Radiation and Scattering of Electromagnetic Waves
Accelerate a charge and it radiates. That single sentence, made quantitative, accounts for the light of every antenna, every synchrotron, every atom and every star, and it also destroys the classical atom — which is why this chapter is both the last word of classical electrodynamics and one of the doors into Part VIII — The Transition to Quantum Physics. The chapter starts from the retarded solutions of the wave equations of The Maxwell Equations, first written down by Lorenz [Lorenz:1867], specializes them to a point charge in arbitrary motion — the Liénard–Wiechert potentials [Lienard:1898] [Wiechert:1900] — and extracts from them the radiated power [Larmor:1897], the angular distribution, and the multipole hierarchy that organizes every emitter of finite size [Hertz:1889].
Three further themes follow, each with its own evidence. Radiation carries momentum away from its source, so an accelerated charge feels a back-reaction: the Abraham–Lorentz force [Abraham:1904] [Lorentz:1909], its covariant completion by Dirac [Dirac:1938a], their notorious runaway and pre-accelerating solutions, and the laser–electron collisions in which the effect has finally been measured [Cole:2018] [Poder:2018]. Radiation incident on a charge is re-radiated, which is scattering: Thomson's cross-section [Thomson:1906], Rayleigh's \(\lambda^{-4}\) law and the measured blue of the sky [Rayleigh:1871] [Rayleigh:1899], Mie's exact solution for a sphere [Mie:1908], and the Compton shift [Compton:1923] at which the classical account provably fails. And a charge moving through matter radiates in ways that a charge in vacuum cannot: bremsstrahlung [Kramers:1923], synchrotron emission [Schott:1912] [Elder:1947] and the Cherenkov cone [Cherenkov:1934] [Frank:1937], which are the working tools of the detectors of Cosmic Rays and Astroparticle Physics. Standard treatments are [Jackson:1999] [Landau:1975].
Radiation and Scattering of Electromagnetic Waves: all derivations of this chapter are pending.
Fields of a charge in arbitrary motion
Retarded potentials
[Reserved: the wave equations for the potentials in Lorenz gauge and their retarded solutions [Lorenz:1867], with the source evaluated at \(t-\abs{\vect{x}-\vect{x}'}/c\); Riemann's independent retarded-potential paper [Riemann:1867]; the retarded Green function of the d'Alembertian, taken from Partial Differential Equations, and its support on the light cone; the advanced solution as an equally valid solution of the same equation, so that the choice of the retarded one is a boundary condition and not a theorem — the thermodynamic reading of that choice, and the Wheeler–Feynman absorber alternative [Wheeler:1945], are to be stated honestly here.]
The Liénard–Wiechert potentials
[Reserved: the potentials of a point charge in arbitrary motion, \(\varphi=\frac{q}{4\pi\epsilon_{0}}\left[\frac{1}{R(1-\vect{n}\cdot \vect{\beta})}\right]_{\text{ret}}\) and the corresponding vector potential, obtained independently by Liénard [Lienard:1898] and Wiechert [Wiechert:1900]; the origin of the factor \((1-\vect{n}\cdot\vect{\beta})^{-1}\) as a Jacobian of the retarded-time condition rather than a Doppler factor; and the covariant one-line form in terms of the four-velocity, which makes the Lorentz transformation properties of Relativistic Dynamics manifest.]
The velocity and acceleration fields
[Reserved: the field strengths obtained by differentiating the potentials, splitting into a velocity field falling as \(1/R^{2}\) — a boosted Coulomb field, carrying no energy to infinity — and an acceleration field falling as \(1/R\) and proportional to \(\vect{a}\); the theorem that only the \(1/R\) part contributes to the flux through a sphere of growing radius, which is what defines radiation; the transverse and mutually orthogonal character of \(\vect{E}\) and \(\vect{B}\) in the wave zone [Jackson:1999].]
Radiated power
The Larmor formula
[Reserved: Larmor's result \(P=q^{2}a^{2}/6\pi\epsilon_{0}c^{3}\) for a slowly moving charge [Larmor:1897], derived from the Poynting flux of the acceleration field; the \(\sin^{2}\theta\) angular distribution about the acceleration; the numerical consequence that an electron on a Bohr-radius orbit radiates its energy away in about \(10^{-11}\,\mathrm{s}\), so that the classical atom cannot exist — the sharpest classical failure in this treatise, taken up in Atomic Models and Spectra.]
