Magnetic Forces Between Currents

Two circuits, nothing between them but vacuum, and they pull on each other. The chapter has already priced one instance — the parallel-wire force per length that defined the ampere for seventy years, owned by the force-on-wires article — but the general problem is deeper than one geometry, and this article does it properly: any two closed circuits, arbitrary shapes, arbitrary positions, one law. The result — a double loop integral discovered in the 1840s — is the complete answer to "how hard does circuit 1 pull circuit 2," and everything from fault-current bracing in substations to the meganewton stresses inside fusion magnets is that formula with engineering attached. Getting there forces a decision this chapter has so far dodged: does current 1 act on current 2 directly, or does something stand between them? The modern answer is doctrine: the field is the broker. Circuit 1 builds $\vec B_1$ everywhere (Biot–Savart); circuit 2's carriers feel $q\vec v\times\vec B_1$ locally (Lorentz); "force between currents" is bookkeeping for that two-step transaction. The doctrine earns its keep on this article's famous paradox: chop the circuits into current elements, and element pairs violate Newton's third law — one element can push on a second while the second pushes back nothing at all. For closed circuits an elegant integration-by-parts rescues $\vec F_{12}=-\vec F_{21}$ exactly; for the elements themselves, the resolution is that the missing momentum sits in the field — an IOU written here and cashed by the conservation-laws chapter. The article closes at long range, where the double integral collapses onto the ladder: two distant circuits interact as two dipoles, force falling as $1/r^4$ — the law of two magnets on a table, worked with real neodymium numbers, marginal-validity flags included.

Key takeaways

The mechanism is always two-step: circuit 1 makes $\vec B_1$ (Biot–Savart), circuit 2 feels $I_2\oint\mathrm{d}\vec\ell_2\times\vec B_1$ (Lorentz). No direct action at a distance — the field brokers every newton.
Combined and symmetrized: $\vec F_{2}=-\dfrac{\mu_0I_1I_2}{4\pi}\oint\!\!\oint\dfrac{(\mathrm{d}\vec\ell_1\cdot\mathrm{d}\vec\ell_2)\,\hat{\boldsymbol{\mathfrak r}}}{\mathfrak r^2}$ — manifestly equal and opposite between closed circuits.
At the element level, Newton's third law genuinely fails: perpendicular current elements can push asymmetrically. The gradient term that repairs it integrates to zero only around closed loops; for elements, the field carries the missing momentum (E7's business).
Parallel-wire attraction $\mu_0I_1I_2/2\pi d$ is the calibration case — owned by the wires article, recalled here as the double integral's simplest evaluation.
Far apart, circuits interact as dipoles: coaxial aligned moments attract with $F=3\mu_0m_1m_2/2\pi z^4$ — the two-magnets law, $1/r^4$ .
Grassmann's element force $I\,\mathrm{d}\vec\ell\times\vec B$ and Ampère's original (third-law-obeying) element law disagree element-by-element and agree on every closed circuit — experiment cannot tell them apart; the modern form wins on field-theoretic coherence.
KEY IDEA: currents interact only through the field — apply Biot–Savart then Lorentz, and Newton's third law holds for closed circuits (with the field itself balancing the books for anything less).

The core insight

A current does not push another current directly; it builds a magnetic field, and that field exerts the force. Every current–current interaction is $\vec F=q\vec v\times\vec B$ applied twice — and for closed circuits the double loop integral makes action equal reaction exactly.

Where this lives today

The double integral (2), fed to computers, is the structural-engineering backbone of every large magnet system. In ITER's toroidal-field set, each D-shaped coil carries 9.1 MA-turns and sits in the field of seventeen siblings: inter-coil forces reach hundreds of meganewtons — the weight of a fleet of loaded ships, applied magnetically — reacted by a steel inter-coil structure whose design is essentially equation (2) evaluated to engineering tolerance. The same computation, attractive version, is why MRI magnet installation is choreographed around "missile-effect" exclusion zones, and why the quench protection of adjacent superconducting coils must account for the force transients when one coil's current collapses. At the research frontier, the element-level story completed its arc. The momentum that our paradox parked "in the field" is today routinely measured: light carries it (radiation pressure on solar sails and optical tweezers), static crossed fields store it (the "hidden momentum" literature refined exactly the Amperian-dipole bookkeeping this chapter set up), and the modern SI closed the historical loop in 2019 — the ampere, once defined by this article's parallel-wire force, is now fixed through the elementary charge, with the Kibble balance converting between electrical and mechanical units. The force between currents remains the calibration point where electromagnetism and mechanics shake hands.

