The distance where a field changes its mind

The rung below this one took a charge, changed its motion once, and found the field around it in two pieces: a velocity field falling as the inverse square that stays with the charge, and a radiation field falling as the inverse first power that leaves and never comes back. That essay ended by admitting that something had been left out. Between the two there is a third term, and it is where antenna design lives.

Let the source oscillate rather than change once — which needs the term Maxwell added for the field to propagate at all — and the exact field of a dipole has three terms rather than two. Their sizes at a distance $r$, in terms of $u = kr$ with $k = 2\pi/\lambda$, are $1/u^3$, $1/u^2$ and $1/u$.

One crossing, and what is not in it #

Three curves in a plane cross pairwise at three points in general. These cross at one, and the reason is that they are consecutive powers of one variable: $1/u$, $1/u^2$ and $1/u^3$ are all equal to one when $u$ is one, whatever else is true.

So the boundary between near and far is a single radius, and its condition is $kr = 1$, which is

$$r = \frac{\lambda}{2\pi}.$$

Look at what is not in that expression. There is no length of the antenna, no current, no power, no material, no geometry. A one-millimetre loop and a one-metre loop driven at the same frequency have the same crossing radius. It is a property of the frequency — of how far light gets in one radian of phase — and the sphere of that radius has a name, the radian sphere, which is a name for a fact about time rather than about anything present.

That is worth holding against the ordinary intuition, which is that an antenna’s near field is somehow the region it fills. It is not. A pocket radio and a broadcast mast at the same frequency have identical radian spheres, and the mast merely happens to be larger than one.

Which derivative each term is #

The three terms are usually introduced as three powers of distance, which makes them look like a mathematical accident. They are not. Each one is a different time derivative of the source, and naming them that way says why there are exactly three.

Written in terms of the dipole moment $p(t)$ evaluated at the retarded time, the field of an oscillating dipole is

$$\mathbf{E} \sim \frac{p}{r^3} + \frac{\dot{p}}{c\,r^2} + \frac{\ddot{p}}{c^2 r}.$$

The first term is the electrostatic field of the charge separation as it is right now — Coulomb’s inverse square applied to a dipole, which for two opposite charges falls one power faster than for one. The second is what the current makes: a dipole moment changing is a current, and a current’s field is the induction field, which is the one a changing flux drives an emf with. The third is what the change in the current makes, and it is the only one that carries energy away — which is the same statement as the rung below’s, that radiation comes from acceleration and not from motion.

Each derivative costs a factor of $\omega/c = k$ and buys a factor of $1/r$, which is why consecutive terms differ by exactly $kr$ and why they cross where $kr = 1$. The crossing is not a coincidence about three curves; it is the statement that one factor of $k$ and one factor of $1/r$ are the same size at that radius.

There are exactly three because a dipole has exactly two derivatives that matter: the source is characterised by $p$, and $\dot{p}$ and $\ddot{p}$ are what the retarded solution’s expansion produces before the series terminates for a point source. A quadrupole has four terms, an octupole five, and each higher multipole’s terms cross at the same radius for the same reason — which is why the radian sphere is a property of the frequency even for sources with complicated shapes.

The same statement, in power #

The field picture is the standard one and it is not the cleanest available. Multiplying the electric field by the conjugate of the magnetic field, which is what the complex Poynting vector does, gives something exact and short. With $A$ and $A'$ the two prefactors,

$$E_\theta H_\phi^{*} = A A' \sin^2\theta \left[\frac{1}{u^2} + \frac{i}{u^5}\right].$$

Every cross term has cancelled. What is left is a real part falling as the inverse square and an imaginary part falling as the inverse fifth power, and nothing in between.

The real part is radiated power. It falls as $1/r^2$, so the total through any sphere is the same, which is what radiation means, and is what separates a field that leaves from one whose falloff is decided by the shape of its source. The imaginary part is reactive power: energy that the source pushes out into the field on one quarter cycle and takes back on the next, with nothing transported anywhere. It falls as $1/r^5$, so it is enormous close in and gone almost immediately.

Their ratio is exactly $u^{-3}$. Equal at $u = 1$; a thousand to one in favour of storage at a tenth of that radius; a thousand to one in favour of radiation at ten times it. The transition is not gentle, and calling the near field a region where the far-field formula is not yet accurate understates it by three powers.

The two descriptions — three terms in the field, two parts in the power — are the same physics and they cross at the same place, which is the check that the crossing is real rather than an artefact of how the field was decomposed. Decompositions can be chosen. A ratio of real to imaginary power flux cannot.

What the radius actually is, for things that have one #

That figure is the essay’s practical content, and the numbers on it explain a set of engineering facts that otherwise look unrelated.

