Current Loops as Magnetic Dipoles: Torque & Moment

Put a compass needle in a magnetic field and it swings to attention. Put a loop of current-carrying wire in the same field and it does exactly the same thing: twists until its face is square-on to the field, overshoots, oscillates, settles. The loop has no iron, no "magnetic material" at all — just moving charge — yet it behaves, in every respect that can be measured from outside, like a small bar magnet. One face acts as a north pole, the other as a south pole, and far away its field is indistinguishable from a compass needle's. André-Marie Ampère saw where that logic had to end and, in 1821, made one of the boldest claims in the history of physics: there is no separate "magnetic substance" anywhere. Every magnet — the compass needle, the fridge magnet, the Earth — is nothing but circulating currents, all the way down. Two centuries later we know he was right: the currents are electrons orbiting and spinning inside atoms. This module builds the machinery that makes the loop-magnet equivalence quantitative. One vector, the magnetic dipole moment $\vec m = NI\vec A$ , packages everything about the loop; the torque $\tau = mB\sin\theta$ and the energy $U = -mB\cos\theta$ then run its mechanics. These three formulas power the next two modules directly — the moving-coil galvanometer is this torque tamed by a spring, and the DC motor is this torque kept forever off balance.

Key takeaways

A current loop is a magnetic dipole: its strength-and-orientation card is the magnetic moment $\vec m = NI\vec A$ , direction by the right-hand curl (fingers with the current, thumb along $\vec m$ ).
In a uniform field the loop feels zero net force but a torque $\tau = mB\sin\theta = NIAB\sin\theta$ , where $\theta$ is the angle between $\vec m$ (the loop's normal) and $\vec B$ — never the angle of the plane.
Vector form: $\vec\tau = \vec m \times \vec B$ — the torque swings $\vec m$ toward $\vec B$ and vanishes at alignment.
Orientation energy: $U = -\vec m\cdot\vec B = -mB\cos\theta$ — minimum $-mB$ aligned (stable), maximum $+mB$ anti-aligned (unstable); flipping costs $2mB$ .
Everything mirrors the electric dipole: $\vec p \leftrightarrow \vec m$ , $\vec\tau = \vec p\times\vec E \leftrightarrow \vec m\times\vec B$ , $U = -\vec p\cdot\vec E \leftrightarrow -\vec m\cdot\vec B$ .
KEY IDEA: one vector m = NIA, normal to the loop, makes a current loop mathematically identical to a bar magnet — τ = m×B twists it, U = −m·B ranks its orientations.

The core insight

A current loop of $N$ turns, current $I$ and area $A$ is a magnetic dipole with moment $\vec m = NI\vec A$ along the loop's normal; a uniform field exerts no net force on it but a torque $\vec\tau = \vec m\times\vec B$ with energy $U = -\vec m\cdot\vec B$ .

Where this lives today

The two devices next door in this chapter are equation (2) industrialised: the moving-coil galvanometer is $\tau = NIAB$ balanced against a hairspring, and the DC motor is the same torque cheated out of its equilibrium every half turn. But the formula's biggest modern workload is microscopic. An MRI scanner manipulates the magnetic moments of protons ( $1.41\times10^{-26}\ \mathrm{A\,m^2}$ apiece, about $10^{27}$ of them in a human body): the machine's field defines the energy ladder $U = -mB$ , radio pulses pay the flip costs, and the returning whispers are Fourier-transformed into anatomy. Every hard-disk bit that stored the previous generation's data was a patch of aligned moments whose $2mB$ barrier kept your files from thermally erasing themselves — magnetic storage engineering is largely the art of setting that barrier high enough to trust, low enough to write. And the equivalence Ampère guessed is now metrology's sharpest instrument: the electron's magnetic moment has been measured to about one part in $10^{13}$ — the most precisely verified prediction in all of physics — by watching a single electron's moment precess in a trap. The humble $\vec m$ of a classroom coil and that record-holding number are the same quantity, twenty-three orders of magnitude apart.

