Mutual Inductance

The first electrical machine to move energy between two circuits that never touch was an iron ring on Michael Faraday's bench in August 1831 — primary winding on one side, secondary on the other, a galvanometer kicking every time the primary's current changed. Nothing crosses the gap but field. Two centuries later the same trick, at scale, is the reason electricity has a grid at all: transformers move terawatts between windings joined only by flux. This article builds the constant that quantifies the trick. Self-Inductance taught a circuit to feel its own flux: $\Phi = LI$ . Mutual inductance is the off-diagonal version: circuit 1's current threads flux through circuit 2 in strict proportion, $\Phi_2 = MI_1$ , with $M$ — like $L$ — a constant of pure geometry. But $M$ carries a surprise its diagonal sibling lacks. Compute "flux through 2 per ampere in 1" and "flux through 1 per ampere in 2" — two completely different-looking calculations, one often vastly harder than the other — and the answers are always identical. $M_{12} = M_{21}$ , exactly, for any two loops in any configuration whatsoever. This reciprocity theorem, falling out of the elegant double-loop integral of the Neumann formula, is the single most labor-saving symmetry in applied electromagnetism: it lets you compute the easy direction and bank the hard one. The article proves both structural facts — reciprocity, and the energy-backed bound $M^2 \leq L_1 L_2$ that caps how tightly two circuits can couple — then puts numbers on the classic geometries. The payoff downstream is immediate: the coupled-energy expression assembled here is raw material for Energy Stored in Magnetic Fields, and the transformer physics sketched here powers everything the AC world does.

Key takeaways

Two circuits couple through geometry: $\Phi_2 = M I_1$ , and changing $I_1$ drives $\mathcal{E}_2 = -M\,\mathrm{d}I_1/\mathrm{d}t$ in circuit 2 — energy transfer with no contact, the transformer principle.
The Neumann formula $M = \frac{\mu_0}{4\pi}\oint\oint \frac{\mathrm{d}\boldsymbol\ell_1\cdot\mathrm{d}\boldsymbol\ell_2}{|\mathbf{r}_1 - \mathbf{r}_2|}$ writes $M$ as a double loop integral — manifestly geometric, and manifestly symmetric.
Reciprocity $M_{12} = M_{21}$ : flux through 2 per ampere in 1 equals flux through 1 per ampere in 2, always. Compute whichever direction is easy; the theorem hands you the other.
Coupled circuits store energy $W = \tfrac12 L_1I_1^2 + \tfrac12 L_2I_2^2 + MI_1I_2$ ; demanding $W \geq 0$ for all currents forces $M^2 \leq L_1L_2$ .
The coupling coefficient $k = M/\sqrt{L_1L_2}$ grades every coupled pair from $k \approx 0$ (stray crosstalk) to $k \to 1$ (ideal transformer, all flux shared).
KEY IDEA: mutual inductance $M = \Phi_2/I_1$ is a geometry-only constant, symmetric between the two circuits ( $M_{12} = M_{21}$ ) and bounded by their self-inductances ( $M^2 \leq L_1L_2$ ) — the complete grammar of flux coupling.

The core insight

Mutual inductance $M = \Phi_2/I_1$ measures how much flux one circuit's current threads through another; it is fixed by geometry alone, obeys the exact reciprocity $M_{12} = M_{21}$ , and is capped by $M^2 \leq L_1L_2$ — the constant behind transformers, wireless power, and every coupled coil.

