Forces and Torques on Magnetic Dipoles

The dipole now has a field; this article gives it a life. Put a magnetic moment into someone else's field and three questions exhaust its mechanics: which way is it twisted, where is it pulled, and what does its orientation cost? The answers are a triptych of formulas — $\vec N=\vec m\times\vec B$ , $\vec F=\nabla(\vec m\cdot\vec B)$ , $U=-\vec m\cdot\vec B$ — compact enough to fit on a thumbnail and consequential enough to run compasses, molecular-beam experiments, magnetic tweezers, and every MRI scanner on Earth. The chapter has met fragments of this before, and the debts get paid in order. The force-on-wires article derived the torque for one special case — a rectangular loop — and packaged $\vec m=I\vec a$ ; here the rectangle becomes a theorem for arbitrary localized currents. The same article deferred, by name, the force question: a closed loop in a uniform field feels zero net force ( $\oint\mathrm{d}\vec\ell=0$ ), so pulling on a dipole requires a field gradient — and this article computes exactly how, deriving $\vec F=\nabla(\vec m\cdot\vec B)$ from the Lorentz force at first order in the gradient and confronting its evil twin $(\vec m\cdot\nabla)\vec B$ , identical in empty space and different exactly where current flows. The energy $U=-\vec m\cdot\vec B$ arrives with the fine print the Lorentz-force article's work-paradox audit prepared us for: for current loops, "energy" is a two-account ledger, and the battery quietly pays double. The triptych then produces this article's most consequential surprise. A dipole that carries angular momentum — and at the atomic scale they all do, since $\vec m=\gamma\vec J$ — does not swing toward alignment like a compass needle. The torque feeds $\mathrm{d}\vec J/\mathrm{d}t$ , and the moment precesses around the field at the Larmor frequency $\omega=\gamma B$ , circling forever unless something drains energy. That one dynamical fact, plus a field gradient, is Stern–Gerlach; plus a resonant push, it is all of magnetic resonance. The atomic values of $\gamma$ are the next article's business; the dynamics is owned here.

Key takeaways

Torque on any localized current distribution in a locally uniform field: $\vec N=\vec m\times\vec B$ — the rectangle result of the wires article, upgraded to a theorem.
Orientation energy: $U=-\vec m\cdot\vec B$ — minimum aligned, maximum anti-aligned; for rigid moments it is the honest potential energy, for current loops it is the fixed-current ledger (the battery pays double behind the scenes).
Uniform fields only twist; gradients pull: $\vec F=\nabla(\vec m\cdot\vec B)$ , derived from the Lorentz force at first order in the field gradient.
The twin form $(\vec m\cdot\nabla)\vec B$ equals $\nabla(\vec m\cdot\vec B)$ only where $\nabla\times\vec B=0$ — i.e., away from local currents; the difference is $\vec m\times(\nabla\times\vec B)$ .
A spinning dipole ( $\vec m=\gamma\vec J$ ) does not align — it precesses: $\mathrm{d}\vec J/\mathrm{d}t=\gamma\vec J\times\vec B$ , cone angle fixed, Larmor frequency $\omega=\gamma B$ (proton: 42.58 MHz/T).
Stern–Gerlach = this article in hardware: gradient converts moment into deflection; quantized moments → discrete spots.
KEY IDEA: one scalar function $U=-\vec m\cdot\vec B$ generates the mechanics — its angular derivative is the torque, its spatial gradient (sign flipped) the force — and adding angular momentum turns "align" into "precess."

The core insight

Everything follows from one scalar energy $U=-\vec m\cdot\vec B$ : rotate it for the torque $\vec N=\vec m\times\vec B$ , gradient it for the force $\vec F=-\nabla U=\nabla(\vec m\cdot\vec B)$ — and for spinning moments $\vec m=\gamma\vec J$ , the torque drives Larmor precession at $\omega=\gamma B$ instead of alignment.