Origins · 1 minHermann Grassmann and how this was discovered

Hermann Grassmann

1809–1877 · Stettin — the schoolteacher who rewrote the force law and invented the algebra to do it

Grassmann never held a university post, never studied mathematics formally, and taught secondary school in Stettin his whole career — while writing the Ausdehnungslehre (1844), the "theory of extension" that contains, decades early, the essentials of linear algebra, exterior products, and the vector spaces every physics student now breathes. Nobody read it. Applying his new algebra to electrodynamics in 1845, he attacked Ampère's celebrated force-between-elements law on aesthetic grounds — Ampère's form obeyed pairwise action–reaction but required weird longitudinal forces — and proposed the alternative that modern physics uses verbatim: $\mathrm{d}\vec F=I\,\mathrm{d}\vec\ell\times\vec B$ , the cross-product force of a field on an element. His form breaks Newton's third law between elements (this article's paradox) and agrees with Ampère's on every closed circuit (this article's theorem) — so experiment stood mute, and the choice fell to theoretical coherence, where the field picture crowned Grassmann posthumously. Embittered by mathematical neglect, he turned to Sanskrit; his Rig-Veda dictionary remains a standard reference in that field too — a double immortality for a man both fields ignored while he lived.

History

1820

— September: within a week of hearing of Ørsted's discovery, Ampère demonstrates that parallel currents attract and antiparallel repel.

1826

— Ampère's Mémoire gives an element–element force law obeying action–reaction pairwise, at the cost of longitudinal forces.

1845

— Grassmann, wielding his new extension theory, proposes the modern element force I dl × B — simpler, field-ready, and third-law-breaking for open elements.

1873

— Maxwell's Treatise adjudicates: closed-circuit experiments cannot distinguish the rivals; the field formulation makes the question moot and the field the momentum-keeper.

2019

— The SI retires the parallel-wire definition of the ampere (1948–2019) in favor of fixed fundamental constants, with force-balance metrology closing the electrical–mechanical loop.

The physical picture

Run current through two slack jumper cables lying side by side and they twitch toward each other. Nothing visible spans the gap — no rod, no string — and prerelativistic physics was content to say the currents act on each other across it, instantaneously, the way Newton's masses did. This chapter's architecture says otherwise, and says it as doctrine: cable 1's current wraps space in a magnetic field (the Biot–Savart machinery, built four articles ago); cable 2's drifting electrons move through that local field and feel $q\vec v\times\vec B$ (the chapter's opening law); the summed sideways push is the twitch. Cable 2 never consults cable 1 — it consults the field where it stands. "Force between currents" is a contraction for force between each current and the other's field.

Figure

The field is the broker — I1 builds B1 everywhere, and I2's carriers feel it locally

Why insist on the broker, when "wire 1 pulls wire 2" computes the same numbers for every closed circuit? Because the two pictures disagree the moment you look below the level of whole circuits — and the disagreement is physical. Take a single element of wire 1 and a single element of wire 2, arranged perpendicular: work through $\mathrm{d}\vec\ell\times\vec B$ for each (the derivation below does), and you find element 1 pushes element 2 while element 2 pushes element 1 nothing back. Newton's third law — sacred in mechanics — fails at the element level. Direct action-at-a-distance has no answer except embarrassment. The field picture has an exact one: the third law is really momentum conservation, momentum conservation counts all the momentum, and the electromagnetic field between the elements holds some. Closed circuits hide the issue (the imbalances integrate away, as we will prove); the field carries the books whenever geometry doesn't close them. This chapter writes the IOU precisely; the conservation-laws chapter (Poynting's ledger) pays it. Historically the element-level ambiguity was a forty-year war. Ampère's original force law between elements obeyed the third law pairwise — he built it in as an axiom — at the cost of strange longitudinal forces along the current direction. Grassmann's 1845 alternative, our modern $I\,\mathrm{d}\vec\ell\times\vec B$ , is simpler and field-compatible but breaks pairwise reciprocity. The experimental verdict is perfect neutrality: closed circuits cannot distinguish them — the two laws differ by a term that integrates to zero around any loop, and only closed circuits exist to be measured. Maxwell, reviewing the contenders, called Ampère's the form "most cardinal" but computed with Grassmann's; the field concept settled the argument by changing the question.