Where this lives today

The grid is mutual inductance at civilization scale: hundreds of thousands of transformers, from gigawatt substation units ( $k > 0.99$ , oil-cooled, humming at twice mains frequency) down to the millimetre transformers inside every USB charger, chained so that generation, transmission, and consumption can each run at their own best voltage. Around them has grown the newer $k \ll 1$ economy: Qi wireless charging pads ( $k \sim 0.3$ , resonance-assisted to move ~10 W across 5 mm), electric-vehicle charging plates, and implanted-device links — cochlear implants and pacemakers powered through intact skin by exactly worked example 1's physics, with the body as the air gap. The same constant, unwanted, is crosstalk: adjacent traces on a PCB, adjacent pairs in a cable bundle, adjacent windings in a motor are all coupled pairs, and signal-integrity engineering is largely the art of driving their $M$ toward zero — twisted pairs (alternating the sign of $M$ segment by segment so it cancels), guard traces, orthogonal routing, and the $1/d^3$ law of worked example 2. And at the precision frontier, mutual inductance is a standard: cryogenic current comparators and SQUID-coupled input coils measure with $M$ 's geometric stability, precisely because a flux ratio between rigid superconducting windings depends on nothing but shape — machinable, calculable, and immune to drift in a way resistors can only envy.

Origins · 1 minOliver Heaviside and how this was discovered

Oliver Heaviside

1850–1925 · London, Newcastle, Torquay

Heaviside never held an academic post, never took a degree, and wrote some of the most consequential electromagnetism of his century from rented rooms — a self-taught ex-telegrapher (nephew-in-law of Wheatstone) who read Maxwell's Treatise and decided it needed rebuilding. The rebuild gave us our subject's language: the four vector Maxwell equations (Maxwell's Equations in Vacuum tells that story), the operational calculus behind transient analysis — and the very words of this chapter. "Inductance," "self-inductance," "mutual inductance," "impedance," "admittance": all Heaviside coinages from the 1880s, minted because telegraph engineering was drowning in effects it could not name. His masterpiece application is pure mutual-and-self-inductance physics: the theory of the distortionless transmission line (1887), which showed — against fierce official resistance — that adding inductance to telephone cables would let signals travel undistorted. Loading coils built on his condition carried long-distance telephony for half a century. The energy method of this article's Derivation B, the $k$ -bookkeeping of coupled lines, the field-first view of circuits: all bear his fingerprints, acknowledged today in the term Hertz–Heaviside equations.

History

1831

— Faraday's iron ring: changing current in one winding induces current in another — mutual induction discovered

1832

— Henry, independently, couples coils in Albany; the American thread of induction begins

1845

— Franz Neumann publishes the double-integral formula: M as pure geometry, reciprocity manifest

1885

— The ZBD engineers and Stanley build practical closed-core transformers; k approaches 1 commercially

1886

— Heaviside coins "inductance" and its self/mutual taxonomy while founding transmission-line theory

1888

— Westinghouse–Tesla AC systems win the currents war on the transformer's ability to step voltage

2008

— MIT's resonant "WiTricity" experiments revive k << 1 power transfer; wireless charging standards follow

The physical picture

Set two loops of wire on a table, a hand-span apart, connected to nothing shared. Drive a changing current through the first. Its field sprawls through space, and some of those field lines — a fraction set entirely by sizes, shapes, distance, and orientation — happen to thread the second loop. Changing current means changing threaded flux, and Faraday's Law of Induction does the rest: an EMF appears in loop 2, driven across empty space. The fraction-of-flux-captured bookkeeping is what $M$ packages: assume one ampere in loop 1; however many webers land in loop 2, that number is $M$ .

Figure

Two circuits joined only by field: the flux one current threads through the other loop defines M

Why should $M$ be a clean constant? Same argument as for $L$ : Biot–Savart makes loop 1's field everywhere proportional to $I_1$ with a purely geometric kernel, so the flux captured by any fixed second contour is (geometry) $\times\, I_1$ . All the physics of coupling — how near, how coaxial, how large, whether an iron core funnels the flux — is compressed into one number with units of henries. Bring the loops closer, $M$ grows; turn one perpendicular, $M$ can pass through zero (as much flux enters as leaves); flip it, $M$ changes sign. And to maximize it, do what Faraday did: wind both coils on a shared iron ring so that essentially every line threading one threads the other. That limit — all flux shared — is the ideal transformer, and the coupling bound derived below says precisely that it is the limit: $k = M/\sqrt{L_1L_2} \to 1$ and no further. The reciprocity claim deserves a moment of honest astonishment before it is proved. Consider a tiny 50-turn coil sitting inside a long solenoid (worked example 1). Direction A: the solenoid's field is uniform and known — the flux through the little coil is a one-line calculation. Direction B: the little coil's field is a hideous near-field dipole pattern, and its flux through every one of the solenoid's thousands of turns is an integral nobody wants. The theorem says the two answers are equal before you start. In circuit language: the coupling between any two circuits is one number, not two — a fact engineers exploit daily and this chapter now owes a proof.

