Paying back the "honest line" of Episode 1 — a field that's lagging yet carries no news, and when it finally starts to carry news
In Episode 1 we said a force points at the source's "just now" — it arrives with a lag. But in the closing "honest line" we left one piece of homework — the field of a source in uniform straight-line motion, though lagging, somehow points precisely at the "present position" and carries no news. This is where fields get most interesting. As long as it moves quietly (uniformly), a field carries no information. A field only truly flings a report — news — at the speed of light when the source accelerates. That flying report is radio waves, light, gravitational waves — that is, radiation. This bonus episode pays back Episode 1's homework while sorting "the quiet field" and "the shaking field" with a single ratio, \(r/\lambda\).
Consider a charge moving straight at constant speed. Measure its electric field and, astonishingly, though there should be a light-lag, it points precisely at the charge's "present position". This is because relativistic effects exactly correct for the lag by looking ahead (for uniform motion the future position is fully predictable, so nature "extrapolates" to make things consistent).
As a result, from the field of a quietly moving source you cannot extract the information "the source moved." The same pattern simply translates along in step with the source. It lags, but no news rides on it. That's why in Episode 1 we noted the figure was "schematic."
So the knowing in "if the Sun vanished, Earth would know 8 min 20 s later" only holds once there is a Sun vanishing = an abrupt change = acceleration. A source that merely exists quietly emits no news.
So what if the source accelerates? There is a famous thought experiment. Suppose a charge that had been moving straight at constant speed suddenly stops at some instant. The news that it stopped can only spread at the speed of light \(c\). So the field inside the sphere of radius \(r=ct\) knows the new fact "it already stopped" and points at the new position, while the field outside still "doesn't know" and keeps pointing at the future position it would have had if it hadn't stopped. Inside and outside, the field directions disagree — and in the thin shell at that boundary, a sideways kink forms in the field lines.
Here the 1/r² of Episode 5 comes into play. The bound field clinging to the source (the Coulomb field) thins out as \(1/r^2\), just as in Episode 5. But the radiation field born of acceleration thins out only as \(1/r\) — it fades much more slowly with distance.
The farther you go, the more the \(1/r\) radiation inevitably overtakes the \(1/r^2\) binding. So the far reaches belong to radiation. Starlight reaches us across billions of light-years precisely because radiation is \(1/r\). In terms of energy the intensity is the square, \(1/r^2\) — it thins only as much as the spreading over a spherical surface (Episode 5), and no more. The news crosses the universe with almost no loss.
Where does the lead role hand over from binding to radiation? The boundary is decided, again, by a ratio. If the source shakes at frequency \(f\), the radiation's wavelength is \(\lambda=c/f\). The ratio of the distance \(r\) to this \(\lambda\), \(r/\lambda\), is ——
For \(r\ll\lambda\), the quiet force near the source; for \(r\gg\lambda\), the flying radiation. The single ratio \(r/\lambda\) sorts the two faces of one field.
The figure below. Turn acceleration on and the central charge (green) shakes up and down, and radiation wavefronts (news) spread outward at the speed of light. The dashed circle is the boundary \(r=\lambda\) — inside is the near zone (the quiet bound field \(1/r^2\)), outside is the wave zone (radiation \(1/r\)). The frequency slider changes \(\lambda=c/f\), stretching or shrinking the boundary circle.
Turn acceleration off and the charge is at rest (representing uniform motion). No wavefronts come out at all — the field merely clings quietly around the source and carries no news. Switch on and off and feel in your bones that "only acceleration radiates."
How much radiation comes out (the radiated power) is proportional to the square of the acceleration \(a\) (Larmor's formula).
Double the acceleration and the radiation is fourfold. The harder you shake it, the more violently it scatters news. An antenna emits radio waves because it makes electrons oscillate = accelerate at high speed. The Sun and a light bulb shine because heat accelerates charged particles roughly. Rainbows, X-rays, the blue-white light of a synchrotron — all of it is radiation from accelerating charges. At constant speed, it never shines.
Episode 1's homework closes here. A change in force (news) always travels at the speed of light \(c\) — but what flings it is acceleration. A quiet (uniform-motion) field, even lagging, carries no news and merely clings to the source as \(1/r^2\). Acceleration carves a kink into the field and releases it at \(c\) as \(1/r\) radiation. The subject of "somehow it travels at the speed of light" was not the quiet force but the shaking field.
That the field of a uniformly moving point charge effectively points at the present (extrapolated) position and carries no information; that acceleration produces radiation (a transverse radiation field) which goes as \(1/r\) (intensity \(1/r^2\)) and prevails far away over the bound field \(1/r^2\); that the boundary between the near field and the radiation field lies at \(r\sim\lambda\) (\(\lambda=c/f\)); that the radiated power is Larmor's \(P=q^2a^2/6\pi\varepsilon_0c^3\propto a^2\); and that similarly, for gravity, an accelerating mass emits gravitational waves — all of these are established standard physics.
The "kink of a suddenly stopped charge" is the classic picture for grasping the origin of radiation (since J.J. Thomson), but strictly it is an idealization involving infinite acceleration; in reality a finite-time acceleration gives a smooth wave packet. "Field lines" are a visualization aid (the real body is the field vector), and the wavefronts in the figure are a schematic of the spread of the radiation news (real dipole radiation has an angular dependence — weak along the oscillation axis and strong transversely — but the figure shows isotropic circles for simplicity). Also, "uniform motion carries no information" concerns the idealization of eternally continuing uniform motion; in a real process that accelerates and then enters uniform motion, radiation is emitted once, during that acceleration.
Paying back Episode 1's homework. The field of a source in uniform motion, even lagging, points at the "present position" via the relativistic look-ahead correction and carries no news (information) — a bound field clinging to the source as \(1/r^2\). But when the source accelerates, the field directions disagree between the inside (the region that learned the new fact) and the outside (the region that doesn't know yet), and a sideways kink forms at the boundary. This kink, escaping at the speed of light \(c\), is radiation (electromagnetic waves, light, gravitational waves).
The radiation field thins out only as \(1/r\) (binding is \(1/r^2\)), so the far reaches belong to radiation. The handover of the lead role is sorted by the ratio \(r/\lambda\) (\(\lambda=c/f\)) — \(r\ll\lambda\) near zone (the quiet force), \(r\gg\lambda\) wave zone (the news). The strength of radiation is the square of the acceleration, \(P\propto a^2\) (Larmor). Antennas, stars, and rainbows are all radiation from accelerating charges, and uniform motion never shines. The subject of "somehow it travels at the speed of light" was not the quiet force but the shaking field — Episode 1's "honest line" closes here.
Print / save as PDF: ⌘+P (Ctrl+P on Windows). On screen, "Acceleration: on/off" toggles whether radiation appears, and the frequency slider moves the boundary r=λ. "Show the answer" opens each solution.