Electrodynamics in Matter
Electrodynamics in matter is the macroscopic Maxwell theory: the polarization and magnetization fields, the auxiliary fields \(\vect{D}\) and \(\vect{H}\), the response functions that close the system, and dispersion—the frequency dependence of the response, constrained by causality alone through the Kramers–Kronig relations. The chapter gathers what Electrostatics and Magnetostatics deferred: how the vacuum equations of The Maxwell Equations, exact at the microscopic level, become equations for smoothed fields with matter-dependent sources, and what measured phenomena—conduction, dielectric response, dia-, para- and ferromagnetism, plasma transparency, the skin effect—the response functions encode. The microscopic explanations are quantum and are deferred to the condensed-matter part; here each phenomenon is stated with its founding measurement and its classical model, and the models' honest failures are marked.
Electrodynamics in Matter: all derivations of this chapter are pending.
From microscopic to macroscopic fields
Bound charges and the polarization field
[Reserved: spatial averaging of the microscopic equations, following the electron-theory programme of Lorentz [Lorentz:1909]; the polarization \(\vect{P}\) as dipole-moment density, bound charge \(-\vect{\nabla}\cdot\vect{P}\) and surface charge \(\vect{P}\cdot\hat{\vect{n}}\); the displacement field \(\vect{D}=\epsilon_{0}\vect{E}+\vect{P}\) of Maxwell's Treatise [Maxwell:1873].]
Bound currents and the magnetization field
[Reserved: the magnetization \(\vect{M}\), bound currents \(\vect{\nabla}\times\vect{M}\) and \(\pp\vect{P}/\pp t\), and the field \(\vect{H}=\vect{B}/\mu_{0}-\vect{M}\) [Maxwell:1873]; the macroscopic Maxwell system in matter and what in it is definition versus physics [Jackson:1999].]
Constitutive relations and boundary conditions
[Reserved: linear isotropic media, \(\vect{D}=\epsilon\vect{E}\) and \(\vect{B}=\mu\vect{H}\); permittivity, permeability, susceptibilities and their SI units; interface conditions for \(\vect{D}\) and \(\vect{H}\), extending Electrostatics and Magnetostatics; energy in linear media [Jackson:1999].]
Dielectrics
Polarizability and local fields
[Reserved: molecular polarizability, the Lorentz local-field correction [Lorentz:1909], and the Clausius–Mossotti relation connecting polarizability to permittivity [Mossotti:1850] [Clausius:1879]; its optical form, the Lorentz–Lorenz equation, derived independently by H. A. Lorentz [Lorentz:1880] and L. Lorenz [Lorenz:1880]—two different men, the same near-collision of names as the gauge condition of The Maxwell Equations.]
Polar molecules and relaxation
[Reserved: orientation polarization of permanent molecular dipoles and its temperature dependence, after Debye [Debye:1912b]; Debye relaxation and the frequency dependence of the dielectric constant of polar liquids; dielectric loss.]
Piezo- and ferroelectricity
[Reserved: piezoelectricity, discovered in quartz by the Curie brothers [Curie:1880]; ferroelectricity, discovered in Rochelle salt by Valasek [Valasek:1921], with hysteresis of \(\vect{P}\) mirroring the magnetic case of Section 62.4.1; the microscopic theory is deferred to the condensed-matter part (Phonons and Lattice Dynamics).]
Conduction
Ohm's law
[Reserved: Ohm's empirical law of the galvanic circuit [Ohm:1827], its local form \(\vect{J}=\sigma\vect{E}\), conductivity and resistivity in SI units; the Wiedemann–Franz proportionality of thermal and electrical conductivity of metals [Wiedemann:1853]; Joule heating and its place in the energy balance of The Maxwell Equations.]
In a metallic conductor held at constant temperature the current is strictly proportional to the electromotive force driving it, over a range of many decades, with a coefficient that depends on the material and its temperature and on nothing else — not on the magnitude of the current, not on how long it has been flowing, not on its history [Ohm:1827]. Locally,
with \(\sigma\) the conductivity, measured in \(\mathrm{A}/\mathrm{V}/\mathrm{m}\). The proportionality is a property of a class of materials and not a law of the field, which is why it may be — and in semiconductors, gas discharges and superconductors is — violated without anything in The Maxwell Equations failing.
Derivation. Model the conductor as containing \(n\) free carriers per unit volume, of charge \(q\) and mass \(m\), which are accelerated by the field and lose all memory of their drift in collisions occurring at a mean interval \(\tau\) [Drude:1900a]. Between collisions the equation of motion is \(m\,\dd\vect{v}/\dd t=q\vect{E}\), so a carrier emerging from a collision with a randomly directed velocity has, after a time \(t\), an extra velocity \(q\vect{E}t/m\) along the field. Averaging over the collision statistics, the mean extra velocity — the drift velocity — is
which is constant in time: the field accelerates, the collisions brake, and the steady state is a terminal drift rather than a runaway. The current density is the charge carried across unit area per unit time, \(\vect{J}=nq\avg{\vect{v}}\), so
which is Equation (62.1), with the linearity traced to the single assumption that \(\tau\) does not depend on \(\vect{E}\).