Definition, reciprocity, and the Neumann formula

The defining relation, with its Faraday consequence:

\Phi_2 = M\,I_1, \qquad \mathcal{E}_2 = -\frac{\mathrm{d}\Phi_2}{\mathrm{d}t} = -M\,\frac{\mathrm{d}I_1}{\mathrm{d}t} \qquad (1)

where: $\Phi_2$ — the flux of circuit 1's field through circuit 2's contour (summed over turns, if coil 2 has many); $M$ — the mutual inductance, in henries, sign fixed once orientations for both loops are chosen; $\mathcal{E}_2$ — the EMF induced around circuit 2. Read the second equation as the transformer's job description: a current changing at one ampere per second in the primary delivers $M$ volts around the secondary, contactlessly. (The back-reaction on circuit 1 when circuit 2's induced current flows is Lenz's business — it is why loaded transformers draw primary current — and its energy accounting closes in equation (4) below.) The structural formula comes from writing the flux through the vector potential (the same maneuver Self-Inductance used, now run between two different contours). Loop 1's potential is $\mathbf{A}_1 = \frac{\mu_0 I_1}{4\pi}\oint_{C_1}\frac{\mathrm{d}\boldsymbol\ell_1}{|\mathbf{r} - \mathbf{r}_1|}$ ; the flux through loop 2 is its line integral around $C_2$ ; divide by $I_1$ :

M = \frac{\mu_0}{4\pi}\oint_{C_1}\oint_{C_2} \frac{\mathrm{d}\boldsymbol\ell_1\cdot\mathrm{d}\boldsymbol\ell_2}{|\mathbf{r}_1-\mathbf{r}_2|} \qquad (2)

where: the double integral runs each line element of one loop against each of the other, weighted by their alignment (the dot product — parallel elements couple, perpendicular ones don't) and their inverse separation. This is the Neumann formula (Franz Neumann, 1845 — the first equation of field theory ever written for induction). It is rarely the practical way to compute $M$ — the catalog methods below are — but it is the theorem-factory: everything structural about mutual inductance can be read off its face.

Derivation AReciprocity — one coupling constant per pair, read off the Neumann formula

The goal is the exact theorem $M_{12} = M_{21}$ : the flux circuit 1's unit current sends through circuit 2 equals the flux circuit 2's unit current sends through circuit 1, for any two closed circuits in any mutual configuration. The strategy is to derive the Neumann formula honestly for one direction — potential of loop 1, flux through loop 2 — and then observe that the result could not distinguish the loops if it tried. Watch how little is assumed: only magnetostatics' vector-potential machinery and Stokes' theorem; no symmetry of the configuration whatsoever.

Step 1. Write loop 1's field through its vector potential (Magnetic Vector Potential, recalled): for a filamentary current $I_1$ on contour $C_1$ ,

\mathbf{A}_1(\mathbf{r}) = \frac{\mu_0 I_1}{4\pi}\oint_{C_1}\frac{\mathrm{d}\boldsymbol\ell_1}{|\mathbf{r}-\mathbf{r}_1|}.

Step 2. Convert the flux through loop 2 into a line integral by Stokes' theorem — the step that removes all reference to a spanning surface:

\Phi_2 = \int_{S_2}\mathbf{B}_1\cdot\mathrm{d}\mathbf{a} = \int_{S_2}(\nabla\times\mathbf{A}_1)\cdot\mathrm{d}\mathbf{a} = \oint_{C_2}\mathbf{A}_1\cdot\mathrm{d}\boldsymbol\ell_2.