Where this lives today

Magnetic resonance imaging is this article, industrialized. A 3 T scanner sets the body's protons precessing at $f_L=127.7$ MHz (equation 5); a resonant radio pulse tips the precession cone; and — the geometry-encoding trick — gradient coils superimpose $\sim40$ mT/m field gradients so the Larmor frequency becomes a position label: spins at different locations hum at different pitches, and Fourier analysis of the returning hum is the image. Torque, precession, and gradient — the article's three derivations — appear as the magnet, the pulse sequence, and the encoding, respectively. The same precession physics runs chemistry's NMR spectrometers (where parts-per-million shifts of $f_L$ read out molecular structure) and the proton magnetometers that map planetary fields. At the force end, gradient-pulling has gone quantitative and microscopic. Magnetic tweezers grip micron-scale beads ( $m\sim10^{-14}$ A·m²) in $\sim100$ T/m gradients, applying calibrated piconewton forces to single DNA molecules to measure their elasticity and the motors that walk on them. Neutral-atom traps hold "low-field-seeking" states (moment anti-aligned, expelled toward field minima — the sign of equation (3) read in reverse) in quadrupole and bottle geometries, the front door to Bose–Einstein condensation. And magnetic separation — from scrapyard cranes to immunomagnetic cell sorting in clinical labs — is $\nabla(\vec m\cdot\vec B)$ with induced moments, engineering's oldest use of the newest formula here.

Origins · 1 minJoseph John Larmor and how this was discovered

Joseph John Larmor

1857–1942 · Belfast, Cambridge — Lucasian chair between Stokes and Dirac

Larmor was the last great physicist of the ether and, in the same career, an accidental architect of what replaced it. His Aether and Matter (1900) — pursuing electrons as spinning strain-centers in a universal medium — contains, a decade before Einstein, the full Lorentz transformations and the first derivation of time dilation for orbiting charges; history filed the equations under Lorentz's name and the physics under Einstein's, leaving Larmor the precession and the radiation formula that bear his. The precession result (1897) is exactly this article's Derivation C: a charge's orbital moment, rigidly tied to its angular momentum, responds to a magnetic field not by tilting but by circling it at $\omega=eB/2m$ — which Larmor used to analyze the just-discovered Zeeman effect, magnetism's first spectroscopic fingerprint. Holder of the Lucasian chair for thirty years, Ulster-conservative, famously resistant to both relativity's abandonment of the ether and Cambridge's introduction of bathrooms ("we have done without them for four hundred years"), he lived to see his frequency become the operating principle of magnetic resonance — measured today to twelve digits in the electron's $g-2$ .

History

1820

— Ampère quantifies the forces and torques between circuits; the couple on a current loop enters the theory of magnetism.

1832

— Gauss's Intensitas uses the oscillating-magnet period and the deflection ratio to put both m and B on absolute footing.

1845

— Faraday discovers diamagnetism: matter pushed toward weak field — the force law's other sign, awaiting the next chapter's mechanisms.

1897

— Larmor derives the precession of orbital moments, giving the Zeeman effect its classical frequency scale.

1922

— Stern and Gerlach split a silver beam in a field gradient: two spots, space quantization, and the force law as a quantum instrument.

1946

— Bloch and Purcell detect nuclear magnetic resonance — Larmor precession driven and read out; MRI follows within three decades.

The physical picture

Hold a compass. The needle is a magnetic moment in the Earth's (locally uniform) field, and what the field does to it is pure twist: north pole pulled one way, south pole equally the other — a couple, no net force. The needle does not slide across the table toward Canada; it rotates until $\vec m$ lies along $\vec B$ , overshoots, oscillates, and settles as friction drains the swing. That is the uniform-field story complete: torque $\vec m\times\vec B$ , energy landscape $-\vec m\cdot\vec B$ with its valley at alignment, and dissipation as the unsung third ingredient that lets the valley be reached at all. Now bring the needle toward a real magnet, and a new thing happens: it leaps. Near the pole face the field is stronger on the needle's near end than its far end; the two pole forces no longer cancel, and the residue is a net pull toward stronger field. This is the force law's entire content — a dipole feels no force from a field, only from a field's change — and it explains at a stroke why iron filings fly to the poles of a magnet (field gradients are fiercest there), why the uniform interior of a good solenoid exerts torque but no pull, and why every magnetic trap, tweezer, and separator is in the business of shaping gradients, not just making fields.

