The coupling that is the same both ways

Two loops of wire sit near one another. Run a current in the first and some of its field passes through the second, so changing the current induces a voltage there; the constant relating the two is the mutual inductance.

Do the same in the other direction and the constant is the same number. That is not obvious. The two loops in the figure differ by a factor of twenty-one in area and entirely in shape, one is tilted and offset relative to the other, and there is no geometrical reason why a small loop should catch as much of a big one’s field as the big one catches of the small one’s.

Two integrals with nothing in common #

The claim is easy to state and easy to make vacuous. Reducing both quantities to the same expression and then observing that it is symmetric proves nothing about the physics — it proves that one expression equals itself.

So the figure computes them separately. One is the flux through the tilted rectangle when a unit current runs in the circle: the circle is chopped into segments, each contributing its Biot–Savart field, and the total is integrated over the rectangle’s tilted surface. The other is the flux through the circle’s disc when the same current runs in the rectangle: the rectangle’s four sides are chopped, and the result is integrated over a disc twenty-one times the area.

The two share the positions of the wires and nothing else — different fields, different surfaces, different quadrature grids. They agree to 0.075 per cent across a range of separations, and the residual is the quadrature’s rather than the physics’s.

That is the shape of an honest test of a symmetry claim: compute both sides by routes that could disagree, and see whether they do.

The arrangement was chosen to make disagreement as easy as possible. The two loops have different shapes, so the surfaces integrated over are a flat disc and a tilted rectangle; different sizes, so one quadrature is spread over twenty-one times the area of the other; different orientations, so the field is nearly normal to one surface and obliquely across the other; and the rectangle is offset sideways as well as along the axis, so no symmetry of the pair remains. If the equality depended on anything but the theorem, this is where it would show.

What both calculations do is integrate the normal component of such a field over a surface spanning the other loop. Seen in section, a loop’s field is the field of two opposite currents, and the integral over the far loop picks up whatever of it threads through — which is why the two calculations look so different: one integrates a compact field over a distant surface, the other a spread-out field over a nearby one, and they agree exactly.

Where the symmetry comes from #

Having established it, the reason is worth having, and it is one line of vector calculus.

The flux through loop 2 from a current in loop 1 can be written as a line integral of loop 1’s vector potential round loop 2, by Stokes’s theorem. And the vector potential of a thin loop is itself a line integral round that loop. Putting the two together gives Neumann’s double integral:

$$M = \frac{\mu_0}{4\pi}\oint_1\oint_2 \frac{\mathrm{d}\boldsymbol{\ell}_1\cdot\mathrm{d}\boldsymbol{\ell}_2}{\lvert\mathbf{r}_1 - \mathbf{r}_2\rvert}$$

which is manifestly unchanged by exchanging the labels 1 and 2. The two loops enter identically, so the coupling cannot depend on which is called the source.

The object that makes the symmetry visible is the vector potential. Written in terms of the field, the two couplings are different integrals over different regions with no obvious relation; written in terms of the potential they are the same double integral with the two loops exchanged — and a double integral does not care which order its two variables are named in. The symmetry is manifest in one formulation and hidden in the other, which is the usual reason for preferring a potential.

What that derivation makes visible is that the symmetry belongs to the kernel: one over the distance between two points, which does not care which point is which. Every reciprocity theorem in physics has that structure — a Green’s function symmetric in its two arguments — and the theorems differ only in what the Green’s function is.

The cancellation that seemed miraculous is therefore this: a large loop produces a weaker field per unit current over any given region, in exact proportion to the extra area it offers to a small loop’s field. The two factors are the same factor, seen from the two ends of the same kernel.

What it is a special case of #

The same statement holds far beyond two loops of wire, and the general version is worth knowing because it is used constantly and rarely named.

The theorem covers radiation as well as quasi-static coupling, and for antennas it has a consequence sharp enough to be a design rule: an antenna’s receiving pattern is its transmitting pattern. That is not an approximation or a rule of thumb — it is the same reciprocity, and it is why an antenna is characterised once and used both ways.