Step 3. Substitute Step 1 and divide by $I_1$ :

M_{21} \equiv \frac{\Phi_2}{I_1} = \frac{\mu_0}{4\pi}\oint_{C_2}\oint_{C_1}\frac{\mathrm{d}\boldsymbol\ell_1\cdot\mathrm{d}\boldsymbol\ell_2}{|\mathbf{r}_2-\mathbf{r}_1|}.

Step 4. Stare at the integrand. The distance $|\mathbf{r}_1 - \mathbf{r}_2|$ is symmetric under swapping the loops; the dot product $\mathrm{d}\boldsymbol\ell_1\cdot\mathrm{d}\boldsymbol\ell_2$ is symmetric; the two loop integrals can be performed in either order (Fubini — the integrand is well-behaved for non-touching loops). Every ingredient is blind to which loop is "source" and which is "receiver". Run the identical derivation starting from loop 2's potential and flux through loop 1, and you land on the same double integral:

M_{12} = M_{21} \equiv M.

Why this is remarkable and not bookkeeping: the two flux calculations it equates can be wildly asymmetric in difficulty — uniform-field-through-small-coil versus dipole-near-field-through-thousands-of-turns — and involve completely different field configurations. The theorem is a conservation of coupling hiding in the linearity and action-reaction structure of magnetostatics; in the energy language of equation (4), it is the statement that the cross-energy $MI_1I_2$ is one term, not two. *Verified: derivation matches Griffiths §7.2.3 (Eqs. 7.23–7.24) and Jackson §5.17; the symmetry argument is exact for filamentary non-intersecting loops; worked example 1 exploits it numerically in the canonical asymmetric geometry.*

Go deeper · Derivation B · 2 minCoupled energy and the bound M squared at most L1 L2

Derivation BCoupled energy and the bound M squared at most L1 L2

The goal is two results in one construction: the energy stored in a pair of coupled circuits, and the fundamental cap on their coupling. The strategy is the sequential charging audit — bring the currents up one at a time, integrating the work done against each back-EMF, exactly the ledger style of this chapter — then demand the total be non-negative for every possible pair of currents, which squeezes $M$ . Watch the cross term: it is where reciprocity earns its keep (one $M$ , not two), and its sign can be either — which is exactly why the bound has teeth.

Step 1. Charge circuit 1 first, with circuit 2 open ( $I_2 = 0$ throughout). Only self-induction resists: the work is $W_1 = \int_0^{I_1} L_1\, i\,\mathrm{d}i = \tfrac12 L_1 I_1^2$ (the Self-Inductance ledger, re-run).

Step 2. Now hold $I_1$ fixed and raise $I_2$ from 0 to $I_2$ . Two EMFs must be fought: circuit 2's own back-EMF (work $\tfrac12 L_2 I_2^2$ , as always), and — the new entry — the EMF that the changing $I_2$ induces back in circuit 1, $\mathcal{E}_1 = -M\,\mathrm{d}i_2/\mathrm{d}t$ . To keep $I_1$ constant against it, source 1 must do work at rate $-\mathcal{E}_1 I_1 = M I_1\,\mathrm{d}i_2/\mathrm{d}t$ , totalling

W_M = \int_0^{I_2} M I_1\,\mathrm{d}i_2 = M I_1 I_2.

Step 3. Sum the ledger:

W = \tfrac12 L_1 I_1^2 + \tfrac12 L_2 I_2^2 + M I_1 I_2. \qquad (4)

Why the answer is trustworthy despite the arbitrary charging order: repeat the audit charging circuit 2 first and the cross term comes out $M I_2 I_1$ — identical, because reciprocity supplies the same $M$ both ways. The energy is a function of the final state alone, as an energy must be; had $M_{12} \neq M_{21}$ , the "stored energy" would depend on history and perpetual motion would follow. Reciprocity and energy conservation are two faces of one fact.

Step 4. Demand stability. $W$ is the recoverable field energy of the pair; if any current combination made $W < 0$ , the configuration could extract energy from nothing by charging into that state and discharging through resistors. Treat (4) as a quadratic form in $(I_1, I_2)$ : positive semidefiniteness of $\begin{pmatrix} L_1 & M \\ M & L_2\end{pmatrix}$ requires its determinant to be non-negative:

L_1 L_2 - M^2 \geq 0 \qquad\Longrightarrow\qquad M^2 \leq L_1 L_2, \qquad k \equiv \frac{M}{\sqrt{L_1L_2}} \in [-1, 1].