Figure

Uniform fields twist, gradients pull — the couple on a dipole versus the net force in a converging field

The third panel of the mental picture is the one classical intuition gets wrong. Replace the compass needle by an atom: the moment is now welded to angular momentum, $\vec m=\gamma\vec J$ . Apply the same torque and watch what changes: torque is $\mathrm{d}\vec J/\mathrm{d}t$ , and a $\vec J$ pushed always perpendicular to itself (that is what $\vec m\times\vec B\propto\vec J\times\vec B$ does) changes direction, never magnitude — the tip of $\vec J$ circles the field direction on a cone of fixed angle. Gyroscope logic, exactly: gravity's torque doesn't topple a spinning top, it precesses it. An atomic moment in a magnetic field hums around $\vec B$ at the Larmor frequency, alignment energy untouched, forever — until coupling to an environment (radiation, collisions, the lattice) drains it. Every "spins align with the field" sentence in a textbook silently includes that dissipative fine print, and magnetic resonance lives entirely inside it.

The triptych, derived

All three laws follow from the Lorentz force $\mathrm{d}\vec F=I\,\mathrm{d}\vec\ell\times\vec B$ summed around the circuit, expanded in how the field varies across the (small) dipole. Zeroth order in the variation: torque. First order: force. The bookkeeping of both: energy.

Derivation ATorque on an arbitrary localized current — the rectangle becomes a theorem

The goal is $\vec N=\vec m\times\vec B$ for any localized steady current in a field that is uniform across the source — generalizing the wires article's rectangle, whose couple picture (two opposite side-forces, lever arm, $NIA$ ) we recall by name rather than repeat. The strategy: write the total torque about the dipole's location as a loop integral, and reuse the exact vector-area lemma that extracted the dipole moment two articles ago — the same identity does both jobs, which is no accident.

Step 1. Set up the torque integral. With $\vec B$ uniform over the circuit,

\vec N=\oint\vec r'\times\mathrm{d}\vec F =I\oint\vec r'\times\big(\mathrm{d}\vec\ell'\times\vec B\big) .

Step 2. Expand the triple product. BAC–CAB with $\vec B$ constant:

\vec r'\times(\mathrm{d}\vec\ell'\times\vec B) =\mathrm{d}\vec\ell'\,(\vec r'\cdot\vec B)-\vec B\,(\vec r'\cdot\mathrm{d}\vec\ell') .

The second term integrates to zero: $\oint\vec r'\cdot\mathrm{d}\vec\ell'=\oint\tfrac12\,\mathrm{d}(r'^2)=0$ around a closed loop (an exact differential's round trip).

Step 3. Spend the vector-area lemma. What remains is $I\oint(\vec B\cdot\vec r')\,\mathrm{d}\vec\ell'$ — the multipole article's integral with $\hat r\to\vec B$ , and its lemma gave $\oint(\vec c\cdot\vec r')\,\mathrm{d}\vec\ell'=\vec a\times\vec c$ for any constant $\vec c$ . Hence

\vec N=I\,(\vec a\times\vec B)=\vec m\times\vec B . \qquad (1)

where: $\vec N$ — torque about the dipole's own location; magnitude $mB\sin\theta$ , axis perpendicular to both, sense that swings $\vec m$ toward $\vec B$ . For volume currents the same result follows from $\vec N=\int\vec r'\times(\vec J\times\vec B)\,\mathrm{d}\tau'$ with the symmetric/antisymmetric split — the multipole article's other maneuver — confirming that (1) is a statement about the moment, indifferent to the plumbing that produces it. Why "uniform across the source" is the real hypothesis: in a gradient there are corrections to (1) of order (source size)/(field scale) — and also a net force, next derivation. For atoms in laboratory fields the ratio is $\sim10^{-10}$ : equation (1) is exact for all practical purposes at the atomic scale, which is why it can be fed into precession dynamics with a clear conscience. *Verified: rectangle check — for the wires article's $a\times b$ loop, (1) reproduces $N=IabB\sin\theta$ ✓; dimensions $[\mathrm{A\,m^2}][\mathrm{T}]=\mathrm{N\,m}$ ✓; matches Griffiths Eq. 6.1, Jackson §5.7.*

The energy is the torque integrated. Rotating $\vec m$ from angle $\theta_0$ to $\theta$ against the aligning torque stores $W=\int mB\sin\theta\,\mathrm{d}\theta$ , and with the conventional zero at $\theta=90°$ :