That assumption is also the honest limit of the derivation. It holds when the drift velocity is small compared with the mean thermal speed, which in a metal at ordinary current densities it overwhelmingly is; it fails in a gas discharge, where the field sets the collision rate. Nothing here explains what fixes \(\tau\), and the classical estimate of its magnitude — and with it Drude's account of the electronic heat capacity — is wrong, for reasons that are quantum (Electrons in Solids: Band Theory).
∎The Drude model and the Hall effect
[Reserved: Drude's kinetic model of metallic conduction [Drude:1900a]: DC and AC conductivity, the relaxation time, and its account of Wiedemann–Franz; the Hall effect [Hall:1879] as the classical probe of carrier sign and density, and the model's honest failures (heat capacity, sign of some Hall coefficients), resolved only by the quantum theory of Electrons in Solids: Band Theory; the quantized Hall effect is treated in Experiment: The Quantum Hall Effect.]
A flat conducting strip carrying a steady current, placed in a magnetic field perpendicular to its face, develops a steady potential difference across the strip, perpendicular both to the current and to the field [Hall:1879]. The transverse field is proportional to the product of the current density and the magnetic field,
so the measurement returns both the density and the sign of the charge carriers. That sign is negative in most metals, as the electron requires — but it is positive in others, beryllium and zinc among them, and no classical model of conduction accounts for a positive carrier in a metal.
Derivation. Let the current run along \(x\) with drift velocity \(v_{x}\), and let \(\vect{B}=B_{z}\hat{\vect{z}}\). Each carrier feels the Lorentz force \(q(\vect{E}+\vect{v}\times\vect{B})\), whose magnetic part \(qv_{x}B_{z}\) pushes it along \(-y\) or \(+y\) according to the sign of \(qv_{x}\). Charge therefore accumulates on one edge of the strip until the transverse electric field it creates cancels the magnetic push, after which the current runs straight again. That steady state is the observation, and it is defined by the vanishing of the transverse force:
Eliminating the drift velocity with \(J_{x}=nqv_{x}\) from Equation (62.3) gives \(E_{y}=J_{x}B_{z}/nq\), which is Equation (62.4). The sign of \(R_{H}\) is the sign of \(q\), because the drift velocity reverses with the carrier sign while the current does not, so the two reversals compound rather than cancel — which is why the experiment is a measurement of the carrier sign and why the metals that return the wrong one remained a genuine anomaly for decades.
∎The skin effect
[Reserved: field penetration into a good conductor, the skin depth \(\delta=\sqrt{2/(\mu\sigma\omega)}\), and high-frequency resistance, analysed by Rayleigh [Rayleigh:1886] and Heaviside [Heaviside:1892]; consequences for shielding, waveguides and cables.]
Magnetic materials
Dia-, para- and ferromagnetism
[Reserved: Faraday's discovery that all matter is magnetic— diamagnetic bismuth repelled, paramagnets attracted [Faraday:1846b]; Curie's systematic measurements: temperature independence of diamagnetism, the \(1/T\) law of paramagnetism, and the disappearance of ferromagnetism at a critical temperature [Curie:1895]; ferromagnetic hysteresis, named and mapped by Ewing [Ewing:1885].]
No substance is indifferent to a magnetic field. Suspended in a field strong enough and inhomogeneous enough to make the force visible, every material tested is either drawn towards the region of stronger field or pushed towards the region of weaker field; the second class, of which bismuth is the type case, had not been suspected before it was looked for, and it includes water, glass and flesh [Faraday:1846b]. The division is sharp: the repelled substances are repelled whatever the direction of the field, and their response is very nearly independent of temperature, while the attracted ones weaken as the temperature rises [Curie:1895].
The magnetic response of matter: derive the Bohr–van Leeuwen theorem, that a classical system of charges in thermal equilibrium has identically zero magnetization because the magnetic field enters the classical Hamiltonian only through a shift of the canonical momenta, which is absorbed by a translation of the integration variables in the partition function. The theorem states that the phenomenon above has no classical explanation at all, so every classical model in this section is a parametrization and not a derivation, and the real account is the quantum one of the magnetism chapter.
In iron, nickel, cobalt and their alloys the magnetization is not a function of the applied field. Carrying the field around a closed cycle carries the magnetization around a closed curve of nonzero area: the material retains a magnetization when the field is removed (remanence), requires a reversed field of definite strength to bring that magnetization to zero (coercivity), and approaches a saturation beyond which further field produces almost nothing [Ewing:1885]. The state therefore depends on the history, the cycle dissipates energy, and no number deserves the name of “the” permeability of the material.