A power grid radiates essentially nothing. At 50 hertz the radian sphere is 954 kilometres. Every transmission line on Earth is deep inside its own near field, so the fields around it are the storing kind — which is why a grid can carry gigawatts without becoming a transmitter, and also why the fields near a line are measured in the near-field way, as separate electric and magnetic quantities rather than as a power flux.

A contactless card is read at three centimetres and cannot be read at three metres. At 13.56 megahertz the radian sphere is 3.52 metres, and a card sits at a hundredth of it. The coupling there is inductive: the reader’s coil and the card’s coil are two windings of a very poor transformer, and essentially no energy is radiated. The security consequence follows from the distance law rather than from any cryptography — eavesdropping on a near-field link means being inside the near field.

And a Wi-Fi client across the room is unambiguously a radio. At 2.4 gigahertz the radian sphere is 19.9 millimetres. Everything beyond a few centimetres is far field, which is why link budgets there are done with an inverse-square law and antenna gains, and why the same arithmetic that works for a satellite works for a laptop.

What a source inside its own near field is, electrically #

The near field is a field to a physicist and an impedance to an engineer, and the second view is where the constraint bites.

An antenna presents the transmitter with a resistance and a reactance. The resistance is not resistance in the wire; it is the radiation resistance, defined so that the power it dissipates equals the power radiated. For a short dipole of length $L$ it is $20\pi^2 (L/\lambda)^2$ ohms. The reactance is the near field, counted as energy stored per cycle rather than as a field in space.

Two things about that picture matter more than the numbers.

The slope is exactly two. Halving an antenna quarters what it radiates into, which means the difficulty grows quadratically rather than linearly as devices shrink. That is why the antenna is often the component that stops a product getting smaller, long after the electronics have stopped mattering.

And the mismatch is what makes it narrowband. A matching network that transforms a two-hundredth of an ohm up to fifty ohms is a very high-$Q$ circuit, and a high-$Q$ circuit has a narrow bandwidth by definition — the same relationship between the width of a resonance and how long it rings that appears everywhere else in physics. So a small antenna is not merely inefficient; it is inefficient and narrowband, and the two are one consequence. Chu and Harrington turned that into a theorem in the late 1940s: the minimum $Q$ of any antenna fitting inside a sphere of radius $a$ is fixed by $ka$ alone, so the trade cannot be designed around, only chosen differently.

The same two powers a driven oscillator has #

The split into real and reactive power is not peculiar to antennas, and recognising where else it appears is what makes it easy to reason about.

A mass on a spring driven by an oscillating force does two things with the energy supplied. Part of it goes into the damper and is gone; part of it goes into the spring and the mass, is stored, and is handed back a quarter cycle later. The driven oscillator’s quarter cycle is exactly the phase relationship that distinguishes the two, and the ratio of stored to dissipated energy per cycle is what defines a quality factor.

An antenna is that oscillator with the damper replaced by radiation. The reactive field is the spring; the radiation resistance is the damper; and the $Q$ of the arrangement is $2\pi$ times the stored energy over the energy radiated per cycle. Writing it out for a small antenna gives $Q \approx 1/(ka)^3$, which is the same cubic law the power figure showed, arrived at from the energy rather than from the flux.

That identity is what turns the Chu limit from an engineering rule into a statement about physics. Confining a field to a region smaller than $\lambda/2\pi$ means most of the energy in that region is stored rather than travelling, the stored fraction rises as the inverse cube of the size, and the bandwidth falls with it. There is no arrangement of conductors that escapes it, because the bound is derived from expanding the field outside the sphere in spherical modes and asking how much energy each mode keeps inside — a calculation with no antenna in it at all.

The practical form of the trade is visible in any phone teardown. The antenna occupies a volume, and the volume is what buys the bandwidth. Everything that has made phone antennas better over thirty years — using the chassis as a radiator, splitting the band across several elements, tuning the match electronically as the band changes — is a way of buying volume or of giving up the requirement to cover the whole band at once.

An instrument that works because it is inside the sphere #

Two technologies exist because the near field carries something the far field does not, and both are worth stating because they are usually taught as separate subjects.

Magnetic resonance imaging is a near-field instrument. A clinical scanner at 1.5 tesla drives protons at 64 megahertz, and the radian sphere at that frequency is 75 centimetres. The body coil surrounds the patient and the patient is inside it, so nothing about the transmission is radiative: the coil and the tissue are inductively coupled in the same sense a contactless card and its reader are. That is why the coil is a coil rather than an aperture, why the exposure limit is written as a specific absorption rate — watts per kilogram deposited by induced currents — rather than as an incident power density, and why moving to a 7-tesla machine at 300 megahertz is genuinely difficult: the radian sphere falls to 16 centimetres, the wavelength in tissue becomes comparable to a head, and the field stops being uniform across the sample in a way that no amount of coil design entirely fixes.

