Magnetic Force on Current-Carrying Wires

In September 1820, within a week of hearing that a current deflects a compass needle, André-Marie Ampère stood before the Paris Academy with a sharper claim: two wires, carrying nothing but current, push on each other — attracting when the currents run parallel, repelling when they oppose. No magnets anywhere. Matter that is electrically neutral to exquisite precision, reaching across empty space to move other neutral matter, purely because charge is flowing inside both. Every electric motor since is a refinement of that demonstration. This article turns the Lorentz force into the working mechanics of currents. The conversion dictionary was built two articles back: summing $q\vec v\times\vec B$ over the carriers in a wire packages the force as $\vec F = I\!\int\mathrm{d}\vec\ell\times\vec B$ — that theorem is recalled here, not re-proved, and then spent on the classics. A straight wire in a uniform field: $F = BIL\sin\theta$ , the formula that weighs current in a lab balance. Two parallel wires: $F/L = \mu_0I_1I_2/2\pi d$ , the mutual force that carried the SI definition of the ampere for seventy years. A current loop in a uniform field: zero net force but a stubborn torque $\vec N = \vec m\times\vec B$ — the twist that turns every motor rotor, swings every galvanometer needle, and previews the magnetic dipole moment that the magnetized-matter chapter will build into a theory of magnets. Running underneath is the energy honesty this chapter established at the outset: none of these forces ever does net work at the carrier level. Motors deliver kilowatts anyway — brokered, not donated, by the field, with the supply paying against the back-EMF exactly as the rising-bar audit in Lorentz Force Law laid out. Here that audit is recalled where it belongs: in the machine room.

Key takeaways

A wire carrying steady current $I$ in an external field feels $\vec F = I\!\int\mathrm{d}\vec\ell\times\vec B$ — the carrier-summing theorem owned by Electric Current and Current Density, spent here on applications.
Straight wire, uniform field: $\vec F = I\vec L\times\vec B$ , magnitude $BIL\sin\theta$ ; force per length $I\,\hat{\boldsymbol\ell}\times\vec B$ ; thick conductors take the body force $\vec f = \vec J\times\vec B$ .
Two parallel wires interact with force per length $F/L = \mu_0I_1I_2/2\pi d$ — parallel currents attract, antiparallel repel; at 1 A and 1 m this is $2\times10^{-7}$ N/m, the pre-2019 definition of the ampere.
A closed loop in a uniform field feels zero net force ( $\oint\mathrm{d}\vec\ell = 0$ ) but a torque $\vec N = \vec m\times\vec B$ with $\vec m = I\vec A$ — the magnetic dipole moment, born here as a rectangle's couple.
The torque drives motors (commutator flips $I$ every half-turn) and galvanometers (spring balances $NIAB$ ); force questions in non-uniform fields belong to the dipole articles of the magnetic-matter chapter.
The field's $\vec B$ in the force law is the external field — a wire exerts no net force on itself; and the energy in every magnetic machine comes from the source, not the field (audit in Lorentz Force Law).
KEY IDEA: replace $q\vec v \to I\,\mathrm{d}\vec\ell$ and the Lorentz force becomes wire mechanics — $BIL$ pushes, $\mu_0I_1I_2/2\pi d$ binds parallel currents, and $\vec m\times\vec B$ twists loops into alignment.

The core insight

Replace the per-charge Lorentz force $q\,\vec v\times\vec B$ by the substitution $q\vec v \to I\,\mathrm{d}\vec\ell$ (or $\rho\vec v \to \vec J$ ) to obtain the force on a current as $\vec F = \int I\,\mathrm{d}\vec\ell\times\vec B = \int\vec J\times\vec B\,\mathrm{d}\tau$ .