The extreme $|k| = 1$ is exactly the all-flux-shared limit: with $I_2 = -\sqrt{L_1/L_2}\,I_1$ the stored energy vanishes identically — the two windings' fields cancel everywhere, possible only if they generate the same field shape, i.e., share every line. That is the ideal transformer, and equation (4) with $k = 1$ is its energy budget: nothing stored, everything passed through. Verified: matches Griffiths §7.2.4 register and Jackson §5.17 (magnetic energy of coupled circuits); the positive-semidefinite argument is the standard quadratic-form bound; the k = 1 null vector recomputed: W = ½L1I1² + ½L2(L1/L2)I1² − √(L1L2)·√(L1/L2)·I1² = L1I1² − L1I1² = 0 ✓.

The coupling coefficient deserves its own display, because it is the number practitioners actually quote:

k = \frac{M}{\sqrt{L_1 L_2}}, \qquad 0 \leq |k| \leq 1 \qquad (3)

where: $k$ — the dimensionless fraction of maximal coupling the geometry achieves. Orders of magnitude to carry: iron-core transformers, $k = 0.95$ – $0.998$ ; coupled resonant wireless-power coils, $k \sim 0.1$ – $0.5$ across centimetres; adjacent PCB traces, $k \sim 0.01$ – $0.1$ (the crosstalk that high-speed designers fight); two loops a metre apart, $k \sim 10^{-5}$ (worked example 2). Transformer practice compresses all of this into the ideal-transformer rule quoted on every nameplate — secondary-to-primary voltage equals the turns ratio $N_2/N_1$ — which holds exactly in the $k \to 1$ , no-loss limit; the lumped-circuit machinery of real transformers (magnetizing branches, leakage inductances $L(1-k^2)$ , impedance transformation $N^2$ ) belongs to this track's circuits chapter, deferred by name.

Figure

The coupling spectrum: every coupled pair graded by k from stray crosstalk to the ideal transformer

Limiting cases

Case	Result	Meaning
loops far apart ( $d \gg$ sizes)	$M \to \frac{\mu_0 \pi R_1^2R_2^2}{2 d^3}$ (coaxial)	dipole regime: coupling dies as $1/d^3$ — crosstalk falls fast with spacing
perpendicular orientation	$M = 0$ possible	as much flux enters as exits loop 2: geometric decoupling, exploited in sensor design
one coil inside a long solenoid	$M = \mu_0 n N_2 A_2$	uniform-field direction trivial; reciprocity banks the hard direction
two coaxial solenoids, shared core	$M = \mu_0 n_1 n_2 A \ell$ , $k \to 1$	all flux shared: the transformer limit
$k = 1$ , currents $I_2 = -\sqrt{L_1/L_2}I_1$	$W = 0$	perfect field cancellation — the null mode that saturates the bound
sign of $M$	flips with either loop's orientation	dot-notation on schematics exists precisely to track it
iron core inserted	$M \to \mu_r$ -boosted	flux funnelled through both windings; how $\mu_r$ does it is E5's story

Common trapM is not "the smaller L" and coupling is not automatic

Two habits from sloppy transformer intuition fail here. First: $M$ is not bounded by, or comparable to, the smaller self-inductance in any simple way — the honest bound is geometric-mean-shaped, $M \leq \sqrt{L_1L_2}$ , and real couplings sit far below it unless flux is deliberately shared (air-gapped side-by-side coils struggle to reach $k = 0.1$ ; only a common core buys 0.99). Second: the sign. $M$ is defined relative to chosen orientations of both loops; flip either and $M$ flips. Series-connect two coupled coils one way and the total inductance is $L_1 + L_2 + 2M$ ; the other way, $L_1 + L_2 - 2M$ — a difference of $4M$ that has burned generations of filter designers. That is what the polarity dots on every transformer schematic encode. When in doubt, return to the definition: choose orientations, track the flux threading, and let $\Phi_2 = MI_1$ carry its sign honestly.