The physical picture

Face a current loop and watch which way the current circulates. Anticlockwise: field lines emerge toward you, and you are looking at the loop's north face. Clockwise: they dive away — a south face. That is already the whole equivalence in cartoon form: the loop's own field threads through its middle and fountains around the outside, precisely the pattern of a short bar magnet. Far away, the resemblance becomes exact — the loop's field takes the universal "dipole" shape falling off as $1/r^3$ , the same law a compass needle broadcasts (the near field of the loop was catalogued in Fields of Straight Wires & Circular Loops; the far-field identity is why "dipole" is the right word). The single vector that carries all of this is the magnetic dipole moment:

\vec m = N I \vec A \qquad (1)

where: $N$ is the number of turns, $I$ the current, and $\vec A$ the loop's area vector — magnitude equal to the enclosed area, direction along the normal to the plane, chosen by the right-hand curl rule (curl your right-hand fingers the way the current flows; your thumb is $\vec A$ , and hence $\vec m$ ). Units: ampere-square-metres, $\mathrm{A\,m^2}$ — equivalently joules per tesla, a hint that $mB$ will be an energy. Read the formula physically: more turns, more current, or more enclosed area all mean more circulating charge doing magnetic work, in direct proportion each. A 50-turn coil of radius 4 cm at 2.5 A carries $m = 50\times2.5\times\pi(0.04)^2 = 0.63\ \mathrm{A\,m^2}$ — about a hundredth of a fridge magnet, from a palm-sized coil. Now place that loop in someone else's uniform field $\vec B$ , with $\vec m$ tilted at angle $\theta$ to it. Two things are true at once, and separating them is worth marks on every board. Net force: zero. Each segment of wire feels $F = BIL$ sideways, but around a closed loop in a uniform field the segment forces cancel completely — the loop as a whole goes nowhere. Net torque: not zero. The cancelling forces act along different lines, forming couples that twist the loop about its own axis until $\vec m$ lines up with $\vec B$ .

Figure

A rectangular current loop in a uniform field — equal opposite forces form a couple that twists m toward B

The torque, built face by face

The claim: for any planar loop,

\tau = N I A B \sin\theta = m B \sin\theta, \qquad\text{vector form } \vec\tau = \vec m \times \vec B \qquad (2)

where: $\theta$ is the angle between $\vec m$ — the loop's normal — and $\vec B$ . Read the structure before the derivation. The $\sin\theta$ says the torque is fiercest when the loop's plane contains the field ( $\vec m$ perpendicular to $\vec B$ , $\theta = 90°$ ) and dies exactly at alignment — which is what "the field twists the loop toward alignment" must mean mathematically. And through equation (1), the torque is proportional to each of $N$ , $I$ , $A$ and $B$ separately: four independent amplifier knobs, which is precisely the list the galvanometer designer will tune in the next module.

Derivation ATorque on a rectangular loop from the force on each side

We want equation (2) for a rectangular loop of width $a$ and height $b$ with $N$ turns, using only $F = BIL$ from Force on a Current-Carrying Conductor — cited, not re-derived — applied to the four sides one at a time. The strategy: in a uniform field the four side-forces cancel pairwise as forces, but one pair acts along different lines and survives as a couple; computing its arm gives the torque, and writing the answer in terms of $m = NIA$ shows the rectangle's shape drops out. Watch for where $\sin\theta$ enters — it comes from the lever arm, not from the force.

Step 1. Set the geometry. Let $\vec B$ be horizontal. Tilt the loop so its normal $\vec m$ makes angle $\theta$ with $\vec B$ ; the two sides of length $b$ (call them the vertical sides) stay perpendicular to $\vec B$ throughout the tilt, while the two sides of length $a$ make angle $\theta$ -dependent tilts.

Step 2. Forces on the $a$ -sides. These carry opposite currents at angle to $\vec B$ ; their forces $BIa\sin(\text{tilt})$ are equal, opposite, and collinear — they run along the same line through the loop's axis. They stretch or squeeze the loop but produce neither net force nor torque.