Where this lives today

The torque equation runs civilization's rotating machinery: industrial motors — which consume roughly $45\%$ of the world's electricity — are engineering elaborations of Derivation B, with the commutator (DC) or the rotating field phase (AC induction, synchronous) solving the same problem: keep $\vec m\times\vec B$ from dying at alignment. Voice coils in speakers and hard-drive arms, MEMS mirrors, and the reaction wheels that point space telescopes all run force-on-current mechanics at their own scales. The parallel-wire force, meanwhile, sets real structural budgets: pulsed magnets and tokamak coils must be braced against their own $\vec J\times\vec B$ loads (which scale as $B^2$ and reach hundreds of atmospheres at 20 T), fault currents in switchyards can bend busbars in milliseconds, and the pinch effect — a plasma column squeezed by the mutual attraction of its own parallel current filaments — is both a fusion confinement scheme and the reason lightning channels crimp into beads. Metrology closed a historical loop here. The ampere lived on equation (3) from 1948 to 2019 — realized by exquisite "current balances" weighing the force between coils. The 2019 SI moved the ampere onto the fixed elementary charge, and the modern Kibble balance runs this article's physics in reverse: it balances a weight against the force on a current-carrying coil to realize the kilogram from electrical quantities. The force law that once defined electrical units now defines mass.

Origins · 1 minAndré-Marie Ampère and how this was discovered

André-Marie Ampère

1775–1836 · Lyon, Paris

Ampère heard of Ørsted's compass deflection on 11 September 1820 and delivered his first memoir on the forces between currents one week later — then spent the autumn in a virtuoso duel of experiment and theory, establishing that parallel currents attract, antiparallel repel, and that coils behave as magnets. His force between current elements, published in mature form in 1826, made currents the fundamental magnetic actors — he proposed that all magnetism, including iron's, is circulating current at the molecular scale, a hypothesis vindicated a century later by atomic physics. Self-taught, widowed young, and by all accounts absent-minded to legend, he built the first theory-driven electrodynamics; Maxwell called him "the Newton of electricity." The SI unit of current carried his name through both of its definitions — force-based and charge-based — and the loop-current picture of magnetic matter that this article's torque previews is still called Ampèrian.

History

1820

— Ørsted's compass twitches in April; by late September Ampère demonstrates wire-wire forces and states the parallel/antiparallel rule.

1821

— Faraday builds the first electromagnetic rotor: a wire circling a magnet — the ancestor of every motor.

1826

— Ampère's Mémoire codifies the force between currents; magnetism becomes the science of circulating charge.

1882

— Deprez and d'Arsonval perfect the moving-coil galvanometer — the radial-gap torque meter of worked example 2.

1948

— The SI defines the ampere by the parallel-wire force: 2×10⁻⁷ N/m at one metre.

2019

— The ampere moves to the fixed elementary charge; the Kibble balance turns the force law into a definition of the kilogram.

The physical picture

Picture the wire as a hose full of charge. Each carrier would, on its own, curve into a cyclotron circle — but the lattice holds it in line, so the sideways magnetic kick on every carrier is delivered, through collisions and the wire's internal fields, to the wire as a whole. The macroscopic force you feel is the bookkeeping of some $10^{23}$ tiny $q\vec v\times\vec B$ nudges per metre, all agreeing on a direction. The right-hand rule survives the promotion: fingers along the current, curl toward $\vec B$ , thumb gives the force — and reversing the current flips it, which is exactly how a DC motor's commutator keeps the rotor turning the same way through every half-revolution. The parallel-wire attraction is the piece worth staring at. Two neutral conductors pull together across centimetres of vacuum. Wire 1's current wraps a circulating field around itself (the Biot–Savart business of the next article — here we borrow the result); wire 2's carriers drift through that field, and the cross product points them straight at wire 1. Run the currents opposite and everything flips to repulsion. The force is feeble at bench currents — $2\times10^{-7}$ newtons per metre at one ampere and one metre — but it scales as $I_1I_2$ , and at short range inside machines it becomes the dominant mechanical load: busbar bracing, transformer hum, and the self-crushing pinch of a lightning channel are all this one formula at industrial current.