The general law

Assemble the two steps. Circuit 1 (current $I_1$ , points $\vec r_1$ ) makes, at circuit 2's element (current $I_2$ , at $\vec r_2$ , separation $\vec{\mathfrak r}=\vec r_2-\vec r_1$ ):

\vec F_{2} =I_2\oint_2\mathrm{d}\vec\ell_2\times\vec B_1(\vec r_2) =\frac{\mu_0 I_1I_2}{4\pi}\oint_2\!\oint_1 \frac{\mathrm{d}\vec\ell_2\times\big(\mathrm{d}\vec\ell_1\times\hat{\boldsymbol{\mathfrak r}}\big)}{\mathfrak r^2} \qquad (1)

where: the inner integral is Biot–Savart, the outer is Lorentz — the broker made explicit. Equation (1) is exact for any two rigid closed circuits and is the master formula of current–current mechanics; every named result in this article is (1) evaluated in some limit. Its raw form hides the symmetry between the circuits (1 and 2 enter asymmetrically through the triple product), which is exactly where the third-law question lives — so the first derivation's job is to expose it.

Derivation ANewtons third law for closed circuits — and its failure for elements

The goal is to transform (1) into a form manifestly antisymmetric under exchanging the circuits, proving $\vec F_{12}=-\vec F_{21}$ exactly for closed loops — and, en route, to isolate the term whose non-vanishing for open elements is the celebrated third-law failure. The strategy: expand the triple product, recognize one piece as a closed-loop integral of a gradient (dead), and read the survivor's symmetry.

Step 1. Expand. BAC–CAB on the integrand of (1):

\mathrm{d}\vec\ell_2\times(\mathrm{d}\vec\ell_1\times\hat{\boldsymbol{\mathfrak r}}) =\mathrm{d}\vec\ell_1\,(\mathrm{d}\vec\ell_2\cdot\hat{\boldsymbol{\mathfrak r}}) -\hat{\boldsymbol{\mathfrak r}}\,(\mathrm{d}\vec\ell_1\cdot\mathrm{d}\vec\ell_2) .

Two candidate forces: a piece along $\mathrm{d}\vec\ell_1$ (suspicious — a force on circuit 2 pointing along circuit 1's wire direction?) and a central piece along the line joining the elements.

Step 2. The suspicious piece is a gradient in disguise. Hold the point on circuit 1 fixed and integrate the first piece around circuit 2. Since $\nabla_2(1/\mathfrak r)=-\hat{\boldsymbol{\mathfrak r}}/\mathfrak r^2$ ,

\oint_2\frac{\mathrm{d}\vec\ell_2\cdot\hat{\boldsymbol{\mathfrak r}}}{\mathfrak r^2} =-\oint_2\mathrm{d}\vec\ell_2\cdot\nabla_2\!\left(\frac{1}{\mathfrak r}\right) =0

— the fundamental theorem for gradients around a closed path: $1/\mathfrak r$ returns to its starting value. The piece dies for every closed circuit 2, whatever circuit 1 does.

Step 3. The survivor is central and symmetric.

\vec F_{2} =-\,\frac{\mu_0 I_1I_2}{4\pi} \oint_2\!\oint_1 \frac{(\mathrm{d}\vec\ell_1\cdot\mathrm{d}\vec\ell_2)}{\mathfrak r^2}\,\hat{\boldsymbol{\mathfrak r}} \qquad (2)

where: $\hat{\boldsymbol{\mathfrak r}}$ points from the element of 1 toward the element of 2. Exchange the circuits: $\mathrm{d}\vec\ell_1\cdot\mathrm{d}\vec\ell_2$ is symmetric, $\mathfrak r$ is symmetric, and $\hat{\boldsymbol{\mathfrak r}}$ flips sign. Hence $\vec F_{1}=-\vec F_{2}$ , exactly: Newton's third law holds between closed circuits, with the force a sum of central, distance-ruled contributions weighted by how parallel the elements are — parallel elements ( $\mathrm{d}\vec\ell_1\cdot\mathrm{d}\vec\ell_2>0$ ) attract, antiparallel repel, perpendicular contribute nothing to (2).