A charge in uniform motion carries its field along with it and radiates nothing. A charge whose velocity changes emits energy that never returns. For speeds small compared with \(c\) the energy leaves at a rate proportional to the square of the acceleration and independent of the velocity,
and it leaves with a \(\sin^{2}\theta\) distribution about the direction of the acceleration, so that nothing at all is emitted along that direction [Larmor:1897]. Both the pattern and the polarization were mapped directly around an oscillating dipole with a resonant loop detector [Hertz:1889], and Equation (65.1) is the quantitative content of every antenna, every synchrotron and every classical radiating system in this chapter.
The Larmor formula: derive the fields of a point charge in arbitrary motion from the retarded potentials, separate the velocity field, which falls as the inverse square of the distance and carries no energy to infinity, from the acceleration field, which falls as the inverse first power and is proportional to the acceleration, and integrate the Poynting flux of the acceleration field over a sphere of large radius.
Matter is made of atoms in which light electrons surround a small heavy nucleus, as the large-angle scattering of alpha particles requires [Rutherford:1911]; an electron so bound is in permanent acceleration. Yet atoms do not collapse — they persist indefinitely, and a hydrogen atom today is indistinguishable from one of any other epoch — and when they do radiate they emit sharp lines at fixed frequencies obeying exact numerical relations [Balmer:1885], not a continuum rising in pitch as the electron falls inward. Both facts contradict classical electrodynamics, and the contradiction is not marginal but catastrophic.
Derivation. Take a single electron of charge magnitude \(e\) and mass \(m_{e}\) on a circular orbit of radius \(r\) about a proton. The Coulomb attraction supplies the centripetal acceleration,
and the total energy of the orbit — kinetic plus potential, with the virial relation between them — is \(E=-e^{2}/8\pi\epsilon_{0}r\). Substituting Equation (65.2) into Equation (65.1) and equating the radiated power to the rate of loss of orbital energy, \(\dd E/\dd t=-P\), gives
where \(r_{e}=e^{2}/4\pi\epsilon_{0}m_{e}c^{2}\) is the classical electron radius. Separating and integrating from an initial radius \(r_{0}\) down to the nucleus,
With \(r_{0}\) the Bohr radius \(5.29\times 10^{-11}\,\mathrm{m}\) and \(r_{e}=2.82\times 10^{-15}\,\mathrm{m}\) [Tiesinga:2021], this is about \(1.6\times 10^{-11}\,\mathrm{s}\).
Two predictions follow, and both are false. The atom would disappear in a fraction of a nanosecond; and as \(r\) shrank the orbital frequency would rise continuously, so the emitted spectrum would be a continuous sweep upward in frequency terminating in a burst, rather than the fixed sharp lines that are observed. This derivation therefore establishes not the phenomenon but its irreducibility: within classical electrodynamics there is no stable atom and no line spectrum, and no adjustment of the orbit, the charge distribution or the radiation law repairs it. The resolution is quantum and is taken up in Atomic Models and Spectra.
∎The relativistic generalization
[Reserved: Liénard's relativistic power law \(P=\frac{q^{2} \gamma^{6}}{6\pi\epsilon_{0}c}\left(\dot{\vect{\beta}}^{2} -(\vect{\beta}\times\dot{\vect{\beta}})^{2}\right)\) [Lienard:1898] and its manifestly invariant form \(P=-\frac{q^{2}} {6\pi\epsilon_{0}c^{3}}\,a^{\mu}a_{\mu}\); the sharply different penalties for acceleration parallel and perpendicular to the velocity (\(\gamma^{6}\) against \(\gamma^{4}\)), which is why linear accelerators are long and circular machines lose energy to synchrotron radiation (Section 65.7.1); the invariance of the total radiated energy but not of its angular distribution [Landau:1975].]
Angular distribution and beaming
[Reserved: the angular distribution in the wave zone for arbitrary velocity, its forward collimation into a cone of half-angle \(\sim1/\gamma\), and the distinction between power radiated per unit retarded time and per unit observer time; the observational consequences — the sweep of a synchrotron beam past a detector, and relativistic beaming in astrophysical jets (Cosmic Rays and Astroparticle Physics) [Jackson:1999].]
Multipole radiation
Electric dipole radiation
[Reserved: the oscillating dipole, the first radiating system ever solved, treated by Hertz from Maxwell's theory [Hertz:1889]; the near, intermediate and radiation zones; the total power \(P=\ddot{p}^{2} /6\pi\epsilon_{0}c^{3}\) and the \(\omega^{4}\) scaling that governs both antenna efficiency and Rayleigh scattering (Section 65.6.2); the polarization and \(\sin^{2}\theta\) pattern that Hertz measured with his resonator loop.]