Figure

Wire mechanics from carrier kicks — the rule, and why parallel currents attract

The last classic needs a loop. Close the wire into a rectangle and put it in a uniform field: the four side-forces now come in equal-and-opposite pairs, so the net force is exactly zero — but the pair on the two active sides act along different lines, and a pair of equal, opposite, offset forces is a couple: pure torque. The loop doesn't slide; it turns, twisting until its face is square to the field. That torque, packaged as $\vec m\times\vec B$ with $\vec m = I\vec A$ , is the first appearance of the magnetic dipole moment — the quantity that will organize the entire theory of magnetic materials two chapters from now, and the reason a compass needle (a bundle of atomic current loops) points north.

The force law, spent

The tool was forged in Electric Current and Current Density: summing the magnetic Lorentz force over carriers and collapsing the geometry gives

\vec F = \int I\,\mathrm{d}\vec\ell\times\vec B, \qquad \frac{\mathrm{d}\vec F}{\mathrm{d}\ell} = I\,\hat{\boldsymbol\ell}\times\vec B, \qquad \vec f = \vec J\times\vec B \qquad (1)

where: $\mathrm{d}\vec\ell$ — line element along the current; $\vec B$ — the external field at that element (the field of all other sources — a wire exerts no net force on itself, a point the warning below sharpens); $\vec f$ — force per volume for thick conductors and plasmas. For a straight segment $\vec L$ in a uniform field the integral collapses:

\vec F = I\,\vec L\times\vec B, \qquad F = BIL\sin\theta \qquad (2)

where: $\theta$ — angle between wire and field. Read the scaling: at $B = 0.4\ \mathrm{T}$ (a strong lab magnet), five amperes through two metres of perpendicular wire feels $4$ newtons — the weight of an apple bag, from invisible drift at fractions of a millimetre per second, because the carrier count is astronomical. This is the formula of the current balance, the jumping-wire demonstration, and — run at $5000$ A between self-fed rails — the railgun. Two of the great set-pieces of magnetostatics now follow in a few lines each.

Derivation AThe force between parallel wires — the old ampere

The goal is the mutual force per unit length between two long parallel wires a distance $d$ apart carrying currents $I_1$ and $I_2$ — the interaction Ampère measured in 1820 and the SI enshrined in 1948. The strategy is a clean division of labour: wire 1 makes a field (we quote the result $B_1 = \mu_0I_1/2\pi d$ , derived honestly by integration in the next article, Biot-Savart Law); wire 2's current feels it through equation (1). Watch the geometry do the attraction/repulsion verdict — no signs are put in by hand.

Step 1. The field of wire 1 at wire 2's location. An infinite straight current wraps circular field lines around itself with magnitude

B_1 = \frac{\mu_0 I_1}{2\pi d},

directed azimuthally (right hand: thumb along $I_1$ , fingers curl along $\vec B_1$ ). At wire 2, this field is perpendicular to wire 2's length. Why quoting is legitimate here: the formula is this chapter's most-used field result, owned and derived from first principles next article; using it forward keeps each derivation single-purpose.

Step 2. Apply the force law to wire 2. With $\mathrm{d}\vec\ell$ along $I_2$ and $\vec B_1$ perpendicular to it:

\frac{F}{L} = I_2B_1 = \frac{\mu_0\,I_1I_2}{2\pi d} \qquad (3)

Step 3. Get the direction from the cross product. For parallel currents, $\hat{\boldsymbol\ell}_2\times\vec B_1$ points from wire 2 toward wire 1 — attraction; flip either current and the cross product flips — repulsion. By Newton's third law (or by repeating the argument with roles swapped) wire 1 feels the equal opposite force. Why "like attracts" here when like charges repel: the interaction is not between charges but between currents — the circulating field geometry reverses the intuition. A useful mnemonic with real physics inside: parallel currents share their field between them ways that lower the field energy when they approach.

