ELECTRIC DIPOLE RADIATION

Take the simplest arrangement that can possibly radiate. A tiny rod with a positive charge at the top, a negative charge at the bottom, and a current that sloshes between them at a fixed angular frequency ω. Write the whole thing as a single number that oscillates:

That is an oscillating electric dipole. Its magnitude is the charge on the rod times the separation; in plain words, p is the "how far apart, how much charge" number that measures how strongly the system pulls on a distant test charge. The moment you demand that p changes with time, Maxwell's equations oblige — they produce a field everywhere in space, and that field splits cleanly into two regimes with very different physics.

Close to the dipole (r much smaller than λ) the field looks like a frozen instantaneous dipole field that breathes in place as p flips sign. Energy flows out on one quarter-cycle and back in on the next, exactly the way energy sloshes between a capacitor's plates — it is stored, not radiated. Nothing escapes. This is the near-field zone, and its amplitude falls off as 1 / r³. Cube the distance and you have killed the field; the near-field dies too fast to reach anyone.

Far from the dipole (r much larger than λ) the field looks like an outgoing spherical wave. Energy flows outward radially and never comes back. The amplitude falls as 1 / r — slow enough that integrating the Poynting flux over a sphere of radius r gives a result independent of r, because 1/r² × (area of sphere) = 1/r² × 4πr² = 4π. The same power leaves every concentric shell, forever. This is the far-field zone, the radiation zone, and it is literally light — or radio waves, or microwaves, depending on the frequency.

Between them is a single crossover radius:

Inside r_t the dipole "owns" its field. Outside r_t the field has forgotten where it came from — it is just a wave, heading for the stars.

Inside the transition ring the lilac loops wrap tightly around the dipole and close back onto themselves. Every loop is a section of the instantaneous dipole field at some retarded time — think of it as a photograph of the dipole's field the way a nearby observer would see it, propagated outward at speed c. The loops breathe: they grow, shrink, and flip polarity as the dipole does.

Now look at a loop as it approaches the dashed ring at r = c/ω. Just as it reaches that radius, the next half-cycle of the dipole reverses polarity. A new loop starts forming right at the source with the opposite orientation, and pushes against the old one. The two loops cannot coexist in contact — so the old one pinches off, detaches cleanly, and begins propagating outward. That is the moment fields leave their source. One half-cycle later the same thing happens again with the opposite polarity. The dipole emits an outgoing wavefront every half period, pinched off at the same ring.

That pinch-off is the whole story of radiation. Without oscillation there is no pinching — the quasi-static field just hugs the source forever, which is exactly what a permanently polarised molecule does. Without the transition radius there is no topological handle on where the loop can detach — a slow oscillation gives the loop time to reach infinity before the source changes its mind, and a fast oscillation pinches it off almost immediately. With both, you get a periodic stream of outgoing loops. By the time they reach the eye they have fused into a smooth far-field that the human brain recognises as "a wave", but the origin of every wave is a single loop that once wrapped a source and then let go.

The Maxwell equations made the prediction in 1865: if current can accelerate in a wire, it must radiate, and the radiation must travel at the speed calculated from ε₀ and μ₀ alone. Heinrich Hertz believed it enough to design the experiment, built a spark-gap dipole in 1887, and measured the first radio waves at Karlsruhe in 1888. He spaced a reflector a known distance behind his transmitter, found the nodes of the resulting standing wave, and computed c from the measurement. It agreed with Maxwell's prediction to within his error bars. Three years later Marconi was turning the demonstration into a commercial product, and the whole of twentieth-century radio was born out of the pinch-off you are looking at in the figure above.

Write the exact retarded-time solution of an oscillating dipole and expand in powers of (ωr/c) = 2π r/λ. The result is a sum of terms with different powers of 1/r. Keep only the dominant one in each regime and you get a clean asymptotic picture.

The near-zone, quasi-static limit gives

where p(t) = p₀ cos(ω·t) evaluated at the point's retarded time t − r/c (a signal travelling at speed c takes r/c seconds to arrive, so "now here" is tied to "then there"). These are exactly the components of an electrostatic dipole — dressed up with a slow time dependence. The fall-off goes as 1/r³.

The far-zone, radiation limit gives a single transverse component:

and the magnetic far-field is locked to it by the vacuum triad B_φ = E_θ / c. Those two fields, transverse and in phase, are a plane wave in the local patch of sky — exactly the thing §08.2 described. The fall-off goes as 1/r. Energy flows outward as an honest radial Poynting flux, and no amount of waiting gets it back.

At r = r_t the two terms are comparable; inside, one wins by factors of (λ/r)², outside, the other wins by the same factor. The eye is happy treating the transition as a clean edge even though the exact solution is smooth — the regimes are separated by three orders of magnitude within half a wavelength either side.

Take the time-averaged Poynting vector of the far-field expression and you get the angular distribution of radiated power:

The sin²θ factor is the famous radiation pattern of a Hertzian dipole. It is zero along the dipole axis (no radiation straight up or straight down), and maximum broadside (θ = π/2, around the equator). The 3D surface of constant dP/dΩ is a torus with its hole threaded by the dipole — the "doughnut".

Integrate over the full sphere using ∫ sin³θ dθ dφ = 8π/3 and the total radiated power drops out:

Two features are worth sitting with. First, the dependence on ω is quartic — double the frequency and the radiated power shoots up by a factor of sixteen. That is Rayleigh's ω⁴ law and it is why the sky is blue: air molecules acting as driven dipoles scatter short-wavelength light sixteen-ish times more efficiently than long-wavelength light, so the scattered component of sunlight we see by looking away from the Sun is tilted toward the blue end. Second, the dependence on p₀ is quadratic: double the charge or double the separation and you get four times the power out. That is the classical antenna-design trade-off, and it is the reason FM stations run kilowatt transmitters on tall masts.

The total-power expression is also the Larmor formula in disguise — §10.1 writes it as P = q²a²/(6πε₀c³) for an accelerating point charge. Substitute a = ω²·d (the peak acceleration of a charge q oscillating with amplitude d at frequency ω, so p₀ = q·d) and the two formulas are identical. Every radiator in §10 is Larmor specialised to some particular charge motion.

The oscillating dipole is the simplest complete radiator — source term, near-field, pinch-off, far-field, angular pattern, total power — all of it follows from one time-dependent p(t). §10.3 scales it up into a practical antenna, swapping the idealised point dipole for a half-wavelength wire that accelerates real electrons and picks up a radiation resistance of 73 Ω. §10.4 takes a relativistic charge bent in a circle, where the same machinery produces the sharp forward cone of synchrotron light. §10.5 takes a non-relativistic charge deflected by another, which is bremsstrahlung — a broadband version of the same pinch-off. And §10.6 finally closes the loop by asking what the radiated field does back to the source — the radiation-reaction seam of classical electrodynamics.

For now, the picture to carry forward is the one the money-shot scene draws: a source, a ring at r = c/ω, and a steady stream of wavefronts pinching off at that ring and disappearing into the sky. That is what an antenna does. That is how light is made.