Step 3. Forces on the $b$ -sides. Each is perpendicular to $\vec B$ , so each feels the full $F = BIb$ per turn — one straight up, one straight down (the ×-and-dot pair in the figure). Net force: zero. But their lines of action are separated by the horizontal projection of the loop's width: $a\sin\theta$ . Why the arm is $a\sin\theta$ : when $\vec m$ is aligned with $\vec B$ ( $\theta = 0$ ) the two forces are head-on collinear and the arm vanishes; when the plane contains $\vec B$ ( $\theta = 90°$ ) the full width $a$ separates them. The projection interpolates.

Step 4. Torque of the couple, for all $N$ turns:

\tau = (\text{force})\times(\text{arm}) = (NIbB)(a\sin\theta) = NI(ab)B\sin\theta = mB\sin\theta

with $m = NIA$ and $A = ab$ . The direction of the twist is exactly what $\vec\tau = \vec m\times\vec B$ encodes: perpendicular to both, in the sense that rotates $\vec m$ toward $\vec B$ . Why the shape drops out: only the area survived into the answer. Any planar loop can be tiled by tiny rectangles whose interior sides carry cancelling currents — so equation (2) holds for circles, hexagons, any flat loop, with $A$ the enclosed area. That is why $m = NIA$ deserves the name "moment": it is the only thing about the loop the field can see. *Verified: $N = 50$ , $I = 2.5\ \mathrm{A}$ , $r = 4.0\ \mathrm{cm}$ , $B = 0.30\ \mathrm{T}$ , $\theta = 60°$ : $m = 50\times2.5\times\pi(0.040)^2 = 0.628\ \mathrm{A\,m^2}$ , $\tau = 0.628\times0.30\times0.866 = 0.163\ \mathrm{N\,m}$ — recomputed; dimensions $\mathrm{A\,m^2\times T} = \mathrm{N\,m}$ ✓; matches NCERT Ch. 4 and the general localized-current theorem at UG level.*

The energy of orientation

If the field's torque twists the loop toward alignment, then orientations must be ranked by energy — turning the loop against the torque stores work, exactly as lifting a mass against gravity does. Integrating the torque over angle (add up $\tau\,d\theta$ slice by slice; the standard convention puts the zero at $\theta = 90°$ ) gives

U = -\,m B \cos\theta = -\,\vec m \cdot \vec B \qquad (3)

where: the minus sign carries the physics. At $\theta = 0$ (aligned): $U = -mB$ , the energy valley — stable equilibrium, zero torque, and any nudge brings a restoring twist. At $\theta = 180°$ (anti-aligned): $U = +mB$ , the energy summit — also zero torque, but unstable: the slightest disturbance and the loop flips. In between, at $\theta = 90°$ : $U = 0$ by convention, and the torque is at its maximum $mB$ . The full climb from valley to summit costs

W_{\text{flip}} = U_{180°} - U_{0°} = mB - (-mB) = 2mB,

the "flip cost" that reappears everywhere from magnetic resonance to data storage. More generally, the work an external agent does rotating the moment from $\theta_1$ to $\theta_2$ is $W = mB(\cos\theta_1 - \cos\theta_2)$ — worth memorising as the source of every "work done in rotating the coil" numerical.

Figure

The orientation energy landscape — valley at alignment, summit at anti-alignment, torque steepest between

If this triptych — a moment vector, $\vec\tau = \vec m\times\vec B$ , $U = -\vec m\cdot\vec B$ — feels familiar, it should. Dipole in a Uniform Field: Torque & Energy built the identical mechanics for the electric dipole $\vec p$ in $\vec E$ , and the dictionary is exact:

electric dipole (E1 module)	magnetic dipole (here)
moment $\vec p = q\vec d$ (charges separated)	moment $\vec m = NI\vec A$ (current circulating)
torque $\vec\tau = \vec p\times\vec E$	torque $\vec\tau = \vec m\times\vec B$
energy $U = -\vec p\cdot\vec E$	energy $U = -\vec m\cdot\vec B$
aligned stable, anti-aligned unstable	same
far field falls as $1/r^3$	same
no net force in a uniform field	same

