1. Why an Antenna Radiates: The Accelerating Charge

We met the deepest fact about radiation back in Chapter 0. A charge sitting still has an electric field, but it does not radiate. A charge moving at constant velocity has both an electric field and a magnetic field, but it still does not radiate (the field travels with the charge, not away from it). A charge that is accelerating radiates. That is the entire physical basis of every antenna ever made.

1.1 The intuition: kinks in field lines

Imagine you are standing in space watching a static positive charge. You see field lines fanning radially outward in every direction, like the spokes of a bicycle wheel. The lines are fixed in space because the charge is fixed. Now suddenly jerk the charge a small distance to the right. The field lines have to adjust. But they cannot adjust everywhere instantly. Far from the charge, the field still points to where the charge used to be, because the news that the charge has moved travels outward at the speed of light, no faster.

What you see is a "kink" in the field lines: close to the new position, the lines point at the new location; far from the original position, they still point at the old location; in between, there is a transition region where the lines bend abruptly. That kink propagates outward at $c$, taking energy with it. That is electromagnetic radiation.

The faster you accelerated the charge, the sharper the kink, the more energy it carries. A charge that wiggles back and forth at frequency $f$ produces a continuous train of these kinks, which manifests as an electromagnetic wave at the same frequency $f$ propagating in all directions. The mathematical result, due to Larmor, is that an accelerating charge radiates power proportional to the square of its acceleration:

$P_{rad} = \frac{q^2 a^2}{6\pi\varepsilon_0 c^3}$

This is Larmor's formula. We will not need it in numerical form for antenna design, because we work with currents (not single charges), but it is the seed of everything that follows.

Bell-and-clapper analogy. A bell rings at certain frequencies because, at those frequencies, the metal flexes and unflexes coherently. Strike it at a wrong rhythm and you get a dull thud, not a ring. The clapper is putting energy into the bell, and the bell decides what to do with it: if the rhythm matches a natural mode, the energy radiates as sound; if not, it dissipates as heat. An antenna is the same. Drive it at its resonant frequency and it radiates beautifully, taking energy from your wire and casting it into space. Drive it off-resonance and most of the energy reflects back down the feedline as if the antenna were a brick wall.

1.2 From single charges to currents

In an antenna, we do not have one charge accelerating. We have an enormous number of electrons in a wire, each oscillating back and forth as alternating current pushes them. Every one of those electrons is accelerating (because every time it reverses direction, its velocity changes), and every one is contributing radiation. The total field at a faraway observer is the vector sum of contributions from every accelerating electron in the antenna.

This sum can be either constructive or destructive, depending on the geometry of the antenna and the direction we are looking from. That is why antennas have patterns: the radiation in one direction adds up in phase, in another direction it cancels out, and the result is a directional emission rather than a uniform glow.

The current in a wire is just a coordinated motion of electrons, so antenna theory talks about currents instead of individual charges. The radiation from a current element $I\,dL$ is computed using essentially Larmor's formula scaled up. The math gets technical, but the picture is identical.

1.3 Near field and far field

When you stand right next to an antenna, the fields look complicated. There are stored fields (energy bouncing back and forth between the antenna and nearby space, like the static field of a capacitor) and radiating fields (energy actually traveling away). Close in, both kinds matter, and the math is unpleasant. This region is called the near field.

Far away, the stored fields die off faster than the radiating fields (stored fields fall as $1/r^2$ or even $1/r^3$, while radiating fields fall only as $1/r$). So eventually only the radiation remains. This region is called the far field or Fraunhofer region.

The boundary, conventionally, is

$r > \frac{2D^2}{\lambda}$

where $D$ is the largest physical dimension of the antenna and $\lambda$ is the wavelength. For a 1 m dish at 10 GHz ($\lambda = 3$ cm), the far field begins at about 67 m. For a small WiFi patch antenna ($D \approx 5$ cm) at 2.4 GHz ($\lambda = 12.5$ cm), the far field begins at just 4 cm. Antenna pattern measurements must be done in the far field, which is why anechoic chambers for big dishes are huge.

Why care about this distinction? Because everything we say about gain, directivity, and radiation patterns is a far-field statement. In the near field, the very concept of "radiation pattern" does not apply; the fields look like something between static and radiating. RFID tags, wireless charging, and inductive coupling all live in the near field on purpose, because there the energy stays put rather than radiating away.

1.4 Reciprocity: transmit and receive are the same

There is one more law we need before we can build anything. If antenna A transmits to antenna B, and the received signal is some level $L_{AB}$, then if you swap the roles (B transmits, A receives) you get exactly the same $L_{BA} = L_{AB}$. This is the reciprocity theorem, derivable from Maxwell's equations under linear conditions.

A practical consequence: the antenna pattern, gain, beamwidth, polarization, and impedance for transmit and receive are identical. This is wildly convenient because it means we only need to analyze the antenna in one direction (usually transmit, where we have a known driving current) and the receive answer comes for free.

Reciprocity does not mean the system is reciprocal. The amplifiers behind the antenna may behave very differently on transmit and receive (most receive amplifiers would burn out if you applied a kilowatt to their input), and antennas in nonlinear or anisotropic plasma do not obey reciprocity exactly. But for ordinary metal antennas in air, reciprocity is rock solid.

For a hardware-security thinker, reciprocity has an unsettling implication. If a chip's PCB trace is unintentionally radiating at some clock harmonic, the same trace also unintentionally receives at that frequency. So the same path that leaks information out can also let injected RF energy in, which is one of the underlying mechanisms in fault-injection attacks via electromagnetic pulses (EMFI).