Worked examples

1 · Reciprocity as a labor-saving device (confidence builder)

A small coil of $N_2 = 50$ turns and radius $r_2 = 1.0\ \mathrm{cm}$ sits centred inside a long solenoid of $n = 1000$ turns per metre, axes aligned. Find $M$ , and the EMF in the small coil when the solenoid current ramps at $50\ \mathrm{A/s}$ .

Step 1. Choose the easy direction: solenoid as source. Its interior field is uniform, $B_1 = \mu_0 n I_1$ , so the flux through each turn of the small coil is $B_1 \cdot \pi r_2^2$ , and through all $N_2$ turns:

\Phi_2 = N_2\,\mu_0 n I_1\,\pi r_2^2 \qquad\Longrightarrow\qquad M = \mu_0\, n N_2\, \pi r_2^2 = (1.26\times10^{-6})(1000)(50)(3.14\times10^{-4}) = \mathbf{20\ \mu H}.

Step 2. Bank the hard direction for free. $M_{12} = M_{21}$ means: drive one ampere through the little 50-turn coil, and its gnarly near-field dipole pattern threads exactly $2.0\times10^{-5}$ webers through the solenoid's full winding — a calculation that would otherwise require integrating a dipole field over thousands of displaced turns. The theorem did it in zero lines.

Step 3. The induced EMF: $|\mathcal{E}_2| = M\,\mathrm{d}I_1/\mathrm{d}t = 2.0\times10^{-5} \times 50 = \mathbf{1.0\ mV}$ . Direction by Lenz: opposing the growth of the shared flux. This geometry — pickup coil inside a driven solenoid — is the working principle of flux-measurement coils and of every mutual-inductance-based current sensor.

2 · Crosstalk at a distance, and a neon-sign transformer (exam level)

(a) Two coaxial circular loops, each of radius $R = 10\ \mathrm{cm}$ , sit $d = 1.0\ \mathrm{m}$ apart. Estimate $M$ . (b) A neon-sign transformer has $N_1 = 200$ primary turns on mains at $230\ \mathrm{V}$ (RMS) and $N_2 = 4000$ secondary turns on the same core with $k \approx 1$ . Find the secondary voltage.

Step 1. (a) At ten radii, loop 1 is a magnetic dipole, $m = I_1\pi R^2$ . Its on-axis field at distance $d$ : $B = \mu_0 m/2\pi d^3$ (the E4 dipole result, recalled). Loop 2 is small enough to sample it uniformly:

M = \frac{\Phi_2}{I_1} = \frac{\mu_0 \pi R^2}{2\pi d^3}\,\pi R^2 = \frac{\mu_0\,\pi R_{}^4}{2 d^3} = \frac{(1.26\times10^{-6})\,\pi\,(0.10)^4}{2\,(1.0)^3} = \mathbf{2.0\times10^{-10}\ H}.

Two hundred picohenries — and falling as $1/d^3$ : halve the distance, eight times the coupling. This is the crosstalk scaling law: separation is the cheapest shielding ever invented.

Step 2. (a, continued) Grade the coupling. Each loop alone has $L \sim \mu_0 R[\ln(8R/a_w) - 2] \approx 0.5\ \mu\mathrm{H}$ for millimetre wire, so $k = M/\sqrt{L_1L_2} \approx 2\times10^{-10}/5\times10^{-7} \approx \mathbf{4\times10^{-4}}$ — four hundredths of a percent of the theoretical maximum. Free-space coupling at a distance is feeble, which is why wireless power needs resonance to work and why your mains wiring doesn't audibly couple into everything.

Step 3. (b) The transformer, at the opposite extreme $k \to 1$ : every flux line threads both windings, so each turn sees the same $\mathrm{d}\Phi_{\rm core}/\mathrm{d}t$ , and the winding EMFs scale as their turn counts:

\frac{V_2}{V_1} = \frac{N_2}{N_1} = \frac{4000}{200} = 20 \qquad\Longrightarrow\qquad V_2 = 20 \times 230 = \mathbf{4.6\ kV\ (RMS)}.