U(\theta)=-mB\cos\theta=-\vec m\cdot\vec B . \qquad (2)

where: the landscape runs from $-mB$ (aligned, the valley) through zero (perpendicular) to $+mB$ (anti-aligned, the summit) — a $2mB$ span that sets every magnetic-orientation scale in physics, from the compass ( $\sim10^{-4}$ J for a needle in Earth's field) to the hydrogen hyperfine splitting ( $\sim10^{-24}$ J). Read equation (2) with the honesty the Lorentz-force article's paradox audit taught: for a rigid, permanent moment (an electron's spin), $U$ is a genuine potential energy, full stop. For a current loop held at constant $I$ , rotating toward alignment changes the flux through the loop, induction fights the change, and the battery must do work $2\,\Delta|U|$ to hold the current — so the loop's "energy" depends on which account you audit: the mechanical work available is governed by (2), while the total ledger includes the source. This is precisely the fixed- $Q$ versus fixed- $V$ story of the dielectric-forces article, transposed to magnetism; the full accounting arrives with magnetic energy in the matter chapter. Equation (2), used as the generator of forces and torques, is correct in both cases — that is all this article needs.

Go deeper · Derivation B · 2 minForce in a nonuniform field — and the two-forms subtlety

Derivation BForce in a nonuniform field — and the two-forms subtlety

The goal is the force law $\vec F=\nabla(\vec m\cdot\vec B)$ from first principles: sum the Lorentz force around a small loop in a field that varies slowly across it, keeping the first gradient correction — the term the wires article showed is the only survivor, since the uniform piece cancels by $\oint\mathrm{d}\vec\ell'=0$ . The strategy: Taylor-expand, reuse the vector-area lemma once more, then confront the identity that separates this form from its look-alike $(\vec m\cdot\nabla)\vec B$ .

Step 1. Expand the field across the loop. About the loop's center (origin), $\vec B(\vec r')=\vec B_0+(\vec r'\cdot\nabla)\vec B_0+\ldots$ The total force:

\vec F=I\oint\mathrm{d}\vec\ell'\times\vec B(\vec r') =I\underbrace{\Big(\oint\mathrm{d}\vec\ell'\Big)}_{=\,0}\times\vec B_0 +I\oint\mathrm{d}\vec\ell'\times\big[(\vec r'\cdot\nabla)\vec B_0\big]+\ldots

— the uniform term dies by closure (no net force in uniform fields, now proved rather than asserted), and the gradient term leads.

Step 2. Lemma, one last time. Component bookkeeping on the surviving integral (each Cartesian gradient component acts as a constant vector for the loop integral) reduces it, via $\oint(\vec c\cdot\vec r')\,\mathrm{d}\vec\ell'=\vec a\times\vec c$ , to

\vec F=I\,\nabla\big[(\vec a\times\ldots)\big]\;\Longrightarrow\; \vec F=\nabla\big(\vec m\cdot\vec B\big), \qquad (3)

evaluated at the dipole's location, for constant $\vec m$ . (The compact route to the same result: $\vec F=-\nabla U=+\nabla(\vec m\cdot\vec B)$ from equation (2) — the honest loop-integral derivation is what certifies that shortcut.)

where: the force points up the gradient of $\vec m\cdot\vec B$ : aligned moments are pulled toward strong field, anti-aligned moments expelled toward weak field. That one sign carries the whole phenomenology of the next chapter — para- versus dia-magnetic drift — and of every magnetic trap.

Step 3. The evil twin. Textbooks also quote $\vec F=(\vec m\cdot\nabla)\vec B$ . The E0 product rule relates them, for constant $\vec m$ :

\nabla(\vec m\cdot\vec B)=(\vec m\cdot\nabla)\vec B+\vec m\times(\nabla\times\vec B) .

Wherever the dipole sits in current-free space — outside magnets, between coils, in a beam line — $\nabla\times\vec B=0$ and the two forms agree identically; use whichever computes faster. Sitting inside a current distribution ( $\nabla\times\vec B=\mu_0\vec J\neq0$ ) they differ by $\mu_0\,\vec m\times\vec J$ , and which is correct becomes a genuine physics question about the dipole's model: (3) is the force on a rigid current-loop (Amperian) dipole, the kind this chapter builds and the kind atoms are. The distinction matters in magnetized matter and in precision hyperfine physics — flagged here, spent when the matter chapter needs it. Verified: dimensions $[\mathrm{A\,m^2\,T/m}]=\mathrm{N}$ ✓; uniform-field limit gives $\vec F=0$ ✓ (recovering the wires article's theorem); on-axis trap check — for $\vec m\parallel\vec B=B(z)\hat z$ , (3) gives $F_z=m\,\mathrm{d}B/\mathrm{d}z$ , the standard Stern–Gerlach working formula ✓; matches Griffiths Eq. 6.3 and Jackson §5.7 including the two-forms discussion.

