Antenna Radiation: From the Hertzian Dipole to Phased Arrays

An accelerating charge is the whole story

A charge sitting still produces only a static field. A charge moving at constant velocity adds a static magnetic field in the lab frame — but still nothing escapes. Only an accelerating charge radiates: push current back and forth in a wire and Maxwell's equations guarantee a self-sustaining electromagnetic wave detaches from the conductor and propagates outward, carrying real energy that never returns. That single fact is the entire physical basis of every antenna built since Heinrich Hertz's spark-gap transmitter in 1887 — an alternating current is just a convenient way to continuously accelerate and decelerate free electrons.

The Hertzian dipole and its doughnut pattern

The simplest radiator is the Hertzian dipole: a wire segment of length dl much shorter than the wavelength, carrying a uniform oscillating current I(t) = I₀cos(ωt). Solving Maxwell's equations for this current gives the far-field electric field:

E_θ = (η₀·k·I₀·dl·sinθ) / (4πr) · sin(ωt − kr)
η₀ ≈ 377 Ω  (impedance of free space)
k = 2π/λ,  r = distance,  θ = angle from the wire

Two things fall out immediately. The field decays as 1/r, not 1/r² — that slower decay is what lets energy escape to infinity rather than staying bound near the source, unlike an electrostatic field. And the field carries a factor of sinθ: radiation peaks broadside to the wire and vanishes exactly along its own axis. Rotate that pattern around the wire and you get the shape every antenna textbook draws — a torus, or doughnut, with the dipole running through the hole.

Near field, far field, and where the pattern stops changing

Close to any antenna the field is a mess of three overlapping terms scaling as 1/r, 1/r² and 1/r³. Within about λ/2π, energy genuinely sloshes back and forth each cycle rather than escaping — the reactive near field, which behaves like the field of a capacitor or inductor and sets the antenna's input reactance. Farther out, in the radiating near field, the pattern still depends on distance as well as angle. Only beyond the far-field, or Fraunhofer, distance:

r_ff = 2D² / λ
Example: a 1 m dish at 10 GHz (λ = 3 cm) → r_ff = 2(1)²/0.03 ≈ 67 m

— does the angular shape of the radiation pattern stop changing with distance and settle into the clean 1/r decay that defines "far field." This is why antenna test ranges need real estate, and why compact test ranges use large reflectors to fake a plane wave at short physical distances.

Directivity, gain, and why the half-wave dipole won

Directivity D compares an antenna's peak radiated intensity to what a lossless isotropic source would produce with the same total power: D = 4π·U_max/P_rad. The Hertzian dipole scores a modest D = 1.5 (1.76 dBi); a satellite dish can reach 35–40 dBi, concentrating power thousands of times more effectively in its main beam. Real antennas trade beamwidth for directivity roughly as D ≈ 4π/(θ_HPBW·φ_HPBW) — narrower beams need physically larger apertures.

Gain adds efficiency to the picture: G = η_rad·D, where η_rad = R_rad/(R_rad+R_loss) compares radiation resistance to ohmic loss. The workhorse half-wave dipole (length λ/2, fed at the centre) has D = 1.64 (2.15 dBi, the dBd reference) and a centre-feed radiation resistance of about 73 Ω — close enough to standard 50–75 Ω feedlines that it needs little or no matching network, which is precisely why it became RF engineering's reference antenna.

The Friis equation and phased arrays

Antenna gain feeds directly into link budgets through the Friis transmission equation: P_r = P_t·G_t·G_r·(λ/4πR)². The (λ/4πR)² term is pure geometry — the wave spreading over an ever-larger sphere — so received power falls with the square of both distance and frequency. That is why AM radio at ~1 MHz travels farther per watt than 28 GHz millimetre-wave 5G, which compensates with directional, high-gain arrays.

A phased array feeds N identical elements the same signal with a controllable phase offset β between neighbours; the total far field is the single-element pattern multiplied by an array factor whose main peak direction θ₀ = arccos(−β/kd) depends only on β. Sweeping the phase electronically swings the beam through space in nanoseconds with no moving parts — the basis of AESA radar and 5G massive-MIMO. Push element spacing past d = λ/2 and spurious full-strength grating lobes appear at other angles, so d ≤ λ/2 is the standard constraint.

Frequently asked questions

Try it live

Everything above runs in your browser — open Antenna Radiation and watch the field detach and propagate at the speed of light, with no radiation along the dipole's axis. Nothing is installed, nothing is uploaded.