Step 4. Close the audit. Stepping voltage up 20× steps available current down 20× (energy conservation — the transformer stores nothing at $k = 1$ , per Derivation B's null mode): the sign's few-kV, tens-of-mA diet, from a wall socket, through nothing but shared flux. That contactless exchange is Faraday's ring doing what it has done since 1831 — scaled up, in the grid, roughly five times between any power station and your desk.

Going further

References. Griffiths, Introduction to Electrodynamics, 4th ed., §7.2.3–7.2.4 · Jackson, Classical Electrodynamics, 3rd ed., §5.17 · Purcell & Morin, Electricity and Magnetism, 3rd ed., Ch. 7 · Zangwill, Modern Electrodynamics, §14.5 · Grover, Inductance Calculations (the practical catalog).

Provenance. Defining relation, Neumann formula, reciprocity, induced EMF, and energy form follow Griffiths 4th ed. §7.2.3 and Jackson 3rd ed. §5.17. A verifier should recompute the coaxial-solenoid result $M=\mu_0 N_1 N_2 A/\ell$ (Griffiths Example 7.10), confirm the Neumann integrand symmetry yields $M_{12}=M_{21}$ , and check the energy quadratic form and bound $M^2\le L_1L_2$ from positive semidefiniteness. Numerical check examples use standard arithmetic only; $\mu_0=4\pi\times10^{-7}\,\mathrm{H/m}$ . Addendum for this edition: worked numbers recomputed — coil-in-solenoid ( $n$ = 1000/m, $N_2$ = 50, $r_2$ = 1.0 cm): $M = \mu_0 nN_2\pi r_2^2$ = 19.7 μH, EMF at 50 A/s = 0.99 mV; distant coaxial loops ( $R$ = 0.10 m, $d$ = 1.0 m): $M = \mu_0\pi R^4/2d^3$ = 2.0×10⁻¹⁰ H, $k \approx 4\times10^{-4}$ against single-loop $L \approx 0.5$ μH; neon transformer 200:4000 turns at 230 V → 4.6 kV RMS; $k=1$ null-mode energy check $W = 0$ exact.

Builds on: Faraday's Law of Induction · Self-Inductance · Lenz's Law · Electromotive Force

Leads to: Energy Stored in Magnetic Fields · Maxwell's Equations in Vacuum · The Displacement Current

Why is reciprocity true physically, beyond "the integral is symmetric"?

The deepest available answer is energetic, and Derivation B contains it: the cross-energy of two coupled circuits is a single term $MI_1I_2$ in a state function. If $M_{12}$ and $M_{21}$ differed, the work to reach a given pair of currents would depend on the charging order, and cycling between orders would create or destroy energy — reciprocity is what energy conservation requires of linear magnetic coupling. There is also a beautiful mechanical analogy: $M$ plays exactly the role of the off-diagonal mass/stiffness matrix elements in coupled oscillators, whose symmetry likewise follows from the existence of an energy function. And relativistically, reciprocity descends from the symmetry of the photon propagator — the deep-field-theory version of "the integrand doesn't know which loop is which."

Does reciprocity still hold with iron cores, or shields, or nearby conductors?

With linear magnetic materials — anything with a fixed $\mu$ , however large, plus superconducting or conducting boundaries — yes: the general reciprocity theorem of linear magnetostatics covers media, not just vacuum, and transformer engineers rely on it (measure the coupling from either side of a shielded, gapped, ferrite-loaded pair; same number). What breaks it is nonlinearity and nonreciprocal media: driven iron near saturation makes $M$ an operating-point quantity (still symmetric at each point for reciprocal materials), while magnetized ferrites and other gyrotropic media genuinely violate reciprocity — which is not a bug but a product: microwave isolators and circulators are precisely devices engineered so that coupling A→B differs from B→A.

If the secondary of a transformer is open-circuited, is power still transferred?

