THE LARMOR FORMULA

§08 and §09 walked a plane wave from free space through a dielectric and out again. Neither section answered the obvious follow-up: where does the wave come from in the first place? Maxwell's equations allow electromagnetic waves; they do not, by themselves, tell you which motion of charge actually launches one.

This is the content of §10. Every broadcast, every X-ray tube, every synchrotron beam line, every star's luminosity — all of it is one result, written down in 1897 by the Irish mathematician Joseph Larmor:

That is the total electromagnetic power radiated by a point charge q with non-relativistic acceleration magnitude a. Three features worth staring at. First, P is proportional to acceleration squared, not velocity — a coasting charge radiates nothing. Second, there is no reference to how far away you measure; P is a global number, the total flux through any enclosing sphere, and the sphere's radius drops out. Third, everything that varies between topics in this module hides inside that a². Circular acceleration gives synchrotron radiation; sinusoidal gives dipole radiation; sudden gives bremsstrahlung. Same formula, different stories about how a is produced.

The cleanest derivation starts from the retarded potentials, pulls out the far-field limit, and invokes the Poynting vector to book the energy flux. For an accelerating point charge, the electric field at a distant observer splits into two pieces: a "velocity field" (quasi-static Coulomb, 1/r², locked to the retarded position) and an "acceleration field" proportional to $a_\perp$ that falls off only as 1/r.

Here $\hat{\mathbf{n}}$ points from the charge's retarded position to the observer, and $\mathbf{a}_{\text{ret}}$ is the charge's acceleration at the retarded time $t_{\text{ret}} = t - r/c$. The double cross-product is just the projection of $\mathbf{a}$ onto the plane perpendicular to $\hat{\mathbf{n}}$ (with a sign flip); what survives in magnitude is $a\sin\theta$, where $\theta$ is the angle between $\hat{\mathbf{n}}$ and the acceleration.

Because $|\mathbf{E}_{\text{rad}}|\sim 1/r$, the Poynting flux $|\mathbf{S}| = |\mathbf{E}|^{2}/\mu_{0}c$ scales as $1/r^{2}$, and integrated power through a sphere is independent of r. That is what radiation means operationally — the far-field contribution that survives to infinity. The velocity field's flux falls off as $1/r^{4}$ and integrates to zero at the horizon.

Substituting $|\mathbf{E}_{\text{rad}}| = (q\,a\sin\theta)/(4\pi\varepsilon_{0}c^{2}r)$ and integrating over the sphere:

The angular integral is standard: $\int_{0}^{\pi}\sin^{3}\theta\,d\theta = 4/3$ and $\int_{0}^{2\pi}d\varphi = 2\pi$, for a solid-angle factor $8\pi/3$. So

One line of algebra after the far-field substitution, and Larmor falls out.

A picture is cheaper than the algebra for remembering the shape.

The lobe is rotationally symmetric about the acceleration axis in 3D; this 2D slice is the cross-section edge-on to the charge. The null along $\mathbf{a}$ is exact — a charge accelerating straight at you radiates nothing in your direction — the maximum is broadside, and the integral of sin²θ over the full sphere (with the sin θ from dΩ) is 8π/3, not 4π.

The pattern is referenced to the instantaneous acceleration vector, not the velocity. For a charge moving in a circle, the acceleration is centripetal and the lobe sweeps with it. Above β ≈ 0.3 a second effect — Doppler beaming in the direction of motion — tilts the lobe forward along $\mathbf{v}$, which is how synchrotron light gets its narrow forward cone (§10.4).

Plug in an electron's charge and an acceleration of exactly 1 m/s², and Larmor returns $P \approx 5.71\times10^{-54}$ W — roughly $10^{36}$ times smaller than a dim LED. The cube of the speed of light in the denominator kills the prefactor; classical radiation at human-scale accelerations is unmeasurable. You need astronomical accelerations to see astronomical power.

The benchmark plot sets the scale. On a log-log axis log P is a straight line of slope 2 against log a.