One dictionary, two chapters — learn the mechanics once, apply it twice. (The matching far-field claim for the loop is the $1/r^3$ tail quoted in Fields of Straight Wires & Circular Loops.) And Ampère's hypothesis closes the circle: if a loop of current is a magnet, perhaps every magnet is loops of current. He proposed that magnetic materials are filled with microscopic "molecular currents" — each molecule a tiny current loop with its own $\vec m$ — aligned in a magnetised bar, randomised in plain iron. Modern physics vindicated him almost embarrassingly well: electrons orbit (a current loop) and spin (an intrinsic moment, $9.27\times10^{-24}\ \mathrm{A\,m^2}$ each — the Bohr magneton, the atom-scale unit of $m$ ), and iron's magnetism is these moments voting together. The chapter on Magnetism & Matter builds on exactly this idea.

Limiting cases

case	result	meaning
$\theta = 0$ ( $\vec m \parallel \vec B$ )	$\tau = 0$ , $U = -mB$	stable equilibrium — the compass rest position
$\theta = 90°$ (plane contains $\vec B$ )	$\tau = mB$ (max), $U = 0$	strongest twist — the galvanometer's favourite geometry
$\theta = 180°$	$\tau = 0$ , $U = +mB$	unstable summit; flip cost $2mB$ from the valley
uniform field, any $\theta$	net force $= 0$	couples twist; only gradients pull (later modules)
$N$ turns	$m$ , $\tau$ , $U$ all $\times N$	the galvanometer's first amplifier knob
loop shape changed, area kept	nothing changes	only $A$ enters — shape is invisible
far from the loop	field $\propto m/r^3$	the bar-magnet impersonation is exact at range

Common trapReading the angle from the plane of the coil instead of its normal

The most reliable mark-harvester in this topic. The $\theta$ in $\tau = mB\sin\theta$ and $U = -mB\cos\theta$ is measured between the field and the loop's normal ( $\vec m$ ) — but problems love to say "the plane of the coil makes 30° with the field." The plane's angle and the normal's angle are complements: plane at 30° means $\theta = 60°$ . Get this backwards and both formulas silently swap their $\sin$ and $\cos$ — worse, the answer is still a plausible number, so nothing warns you. The drill: on reading any coil-in-field problem, draw the normal first, label $\theta$ from the normal to $\vec B$ , and translate the given angle before any formula. Corollary worth engraving: torque is maximum when the plane is parallel to the field ( $\theta = 90°$ ), zero when the plane is perpendicular — phrased that way precisely because it sounds backwards until you think in normals.

Worked examples

Board level — moment, torque and energy of a coil (CBSE/ISC pattern)

A circular coil of 50 turns and radius 4.0 cm carries a current of 2.5 A. It is placed in a uniform magnetic field of 0.30 T with its magnetic moment at 60° to the field. Find (a) the magnetic dipole moment of the coil, (b) the torque on it, and (c) its potential energy in this orientation.

Step 1. Area of one turn: $A = \pi r^2 = \pi(0.040)^2 = 5.03\times10^{-3}\ \mathrm{m^2}$ . (Radius to metres before squaring — the cm² slip scales the answer by 10⁴.)

Step 2. Moment, equation (1): $m = NIA = 50 \times 2.5 \times 5.03\times10^{-3} = \boxed{0.63\ \mathrm{A\,m^2}}$ (0.628 kept for the chain).

Step 3. Torque, equation (2): $\tau = mB\sin 60° = 0.628 \times 0.30 \times 0.866 = \boxed{0.16\ \mathrm{N\,m}}$ (0.163 unrounded), axis perpendicular to both $\vec m$ and $\vec B$ , twisting the moment toward the field.