Magnetic dipole and electric quadrupole
[Reserved: the next terms in the expansion, suppressed relative to the electric dipole by one power of \((\text{size}/\lambda)\); their distinct angular patterns and parity, connecting to the selection rules of Atomic Models and Spectra and Discrete Symmetries and CPT; the observational contrast with gravitation, where charge conservation and momentum conservation forbid the monopole and dipole terms outright and the leading emission is quadrupole (Gravitational-Wave Theory) [Jackson:1999].]
The multipole expansion of the radiation field
[Reserved: the systematic expansion in vector spherical harmonics, with electric and magnetic multipoles of each order, their radiated power and angular momentum content; the expansion parameter and its failure for sources comparable with the wavelength; the connection to the spherical-harmonic machinery of Ordinary Differential Equations and Sturm–Liouville Theory and its reappearance for photon angular momentum in Quantum Optics and the Photon [Jackson:1999].]
Antennas
Radiation resistance
[Reserved: the short dipole treated as a circuit element, with the radiated power written as \(\tfrac{1}{2}I^{2}R_{\text{rad}}\) and \(R_{\text{rad}}=\tfrac{2\pi}{3}Z_{0}(\ell/\lambda)^{2}\) in terms of the impedance of free space \(Z_{0}=376.730\,\mathrm{\Omega}\); the half-wave dipole with its measured \(73\,\mathrm{\Omega}\); radiation as a genuine loss term in the energy balance of a circuit, which is how Electrodynamics in Matter meets this chapter [Hertz:1889] [Jackson:1999].]
Arrays, directivity and reciprocity
[Reserved: linear arrays and the array factor, directivity and gain in dimensionless units, beam steering by phasing; the reciprocity theorem making the transmitting and receiving patterns of an antenna identical; aperture antennas and the diffraction limit shared with Experiment: Wave Optics; the practical instruments this section must cover are radio telescopes and interferometric arrays, whose resolution argument is the same one used for stellar interferometry there [Jackson:1999].]
Radiation reaction
The Abraham–Lorentz force
[Reserved: the self-force \(\vect{F}_{\text{rad}}=\frac{q^{2}} {6\pi\epsilon_{0}c^{3}}\dot{\vect{a}}\) obtained from the work–energy balance over a period, by Abraham [Abraham:1904] and Lorentz [Lorentz:1909]; the characteristic time \(\tau=q^{2}/6\pi \epsilon_{0}mc^{3}\), equal to about \(6.3\times 10^{-24}\,\mathrm{s}\) for an electron, and the length \(c\tau\) far below any distance at which classical electrodynamics is tested; the divergent self-energy of a point charge and the failure of every classical extended-electron model to remove it.]
The Lorentz–Abraham–Dirac equation
[Reserved: Dirac's covariant derivation from conservation of
energy–momentum in a tube around the world line, with mass
renormalization absorbing the divergent piece [Dirac:1938a]; the
resulting third-order equation of motion and the Schott term; the
statement that this is a classical, not quantum, renormalization, and
the honest note that the key Dirac:1938 in this treatise's
bibliography is the unrelated large-numbers paper.]
Runaways, pre-acceleration and the domain of validity
[Reserved: the exponentially growing free solutions \(a\propto\ee^{t/\tau}\); the acausal pre-acceleration that appears when they are excluded by a final condition; the argument that both pathologies live at times of order \(\tau\), where the classical theory has no claim to validity because pair creation and the Compton wavelength intervene (Quantum Electrodynamics and Renormalization); the standing of radiation reaction as an effective, not fundamental, statement [Landau:1975].]
The Landau–Lifshitz reduction of order
[Reserved: the substitution of the zeroth-order equation of motion into the small radiation-reaction term, producing a second-order equation free of runaways and pre-acceleration and accurate to the same order in \(\tau\) [Landau:1975]; the criterion for the reduction to be legitimate; and this equation as the one actually integrated in accelerator and laser-plasma modelling.]
Radiation reaction observed
[Reserved: radiation damping as a routine engineering fact in electron storage rings, where the damping times and the equilibrium emittance follow from the same self-force; and the direct laboratory observation in the collision of a laser-wakefield electron beam with an intense laser pulse, where the measured electron energy loss departs from the radiation-free prediction [Cole:2018] [Poder:2018]; what these data do and do not distinguish among the competing equations of Sections 65.5.2 and 65.5.4.]