Step 4. Put in the definition-grade numbers. $I_1 = I_2 = 1\ \mathrm{A}$ , $d = 1\ \mathrm{m}$ :

\frac{F}{L} = \frac{(4\pi\times10^{-7})(1)(1)}{2\pi(1)} = 2\times10^{-7}\ \mathrm{N/m}.

Why this number mattered: from 1948 to 2019 it was the ampere — the SI defined 1 A as the current producing exactly this force, which is also why $\mu_0$ was exactly $4\pi\times10^{-7}$ by decree. The 2019 redefinition moved the ampere onto the fixed elementary charge, demoting $\mu_0$ to a measured constant (it still agrees with $4\pi\times10^{-7}$ to a part in $10^{10}$ ), but equation (3) remains the conceptual bridge between current and mechanical force. *Verified: matches Griffiths Eq. 5.40 and the historical definition; magnitude recomputed; scale check — 100 A busbars 2 cm apart feel 0.10 N/m, the origin of busbar bracing specs and 100/120 Hz transformer hum (the force pulses at twice the AC frequency, $\propto I^2$ ).*

Figure

The loop couple: equal, opposite, offset — zero force, pure torque

Go deeper · Derivation B · 2 minTorque on a current loop — the dipole moment is born

Derivation BTorque on a current loop — the dipole moment is born

The goal is the behaviour of a closed rectangular loop (sides $a$ and $b$ , current $I$ ) in a uniform field: zero net force, but a torque $\vec N = \vec m\times\vec B$ where $\vec m = I\vec A$ — the definition and debut of the magnetic dipole moment. The strategy is direct assembly from equation (2) applied side by side; the structural insight to watch is why the net force dies while the torque survives: equal-and-opposite forces acting along different lines.

Step 1. Kill the net force in full generality. For any closed loop in a uniform field,

\vec F = I\left(\oint\mathrm{d}\vec\ell\right)\times\vec B = \vec 0,

because the constant $\vec B$ factors out of the integral and the closed sum of displacement elements vanishes identically. Why this is worth its own step: it holds for loops of any shape, so uniform fields can only ever twist closed circuits, never translate them. Net translation of loops requires field gradients — the force $\nabla(\vec m\cdot\vec B)$ that the magnetic-matter chapter derives for dipoles.

Step 2. Set the stage for the torque. Tilt the loop so its normal $\hat{\vec n}$ (right-hand rule from the current sense) makes angle $\theta$ with $\vec B$ . The two sides of length $a$ perpendicular to the tilt axis carry forces of magnitude

F = BIa,

equal, opposite — and offset: they act at the two ends of the sides of length $b$ .

Step 3. Compute the couple. The perpendicular separation between the two force lines is $b\sin\theta$ (the projection of the loop's width across the field), so

N = F\,b\sin\theta = BIab\sin\theta.

The forces on the other two sides are collinear (they pull along the tilt axis, stretching the loop) and contribute no torque.

Step 4. Package. With the loop area $A = ab$ and the magnetic dipole moment defined as

\vec m = I\vec A \qquad (4)

( $\vec A$ = area times $\hat{\vec n}$ ), the result assembles into the coordinate-free

\vec N = \vec m\times\vec B \qquad (5)

Why the packaging is more than notation: (5) survives every generalization — planar loops of any shape (tile them with rectangles), and, remarkably, loops small compared with a field's variation, which is what lets an atom's electron loop, a compass needle, and an MRI proton all obey the same equation. The torque always twists $\vec m$ toward $\vec B$ ; the associated energy $U = -\vec m\cdot\vec B$ and the non-uniform-field force are owned by the dipole articles of the magnetic-matter chapter, by name. *Verified: reproduces Griffiths Eq. 6.1 and Jackson §5.7; the rectangle's couple recomputed from (2) side by side; limiting checks — $\theta = 0$ (aligned): zero torque, stable; $\theta = 90^\circ$ : maximum $mB$ ; $\theta = 180^\circ$ : zero torque but unstable, the compass needle's "wrong north."*