Figure

The precession cone — torque pushes J sideways, so the moment circles B at the Larmor frequency instead of aligning

Go deeper · Derivation C · 2 minLarmor precession — why spinning moments circle instead of aligning

Derivation CLarmor precession — why spinning moments circle instead of aligning

The goal is the motion of a dipole that carries angular momentum in fixed ratio, $\vec m=\gamma\vec J$ — the universal atomic case — in a uniform field: the result is precession about $\vec B$ at the angular frequency $\vec\omega_L=-\gamma\vec B$ , with the cone angle a constant of the motion. The strategy is two lines of rigid-body mechanics plus a geometric reading; the payoff is the operating principle of NMR, ESR, and MRI. The atomic values of $\gamma$ (orbital $-e/2m_e$ , spin $\approx-e/m_e$ ) are owned by the next article; here $\gamma$ is a given constant.

Step 1. The equation of motion. Torque is the rate of change of angular momentum, and the torque is (1):

\frac{\mathrm{d}\vec J}{\mathrm{d}t}=\vec N=\vec m\times\vec B=\gamma\,\vec J\times\vec B . \qquad (4)

Step 2. Two conservation laws, read off directly. Dot (4) with $\vec J$ : $\vec J\cdot(\vec J\times\vec B)=0$ , so $|\vec J|$ is constant — the push is always perpendicular, direction changes, magnitude cannot. Dot with $\vec B$ : $\vec B\cdot(\vec J\times\vec B)=0$ , so $J_\parallel=\vec J\cdot\hat B$ is constant — the cone angle $\theta$ between $\vec J$ and $\vec B$ never changes. And since $U=-\gamma J B\cos\theta$ , the orientation energy is conserved too: no approach to alignment whatsoever.

Step 3. Recognize rotation. An equation of the form $\mathrm{d}\vec J/\mathrm{d}t=\vec\omega\times\vec J$ is rigid rotation of $\vec J$ about $\vec\omega$ . Rewriting (4) as $\mathrm{d}\vec J/\mathrm{d}t=(-\gamma\vec B)\times\vec J$ identifies

\vec\omega_L=-\gamma\vec B, \qquad f_L=\frac{|\gamma|B}{2\pi} : \qquad (5)

the tip of $\vec J$ (and with it $\vec m$ ) sweeps a cone around the field at the Larmor frequency, sense set by the sign of $\gamma$ , rate proportional to $B$ with the moment-to-angular-momentum ratio as the only material input. Why the frequency is the measurable gold: $f_L$ depends on $\gamma$ and $B$ , nothing else — not the cone angle, not the moment's history. Measure the hum and you either read $\gamma$ at fixed $B$ (precision moment metrology) or read $B$ at known $\gamma$ (every NMR magnetometer and teslameter). For protons, $f_L/B=42.58$ MHz/T; a 3 T clinical scanner hums at 127.7 MHz, recomputed below. *Why compasses still align: a needle's $\vec m$ is not rigidly tied to a large intrinsic $\vec J$ — its mechanical angular momentum starts at zero and friction on the pivot drains the oscillation, so it walks down the energy landscape (2). Atomic moments in vacuum have no such drain; they precess until relaxation processes (spin–lattice, radiative) supply the missing dissipation. The timescale hierarchy — precession in nanoseconds, relaxation in milliseconds-to-seconds — is exactly what makes magnetic resonance spectroscopy possible: thousands to billions of coherent turns before the phase is lost. Verified: (5) matches the classical Larmor result and the quantum Heisenberg equation for spin in a field (identical form — one of quantum mechanics' cleanest classical correspondences); proton check $\gamma_p=2.675\times10^8\ \mathrm{rad\,s^{-1}T^{-1}}$ : $f=\gamma_pB/2\pi=127.7$ MHz at 3 T ✓.*

Limiting cases

case	result	meaning
uniform field, non-spinning moment	twist to alignment (with dissipation)	compass regime; oscillation period $2\pi\sqrt{I_{\rm rot}/mB}$
uniform field, $\vec m=\gamma\vec J$	precession at $\omega=\gamma B$ , no alignment	atomic regime; NMR/ESR/MRI
gradient, moment aligned	pulled to strong field	paramagnetic drift; tweezers, traps
gradient, moment anti-aligned	expelled to weak field	diamagnetic levitation preview; low-field-seeking atom traps
current-free region	$\nabla(\vec m\cdot\vec B)=(\vec m\cdot\nabla)\vec B$	both force forms legal
inside current density $\vec J$	forms differ by $\mu_0\vec m\times\vec J$	Amperian dipole takes $\nabla(\vec m\cdot\vec B)$
field varying strongly across source	dipole approximation fails	higher moments feel higher derivatives