Scattering
Thomson scattering
[Reserved: a free charge driven by a plane wave re-radiating as an oscillating dipole, giving the frequency-independent cross-section \(\sigma_{\text{T}}=\tfrac{8\pi}{3}r_{e}^{2}=6.6525\times 10^{-29}\,\mathrm{m}^{2}\) with \(r_{e}=q_{e}^{2}/4\pi\epsilon_{0}m_{e}c^{2}\) [Thomson:1906]; the \((1+\cos^{2}\theta)\) angular dependence for unpolarized light and the complete polarization of the scattered beam at \(90\) degrees; the observational uses — the solar K-corona, the electron density diagnostics of Plasmas and Magnetohydrodynamics, and the Thomson scattering that sets the last-scattering surface of Experiment: The Cosmic Microwave Background.]
A beam of radiation traversing a gas is scattered by the free electrons it contains, and at low photon energy the cross-section per electron is independent of the frequency of the radiation:
a fixed area attached to the electron and to nothing else [Thomson:1906]. The scattered radiation from an unpolarized beam follows a \(1+\cos^{2}\theta\) angular distribution and is completely polarized at right angles to the beam. Because the cross-section scales as the inverse square of the mass, scattering by nuclei is negligible by six orders of magnitude, and a Thomson measurement counts electrons.
Derivation. A free charge in the field of a plane wave is driven with acceleration \(\vect{a}=q_{e}\vect{E}/m_{e}\), the magnetic force being smaller by \(v/c\) and negligible for a non-relativistic response. Substituting into the Larmor formula Equation (65.1) and taking the time average,
The incident irradiance of the same wave is \(I=\epsilon_{0}c\avg{E^{2}}\), and the cross-section is by definition the ratio of the power removed from the beam to the power per unit area carried by it:
the last step by substituting \(r_{e}\) from Equation (65.5). The mean-square field \(\avg{E^{2}}\) cancels between numerator and denominator, so the answer does not depend on the intensity; and no frequency ever entered, because a free charge has no internal time scale — which is precisely why the cross-section is frequency-independent, and why the failure of that independence, in Section 65.6.4, is a failure of the classical account rather than a refinement of it.
∎Rayleigh scattering and the colour of the sky
[Reserved: scattering by bound charges below resonance, with \(\sigma\propto\omega^{4}\propto\lambda^{-4}\) from the driven-oscillator polarizability [Rayleigh:1871]; Rayleigh's later demonstration that the molecules of the air themselves suffice, so the blue of the sky needs no dust [Rayleigh:1899]; the fluctuation account that explains why a dense but uniform medium does not scatter at all [Einstein:1910]; the measured evidence — the spectrum of skylight, its strong polarization at \(90\) degrees from the Sun, and the reddening of the setting Sun.]
The light of a clear daytime sky arrives from every direction and not merely from the Sun, and it is blue: measured against the solar spectrum it is enriched at short wavelengths in a ratio that rises steeply towards the violet, the scattering coefficient of the air varying as the inverse fourth power of the wavelength [Rayleigh:1871]. The same light is strongly polarized, most strongly along the great circle \(90\) degrees from the Sun. The molecules of the air suffice to produce the whole effect; no suspended dust is required, and the observed brightness agrees with what the known number of molecules gives [Rayleigh:1899]. The setting Sun is red for the complementary reason: the short wavelengths have been scattered out of the long atmospheric path.
Derivation. A molecule is not a free charge but a bound one, and well below its resonances the field of the wave induces a dipole moment \(\vect{p}=\alpha\vect{E}\) with a polarizability \(\alpha\) that is very nearly constant in frequency — this is the same low-frequency limit of the oscillator response that makes the static permittivity of a gas frequency-independent. Its second derivative is not constant: for \(\vect{E}\propto\ee^{-\ii\omega t}\), \(\ddot{\vect{p}}=-\omega^{2}\alpha\vect{E}\). Feeding this into the dipole form of the Larmor formula, \(P=\avg{\ddot{p}^{2}}/6\pi \epsilon_{0}c^{3}\), and dividing by the incident irradiance \(I=\epsilon_{0}c\avg{E^{2}}\) as in Equation (65.7),
The whole of the colour is in the two time derivatives, each contributing one power of \(\omega\) to the amplitude and hence two to the cross-section: an oscillating dipole radiates in proportion to its acceleration, not its displacement. Numerically, blue light near \(450\,\mathrm{nm}\) is scattered some four times more strongly than red near \(650\,\mathrm{nm}\), which is the observed ratio.