Lorentz reciprocity says that for any linear reciprocal medium, a current distribution A driving a field measured at B gives the same result as the same current at B measured at A. Mutual inductance is that theorem in the quasi-static limit.

The version with teeth is the antenna one. An antenna’s receiving pattern is identical to its transmitting pattern, which means a directional antenna receives from exactly the directions it transmits into, with the same gain. That is why a receiving antenna’s performance is never measured directly — it is measured by transmitting from it, which is far easier — and why the whole vocabulary of gain, beamwidth and sidelobes is used for both functions without distinguishing them.

Circuit reciprocity is the same theorem reduced to a network: putting a voltage source at one port and measuring the current at another gives the same ratio as the two exchanged, which is why an impedance matrix built from linear passive elements is symmetric. That symmetry is assumed by every network analyser calibration and is checked as a matter of routine, because a measured asymmetry means either a nonlinearity or a magnetic component somewhere.

Acoustic reciprocity is the same theorem in a different field: a source at A and a microphone at B give the same transfer function as the two exchanged. Concert-hall measurements exploit it constantly, because a loudspeaker on stage and a microphone in a seat can be swapped when one of the two positions is awkward.

The flux rule is what turns any of these couplings into a measurable voltage, and it is worth remembering that the mutual inductance is a purely geometrical quantity until something changes: two loops sitting still with a steady current in one have a mutual inductance and no voltage anywhere.

And in optics the same principle says that a ray path is reversible, which is the assumption behind the argument that makes minimum deviation symmetric and behind half the constructions in geometrical optics.

Why the flux was the right thing to integrate #

There is a step in both calculations that deserves attention, because it is where most of the difficulty in an inductance problem lives.

Flux was the right thing to write down because it depends only on the loop’s boundary and not on which surface spanning that boundary is chosen. That independence is what makes a mutual inductance a property of two curves rather than of two surfaces — and it is why the number can be computed from the wires alone, without deciding anything about what is stretched between them.

The flux through a loop is defined as an integral over a surface, and there are infinitely many surfaces bounded by the same loop. A flat disc, a hemisphere, a crumpled bag — all have the same rim. The answer had better not depend on which is chosen, and it does not, because the magnetic field has no divergence: the flux out of the closed volume between any two such surfaces is zero, so the flux through them is the same.

That is the statement that magnetic field lines have no ends doing quiet work. Without it, a mutual inductance would not be a number at all; there would be one for each choice of surface, and no reason to prefer any.

It also explains why the tilted rectangle in the figure could be integrated over its own flat surface without any care. The rectangle’s plane is not perpendicular to the field anywhere, the field varies across it, and none of that matters — what is being computed is a property of its rim.

And it is why the theorem’s practical form is usually stated with the vector potential rather than the field: a line integral round the rim has the surface independence built in, and needs no choice to be made at all.

What symmetric does not mean #

Reciprocity is regularly over-read, and two of the over-readings are common enough to be worth heading off.

Symmetric does not mean equal, and the warning is worth making explicit. The mutual capacitance between two conductors is symmetric in exactly the way an inductance is, and in both cases a symmetric coupling can still be tiny — reciprocity says the two directions match, not that either is large. A badly coupled pair is badly coupled both ways round, which is a guarantee about symmetry and not about performance.

A transformer is not symmetric. The mutual inductance is, but the voltage ratio involves the self-inductances too, and a transformer with a hundred turns on one side and ten on the other steps up in one direction and down in the other. What is symmetric is the coupling coefficient — the mutual inductance over the geometric mean of the two self-inductances — and the power transfer. Confusing the two is the commonest misuse of the theorem.