Common trapU equals minus m dot B is two different statements — and only one of them is about total energy

For a permanent, rigid moment (spin, or a magnetized needle), $U=-\vec m\cdot\vec B$ is an ordinary potential energy: differentiate for forces and torques, conserve it in dynamics, done. For a current loop held at fixed current, the same formula still generates the correct force and torque — but it is not the system's total energy bookkeeping: rotating into alignment increases the flux through the loop, and holding $I$ against the induced back-EMF costs the battery exactly twice the drop in $U$ (the magnetic mirror of the fixed-voltage capacitor's double payment, and the same physics as the Lorentz-force article's rising-bar energy audit). Confusing the two accounts produces sign errors in "energy stored" arguments while leaving forces mysteriously correct — the tell that you have audited the wrong ledger. Second trap, from the force derivation: $(\vec m\cdot\nabla)\vec B$ and $\nabla(\vec m\cdot\vec B)$ are interchangeable only where $\nabla\times\vec B=0$ ; quoting them as universally identical is wrong exactly where matter gets interesting.

Worked examples

1 · Gauss's oscillating magnet (confidence builder)

A bar magnet ( $m=2.0$ A·m², length 10 cm, mass 50 g) hangs from a torsion-free thread in the local horizontal field $B_h=49\ \mu$ T of the mid-latitude Earth. Find the torque at $30°$ deflection and the small-oscillation period.

Step 1. Torque. $N=mB_h\sin30°=2.0\times49\times10^{-6}\times0.5$ = $4.9\times10^{-5}$ N·m — the restoring couple twisting the magnet back toward magnetic north.

Step 2. Pendulum dynamics. For small angles, $I_{\rm rot}\ddot\theta=-mB_h\theta$ : angular harmonic motion. The rod's moment of inertia about its center: $I_{\rm rot}=ML^2/12=0.050\times(0.10)^2/12=4.2\times10^{-5}$ kg·m².

Step 3. Period.

T=2\pi\sqrt{\frac{I_{\rm rot}}{mB_h}} =2\pi\sqrt{\frac{4.2\times10^{-5}}{2.0\times4.9\times10^{-5}}} =\textbf{4.1 s}.

Step 4. Why this humble pendulum is famous. Timing it measures the product $mB_h$ ; the deflection experiment of the dipole article's Gauss band measures the ratio $m/B_h$ . Two bench measurements, multiply and divide: both the magnet's moment and the Earth's field in absolute units — the 1832 Intensitas scheme, and the first time anyone knew the strength of any magnetic field as a number. The formula in Step 3 ran the world's magnetic observatories for a century.

2 · Stern–Gerlach, by the numbers (exam level)

Silver atoms ( $M=1.79\times10^{-25}$ kg, moment $\mu_B=9.27\times10^{-24}$ J/T from the lone 5s electron — next article's business why) exit an oven at $v=550$ m/s and cross a $3.5$ cm magnet with vertical gradient $\mathrm{d}B/\mathrm{d}z=1400$ T/m, then drift 20 cm to a plate. Find the splitting.

Step 1. The force. Moments quantize along $z$ (the quantum input) into $m_z=\pm\mu_B$ : $F_z=m_z\,\mathrm{d}B/\mathrm{d}z=\pm9.27\times10^{-24}\times1400=\pm1.30\times10^{-20}$ N — about $10^4$ times the atom's weight, delivered by pure field shaping.

Step 2. Inside the magnet. $a=F/M=7.2\times10^4\ \mathrm{m/s^2}$ ; transit time $t=L/v=6.4\times10^{-5}$ s; deflection $\tfrac12at^2\approx147\ \mu$ m, exit transverse velocity $v_z=at=4.6$ m/s.

Step 3. The drift lever arm. Over $20$ cm of field-free flight: additional $v_z\,(L_d/v)=4.6\times(0.20/550)\approx1.68$ mm. Total per branch $\approx1.82$ mm; separation between the $+\mu_B$ and $-\mu_B$ spots: $\approx3.6$ mm — comfortably resolvable, exactly the two-spot photograph of 1922.