Step 4. Energy, equation (3): $U = -mB\cos 60° = -0.628\times0.30\times0.500 = \boxed{-0.094\ \mathrm{J}}$ . Negative: the coil sits below the $\theta = 90°$ reference — already partway down the valley toward alignment.

Competitive level — the plane-angle ambush plus a flip (JEE-Main pattern, ~2.5 minutes)

A 50-turn circular coil of radius 4.0 cm carrying 2.0 A sits in a 0.30 T field with the <strong>plane of the coil</strong> making 30° with the field. Find (a) the torque on the coil, (b) the maximum torque available at this current, and (c) the work required to rotate the coil from its most stable to its least stable orientation.

Step 1. Translate the angle first: plane at 30° to $\vec B$ means the normal — and $\vec m$ — is at $\theta = 90° - 30° = 60°$ . (This translation is the question; everything after is arithmetic.)

Step 2. Moment: $m = NIA = 50\times2.0\times\pi(0.040)^2 = 0.503\ \mathrm{A\,m^2}$ .

Step 3. Torque: $\tau = mB\sin 60° = 0.503\times0.30\times0.866 = \boxed{0.13\ \mathrm{N\,m}}$ (0.131). Maximum torque, at $\theta = 90°$ : $\tau_{\max} = mB = 0.503\times0.30 = \boxed{0.15\ \mathrm{N\,m}}$ (0.151).

Step 4. Flip work, valley to summit: $W = 2mB = 2\times0.503\times0.30 = \boxed{0.30\ \mathrm{J}}$ (0.302). Note it doesn't depend on the path or speed of rotation — only the endpoints of $U = -mB\cos\theta$ . Timing note: 2.5 minutes, of which the first 20 seconds — converting the plane's 30° into the normal's 60° — decide all the marks. If a coil-torque answer of yours ever matches the "wrong complement," assume the ambush and re-read the geometry sentence.

Exam traps

Plane angle vs normal angle. Translate before computing (warning box above). The examiner's sentence "plane makes angle α with B" always means $\theta = 90° - \alpha$ in the formulas.

Dropping $N$ . The moment of a coil is $NIA$ , not $IA$ . Fifty turns is a factor of fifty — an error too large to survive, yet the most common one-line slip in board scripts.

Area in cm². $\pi(4\ \mathrm{cm})^2 = 50\ \mathrm{cm^2} = 5\times10^{-3}\ \mathrm{m^2}$ . Forgetting the $10^{-4}$ in the conversion inflates torque ten-thousand-fold.

$\sin$ / $\cos$ swap between τ and U. Torque carries $\sin\theta$ (zero at alignment); energy carries $-\cos\theta$ (minimum at alignment). If your aligned coil has maximum torque or zero energy gap, the pair got swapped.

"Uniform field exerts no force, so nothing happens." No net force — but a torque, unless aligned. Conversely, "the coil experiences a force toward the magnet" is wrong in a uniform field; pulling requires a field gradient (a later module's business). Assertion-reason setters love this pair.

Work to flip vs work to reach 90°. From aligned: to $90°$ costs $mB$ ; to $180°$ costs $2mB$ . Read which rotation the problem asks for.

Units of $m$ . $\mathrm{A\,m^2}$ , or equivalently $\mathrm{J/T}$ — both appear in options; they are the same thing (since $U = -mB$ must be joules).

Constants: $\mu_0$ isn't needed anywhere in this module's mechanics — a tell that a question quoting $\mu_0$ at you here is really asking about the loop's own field (that's Fields of Straight Wires & Circular Loops' territory: centre field $\mu_0 I/2R$ ).