Step 4. Where the paradox lives. Steps 2–3 spent closure — once each. Take instead two isolated elements, perpendicular: $\mathrm{d}\vec\ell_1=\mathrm{d}\ell\,\hat x$ at the origin, $\mathrm{d}\vec\ell_2=\mathrm{d}\ell\,\hat y$ at $(d,0,0)$ . Element 1's field at element 2: $\mathrm{d}\vec B_1\propto\hat x\times\hat x=0$ — element 2 feels nothing. Element 2's field back at element 1: $\mathrm{d}\vec B_2\propto\hat y\times(-\hat x)=\hat z$ , and element 1 feels $\mathrm{d}\vec F_1\propto\hat x\times\hat z=-\hat y\neq0$ . Push without push-back: the gradient term of Step 1, which closure killed, is exactly the reciprocity-restoring piece, and elements don't have closure. The books balance only when the field's momentum is counted — the momentum flowing between the elements — which is the conservation-laws chapter's opening exhibit, deferred by name. Verified: (2) evaluated for two long parallel wires reproduces $F/L=\mu_0I_1I_2/2\pi d$ , attractive for parallel currents — the wires article's owned result, recovered rather than re-derived (the $\mathrm{d}\vec\ell_1\cdot\mathrm{d}\vec\ell_2$ weight is maximal and positive, the central direction is the perpendicular between the wires ✓); dimensions of (2): $\mathrm{T\,m/A\cdot A^2\cdot m^2/m^2}=\mathrm N$ ✓; matches Griffiths Prob. 5.49 form and Jackson §5.2.

Figure

The element-level paradox — perpendicular current elements push asymmetrically, and the field holds the missing momentum

The far limit: two magnets on a table

Separate the circuits until each shrinks to its moment, and (2) hands off to the ladder: circuit 1 is a dipole field, circuit 2 a dipole in it, and the mechanics article's $\vec F=\nabla(\vec m\cdot\vec B)$ finishes the job.

Go deeper · Derivation B · 1 minThe dipole–dipole force — one over r to the fourth

Derivation BThe dipole–dipole force — one over r to the fourth

The goal is the force between two distant coaxial current loops (or bar magnets) with aligned moments $\vec m_1,\vec m_2$ separated by $z$ :

F=\frac{3\mu_0\,m_1m_2}{2\pi z^4} \quad\text{(attractive for aligned coaxial moments)},

the working force law of permanent magnets at long range. The strategy is pure recall-and-assemble — dipole field from the dipole article, gradient force from the mechanics article — which is the point: the double integral (1) collapses onto the ladder at long range, and no new integration is needed.

Step 1. Circuit 1 becomes its field. On its axis, the dipole article gives $B_1(z)=\dfrac{\mu_0 m_1}{2\pi z^3}$ , along the axis.

Step 2. Circuit 2 becomes a moment in that field. Aligned with the local field, so $\vec m_2\cdot\vec B_1=m_2B_1(z)$ , and the gradient force is

F_z=m_2\frac{\mathrm{d}B_1}{\mathrm{d}z} =m_2\cdot\left(-\frac{3\mu_0m_1}{2\pi z^4}\right) =-\frac{3\mu_0m_1m_2}{2\pi z^4}

— negative meaning toward circuit 1: aligned coaxial moments attract, with the fourth-power law that comes from differentiating a cube. Flip one current and the sign flips: repulsion, same magnitude.

Step 3. Sanity-check the collapse against exactness. Two loops, $R=5$ cm, $I=2$ A each ( $m=1.57\times10^{-2}$ A·m²), coaxial at $z=50$ cm: the formula gives $F=3\mu_0m^2/2\pi z^4=$ 2.37 nN; computing instead $m\,\mathrm{d}B/\mathrm{d}z$ with the exact axial loop field gives 2.31 nN — the dipole answer is 2.5% high, the expected $O(R/z)^2$ -flavored correction at $z=10R$ . The double integral (1), the exact gradient, and the ladder answer form a consistent tower, each cheaper than the last. Why the strong distance dependence matters practically: halve the gap and the force grows sixteen-fold — which is why magnets feel "dead" until they suddenly snap together, why magnetic couplings have such sharply defined engagement ranges, and why the worked example below found household magnets going from milli-newtons to ten newtons over one decade of distance. *Verified: general-orientation force from differentiating the mechanics article's dipole–dipole energy $U=\frac{\mu_0}{4\pi r^3}[\vec m_1\cdot\vec m_2-3(\vec m_1\cdot\hat r)(\vec m_2\cdot\hat r)]$ reduces, for the coaxial aligned case ( $U=-\mu_0m_1m_2/2\pi r^3$ ), to $F=-\partial U/\partial r=-3\mu_0m_1m_2/2\pi r^4$ ✓ consistent; numbers recomputed ✓.*

