Accelerated Charges as Sources
Every time a charged particle accelerates, it disturbs the electric and magnetic fields around it, sending ripples outward through space at the speed of light. These ripples are electromagnetic (EM) waves — the same family that includes radio waves, visible light, X-rays, and gamma rays. The key insight is that a charge sitting still, or even moving at constant velocity, does not radiate energy. Only acceleration produces radiation. This is one of the most profound results in classical physics, connecting the motion of matter to the propagation of light itself. A practical way to generate EM waves is to force charges to oscillate back and forth in an antenna. When electrons in a wire are driven by an alternating voltage, they accelerate periodically, creating oscillating electric and magnetic fields that detach from the antenna and travel outward as a wave. The frequency of the radiated wave exactly matches the frequency at which the charges oscillate — tune the oscillation and you tune the emission. This is precisely how radio transmitters, Wi-Fi routers, and cell towers work. Understanding accelerated charges as sources of EM radiation also explains natural phenomena: the glow of hot objects (thermal radiation from thermally agitated charges), the emission of light by atoms (electrons changing energy levels undergo rapid acceleration), synchrotron radiation from electrons bent in magnetic fields, and even the 'antenna' behavior of individual molecules. Mastering this concept is the gateway to understanding almost every source of light in the universe.
- Static and Constant-Velocity Charges Do Not Radiate
- Acceleration Kinks the Field Lines
- Oscillating Charges and Antennas
- The Larmor Formula: Power Radiated
- Direction and Polarisation
Static and Constant-Velocity Charges Do Not Radiate
A stationary charge creates a static electric field — field lines point radially outward and nothing changes with time, so no wave is produced. A charge moving at constant velocity carries its field with it, but because the field pattern moves uniformly without oscillating, it still does not radiate energy away. Think of dragging a perfectly smooth boat through still water at constant speed: you produce no persistent wake of spreading waves.
Acceleration Kinks the Field Lines
When a charge suddenly accelerates, the 'news' of its new position and velocity can only travel outward at the speed of light c. Regions of space far from the charge still 'see' the old field configuration, while regions close to the charge already reflect the new one. This mismatch creates a kink — a sharp bend in the electric field lines. That kink propagates outward at c. The propagating kink is the electromagnetic wave. The greater the acceleration, the sharper and larger the kink, meaning more energy is radiated.
Oscillating Charges and Antennas
If a charge oscillates sinusoidally with frequency f and amplitude A, it undergoes continuous periodic acceleration a = −(2πf)²x (simple harmonic motion). Each cycle of oscillation generates one complete cycle of the EM wave. In a half-wave dipole antenna, electrons surge back and forth along a conductor of length λ/2, where λ = c/f is the wavelength. The antenna radiates most strongly perpendicular to its axis and not at all along its axis — a pattern called the radiation pattern or antenna beam pattern.
The Larmor Formula: Power Radiated
The total power P radiated by a non-relativistic accelerating point charge is given by the Larmor formula: P = q²a²/(6πε₀c³). Notice the strong dependences: doubling the charge quadruples the radiated power, and doubling the acceleration also quadruples it. The factor of c³ in the denominator means that radiation is an inherently relativistic phenomenon — it exists because the speed of light is finite and field updates cannot travel instantaneously.
Direction and Polarisation
The electric field of the radiated wave oscillates in the plane containing the acceleration vector and the direction of propagation — this defines the polarisation. Radiation is zero directly along the acceleration axis and maximum perpendicular to it (the sin²θ dependence in the angular power distribution). The magnetic field is always perpendicular to both E and the propagation direction. Both fields oscillate in phase and in step with each other, carrying energy away from the source.
The Key Equation
- P — Radiated Power (W (watts)): Total electromagnetic power emitted per unit time by the accelerating charge.
- q — Electric Charge (C (coulombs)): The net charge of the particle; appears squared so sign does not matter — both positive and negative charges radiate.
- a — Acceleration (m s⁻²): The instantaneous acceleration of the charge. Radiation is proportional to a², so larger accelerations produce dramatically more power.
- \varepsilon_0 — Permittivity of Free Space (C² N⁻¹ m⁻²): The electric constant of vacuum, approximately 8.85 × 10⁻¹² C² N⁻¹ m⁻².
- c — Speed of Light (m s⁻¹): The speed of electromagnetic waves in vacuum, approximately 3 × 10⁸ m s⁻¹. Appears cubed, making the radiated power very sensitive to this fundamental constant.
Worked Example
Problem. An electron (charge q = 1.6 × 10⁻¹⁹ C, mass m = 9.11 × 10⁻³¹ kg) in a simple radio antenna oscillates with a peak acceleration of a = 1.0 × 10¹² m s⁻². Using the Larmor formula, estimate the peak radiated power. (ε₀ = 8.85 × 10⁻¹² C² N⁻¹ m⁻², c = 3.0 × 10⁸ m s⁻¹)
Answer. Peak radiated power per electron ≈ 5.7 × 10⁻³⁰ W
Origins · 1 minHeinrich Hertz and how this was discovered
Hertz was the first person to experimentally produce and detect electromagnetic waves deliberately in the laboratory (1887–1888), confirming Maxwell's theoretical prediction. He built a spark-gap transmitter — essentially an oscillating charge source — and detected the radiated waves with a loop antenna several metres away. He measured the wavelength and showed the waves reflected, refracted, and polarised just like light, proving light itself is an EM wave. The SI unit of frequency, the hertz (Hz), is named in his honour. Hertz famously wrote that his discovery of radio waves had 'no use whatsoever' — he saw it as pure science. Within a decade, Marconi used the same waves to transmit signals across the Atlantic Ocean.
Related Concepts
- Structure of Electromagnetic Waves — Understanding how E and B fields are arranged in a propagating EM wave clarifies what the kink in field lines from an accelerated charge actually looks like as it travels outward.
- The Electromagnetic Spectrum — All regions of the EM spectrum — from radio waves to gamma rays — are produced by accelerating charges at different frequencies, making the spectrum the direct catalogue of charge oscillation phenomena.
- Oscillating Electric Charges and AC Circuits — Antenna operation relies on alternating current, where charges undergo periodic acceleration — connecting circuit theory directly to wave generation.
- Atomic Emission Spectra — Light emitted by atoms arises because electrons undergo rapid acceleration (deceleration and transition between orbits), making atomic spectra a quantum-mechanical extension of the accelerated-charge principle.