And the theorem has a condition. The medium has to be linear and reciprocal. A magnetised ferrite is not: its permeability is a tensor that is not symmetric, and in such a medium the coupling from A to B genuinely differs from the coupling from B to A. That failure is not a nuisance, it is a component — a circulator is built out of exactly it, and it is the only way to make a device that passes a signal one way round a ring and not the other. Every radar transmitter that shares an antenna with its receiver depends on that non-reciprocity.

So the theorem’s usefulness and the exceptions to it are the same fact seen twice: reciprocity holds unless something breaks time-reversal symmetry, and the things that break it are worth building.

Reading it as a measurement #

The symmetry is not only a curiosity to be verified; it is a way of measuring things that are otherwise inaccessible.

A field measured where no probe fits. To find the field a coil would produce at an inaccessible point, put a small loop at that point and measure the voltage induced in the coil when the small loop is driven — which is often much easier than reaching in. Reciprocity says the two are the same coupling.

An antenna’s pattern in a place it cannot be put. The same trick, at radio frequencies: measure the pattern by transmitting from the antenna into a known receiver, and use it as a receiving pattern.

A loop antenna’s response, from its field. The same argument extends to any transducer that is linear and reciprocal, which is why a coil that produces a field can be characterised as a receiver without ever putting a signal into it, and why the sensitivity of a search coil is quoted as an effective area — a purely geometrical number that would otherwise have to be calibrated against a known field.

And a detector’s sensitivity map. In magnetic resonance imaging the coil’s sensitivity to spins at a point equals the field it would produce at that point per unit current — the principle of reciprocity applied to a receive coil — and the whole of coil design rests on it.

Each of those replaces a hard measurement with an easy one, and the licence for the replacement is the theorem this essay’s opening figure tests.

The self-inductance the same integral does not give #

It is worth noticing where the method stops, because the failure is instructive rather than technical.

Most of a coil’s behaviour depends on its self-inductance rather than its mutual one, and the same double integral gives it — with both loops taken to be the same loop. That is where the integral becomes awkward, because the two variables can coincide and the integrand diverges; the standard treatments handle it by giving the wire a radius, which is an admission that a filament is an idealisation the self term cannot tolerate.

Neumann’s double integral applied to a single loop — the same expression with both labels referring to the same wire — diverges. The kernel is one over the distance between two points on the loop, and when the two points coincide the integrand is infinite. So the self-inductance of an infinitely thin loop is infinite, which is not an approximation error but the correct answer to a badly posed question.

The repair is to give the wire a radius, at which point the internal and external contributions separate and the answer contains a logarithm of the loop radius over the wire radius. That logarithm is why inductance is such a weak function of geometry: a wire ten times thinner has only about 20 per cent more inductance in a loop of the same size, which is why inductors are hard to make small and easy to estimate roughly.

The mutual case has no such difficulty because the two loops do not touch, so the kernel is bounded everywhere and the integral converges. That asymmetry between the two calculations is a good reminder that a mutual inductance is a cleaner object than a self-inductance — it is a property of two well-separated objects, whereas a self-inductance is a property of a wire’s own cross-section as much as of its path.

The signal that appears before anything arrives #

The most consequential use of a reciprocity theorem in instrumentation is one that sounds wrong when it is first stated, and it is worth working through because it corrects a picture almost everybody starts with.

A radiation detector is a block of semiconductor or gas with electrodes on it. A particle passes through, liberates charge, and a field sweeps that charge to an electrode. The obvious account of the signal is that current flows when the charge arrives.

It does not. Current flows the entire time the charge is moving, and by the time it arrives the signal is over. The instantaneous current induced on a given electrode is

$$i = q\,\mathbf{v}\cdot\mathbf{E}_w,$$

where $\mathbf{E}_w$ is not the real field but a weighting field — the field that would exist if that one electrode were raised to unit potential, every other electrode grounded, and all the charges removed. It is a purely geometrical construction with no physical existence, and it decides the signal.

That is a reciprocity theorem: the coupling between the moving charge and the electrode is being computed by driving the electrode and asking what field appears at the charge, which is exactly the exchange this essay’s opening figure tests, in its electrostatic form.