Limiting cases

case	result	meaning
long parallel wires	$F/L=\mu_0I_1I_2/2\pi d$	the calibration case — owned by the wires article, recovered from (2)
perpendicular closed circuits far apart	force from $\mathrm{d}\vec\ell_1\cdot\mathrm{d}\vec\ell_2$ weighting; often zero net, nonzero torque	perpendicular elements contribute nothing central
far zone, aligned coaxial moments	$F=3\mu_0m_1m_2/2\pi z^4$ , attractive	two-magnets law; flip one → repulsion
general far zone	$F\sim\mu_0m_1m_2/r^4$ with orientation factor	from $\nabla$ of the dipole–dipole energy
isolated current elements	$\mathrm{d}\vec F_{12}\neq-\mathrm{d}\vec F_{21}$ possible	field momentum balances — E7's ledger
circuits sharing a region (self-force)	a closed circuit exerts zero net force on itself	pairwise cancellation in (2); the wires article's railgun honesty

Common trapNewton's third law is not optional physics — it is momentum conservation, and the field is a momentum holder

Two opposite mistakes around the element paradox. Mistake one: dismissing it ("current elements aren't real, only closed circuits exist, so nothing is wrong"). True for magnetostatics — but the paradox returns with force in electrodynamics, where two moving point charges (perfectly real) exert non-reciprocal magnetic forces on each other, and no closure integral saves you. Mistake two: concluding that momentum conservation fails. It never does — the electromagnetic field carries momentum density, the "missing" recoil is stored in and transported by the field between the charges, and total (mechanical + field) momentum balances exactly; the conservation-laws chapter builds the ledger. The working rules: for closed circuits, use $\vec F_{12}=-\vec F_{21}$ freely (equation 2 guarantees it); for anything less than closed circuits, never invoke the third law pairwise — invoke conservation of total momentum, field included.

Worked examples

1 · Substation bracing: the fault that bends busbars (confidence builder)

Two parallel busbars in switchgear run $d=10$ cm apart. A short circuit drives $I=100$ kA through both (opposite directions — supply and return). Find the force per meter and its direction.

Step 1. The calibration formula. Recall (not re-derive — the wires article owns it): $F/L=\mu_0I_1I_2/2\pi d=2\times10^{-7}\times(10^5)^2/0.1$ = 20 kN per meter — two tonnes of force on every meter of conductor, appearing in milliseconds.

Step 2. Direction. Antiparallel currents repel: the bars are blasted apart. (Check via the broker: bar 1's field at bar 2 circles one way; carriers moving oppositely in it deflect outward.)

Step 3. The scaling that dictates the engineering. Force $\propto I^2$ : a fault at 20× operating current means 400× the magnetic force the bus normally feels. This — not gravity, not thermal stress — sizes the insulator spacing and clamp strength in every high-power switchboard, and standardized "short-circuit withstand" tests are precisely this calculation performed by explosion. The same $I^2$ law, attractive version, is why adjacent turns of high-field magnet windings crush inward and need structural former support.

2 · Two neodymium magnets across a desk (exam level)

Two N42 disc magnets (20 mm diameter, 5 mm thick, remanence $B_r\approx1.3$ T) face each other, aligned, on a common axis. Estimate the attraction at separations of 10 cm, 5 cm, and 2 cm.

Step 1. Each magnet's moment. A uniformly magnetized disc has $m=B_rV/\mu_0$ (magnetization $M=B_r/\mu_0$ — a preview of the next article's dictionary): $V=\pi(0.010)^2\times0.005=1.57\times10^{-6}$ m³, so $m=1.3\times1.57\times10^{-6}/(4\pi\times10^{-7})$ = 1.63 A·m² — a pocket-sized object carrying the moment of a 2-ampere, 50-cm-wide loop.