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

André-Marie Ampère

1775–1836 · Lyon and Paris

Ampère heard of Ørsted's twitching compass needle in September 1820 and, within a single week, stood before the French Academy demonstrating that two parallel currents attract — magnetism made and felt by wires alone. But his deepest move came the following year, and it is this module's move: if a coil of current is experimentally indistinguishable from a bar magnet, then perhaps the bar magnet is the impostor. His molecular current hypothesis declared every magnetic material to be a swarm of microscopic, perpetual current loops — no magnetic "substance," no poles as things, just circulating electricity at a scale too small to see. Colleagues found it extravagant: perpetual currents, in every molecule, forever? Yet that is precisely what quantum mechanics delivered a century later — electrons orbiting and spinning without resistance, each a tiny $\vec m$ , their alignment the whole story of iron. (Ampère the lawgiver of circulation — the circuital law bearing his name — is celebrated in the previous module; this is Ampère the unifier, who guessed the electron's job description eighty years before the electron.) A grieving, self-taught polymath who had lost his father to the guillotine and taught himself Latin to read Euler, he asked that his tombstone carry the words tandem felix — happy at last. The SI unit of current is his.

History

1820

— Ørsted's compass twitch reaches Paris; within a week Ampère demonstrates wire–wire forces and begins the mathematics of currents

1821

— Ampère's molecular-current hypothesis: all magnetism is circulating current, magnets included

1832

— Gauss's absolute measurements make magnetic moment a quantity with units, via oscillation and deflection experiments

1922

— Stern and Gerlach split a silver-atom beam: atomic magnetic moments are real, quantised, and measurable one atom at a time

1946

— Bloch and Purcell detect nuclear magnetic resonance — flipping nuclear moments across the 2mB gap with radio waves; MRI follows within three decades

Going further

Texts. NCERT Physics Class 12, Ch. 4 §4.7 (torque on a current loop, magnetic dipole) · H C Verma, Concepts of Physics II, Ch. 34–35 · Griffiths, Introduction to Electrodynamics §5.4.3 and §6.1 for the general dipole field and the localized-current torque theorem (UG deep dive).

Provenance. Standard results per NCERT Ch. 4 and Griffiths §6.1. A verifier should recompute: $A = \pi(0.040)^2 = 5.03\times10^{-3}\ \mathrm{m^2}$ ; board example $m = 50\times2.5\times5.03\times10^{-3} = 0.628\ \mathrm{A\,m^2}$ , $\tau = 0.628\times0.30\times\sin 60° = 0.163\ \mathrm{N\,m}$ , $U = -0.628\times0.30\times\cos 60° = -0.094\ \mathrm{J}$ ; competitive example $m = 0.503\ \mathrm{A\,m^2}$ , $\tau(60°) = 0.131\ \mathrm{N\,m}$ , $\tau_{\max} = 0.151\ \mathrm{N\,m}$ , $W_{\text{flip}} = 2mB = 0.302\ \mathrm{J}$ ; Bohr magneton $9.27\times10^{-24}\ \mathrm{A\,m^2}$ ; Earth's moment $8\times10^{22}\ \mathrm{A\,m^2}$ .

Prerequisites. Force on a Current-Carrying Conductor · Dipole in a Uniform Field: Torque & Energy. Leads to. The Moving-Coil Galvanometer (this torque, spring-balanced) · The DC Motor & Its Cousins (this torque, commutated) · Coils in Fields — Torque & Equilibrium (coming up in this chapter's problem gym) · Magnetic Dipole and Magnetic Moment (UG deep dive).

Why is the net force zero but the torque not? Both come from the same forces.

Because force and torque ask different questions of the same set of pushes. Net force asks: do the pushes cancel as vectors? In a uniform field they must — opposite sides of the loop carry opposite currents through the same field, so their forces are equal and opposite. Net torque asks: do the pushes act along the same lines? Generally not — the up-force and down-force in the derivation act at opposite edges of the loop, separated by the arm $a\sin\theta$ , and an equal-opposite pair on different lines is a couple: pure twist, no shove. The moment the field becomes non-uniform, the "equal" part fails too, and a net force appears — that is how magnets attract iron, and it belongs to a later module.

Is θ measured from the plane or the normal? I keep seeing both in solutions.

