Larmor's Formula and the Radiating Dipole

How much power? Larmor's formula

Take the radiation field E_{\text{rad}}=\mu_0 q a\sin\theta/(4\pi r), form the Poynting flux S=E_{\text{rad}}^2/(\mu_0 c), and integrate over a large sphere. Every 1/r^2 in S is cancelled by the sphere's r^2 area, leaving a finite total power independent of radius — the Larmor formula.

P = \frac{\mu_0 q^{2} a^{2}}{6\pi c} = \frac{q^{2}a^{2}}{6\pi\varepsilon_0 c^{3}}

Larmor's formula: radiated power grows as the square of the acceleration and of the charge.

The radiation pattern: the doughnut

The angular distribution carries the \sin^2\theta factor, with \theta measured from the acceleration axis. So there is no radiation along the direction of acceleration and maximum radiation broadside to it. Spun into three dimensions, the pattern is a doughnut (torus) whose axis is \mathbf{a} and whose hole opens along the line of oscillation.

\frac{dP}{d\Omega} = \frac{\mu_0 q^{2} a^{2}}{16\pi^{2} c}\,\sin^{2}\theta

The angular distribution of dipole radiation; \theta is the angle from the acceleration.

Worked example: the classical atom must collapse

Rutherford's planetary atom has an electron orbiting a proton. Circular motion is acceleration, so by Larmor the electron must radiate continuously, lose energy, and spiral inward. Let us estimate how long it survives — and watch classical physics annihilate the atom, the very crisis that forced quantum mechanics.

Oscillating dipoles and antennas

Most real radiators are oscillating electric dipoles: charge sloshing back and forth, p(t)=p_0\cos\omega t. The acceleration goes as \omega^2 p_0\cos\omega t, so the radiated power scales as \omega^4 — the dramatic frequency dependence behind why the sky is blue (Rayleigh scattering favours high frequencies) and why antennas only work efficiently near resonance.

\langle P\rangle = \frac{\mu_0\, p_0^{2}\,\omega^{4}}{12\pi c}

Time-averaged power of an oscillating electric dipole — note the fierce \omega^4.

A real dipole antenna converts drive power into radiation as though through a resistor — its radiation resistance. For a short antenna of length d\ll\lambda the idealized value is R_{\text{rad}}\approx 790\,(d/\lambda)^2\,\Omega: efficiency climbs steeply with d/\lambda, which is exactly why usable antennas are sized to a substantial fraction of the wavelength they broadcast.

R_{\text{rad}} \approx \frac{2\pi}{3}\sqrt{\frac{\mu_0}{\varepsilon_0}}\left(\frac{d}{\lambda}\right)^{2} \approx 790\left(\frac{d}{\lambda}\right)^{2}\ \Omega

Radiation resistance of an idealized short, thin, centre-fed dipole; \sqrt{\mu_0/\varepsilon_0}\approx 377\,\Omega is the impedance of free space.