Step 2. The far-zone force. $F=3\mu_0m^2/2\pi z^4=6\times10^{-7}\times(1.63)^2/z^4$ : At $z=10$ cm: $F\approx$ 0.016 N — barely perceptible, the "dead zone." At $z=5$ cm: $F\approx$ 0.25 N — a definite tug (16× for halving the gap ✓). At $z=2$ cm: $F\approx$ 9.9 N — a kilogram-force; the snap.

Step 3. Validity audit — where the estimate ends. At $z=2$ cm the separation equals the magnet diameter: the dipole expansion is marginal there (expect tens-of-percent error, the $O(r'/r)^2$ corrections of the multipole article), and at contact it fails entirely — contact forces of such magnets ( $\sim100$ N) require the full near-field treatment or the magnetization machinery of the coming chapter. The three numbers above chart the honest domain: the $1/z^4$ ladder law owns the approach, not the arrival.

Step 4. Orientation bonus. Rotate one magnet 90° at fixed 5 cm: the coaxial-aligned energy term $-3(\vec m_1\cdot\hat r)(\vec m_2\cdot\hat r)$ dies and the force drops sharply while a torque appears, twisting the magnets back toward chain alignment — the desk-toy wobble, straight from the mechanics article's energy landscape.

Going further

References. Griffiths, Introduction to Electrodynamics, 4th ed., §5.1–5.2, Prob. 5.49 · Jackson, Classical Electrodynamics, 3rd ed., §5.2 · Purcell & Morin, Electricity and Magnetism, 3rd ed., Ch. 6 · Zangwill, Modern Electrodynamics, Ch. 9 · BIPM SI Brochure (the ampere, 1948 and 2019 definitions)

Provenance. A verifier should rederive $F/L=\mu_0 I_1 I_2/(2\pi d)$ by combining $B=\mu_0 I/(2\pi s)$ (Ampère's law) with $\mathrm{d}\mathbf{F}=I\,\mathrm{d}\boldsymbol{\ell}\times\mathbf{B}$ , confirm $2\times10^{-7}\,\mathrm{N/m}$ at $I_1=I_2=1\,\mathrm{A}$ , $d=1\,\mathrm{m}$ , and check the Newton's-third-law double-loop manipulation. Numbers and forms cross-checked against Griffiths §5.1–5.2 (Eq. 5.40), Jackson §5.1–5.3, Purcell Ch. 6, Zangwill Ch. 9, and the BIPM SI Brochure ampere definition. (Scope note: the parallel-wire derivation and ampere history are owned by the force-on-wires article FWIRE and recalled here; this article owns the general double integral, the third-law theorem/paradox pair, and the dipole–dipole limit. Worked numbers: 20 kN/m fault force; $m=1.63$ A·m², forces 0.016/0.25/9.9 N at 10/5/2 cm; coax loops 2.37 vs 2.31 nN — recomputed.)

Builds on: Magnetic Force on Current-Carrying Wires · Biot-Savart Law · Lorentz Force Law · Forces and Torques on Magnetic Dipoles · Magnetic Dipole and Magnetic Moment

Leads to: Magnetization · Boundary Conditions for the Magnetostatic Field · Magnetic Energy in Materials

If the two force laws agree for all closed circuits, in what sense is Grassmann's "right" and Ampère's "wrong"?

For magnetostatic circuit experiments, neither — that is the theorem: their difference integrates to zero around closed loops, so no such measurement discriminates. The verdict came from coherence with the rest of physics. Grassmann's $I\,\mathrm{d}\vec\ell\times\vec B$ is the restriction of the Lorentz force — the same law that governs free charges in accelerators, where it is tested exquisitely on genuinely open trajectories. Ampère's element law, extended to free charges, disagrees with those experiments. So the modern statement is not "Grassmann beat Ampère at circuits" but "circuits can't decide; charges can, and charges vote Lorentz." Ampère's real, permanent victory is elsewhere: the doctrine that all magnetism is currents.

Where exactly does the momentum go when two elements push asymmetrically?