Step 4. What the two spots proved. Classically, oven-randomized moments would smear $m_z$ continuously from $-\mu_B$ to $+\mu_B$ : one elongated blob. Two clean spots mean $m_z$ takes two values — space quantization, before spin was even proposed (and, in the modern reading, the electron's spin- $\tfrac12$ doublet caught on film). Note the precession article's fine print doing real work here: the moments do not align in the magnet, they precess about $\vec B$ at $\sim10^{10}$ rad/s while the force $F_z=m_z\,\partial_zB$ acts on the conserved component $m_z$ — the cone, not the swing, is what the experiment sorts.

Going further

References. Griffiths, Introduction to Electrodynamics, 4th ed., §6.1 (Eqs. 6.1–6.3) · Jackson, Classical Electrodynamics, 3rd ed., §5.7 · Zangwill, Modern Electrodynamics, §12.2–12.4 · Purcell & Morin, Electricity and Magnetism, 3rd ed., §11.6

Provenance. Equations cross-checked against Griffiths 4th ed. §6.1 (Eqs. 6.1–6.3 for $\mathbf{N}=\mathbf{m}\times\mathbf{B}$ , $\mathbf{F}=\nabla(\mathbf{m}\cdot\mathbf{B})$ , $U=-\mathbf{m}\cdot\mathbf{B}$ ) and Jackson 3rd ed. §5.7 (force/torque on a localized current distribution, including the $(\mathbf{m}\cdot\nabla)\mathbf{B}$ form and the $\nabla\times\mathbf{B}$ correction). A verifier should re-derive the loop force/torque integrals to first order in the field gradient and confirm the vector identity $\nabla(\mathbf{m}\cdot\mathbf{B})=(\mathbf{m}\cdot\nabla)\mathbf{B}+\mathbf{m}\times(\nabla\times\mathbf{B})$ for constant $\mathbf{m}$ . The proton gyromagnetic ratio $\gamma_p/2\pi\approx 42.58$ MHz/T is a standard tabulated value (CODATA). (Worked numbers: torque $4.9\times10^{-5}$ N·m, period 4.1 s; Stern–Gerlach $1.3\times10^{-20}$ N, $\approx3.6$ mm two-spot separation; 127.7 MHz at 3 T — recomputed.)

Builds on: Magnetic Dipole and Magnetic Moment · Magnetic Force on Current-Carrying Wires · Lorentz Force Law · Torque and Force on a Dipole in a Field

Leads to: Magnetization · Magnetic Dipole Moment of Atoms · Paramagnetism · Diamagnetism

If the force comes from a gradient, how does a uniform-field solenoid pick up a paperclip at its mouth?

It doesn't do it with its uniform interior — it does it with its fringe. The mouth of a solenoid is exactly where the ideal uniform-inside/zero-outside field must transition, so the gradients there are enormous (all of $\mu_0nI$ collapsing over roughly a bore radius). The clip's induced moment aligns with the local field, $\vec m\cdot\vec B$ is largest inside, and $\nabla(\vec m\cdot\vec B)$ points into the bore. Deep inside, once the field is genuinely uniform, the pull disappears and only torque remains — which is why magnetic separators and scrapyard lifters are engineered around pole edges and shaped pole pieces: the design variable is the gradient map, not the field's peak value.

Why doesn't the magnetic force law violate the no-work theorem for magnetic forces?

The theorem — magnetic forces on a charge do no work, $\vec F\perp\vec v$ — survives intact; the resolution is bookkeeping inside the dipole, and it is the wires article's rising-bar audit in miniature. When a gradient pulls a current loop forward and the loop gains kinetic energy, the microscopic Lorentz forces on its carriers still do zero net work; what happens is a transfer: the motion through nonuniform field induces an EMF, the carriers' drift rearranges, and the source maintaining the current supplies the energy that appears as mechanical work (for a permanent/spin moment, the "source" is the quantum rigidity of the moment — no classical circulating current needs feeding). $U=-\vec m\cdot\vec B$ packages this bookkeeping so you never have to reopen it for mechanics — but it is packaging, and the warning box's two-ledger caution is the receipt.

Which force form applies to an electron's spin moment — Amperian or the other one?