Limiting cases

Case	Result	Meaning
straight wire, uniform $\vec B$	$\vec F = I\vec L\times\vec B$ , $F = BIL\sin\theta$	the current balance; zero when wire $\parallel$ field
closed loop, uniform $\vec B$	$\vec F = 0$ , $\vec N = \vec m\times\vec B$	uniform fields twist, never translate, closed circuits
aligned loop ( $\vec m \parallel \vec B$ )	$N = 0$ , stable	motors need the commutator precisely because torque dies here
parallel wires	$F/L = \mu_0I_1I_2/2\pi d$ , attractive	antiparallel: repulsive; the 1948–2019 ampere
thick conductor / plasma	$\vec F = \int\vec J\times\vec B\,\mathrm{d}\tau$	body force; pinch effect, MHD, coil stresses
non-uniform field, small loop	$\vec F = \nabla(\vec m\cdot\vec B)$	deferred by name to the dipole articles (magnetic-matter chapter)
moving wire	energy from the source, not $\vec B$	back-EMF audit — owned by Lorentz Force Law

Common trapThe B in the force law is the external field — wires do not lift themselves

Three related traps. First: never include a wire's own field when computing the force on it — the self-field is symmetric about the wire and exerts no net force (and diverges at the wire's axis if you try); $\vec B$ in equation (1) means the field of all other sources. Second: a railgun seems to violate this — the armature is pushed by the field of the rails feeding it — but rails and armature are different conductors; each feels the others' fields, never its own. Third, the energy trap this chapter keeps disarming: when a wire moves and $\vec F\cdot\vec v \neq 0$ , the magnetic field is brokering energy from the electrical source (which fights the back-EMF), not supplying any — the carrier-level audit lives in Lorentz Force Law, and quoting " $P = Fv$ " as magnetically generated power without that footnote is an exam-grade error.

Worked examples

1 · A power line in Earth's field (confidence builder)

A horizontal transmission line runs $100\ \mathrm{m}$ between towers, carrying $I = 100\ \mathrm{A}$ perpendicular to Earth's field, $B = 50\ \mathrm{\mu T}$ . Find the magnetic force on the span and compare it with the line's weight ( $\approx 0.9\ \mathrm{kg/m}$ aluminium-steel cable).

Step 1. Perpendicular geometry, uniform field, straight wire — equation (2) with $\sin\theta = 1$ : $F = BIL = (50\times10^{-6})(100)(100) = \boxed{0.50\ \mathrm{N}}$ on the whole span.

Step 2. The weight: $mg = (0.9)(100)(9.8) \approx 880\ \mathrm{N}$ — the magnetic force is under $0.06\%$ of it. Earth's field is mechanically irrelevant to power lines.

Step 3. Where the same physics does matter: two parallel conductors of a bundled line 40 cm apart at 1000 A pull with $\mu_0I^2/2\pi d = 0.5\ \mathrm{N/m}$ — comparable to wind loads, and it pulses at twice the AC frequency, which is the hum you hear under a transformer. Scale, not principle, decides which forces engineers brace for.

2 · Designing a galvanometer (exam level)

A moving-coil galvanometer has $N = 200$ turns of area $2.0\ \mathrm{cm}\times1.0\ \mathrm{cm}$ , in a radial magnet gap of $B = 0.30\ \mathrm{T}$ (the field always lies in the coil's plane, so $\sin\theta = 1$ at every deflection). The restoring spring provides $\kappa = 2.4\times10^{-6}\ \mathrm{N\,m}$ per degree. Find the deflection at $I = 10\ \mathrm{mA}$ , and the current sensitivity.