Ferromagnetic hysteresis: derive the energy dissipated per unit volume per cycle as the area enclosed by the loop traced in the plane of applied field against magnetization, and identify precisely which hypothesis behind a linear constitutive relation the loop violates. The microscopic origin — domains, exchange coupling and magnetocrystalline anisotropy — is quantum and belongs to the magnetism chapter.
Langevin theory and the Curie–Weiss law
[Reserved: Langevin's statistical theory of dia- and paramagnetism [Langevin:1905], the Langevin function and the Curie law; Weiss's molecular field, the Curie–Weiss law and spontaneous magnetization [Weiss:1907]; what the molecular field really is (exchange) awaits Magnetism in Matter.]
Measurements of magnetic susceptibility carried across a wide range of temperature separate three behaviours [Curie:1895]. Diamagnetic susceptibility is negative and nearly independent of temperature. Paramagnetic susceptibility is positive and falls as the reciprocal of the absolute temperature,
with \(C\) a constant of the substance. And a ferromagnet loses its ferromagnetism entirely above a temperature characteristic of the material, above which it behaves as an ordinary paramagnet. The transition is sharp, and the temperature at which it happens is a reproducible property of the substance.
Derivation. Suppose the substance to contain \(n\) permanent magnetic dipoles per unit volume, each of fixed magnitude \(\mu\) and free to take any orientation [Langevin:1905]. A dipole at angle \(\theta\) to the applied field has energy \(-\mu B\cos\theta\), so in thermal equilibrium at temperature \(T\) the orientations are distributed as \(\ee^{\mu B\cos\theta/k_{B}T}\) and the mean component along the field is
the angular integrals being elementary in the variable \(\cos\theta\). The magnetization is \(M=n\mu L(x)\), which saturates at \(n\mu\) for \(x\gg1\) — all dipoles aligned — and for \(x\ll1\) has the expansion \(L(x)=x/3+O(x^{3})\). Ordinary laboratory conditions are deep in the second regime, so
which is Equation (62.6) with \(C=\mu_{0}n\mu^{2}/3k_{B}\). The \(1/T\) is the competition of two energies and nothing else: alignment is worth \(\mu B\) and disorder is worth \(k_{B}T\).
The derivation is not, however, a classical one, and saying so is the point of Phenomenon 62.3. It postulates dipoles of fixed magnitude that cannot be destroyed by the field or by the thermal motion, and classical mechanics supplies no such object; without that postulate the equilibrium magnetization is exactly zero. The Curie temperature is outside the model altogether — independent dipoles have no cooperative transition — and requires the molecular field of Section 62.4.2, whose own origin is quantum (Magnetism in Matter).
∎The Bohr–van Leeuwen theorem
[Reserved: classical statistical mechanics yields exactly zero equilibrium magnetization [vanLeeuwen:1921]—the theorem that makes every classical model of this section provisional and magnetism intrinsically quantum; statement, scope, and its role as a signpost to Magnetism in Matter.]
Superconductors: deferred
[Reserved: perfect conductivity and the Meissner–Ochsenfeld expulsion of magnetic flux [Meissner:1933] are named here only as the extreme magnetic response of matter; the London phenomenology, the microscopic theory and the experimental record are treated in Superconductivity and Superfluidity and Experiment: Superconductivity.]
Dispersion and causality
The Lorentz oscillator model
[Reserved: bound electrons as damped driven oscillators [Lorentz:1909]; normal and anomalous dispersion, absorption lines, and the empirical Sellmeier form recovered ([Sellmeier:1871], cf. Electromagnetic Waves and Optics); oscillator strengths and the f-sum rule; the quantum reinterpretation is noted and deferred to Part IX.]
The Kramers–Kronig relations
[Reserved: causality of the response function alone implies dispersion relations connecting the real and imaginary parts of \(\epsilon(\omega)\), derived by Kronig [Kronig:1926] and Kramers [Kramers:1927]; sum rules; their derivation uses the complex analysis of Complex Analysis, and their reach extends far beyond optics (scattering amplitudes, Scattering Theory).]
In a spectral region where a substance is transparent, its refractive index rises with frequency, smoothly and monotonically, and the measured curve is reproduced across the whole visible range by an empirical formula with one term for each absorption band lying outside it [Sellmeier:1871]. Wherever the substance does absorb, that behaviour reverses: within the absorption band the index falls with increasing frequency. Anomalous dispersion is found where and only where there is absorption, never elsewhere, and the strength of the one tracks the strength of the other. The refractive index and the absorption coefficient are therefore not independent properties of a material, and measuring either over all frequencies determines the other.
