave in any useful sense but a diffusing field pattern — consistent with the diffusion-equation reading of the good-conductor limit.
For copper at \( 1\,\mathrm{MHz} \), find the ratio \( |B_0|/|E_0| \) inside the metal, compare it with the vacuum value \( 1/c \), state the phase relation between the fields, and hence show that the time-averaged energy density is almost entirely magnetic.
Solution
\( |\tilde{k}| = \sqrt{\mu_0\sigma\omega} = \sqrt{(1.257\times10^{-6})(5.96\times10^{7})(6.28\times10^{6})} = \sqrt{4.71\times10^{8}} = 2.17\times10^{4}\,\mathrm{m^{-1}} \). So \( |B_0|/|E_0| = |\tilde{k}|/\omega = (2.17\times10^{4})/(6.28\times10^{6}) = 3.5\times10^{-3}\,\mathrm{s\,m^{-1}} \), versus \( 1/c = 3.3\times10^{-9}\,\mathrm{s\,m^{-1}} \) in vacuum: the magnetic field is relatively enhanced by a factor \( \sim10^{6} \). \( \vec{B} \) lags \( \vec{E} \) by \( \arg\tilde{k} = 45^\circ \). The energy ratio is \( \dfrac{\langle u_B\rangle}{\langle u_E\rangle} = \dfrac{|B_0|^2/2\mu_0}{\varepsilon_0|E_0|^2/2} = c^2\left(\dfrac{|B_0|}{|E_0|}\right)^2 = (9\times10^{16})(3.5\times10^{-3})^2 \approx 1.1\times10^{12} \) — which equals \( \sigma/(\varepsilon_0\omega) \), confirming that in a good conductor the field energy is essentially all magnetic.
Show from the exact result of Step 7 that for a poor conductor ( \( \sigma \ll \varepsilon\omega \) ) the attenuation constant is \( \kappa \approx \frac{\sigma}{2}\sqrt{\mu/\varepsilon} \), independent of frequency, and evaluate the \( e \)-folding attenuation length for fresh water ( \( \sigma = 5\times10^{-3}\,\mathrm{S\,m^{-1}} \), \( \varepsilon = 81\,\varepsilon_0 \), \( \mu = \mu_0 \) ) at \( 100\,\mathrm{MHz} \), first verifying the regime.
Solution
With \( x \equiv \sigma/(\varepsilon\omega) \ll 1 \), expand Step 7: \( \sqrt{1+x^2} - 1 \approx x^2/2 \), so \( \kappa \approx \omega\sqrt{\mu\varepsilon/2}\,\cdot x/\sqrt{2} = \frac{\omega\sqrt{\mu\varepsilon}}{2}\cdot\frac{\sigma}{\varepsilon\omega} = \frac{\sigma}{2}\sqrt{\frac{\mu}{\varepsilon}} \): \( \omega \) cancels. Regime check at \( 100\,\mathrm{MHz} \): \( \varepsilon\omega = (81)(8.85\times10^{-12})(6.28\times10^{8}) = 0.45\,\mathrm{S\,m^{-1}} \), so \( x = 5\times10^{-3}/0.45 = 0.011 \ll 1 \) — poor-conductor regime confirmed (note that at \( 100\,\mathrm{kHz} \) the same water has \( x \approx 11 \) and would need the exact formula). Then \( \sqrt{\mu_0/\varepsilon} = \sqrt{\mu_0/\varepsilon_0}/\sqrt{81} = 377\,\Omega/9 = 41.9\,\Omega \), giving \( \kappa = (2.5\times10^{-3})(41.9) = 0.105\,\mathrm{m^{-1}} \) and attenuation length \( 1/\kappa \approx 9.5\,\mathrm{m} \). Fresh water passes VHF far better than seawater passes anything — its thousandfold lower conductivity, not its permittivity, is what matters.