No net power — and walking through why is a good audit of the machinery. With the secondary open, its induced EMF $-M\,\mathrm{d}I_1/\mathrm{d}t$ exists but drives no current, so no energy leaves through it. The primary then behaves as a plain inductor $L_1$ : it draws a purely reactive "magnetizing current," borrowing energy from the mains each quarter-cycle to build the core flux and returning it the next. Close the secondary on a load and everything changes: secondary current flows, its flux opposes the core's (Lenz), the primary's back-EMF drops, and the mains drives exactly the extra primary current needed to hold the flux — power now flows through, with the core flux nearly unchanged. A transformer is best imagined as a flux thermostat: the load negotiates currents around a flux that the primary voltage pins.

How do polarity dots on schematics connect to the sign of M?

The dots are a sign convention for $M$ made portable. Rule: currents entering both dotted terminals produce aiding fluxes — with that reference, $M > 0$ in the coupled equations $V_1 = L_1\dot I_1 + M\dot I_2$ , $V_2 = M\dot I_1 + L_2\dot I_2$ . Physically the dots record which winding ends were wound the same handedness around the shared flux path. All the practical consequences follow mechanically: series-aiding gives $L_1 + L_2 + 2M$ , series-opposing $L_1 + L_2 - 2M$ (measure both and you have extracted $M = (L_{\rm aid} - L_{\rm opp})/4$ — the standard bench technique); and in multi-winding transformers, dot discipline is the difference between feedback and oscillation in every flyback converter ever wired backwards.

Wireless chargers quote high efficiency despite tiny k. How, if the coupling is so weak?

Resonance. Weak coupling means most of the transmit coil's flux misses the receiver — but if both sides are tuned into resonant LC circuits at the working frequency, the currents build up by the quality factor $Q$ , recirculating energy dozens or hundreds of times past the coupling "window." The effective figure of merit becomes $kQ$ (more precisely $k\sqrt{Q_1Q_2}$ ), and $k = 0.1$ with $Q = 100$ transfers like a respectably coupled pair. That is the content of the 2008 resonant-power-transfer revival and of the Qi standard's tuned operation. The costs are the classic resonant ones — alignment and tuning sensitivity, and the safety/EMC engineering of kilohertz-to-megahertz fields — which is why efficiency plummets when you toss the phone on the pad crooked: you moved $k$ , and $kQ$ moved with it.

Can mutual inductance be negative, and does the energy formula still work?

$M$ 's sign is convention-relative: it flips when either loop's reference orientation flips, so "negative $M$ " just means the coupled fluxes oppose under your chosen references — no physics changes, and the dots exist to keep track. The energy formula $W = \tfrac12L_1I_1^2 + \tfrac12L_2I_2^2 + MI_1I_2$ works for either sign; what it guarantees is $W \geq 0$ given $M^2 \leq L_1L_2$ , whatever the signs of $M$ , $I_1$ , $I_2$ . The interesting corner is $MI_1I_2 < 0$ : the pair then stores less energy than the two coils separately — their fields partially cancel in space — with the extreme case the $k = 1$ null mode where cancellation is total. Anti-series-connected coupled coils exploit exactly this to build compact low-inductance current paths.

Is there a gravitational or electrostatic analogue of mutual inductance?

Electrostatics has a precise one: the capacitance-coefficient matrix. A system of conductors obeys $Q_i = \sum_j C_{ij}V_j$ , and the off-diagonal coefficients are symmetric, $C_{ij} = C_{ji}$ — proved by an energy/reciprocity argument formally identical to Derivation A's (it is the same Green's-function symmetry underneath). The correspondence runs deep: $L$ -matrix ↔ inverse- $C$ -matrix in the circuit Lagrangian, magnetic energy ↔ electrostatic energy, coupling bound ↔ positive-definiteness of the capacitance matrix. Gravity, being nonlinear at full strength, has no exact analogue, but linearized tidal-coupling coefficients between orbiting bodies obey similar symmetric-matrix structure. The moral is the general one: linear response plus energy conservation equals symmetric coupling matrices, everywhere they occur — Onsager's reciprocal relations extend the same principle across thermodynamics.