The polarization follows from the \(\sin^{2}\theta\) pattern of Phenomenon 65.1. The induced dipole lies in the plane transverse to the incident ray, so an observer looking at right angles to the beam sees only the dipole component perpendicular to the plane containing the incident and scattered rays — the other component points at the observer and radiates nothing that way. The scattered light is therefore completely polarized at \(90\) degrees for an ideal dipole scatterer, and strongly but not completely polarized in the real sky, where multiple scattering and molecular anisotropy dilute it.
∎Mie scattering
[Reserved: the exact solution for a homogeneous sphere of arbitrary size and refractive index [Mie:1908]; the crossover from the Rayleigh regime to the size-independent, colour-neutral regime that makes clouds and fog white; forward-peaked scattering, the resonances of the series, and their use in aerosol sizing and lidar; the connection to the boundary-value machinery of Electrodynamics in Matter.]
Where the classical account fails: Compton scattering
[Reserved: the measured wavelength shift \(\Delta\lambda= \frac{h}{m_{e}c}(1-\cos\theta)\) [Compton:1923], which classical scattering forbids outright since a driven charge re-radiates at the driving frequency; the Klein–Nishina cross-section that replaces Thomson's at photon energies approaching \(m_{e}c^{2}\) [Klein:1929a]; the statement of the boundary — Thomson scattering is the low-energy limit of a quantum process — with the experiment itself treated in The Photon: Photoelectric and Compton Effects and the theory in Quantum Electrodynamics and Renormalization.]
X-rays scattered from light elements contain, besides radiation of the incident wavelength, a component of longer wavelength. The shift grows with the scattering angle, vanishing in the forward direction and reaching its maximum on backscattering, and it depends on nothing else: not on the incident wavelength, and not on the scattering material [Compton:1923]. Its measured value is
the constant being the Compton wavelength of the electron [Tiesinga:2021]. Classical scattering forbids the effect outright: a charge driven at frequency \(\omega\) re-radiates at \(\omega\), so the scattered wavelength must equal the incident one at every angle.
Derivation. The derivation is not classical, and that is its content. Treat the radiation as quanta of energy \(hc/\lambda\) and momentum \(h/\lambda\), and let one of them strike an electron of mass \(m_{e}\) at rest, scattering through \(\theta\) while the electron recoils with momentum \(\vect{p}_{e}\) and energy \(E_{e}\). Momentum conservation, squared, gives
and energy conservation gives \(E_{e}=m_{e}c^{2}+hc\left(1/\lambda-1/\lambda'\right)\). Substituting both into the relativistic relation \(E_{e}^{2}=p_{e}^{2}c^{2} +m_{e}^{2}c^{4}\) of Relativistic Dynamics, the terms in \(m_{e}^{2}c^{4}\) cancel, as do the terms in \(h^{2}c^{2}\left(1/\lambda^{2}+1/\lambda'^{2}\right)\), and what remains is
Writing \(1/\lambda-1/\lambda'=(\lambda'-\lambda)/\lambda\lambda'\), the factor \(\lambda\lambda'\) cancels from both sides and Equation (65.9) follows immediately, with \(\Delta\lambda=\lambda'-\lambda\).
Every step used conservation of energy and momentum and nothing else. What it did not use is any property of the electron beyond its mass, which is why the shift is independent of the material; and what it did use is that the radiation carries energy and momentum in localized quanta with \(E=pc\), which classical electrodynamics does not provide. The classical result is recovered where the quantum of energy is negligible against \(m_{e}c^{2}\): there \(\Delta\lambda/\lambda\to0\), the shift disappears, and the cross-section returns to Thomson's Equation (65.5).
∎Radiation from relativistic charges and in matter
Synchrotron radiation
[Reserved: emission by a charge on a circular orbit, worked out classically by Schott [Schott:1912] long before it was seen; the critical frequency \(\omega_{c}\simeq\tfrac{3}{2}\gamma^{3}c/\rho\), the broad power-law spectrum below it, the \(\gamma^{4}\) energy loss per turn and the linear polarization in the orbital plane; the first direct observation, in the General Electric \(70\,\mathrm{MeV}\) synchrotron [Elder:1947]; the two consequences that matter — synchrotron light sources as instruments, and synchrotron emission as the diagnostic of relativistic electrons in cosmic magnetic fields (Plasmas and Magnetohydrodynamics and Cosmic Rays and Astroparticle Physics).]