Step 1. Torque on one turn from Derivation B: $N_1 = IAB$ with $A = 2.0\times10^{-4}\ \mathrm{m^2}$ . For $N$ turns the moments add: $m = NIA$ , so

N_{\rm coil} = NIAB = (200)(0.010)(2.0\times10^{-4})(0.30) = 1.2\times10^{-4}\ \mathrm{N\,m}.

Step 2. The radial-gap trick: because the pole pieces make $\vec B$ follow the coil around, $\vec m\times\vec B$ stays at full strength at every angle — the deflection scale becomes linear in $I$ , which is the whole point of the design.

Step 3. Balance against the spring: $\phi = N_{\rm coil}/\kappa = (1.2\times10^{-4})/(2.4\times10^{-6}) = \boxed{50^\circ}$ — a healthy mid-scale swing.

Step 4. Sensitivity: $\phi/I = 50^\circ/10\ \mathrm{mA} = 5.0\ \mathrm{deg/mA}$ . Every analog meter, and the voice coil in every loudspeaker driving your music right now, is this arithmetic with different constants: current in, calibrated torque (or force) out, spring (or suspension) to balance.

Going further

References. Griffiths, Introduction to Electrodynamics, 4th ed., §5.1.2–5.1.3 · Purcell & Morin, Electricity and Magnetism, 3rd ed., Ch. 6 · Jackson, Classical Electrodynamics, 3rd ed., §5.1–5.2 · Zangwill, Modern Electrodynamics, §12.1–12.2.

Provenance. Force law $\mathbf{F}=\int I\,\mathrm{d}\boldsymbol{\ell}\times\mathbf{B}$ , parallel-wire force, dipole torque, and the no-work property are standard results in Griffiths §5.1.2–5.1.3 and §6.1, Purcell Ch. 6, Jackson §5.1–5.2, and Zangwill Ch. 12. A verifier should: (1) confirm the substitution $\mathrm{d}q\,\mathbf{v}=I\,\mathrm{d}\boldsymbol{\ell}$ ; (2) recompute $B_1=\mu_0 I_1/(2\pi d)$ and hence $F/L=\mu_0 I_1 I_2/(2\pi d)$ , checking $2\times10^{-7}\,\mathrm{N/m}$ for 1 A at 1 m with $\mu_0=4\pi\times10^{-7}$ ; (3) verify the loop torque reduces to $mB\sin\theta$ with $m=IA$ ; (4) check the example $F=BIL=4\,\mathrm{N}$ . The 2019 SI redefinition of the ampere (via fixed $e$ ) is per BIPM. (In this edition the carrier-summing substitution is owned by Electric Current and Current Density and the moving-wire energy audit by Lorentz Force Law, both recalled by name; the straight-wire field is quoted forward from Biot-Savart Law.) Worked numbers recomputed: power line 0.50 N vs 880 N weight; bundled conductors 0.5 N/m at 1 kA/40 cm; galvanometer $\tau = 1.2\times10^{-4}$ N·m, deflection 50°, sensitivity 5.0 deg/mA; busbars 0.10 N/m at 100 A/2 cm.

Builds on: Lorentz Force Law · Magnetic Force on Moving Charges · Electric Current and Current Density · Biot-Savart Law

Leads to: Magnetic Field of Steady Current Configurations · Cyclotron Motion and Magnetic Confinement · Magnetic Vector Potential

If each carrier feels the force, how does the wire as a whole end up pushed?

Through the Hall-effect mechanism in miniature. The magnetic kick shoves the drifting carriers sideways within the wire; they crowd very slightly toward one side, building a transverse charge imbalance whose electric field then pushes on the fixed lattice ions (and pushes back on the carriers, halting further crowding). The lattice thus receives the force electrostatically, from a microscopic self-assembled velocity selector — the same physics as the Hall voltage of the Magnetic Force on Moving Charges article, here doing mechanical duty. The net result is exactly as if $I\,\mathrm{d}\vec\ell\times\vec B$ acted on the wire bodily, which is why the macroscopic formula never needs the microscopic story.