Always from the normal (the direction of $\vec m$ ) in the formulas $\tau = mB\sin\theta$ and $U = -mB\cos\theta$ . What you're seeing in solutions is the translation step: problems often state the plane's angle, and the solver converts ( $\theta_{\text{normal}} = 90° - \alpha_{\text{plane}}$ ) before substituting. Some textbooks write the torque as $mB\cos\alpha$ using the plane's angle directly — the same physics wearing the complement. Pick one convention and make it a ritual: draw the normal, mark $\theta$ from normal to field, translate the given angle, then compute. Mixed conventions inside one solution are how right methods produce wrong numbers.

Why does only the area matter and not the loop's shape?

Tile any flat loop with tiny rectangles, all carrying the same circulating current: along every interior edge, two neighbouring rectangles run current in opposite directions, cancelling exactly — only the outer boundary survives, which is your original loop. Each tiny rectangle obeys $\tau = I(\delta A)B\sin\theta$ , and torques add, so the total is $IAB\sin\theta$ with $A$ the summed area. Shape information cancels along with the interior edges. The same tiling argument is why $m = NIA$ works for a circle, a hexagon, or a lumpy planar blob — and why the field far away can only depend on $m$ , never on the silhouette.

What happens if I hang the loop at exactly 180°, anti-aligned?

Formally it's an equilibrium: $\sin 180° = 0$ , so the torque vanishes and the loop can balance there — but it's the unstable one, the marble on the summit. Energy tells the story that torque alone hides: $U = +mB$ , the maximum. Any thermal jiggle or vibration tips it off the summit, the torque reappears with a sign that accelerates the fall, and the loop swings down toward alignment, overshooting and oscillating around $\theta = 0$ until friction or eddy damping drains the $2mB$ it released. Exam phrasing to recognise: "in stable equilibrium" means $\theta = 0$ , $U = -mB$ ; "in equilibrium" alone may include the summit — read for the word stable.

Does the loop's own magnetic field push on the loop itself?

No — a closed loop exerts zero net force and zero net torque on itself, which is fortunate, since otherwise coils would spontaneously accelerate and physics would hand out free energy. Every segment's field does act on every other segment, but the internal forces come in balanced pairs (they do create internal stress — high-field magnet windings genuinely try to burst outward, a hoop tension that magnet engineers brace against). When this module computes $\tau = \vec m\times\vec B$ , the $\vec B$ is always someone else's field — external, uniform across the loop. Separating "my field" from "the field I respond to" is the same bookkeeping discipline the parallel-wires module used.

How can an electron have a magnetic moment if it isn't actually a spinning ball of charge?

Honest answer: "spin" is intrinsic angular momentum that quantum particles simply have, like charge or mass — no rotation of a little sphere is involved (modelled as a classical spinning ball, the electron's surface would need to move faster than light). Yet the moment is perfectly real and behaves exactly as this module's $\vec m$ : it feels $\vec\tau = \vec m\times\vec B$ , carries $U = -\vec m\cdot\vec B$ , and its flip cost $2mB$ is what magnetic resonance measures. The value, one Bohr magneton $\approx 9.27\times10^{-24}\ \mathrm{A\,m^2}$ , even matches what a naive orbiting-charge estimate suggests to within a factor of two. Ampère's molecular currents turned out to be partly literal (orbital motion) and partly deeper than he could have guessed (spin) — but the mechanics he'd have written down is unchanged.

Is the Earth a current loop too, then?

In the one-vector summary sense, yes. Earth's magnetism comes from the geodynamo — convecting molten iron in the outer core sustaining electric currents — and the whole messy system, viewed from the surface or from space, is characterised by a single magnetic moment of about $8\times10^{22}\ \mathrm{A\,m^2}$ , tilted roughly 11° from the spin axis. That's the same bookkeeping as your 50-turn coil: complicated currents inside, one $\vec m$ outside. It's also why a compass works: the needle's $\vec m$ sits in Earth's field and obeys exactly this module's torque, settling into the $U = -mB$ valley that points it north. One formula, from lab bench to planet.