Two numbers on that chart are worth reading carefully. The electron in the hydrogen ground state sits at $a \approx 9 \times 10^{22}$ m/s² — the Coulomb attraction at a 0.5 Å radius is that fierce — and Larmor then predicts a continuous $\sim\!10^{-7}$ W of radiated power. If that were correct the electron would spiral into the proton in about $10^{-11}$ s, the atom would not exist, and you would not be reading this. Classical electrodynamics applied to the atom gives a catastrophic prediction; Bohr (1913) and Schrödinger (1926) resolved it in the quantum branch. The formula is not wrong — it is being applied outside its domain. For any charge not bound in a quantum ground state (beam, antenna, pulsar magnetosphere), Larmor is exactly right.

The formula above assumes β = v/c is small enough to drop retardation corrections at leading order. Liénard in 1898 handled the fully relativistic case. For acceleration aligned with or perpendicular to velocity, the answer splits cleanly:

Perpendicular acceleration picks up γ⁴, parallel γ⁶ — a γ² multiplier that says parallel radiates harder. Physically, parallel acceleration compresses the retarded-time pulse of radiation by a factor γ, which appears twice in the Poynting flux and once more in the solid-angle factor, totalling γ⁶.

This gap drives modern accelerator architecture. Protons are heavy enough that circular colliders like the LHC (γ ≈ 7500, acceleration perpendicular) bleed tolerable power. Electrons are so light that a machine of the LHC's size would radiate away the full beam energy in under a millisecond — so circular electron machines are run deliberately as light sources (synchrotron, §10.4) and electron colliders are built straight (parallel acceleration, γ⁶ beaten by nobody bending the beam).

One corollary before the next topic. If a free electron sits in an electromagnetic wave of intensity $I$, the wave's E-field drives the electron sinusoidally at the wave frequency; the electron re-radiates via Larmor. Re-radiated power divided by incoming flux defines a scattering cross-section:

$r_e$ is the classical electron radius — the scale at which a uniform ball of charge would have Coulomb self-energy equal to $m_e c^{2}$. (It is not the "actual radius" of an electron; the electron is point-like to every measurement we have. It is the scale below which the classical theory visibly stops making sense.) Plugging in gives $\sigma_{T} \approx 6.65\times10^{-29}$ m² — the opacity of a plasma of free electrons to low-frequency light, why the Sun's interior is opaque, and one of the handful of cross-sections in physics computable exactly without quantum theory.

Thomson is the low-energy limit of Compton scattering. Above $\sim\!m_{e}c^{2}/10$ photon recoil becomes measurable, the wavelength shift $\Delta\lambda = (h/m_{e}c)(1-\cos\theta)$ kicks in, and the classical picture needs a quantum patch (see §08.3). Below that crossover, $\sigma_T$ is exact.

Every remaining topic in the module specialises the formula. §10.2 plugs in the sinusoidal acceleration of an oscillating dipole and picks out the near-field / far-field split — the moment electromagnetic energy leaves its source. §10.3 turns that dipole into an antenna, adds resonance, and closes the loop from Maxwell (1862) to Hertz (1888) to Marconi (1901). §10.4 bends the charge in a ring, cashes in the γ⁴, and makes X-rays on purpose. §10.5 replaces gentle acceleration with a hard Coulomb deflection and gets the continuous bremsstrahlung spectrum. §10.6 asks the honest next question — if an accelerating charge radiates, the charge itself must feel a drag force — and finds classical electrodynamics does not quite manage to write that force down without runaway solutions or an infinite self-energy. That is the seam in the fabric of classical EM, and we will look at it directly.

Hold onto the one identity: power radiated by an accelerating charge is $q^{2}a^{2}/(6\pi\varepsilon_{0}c^{3})$, with a $\sin^{2}\theta$ lobe about the acceleration axis, boosted by $\gamma^{4}$ or $\gamma^{6}$ at high velocities, and proportional to intensity times $\sigma_{T}$ for scattering off a free electron. The rest of §10 is how the acceleration gets set up.