Absorption and dispersion: derive the Kramers–Kronig relations, in which the real and imaginary parts of the permittivity are each the Hilbert transform of the other. The only physical input is causality — no response before the stimulus — which makes the response function analytic in the upper half of the complex frequency plane, whence a contour integral gives the pair of dispersion relations and the sum rules that follow from the high-frequency limit. The argument needs the complex analysis of the corresponding methods chapter, and it is worth noting that it uses no model of the medium at all.
Signal velocity and the front
[Reserved: anomalous dispersion permits phase and group velocities exceeding \(c\); Sommerfeld and Brillouin's resolution—the signal front never travels faster than \(c\), with forerunner precursors [Sommerfeld:1914] [Brillouin:1914]; consistency with the causal structure of Lorentz Transformations.]
Free-carrier response and plasmas
Plasma frequency and transparency
[Reserved: the free-electron permittivity \(\epsilon(\omega)=\epsilon_{0}(1-\omega_{p}^{2}/\omega^{2})\) from the Drude model [Drude:1900a]; plasma oscillations identified by Tonks and Langmuir [Tonks:1929]; reflection below and transparency above \(\omega_{p}\)—the ultraviolet transparency of metals and the ionospheric reflection of radio; the full dynamics of plasmas belongs to Plasmas and Magnetohydrodynamics.]
Two observations, made in different sciences, are the same phenomenon. A metal reflects visible light almost completely and is opaque to it, yet the alkali metals become transparent in the ultraviolet, each above a threshold frequency of its own. And radio waves directed upwards are returned from the upper atmosphere below a critical frequency, while above it they are lost into space — a reflection whose height was measured directly by timing the returned pulse and by interference between the direct and reflected rays [Appleton:1925]. In both cases the medium contains free charges of number density \(n\), and the dividing frequency is
Below \(\omega_{p}\) the medium reflects; above it, it transmits [Drude:1900a] [Tonks:1929].
Derivation. Take carriers that are free — no restoring force — and, for the moment, collisionless, so that \(m_{e}\,\dd\vect{v}/\dd t=q_{e}\vect{E}\). For a field oscillating as \(\ee^{-\ii\omega t}\) this gives \(\vect{v}=\ii q_{e}\vect{E}/m_{e}\omega\) and a current density \(\vect{J}=nq_{e}\vect{v}=\ii nq_{e}^{2}\vect{E}/m_{e}\omega\). The current that drives the magnetic field in the Ampère–Maxwell law is the sum of this and the vacuum displacement current, and the sum is
with \(\omega_{p}\) as in Equation (62.9): the free carriers act as a medium of permittivity \(\epsilon(\omega)=\epsilon_{0}(1-\omega_{p}^{2}/\omega^{2})\). Repeating the wave-equation derivation of The Maxwell Equations with \(\epsilon_{0}\to\epsilon(\omega)\) gives the dispersion relation
For \(\omega>\omega_{p}\) the wavenumber \(k\) is real and the wave propagates, with a phase velocity above \(c\) and a group velocity below it. For \(\omega<\omega_{p}\), \(k\) is imaginary: the field falls off exponentially into the medium instead of propagating. Since \(\epsilon(\omega)\) is real in this model, nothing is absorbed, and an incident wave that cannot propagate and cannot be absorbed must be returned — total reflection. The dividing frequency is a measurement of \(n\) through Equation (62.9), which is what makes ionospheric sounding a density diagnostic.
∎Magneto- and electro-optics
[Reserved: field-induced anisotropy of media. The Faraday effect—rotation of the plane of polarization by a magnetic field along the ray, the first observed link between light and electromagnetism [Faraday:1846a]; the quadratic electro-optic (Kerr) effect [Kerr:1875]; Verdet constants and applications (optical isolators, polarimetry); connection to the birefringence machinery of Electromagnetic Waves and Optics.]
A block of heavy glass placed in a strong magnetic field rotates the plane of polarization of linearly polarized light sent along the field direction. The rotation is proportional to the field strength and to the path length,
with \(V\) a constant of the material, and its sense is fixed by the field alone: sending the beam back through the same block doubles the rotation instead of undoing it [Faraday:1846a]. The effect is therefore not reciprocal, which distinguishes it sharply from the rotation produced by a sugar solution, and it was the first demonstration that light and magnetism act on one another — the observation that made an electromagnetic theory of light thinkable before there was one.
The Faraday effect: derive the difference between the refractive indices of the two circular polarizations for a medium of bound charges in a magnetic field along the ray, by solving the driven oscillator equation with the magnetic force included; then show that a linear polarization, resolved into two circular components of equal amplitude, emerges rotated by half the accumulated phase difference, which gives the Verdet constant. The non-reciprocity should be derived too, and it follows from the field being an axial vector that does not reverse when the propagation direction does.