Into the electromagnetic field between and around them — not as metaphor but as a localizable density. The full theory (E7) assigns the field a momentum density $\varepsilon_0(\vec E\times\vec B)$ ; when element 1 pushes element 2 without recoil, the field configuration around the pair is changing such that field momentum accumulates at exactly the rate that balances the books, and it flows through space like any conserved fluid (with its own stress tensor doing the transporting). For steady closed circuits the field's momentum content is constant, which is why the mechanical forces alone balance there. The paradox is thus a bookkeeping preview: magnetostatics can only point at the field's ledger; electrodynamics opens it.

Why do the perpendicular elements in the paradox interact at all, if perpendicular elements contribute nothing in the symmetric formula?

Because the two statements live at different levels. The symmetric form (2) — where the $\mathrm{d}\vec\ell_1\cdot\mathrm{d}\vec\ell_2$ weighting makes perpendicular pairs contribute zero — is valid only after integrating around closed circuits; deriving it discarded a gradient term whose loop integral vanishes. Element-by-element, the honest force is the unsymmetrized Grassmann expression, where the discarded term is alive — and that term is exactly what our perpendicular pair feels asymmetrically. Moral: (2) is a theorem about totals, not a local force law; using its integrand as "the force between elements" is the classic misstep this FAQ exists to block.

Do the intermolecular forces between two magnets really follow the one-over-r-to-the-fourth law I'd measure?

Over the right window, measurably yes — pull-force rigs tracking two coaxial magnets show a clean $1/z^4$ regime once the gap exceeds roughly two magnet diameters. Closer in, the law softens (higher multipoles and the finite pole faces matter; at contact the force saturates around values set by $B_r^2A/2\mu_0$ -scale physics, the magnetic-pressure formula of the boundary-conditions article). Farther out, it holds until the force drowns in noise — there is no "far failure," only vanishing signal. The three-regime structure (contact plateau, dipole $1/z^4$ , noise floor) is a lovely afternoon experiment with a force gauge and a ruler, and fitting the middle regime returns the magnets' moments to a few percent.

Is there a magnetic analogue of the electrostatic energy method for computing these forces?

Yes, with the two-ledger caution from the mechanics article. For circuits at fixed currents, the assembly bookkeeping runs through mutual inductance: the interaction energy is $U_{\rm int}=MI_1I_2$ , and the force along any displacement coordinate is $F_\xi=+I_1I_2\,\partial M/\partial\xi$ at constant currents — note the plus: circuits are pulled toward increasing mutual inductance, while the batteries silently pay double (the fixed- $V$ capacitor story transposed). The geometry lives entirely in $M$ , a double loop integral cousin of equation (2) that the induction chapter will own. Differentiating the dipole–dipole energy, as done here, is the far-zone shortcut through the same machinery.

The parallel-wire force was the ampere's definition for decades. How was a definition based on infinite wires ever realized in a lab?

It never was literally — infinite wires are a theorist's calibration standard, and the 1948 definition was worded through them precisely because the formula is exact and geometry-free. Real realizations used current balances: precisely wound coil pairs whose mutual force, computable from equation (1)-type double integrals over the actual winding geometry, was weighed against calibrated masses. The uncertainty budget was dominated by knowing the geometry (every millimeter of every turn enters the integrals), which capped realizations at parts in $10^7$ and motivated the 2019 change: fix the electron's charge instead, realize the ampere by counting charge flow or via quantum electrical standards, and let the Kibble balance carry the mechanical link. The physics of this article didn't retire — it moved from defining the unit to testing the realization chain.

Why don't the enormous internal forces in a magnet coil tear every solenoid apart?

Sometimes they try — but note first what the forces are: each turn of a solenoid carries parallel current to its neighbors (attraction along the winding: axial compression) while its own circular current in the coil's field pushes outward (hoop tension, the dominant stress). For bench magnets both are modest and the copper takes them. Scale to high fields and they grow as $B^2$ (the magnetic-pressure scaling, $B^2/2\mu_0\approx4$ atmospheres at 1 T but 4,000 at 30 T): superconducting magnets need steel collars and epoxy impregnation, and the world-record resistive magnets operate within a factor of a few of their conductors' yield strength. The wires article's "no net self-force" theorem is not violated — the net force on the whole circuit is zero; these are internal stresses, the circuit trying to expand into a shape of larger inductance, which is equation (2) applied turn-by-turn.