And microscopy beats the diffraction limit by staying inside the sphere. The resolution of an ordinary lens is bounded near half a wavelength because the fine detail of an object is carried by field components that do not propagate — the same evanescent components that sit on the far side of a totally internally reflecting surface, decaying rather than travelling. Those components are the near field, and they exist only within a fraction of a wavelength of the object.

So a probe that gets closer than that can read them. A near-field scanning optical microscope drags an aperture tens of nanometres across a few nanometres above a surface and resolves detail at $\lambda/50$, using light that no lens could focus to better than $\lambda/2$. Nothing has been done to the light. The instrument has simply gone to where the information still is, before the near-field terms have decayed away — and the distance available to it is the same $\lambda/2\pi$ that decides everything else in this essay.

The same idea in the radio band is why a spectrum analyser probe held a millimetre above a circuit board reads the currents in individual traces, while an antenna across the room reads only the sum. Detail is a near-field quantity, and the far field is a low-pass filter that has already been applied by the time anything has propagated.

The two ways of getting energy across a gap #

The last figure is the one that turns the crossing radius into a decision.

Two coils can be coupled through the storing field, which is what a transformer, a contactless card and a wireless charger do. Two antennas can be coupled through the radiating field, which is what a radio link does. The first falls as the inverse sixth power of distance in delivered power and the second as the inverse square, so the choice is decided by which side of the radian sphere the receiver is on, and by nothing else.

The sixth power is brutal and is the reason wireless charging has the range it does. Doubling the distance from a charging pad costs a factor of sixty-four, which is why the standards specify millimetres and why a pad that works at four millimetres does not work at a centimetre. Raising the frequency shrinks the radian sphere and would help, except that it also shrinks the coil that fits inside a phone, and the received power depends on both.

Running it the other way explains an old puzzle about radio. Marconi’s first transatlantic signal in 1901 was at a wavelength of several hundred metres, so the radian sphere was tens of metres and the receiver was thousands of kilometres away — deep in the far field, as it had to be. The near-field terms that dominate every laboratory demonstration of induction had fallen below anything measurable, and what was left was the term that falls slowest. Radio is not a strong effect; it is the only effect that survives the distance.

Where this stops being right #

The source has been a point dipole throughout. Every expression here comes from expanding a source in multipoles and keeping the first one, which is exact only in the limit of a source small compared with a wavelength. A source that is not small has extra terms whose retardation is across the source rather than out from it, and that is the next rung.

The crossing is a crossing rather than a boundary. Nothing changes at $r = \lambda/2\pi$; the terms are continuous and all three are present everywhere. What is real is the ratio, and speaking of a near field and a far field as regions is a convenience that the sharpness of the $u^{-3}$ law makes almost honest.

The reactance figure used one wire geometry. A short dipole’s capacitance depends logarithmically on its length-to-radius ratio, so the reactance curve moves by a factor of two or so for a fat antenna against a thin one. The slope does not move, and the slope is the claim.

And the whole treatment is monochromatic. A pulse contains many frequencies and therefore many radian spheres, so a broadband source has no single boundary at all — which matters for ultra-wideband systems, where the near-field and far-field descriptions overlap across the band.

What the pictures cannot show #

Every figure here is drawn against distance, and the subject is a phase. What distinguishes the reactive part of the power flux from the real part is not where it is but when it is: reactive power is in quadrature, out and back within a cycle, and a plot of its magnitude against distance is a plot of the size of something whose whole character is its timing. A still picture cannot show energy going out and coming back.

Nor can any of them show the direction. The dipole’s field has a $\sin\theta$ that has been set to one throughout, so every curve is the equatorial case and the polar case — where the radiation term vanishes and only the near-field terms survive — appears nowhere. Directly above a dipole there is no far field at all, at any distance.

What the ladder has established #

Two rungs stand on retardation. The first found that the retarded field of a charge splits into a piece that stays and a piece that leaves. This one finds that an oscillating source has three pieces rather than two, and that the distance at which the last of them takes over is set by the frequency and by nothing else.

The habit worth carrying away concerns quantities that contain fewer things than they should. When a boundary in a problem turns out not to depend on any property of the object, it is a boundary in time wearing a length’s clothes. The radian sphere is how far light gets in one radian of phase, and calling it a distance is a translation. The same test applied elsewhere identifies the skin depth as a diffusion time, the coherence length as a bandwidth, and the Debye length as a temperature.

What is left on this ladder is the assumption that has been made silently throughout. Everything here treats the source as a point, which is to say it assumes every part of the source is heard at the same retarded time. Dropping that assumption turns one source into a sum over a source, and the sum is the one a diffraction grating performs.