The consequences are not cosmetic. A charge that drifts partway and is then trapped still delivers a signal, proportional to how far it moved in weighting potential rather than to whether it got anywhere. In a thick detector where one carrier is slow — holes in cadmium zinc telluride are the standard nuisance — the pulse height therefore depends on how deep in the crystal the interaction happened, which ruins the energy resolution. And the fix is to change the weighting field rather than the material: divide the anode into small pixels or a coplanar grid, so that the weighting potential is near zero through most of the crystal and rises steeply only near the electrode. The signal then comes almost entirely from the last short stretch of the electron’s drift, and the slow carrier stops mattering.

That is a detector redesigned by editing a field that is not there.

Why the theorem holds at all, and what breaks it #

The essay’s closing warning — that reciprocity fails where something breaks time-reversal symmetry — is the statement of a much larger result, and naming it explains both the rule and every exception at once.

Take any system whose fluxes respond linearly to a set of driving forces, so that $J_i = \sum_j L_{ij} X_j$. Onsager proved in 1931 that the matrix of coefficients is symmetric, $L_{ij} = L_{ji}$, and the proof rests on nothing electromagnetic: it comes from the microscopic equations of motion being unchanged under reversal of time, together with the assumption that a fluctuation decays on average the way a macroscopic disturbance does.

The reach of that is considerable. A temperature difference drives an electric current and a current carries heat; Onsager’s relation says the two cross-coefficients are one number, which is the reason the Peltier coefficient of a material equals its Seebeck coefficient times the absolute temperature. Thomson had guessed that relation from thermodynamics in 1854 with a step he could not justify, and it stood unexplained for three quarters of a century.

The exception is written into the theorem rather than hiding outside it. In a magnetic field the microscopic equations are unchanged under time reversal only if the field is reversed as well, so the correct statement becomes $L_{ij}(\mathbf{B}) = L_{ji}(-\mathbf{B})$. Reciprocity survives, in a form that requires the magnet to be turned round.

That is precisely the magnetised ferrite. A circulator is not a device that evades a theorem; it is a device built in the one place where the theorem relates a measurement to a different experiment rather than to the same one, and every one-way component in optics and microwaves lives there.

What the pictures cannot show #

Everything is quasi-static. The loops are small compared with any wavelength, so the field is computed from the instantaneous current with no retardation. The full theorem survives retardation and the arithmetic does not: the correct statement then involves the retarded Green’s function, which is still symmetric.

The wires are infinitely thin. A real conductor has a cross-section, and at high frequency the current is not uniform across it — which changes the self-inductance considerably and the mutual inductance hardly at all, since the mutual term depends on the separation rather than on the internal distribution.

No magnetic material is present anywhere. A linear isotropic core changes both couplings by the same factor and leaves the symmetry intact; a saturating one makes the problem nonlinear and the theorem does not apply.

Neither loop moves. Reciprocity as stated is about two stationary circuits; a moving one changes its flux for a second reason, and separating the two ways flux can change is a different question from whether the coupling is symmetric.

And nothing here is a transformer. The figure computes a coupling between two loops in air, whose coupling coefficient is a few per cent. A transformer is built to make that coefficient approach one, which needs a shared magnetic circuit, and the design questions there are about leakage rather than about reciprocity.

The ladder from here #

Later rungs on this anchor: the energy method, which derives the symmetry from the requirement that the stored energy be a state function of the two currents; the coupling coefficient and the bound that keeps it below one; Lorentz reciprocity with retardation and its consequences for antenna arrays; and non-reciprocal media, where the theorem fails on purpose and gives isolators, circulators and one-way optical devices.

The neighbouring ladders are the field that makes the other, which is the induction this coupling is a coefficient of, the rule that is two laws wearing one coat, which distinguishes the two ways flux can change, and the field that wraps a current, which is the field being integrated in both directions here.