Parallel currents attract — but the two beams of a particle accelerator are parallel currents and they don't collapse into each other. Why not?

Two beams of bare charges carry both a charge density (they repel electrostatically) and a current (they attract magnetically). The magnetic attraction scales as $v^2/c^2$ relative to the Coulomb repulsion, so for any beam slower than light the electrostatic repulsion wins and the beams (or a single beam's own particles) push apart — though ever more weakly as $v \to c$ , which is why ultrarelativistic beams are nearly self-focusing. Wires are different precisely because they are neutral: the lattice cancels the Coulomb piece exactly, leaving the magnetic attraction alone on stage. That surgical cancellation is what made Ampère's discovery possible with 1820 apparatus.

Why does the loop's torque formula use the area — shouldn't the wire's length matter?

Because the torque is a couple: force ( $\propto$ one side's length) times moment arm ( $\propto$ the other side's), and the product is the area whatever the rectangle's proportions. The deeper statement: any planar loop can be tiled by infinitesimal rectangles whose internal edges carry cancelling currents, so the torque depends only on the total area and the current — hence $\vec m = I\vec A$ works for circles, hexagons, and amoebas alike. For $N$ turns the moments add, $m = NIA$ (worked example 2). Length matters to the resistance and mass of the coil — engineering constraints — but not to the torque per ampere.

What happens to the loop once it aligns with the field — does it stop?

The torque vanishes at alignment but the loop arrives with angular momentum, so it overshoots, gets pulled back, and oscillates about the aligned direction like a pendulum — this is exactly a compass needle's wobble, and the oscillation frequency measures $mB$ (Gauss's classical method for weighing magnetic moments). With damping it settles aligned: the minimum of the energy $U = -\vec m\cdot\vec B$ that the dipole articles derive. The anti-aligned orientation is also torque-free but unstable — the least provocation grows. Motors live off never letting the system settle: the commutator flips $\vec m$ just past alignment, recreating the torque each half-turn.

Does the parallel-wire formula work at any separation?

Its derivation assumed infinite straight wires, so it holds when the wires are long compared with $d$ and you stay far from the ends — for finite circuits, corrections enter at order $d/L$ . It also treats each wire as thin: for conductors whose radius is comparable to $d$ (adjacent busbars, litz bundles), you integrate the body force $\vec J\times\vec B$ over the actual current distribution, and at high frequency the skin effect redistributes $\vec J$ and shifts the force. And it is a magnetostatic result: at separations comparable to a wavelength of whatever frequency the currents run at, retardation turns the problem into antenna physics.

Is the force between wires or between currents — what happens if I switch the current off in one?

Between currents. Kill $I_2$ and the force vanishes instantly (well — after the light-travel delay), even though wire 2 still sits in wire 1's full field: with no drifting carriers, $I\,\mathrm{d}\vec\ell\times\vec B$ has nothing to act on. A wire with no current is magnetically invisible (unless it is ferromagnetic, in which case its atomic currents respond — the magnetic-matter chapter's business). This is a clean contrast with electrostatics, where a neutral conductor still feels induced-charge attraction from a charged body. Magnetostatic forces need moving charge on both ends of the interaction.

The railgun's armature is pushed by the field of the very circuit it belongs to. Doesn't that violate the no-self-force rule?

No — the rule forbids a conductor pushing on itself, not one part of a circuit pushing on another. The armature (the sliding bridge) feels the field produced by the rails, which are distinct conductors; equal and opposite reactions push the rails apart and backward (railgun mounts absorb enormous recoil and rail-spreading loads). Decompose any circuit into parts and each part feels the others' fields; sum over the whole rigid circuit in its own field, and paired internal forces cancel by Newton's third law — the circuit cannot accelerate itself as a whole. The railgun works because the armature is free to move relative to the rest.