Electromagnetic Waves and Radiation

Why Does an Accelerated Charge Radiate?

A static charge creates only the Coulomb field—a static field that decreases as 1/r². But an accelerated charge creates an additional field—the radiation field—which decreases only as 1/r. This is critically important: the energy flux is proportional to E²/r², and if E ~ 1/r, then flux ~ 1/r², and when integrating over a sphere of increasing radius, we get a finite power. An accelerated charge transmits energy to infinity—and this is precisely what is called radiation.

Liénard–Wiechert Retarded Potentials

The electromagnetic fields from a moving charge arrive with a delay: the potential at point r at time t is determined by the position of the charge at the retarded moment tret, from which the signal propagated at speed c:

|r − r(tret)| = c(t − tret)

Retarded potentials: φ(r, t) = q/(4πε₀) · 1/(κR)|_ret, A = φ v/c², where R = r − r(tret) is the vector from the retarded position, κ = 1 − R̂·β, β = v/c—a parameter that takes into account the "compression" of the field in the direction of motion.

For β → 1 (relativistic motion): κ → 0 in the direction of motion—the fields are strongly "bunched" forward. This is used in synchrotron light sources.

Larmor Formula

The power radiated by a nonrelativistic accelerated charge:

P = q²a²/(6πε₀c³)

Breakdown: q—charge (C); a—acceleration (m/s²); c—speed of light; ε₀—electric constant. P ∝ a²—doubling acceleration increases the power fourfold. P ∝ q²—heavier particles (protons) at the same acceleration radiate (mₑ/mₚ)² ≈ 3×10⁻⁷ times weaker.

Relativistic generalization: P = q²/(6πε₀c) · aμaᵘ, where aμ is the 4-acceleration.

Numerical Example: Electron on a Bohr Orbit

A classical electron moves along the first Bohr orbit: r₁ = 0.529 Å, v = c/137 ≈ 2.19 × 10⁶ m/s. Centripetal acceleration: a = v²/r₁ = (2.19 × 10⁶)²/(0.529 × 10⁻¹⁰) ≈ 9.06 × 10²² m/s²

Larmor power: P = (1.6×10⁻¹⁹)²·(9.06×10²²)²/(6π·8.85×10⁻¹²·(3×10⁸)³) ≈ 4.6 × 10⁻⁸ W

The energy of the hydrogen atom in the ground state: E = −13.6 eV ≈ −2.18 × 10⁻¹⁸ J. Time to "fall" onto the nucleus: t = E/P ≈ 2.18×10⁻¹⁸/4.6×10⁻⁸ ≈ 5 × 10⁻¹¹ s—50 picoseconds!

This is the famous catastrophe: classical electrodynamics predicts that the atom should collapse in ≈50 ps. It was precisely this contradiction that required the creation of quantum mechanics. The quantization of orbits (Bohr, 1913) "forbids" radiation at stationary levels.

Dipole Radiation

For an oscillating electric dipole p(t) = p₀ cos(ωt) ẑ the radiated power:

P = ω⁴ p₀²/(12πε₀c³)

The dependence ~ ω⁴ explains the "Blue Sky": Rayleigh scattering by the atmosphere ~ ω⁴, therefore blue light (f ≈ 600 THz) is scattered (600/400)⁴ ≈ 5 times stronger than red light (f ≈ 400 THz). Blue light is scattered across the entire sky—the sky is blue!

Synchrotron Radiation

An electron in a magnetic field moves in a circle: acceleration a = v²/R. By Larmor: P ~ a² ~ v⁴/R². Synchrotron light sources (ESRF in Grenoble, DESY in Hamburg): electrons are accelerated to E ~ several GeV, γ ~ 10⁴–10⁵. Their radiation is a bright, narrow beam in the X-ray range. It is used for studying the structure of proteins, nanomaterials, and in medical diagnostics.

Cherenkov Radiation

If a charged particle moves in a medium faster than the speed of light in that medium (v > c/n), it emits a "Cherenkov cone." Cone angle: cos θ = c/(nv). Application: particle detectors in high-energy physics (Ring Imaging Cherenkov counters, RICH), used in LHCb at the Large Hadron Collider.

Hawking radiation—a quantum effect at the event horizon of a black hole—is mathematically analogous to the thermal radiation of an accelerated observer (Unruh effect). The temperature of the radiation: T_H = ℏc³/(8πGMk_B). A black hole of the mass of the Sun radiates at T_H ≈ 60 nK—unobservable, but fundamentally important for quantum gravity.

Electromagnetic Waves in Engineering and Medicine

The electromagnetic spectrum spans 20 orders of magnitude in frequency and underlies all of modern civilization. Radio waves are used for AM/FM broadcasting, long-distance communication, radar, and navigation. Microwaves ensure the operation of mobile communications, Wi-Fi, satellite systems, and magnetron ovens. Infrared radiation is used in thermal imaging, remote thermometers, and fiber-optic communication lines. Visible light is the basis for laser technologies: retinal surgery, high-precision cutting and welding of metals, optical lithography for microchip production at resolutions of several nanometers. Ultraviolet is used for sterilizing water and medical equipment, as well as in the photolithography of integrated circuits. X-rays are used in medical diagnostics, computed tomography, and crystallography, allowing the determination of the atomic structure of proteins. Gamma rays are used in nuclear medicine: positron emission tomography and gamma knife for treating brain tumors. Synchrotron radiation (resulting from the braking of relativistic electrons) is a powerful research tool in structural biology, materials science, and nanoscience.

The Larmor formula has an important implication for accelerators: synchrotron radiation limits the maximum energy of an electron storage ring. This is why the LHC accelerates protons (1836 times heavier → losses are 10¹³ times smaller), not electrons. Synchrotrons for electrons (ESRF, APS, SPring-8) use these losses as a source of bright X-rays for research into the structure of proteins and materials.

Assignment: (a) For an electron on a Bohr orbit: calculate P by the Larmor formula and the time to lose all energy (−13.6 eV). (b) An antenna—a dipole 1 m long, current 1 A at a frequency of 100 MHz. Find P_radiation. What is the wavelength? (c) In a synchrotron, an electron with E = 1 GeV (γ = 2000) moves in a circle of R = 10 m. Calculate P in the relativistic case (substitute into P = q²γ⁴v⁴/(6πε₀c³R²)).