Electrons circulating at relativistic energy in a magnetic field emit light that was first seen directly, as an intense spot on the orbit visible through the glass wall of the vacuum chamber of a \(70\,\mathrm{MeV}\) machine [Elder:1947]. The emission is not confined to the orbital frequency and its low harmonics, as a non-relativistic calculation would have it: it is spread over an enormous range of harmonics up to a critical frequency far above the orbital one, it is collimated into a narrow cone tangent to the orbit so that a fixed detector sees a brief flash once per turn, and it is linearly polarized in the plane of the orbit. The energy lost per turn grows very steeply with the electron energy, which is the practical limit on circular electron accelerators and the reason such machines are also the brightest laboratory light sources.
Synchrotron radiation: derive the emission of a charge on a circular orbit at arbitrary speed from the acceleration field and the relativistic generalization of the Larmor formula, obtaining the forward collimation into a cone whose half-angle is the reciprocal of the Lorentz factor, the critical frequency proportional to the cube of that factor divided by the orbit radius, the power-law spectrum below it, the energy radiated per turn, and the degree and sense of the polarization.
Bremsstrahlung
[Reserved: radiation emitted when a charge is deflected in the Coulomb field of a nucleus; the classical impact-parameter treatment and Kramers' spectrum for the continuous X-ray background of a tube [Kramers:1923], with its sharp short-wavelength cutoff at the electron kinetic energy — a quantum feature no classical calculation gives; the Bethe–Heitler cross-section and the radiation length [Bethe:1934]; screening; thermal bremsstrahlung as the X-ray emission of hot plasmas (Plasmas and Magnetohydrodynamics).]
Cherenkov radiation
[Reserved: the faint blue light emitted by a charge moving faster than the phase velocity of light in a medium, discovered and characterized by Cherenkov [Cherenkov:1934] after Vavilov's suggestion that it was not fluorescence [Vavilov:1934]; the Frank–Tamm theory [Frank:1937] with the cone condition \(\cos\theta_{c}=1/n\beta\) and the emitted energy per unit path and unit frequency \(\propto\left(1-1/\beta^{2}n^{2}\right)\); the threshold as a velocity measurement, which is what makes ring-imaging Cherenkov detectors and the water tanks of Experiment: Neutrino Oscillations work.]
A fast charged particle traversing a transparent medium makes it glow with a faint blue light. The light is not fluorescence: it is undiminished by quenching agents that suppress fluorescence, it appears in every pure liquid tested, and — decisively — it is not emitted isotropically but along a cone about the particle's track, sharply directed forward, and polarized in the plane containing the track and the direction of emission [Cherenkov:1934]. The half-angle of the cone is fixed by the speed of the particle and the refractive index of the medium,
and there is a sharp threshold: no light at all unless \(\beta>1/n\) [Frank:1937]. Measuring the cone angle therefore measures the speed of the particle directly.
Derivation. The medium is polarized by the passing charge and relaxes behind it, so that each point of the track acts as a momentary source of a wavelet spreading at the phase speed \(c/n\). Consider the wavelet emitted at the moment the particle passes a point \(O\), and the position of the particle a time \(t\) later. The wavelet has expanded to a sphere of radius \(ct/n\) about \(O\), while the particle has advanced a distance \(vt=\beta ct\) along the track. When \(\beta n>1\) the particle has outrun its own wavelets, and the spheres emitted at all earlier points of the track possess a common tangent cone with apex at the particle: the wavelets add in phase on that cone and cancel elsewhere. Its half-angle \(\theta_{c}\), measured from the track, satisfies
which is Equation (65.12), the ratio being independent of \(t\) so that the construction is consistent along the whole track.
The threshold is contained in the same formula: a cosine cannot exceed unity, so a real cone exists only for \(n\beta\ge1\), and at \(n\beta=1\) it degenerates to \(\theta_{c}=0\), the light being emitted straight ahead. Nothing here is peculiar to electrodynamics — it is the construction that gives the Mach cone of a supersonic body — and nothing in it violates relativity, since what is exceeded is the phase speed of light in the medium and never \(c\).
∎Transition radiation
[Reserved: radiation emitted when a charge crosses a boundary between media of different permittivity, predicted by Ginzburg and Frank [Ginzburg:1946]; the total energy proportional to \(\gamma\), which is what makes transition-radiation detectors the standard particle-identification tool at high \(\gamma\); foil stacks and the X-ray yield; used by the detectors of Cosmic Rays and Astroparticle Physics.]