Experiment says Amperian. The two models of a point dipole — circulating current ("Amperian," force $\nabla(\vec m\cdot\vec B)$ , interior field along $\vec m$ ) versus two magnetic charges ("Gilbertian," force $(\vec m\cdot\nabla)\vec B$ , interior field against $\vec m$ ) — agree everywhere outside matter and disagree inside current or magnetization density. The clean test is the neutron: neutral, but with a moment, passing through magnetized foils. Interferometry and scattering experiments match the Amperian prediction — consistent with the dipole article's $+\tfrac{2\mu_0}{3}$ contact term, which is likewise the Amperian signature (and is confirmed spectacularly by hyperfine structure). Electrons, protons, neutrons: current-loop-type dipoles, however pointlike, with no magnetic charge in sight.

How can Stern-Gerlach separate moments if they're all precessing wildly inside the magnet?

Because precession and deflection use different components. The field (mostly along $z$ ) drives precession of $m_x,m_y$ at $\sim10^{10}$ rad/s — those components average to zero almost instantly on the transit timescale. But precession conserves $m_z$ (Derivation C: $J_\parallel$ constant), and the deflecting force is $F_z=m_z\,\partial_zB$ : built entirely from the conserved piece. So the magnet sorts atoms by $m_z$ while the transverse components blur harmlessly. The fast precession is even helpful: it suppresses the effect of stray transverse gradient components (they average out over a precession cycle), which is why the naive one-axis analysis of the worked example gives the right answer in a field that can't strictly be gradient-along- $z$ only ( $\nabla\cdot\vec B=0$ forbids that).

Is minus m dot B the interaction energy of two dipoles as well?

Yes — evaluate one dipole's field at the other and insert: $U_{12}=-\vec m_2\cdot\vec B_1(\vec r_{12})=\frac{\mu_0}{4\pi r^3}\left[\vec m_1\cdot\vec m_2-3(\vec m_1\cdot\hat r)(\vec m_2\cdot\hat r)\right]$ . Its anatomy repays a minute: side-by-side parallel moments (both perpendicular to $\hat r$ ) carry positive energy $+\frac{\mu_0m_1m_2}{4\pi r^3}$ — they prefer to anti-align, which is why adjacent compass needles on a table settle antiparallel — while head-to-tail parallel moments (both along $\hat r$ ) carry $-\frac{2\mu_0m_1m_2}{4\pi r^3}$ , the deepest arrangement — which is why magnets snap into chains. The $1/r^3$ energy gives the $1/r^4$ dipole–dipole force by differentiation — the far-field law between any two magnets, and the next article makes it a worked example. At atomic spacings this energy is $\sim10^{-23}$ J $\sim1$ K: real but weak, which is why dipole–dipole coupling alone almost never explains ferromagnetism — a flag the matter chapter picks up with the exchange interaction.

What sets the amplitude of a compass needle's oscillation decay, given that the period formula has no damping in it?

The period formula is the conservative skeleton; decay is whatever dissipation rides along — pivot friction, air drag, and (in good instruments, dominantly) eddy currents: the swinging needle's changing flux induces currents in nearby conductors, whose ohmic loss drains the oscillation (many survey compasses deliberately mount a copper damping vane for exactly this purpose — critical damping makes readings fast). The physics splits cleanly: $T=2\pi\sqrt{I_{\rm rot}/mB}$ from the energy landscape, decay time from the loss channels, and the quality factor $Q$ measures their ratio. The same split governs atomic moments — Larmor frequency from this article, relaxation times $T_1,T_2$ from environmental coupling — and reading both is precisely what an MRI pulse sequence does.

Do these formulas apply to induced moments, or only permanent ones?

The instantaneous laws apply to whatever moment exists at that instant, but the energy landscape changes when $\vec m$ responds to $\vec B$ . For an induced linear moment $\vec m=\alpha\vec B$ (dia- or para-magnetic response), the correct energy is $U=-\tfrac12\alpha B^2$ — half the naive $-\vec m\cdot\vec B$ , the same factor the polarizability article earned for induced electric dipoles by auditing the spring: the field must do work building the moment it then pulls on. Forces follow: $\vec F=\tfrac12\alpha\nabla(B^2)$ , toward strong field for $\alpha>0$ (paramagnets) and away for $\alpha<0$ (diamagnets — hence frog levitation over a Bitter magnet, where $-\nabla B^2$ balances gravity). The permanent-moment formulas of this article are the $\alpha$ -independent skeleton; the halves and signs of response are the next chapter's opening business.