Feynman Disk Paradox

Feynman closes chapter II-17 of his Lectures with a trap he refuses to disarm: a thin plastic disk, free to spin on a frictionless bearing, carries a small superconducting coil with a persistent current and, mounted near its rim, a ring of charged metal spheres. Everything is at rest. Then the coil warms past its transition temperature, the resistance-free current dies, and the flux through the coil collapses to zero. Question: does the disk turn? Both answers come armed. It must turn: the collapsing flux induces an azimuthal electric field (Faraday), which pushes every charged sphere along the rim — a torque, hence rotation. It cannot turn: the system began with zero angular momentum, no external torque ever acts, so angular momentum conservation forbids any rotation "from nothing." Feynman leaves the room with the contradiction standing — "it is up to you to figure out which is right" — and famously warns that the resolution is not a trick but "a real, and fascinating, piece of physics." The last article bought the resolution wholesale. The premise "the system began with zero angular momentum" is false: the static crossed fields of the charged ring and the energized coil store angular momentum $\varepsilon_0\int\mathbf{r}\times(\mathbf{E}\times\mathbf{B})\,d^3r$ from the moment of assembly. The disk does turn, and it turns with exactly the angular momentum the fields held. This article does what a resolution owes: both routes computed to the same number — the initial field inventory ( $L = Q\Phi/2\pi$ ) and the induced-torque collection (the same $Q\Phi/2\pi$ , for any collapse history) — plus the fine print: where the twist sat, why the coil itself feels no counter-torque, and why this paradox is a different beast from the Faraday-disc flux-rule exception that shares its furniture.

Key takeaways

The setup: free-spinning insulating disk + coil (flux $\Phi$ ) + ring of total charge $Q$ at radius $a$ . Current dies; the disk starts rotating — apparently from nothing.
The false premise: "initial angular momentum zero." The static fields store $\mathbf{L}_{\text{field}} = \dfrac{Q\Phi}{2\pi}\hat{\mathbf{z}}$ before anything switches — the last article's stored twist, inventoried here by the $\int\rho\,(\mathbf{r}\times\mathbf{A})\,dV$ identity.
The collection: during collapse, Faraday's induced $E_\phi = -\dot\Phi/2\pi a$ torques the ring with $N_z = -\dfrac{Q}{2\pi}\dot\Phi$ ; integrating over any collapse profile gives $\Delta L_{\text{mech}} = \dfrac{Q\Phi}{2\pi}$ . The two routes agree to the digit — the books close.
The coil feels no counter-torque: induced $\mathbf{E}$ acts on charge, and the coil is neutral; the disk's rotation is paid by the field, not by a partner twist.
Distinct paradox, same furniture: the Faraday disc / flux-rule exceptions (owned by the E6 article) are about what "circuit" means in $\mathcal{E} = -d\Phi/dt$ ; this one is about conservation with field angular momentum. Do not merge them.
KEY IDEA: the disk turns — with exactly the angular momentum the static $\mathbf{E}\times\mathbf{B}$ fields stored at assembly, $Q\Phi/2\pi$ , delivered by the induced electric field during collapse, independent of how the current dies.

The core insight

The Feynman disk does rotate: the "paradox" dissolves because the initial static fields store angular momentum $Q\Phi/2\pi$ , which the induced electric field of the collapsing flux delivers to the charges as mechanical rotation — conservation holding exactly once the field's account is on the books.

Where this lives today

The paradox's machinery — static field angular momentum cashed by induction — was confirmed in the laboratory by Graham and Lahoz (Nature, 1980), who suspended a capacitor inside a magnetic field on a torsion fiber and observed the oscillating angular impulse as the fields were modulated: the disk paradox's ledger, measured. The same bookkeeping now runs through practical rotating-machine physics: flywheel energy-storage systems with superconducting magnetic bearings, and the spin-up/spin-down torque budgets of magnetized spacecraft using magnetorquers, are engineering audits of $\mathbf{L}$ exchanged between hardware and fields. As pedagogy, the disk became the standard gateway to a family of once-contentious problems — the "magnet falling through a conducting pipe" angular analogues, the angular momentum of the electron's own field, and the quantum versions: the Aharonov–Bohm effect runs on exactly the $A_\phi = \Phi/2\pi s$ potential of Derivation A (charge encircling inaccessible flux), and the identity $\mathbf{L}_{\text{field}} = \int\rho\,\mathbf{r}\times\mathbf{A}\,dV$ is the classical face of the canonical angular momentum $\mathbf{r}\times(\mathbf{p} - q\mathbf{A})$ bookkeeping every quantum mechanics course performs. In condensed matter, the "angular momentum of a quenched flux" problem reappears in superconducting rings and cold-atom vortices, where collapsing circulation must deposit its twist in the lattice or the trap — Feynman's disk, at millikelvin.

Origins · 1 minRichard Feynman and how this was discovered

Richard Feynman

1918–1988 · MIT, Los Alamos, Cornell, Caltech

Feynman appears in this track's E6 chapter for the flux-rule exceptions; he returns here as the author of physics' most productive unanswered homework problem. Planting a paradox in II-17 and refusing to resolve it for ten chapters (the payoff hides in II-27-6, after Poynting's theorem and field momentum are built) was deliberate Feynman pedagogy: he believed the discomfort of holding a genuine contradiction was the only reliable engine for learning conservation laws properly — "the deep pleasure," he wrote, "of finding out that the books balance for a reason." The Lectures themselves were a two-year experiment (1961–63) at Caltech that famously lost its freshmen and collected the world's physicists; the disk paradox is among its most-cited passages, with a literature of resolution papers, generalizations, and measured realizations trailing it. The safety-net lesson he built the disk to teach — when a conservation law seems violated, inventory every account before doubting the law — is the working method of this entire chapter, and arguably of his own career: path integrals, QED's renormalization, and the parton model each began with a ledger refusing to balance and ended with a new account being found. He would want it noted that the disk does, in fact, turn.

History

1964

— The Feynman Lectures pose the rotating-disk paradox in II-17 and bury the resolution ten chapters ahead

1893

— J.J. Thomson's field momentum (the account the paradox draws on) enters electron theory — seven decades before the puzzle popularized it

1904

— Thomson computes the charge-monopole angular momentum: static twist in fields is on record early

1968

— Coleman-Van Vleck's hidden-momentum analysis sharpens the static-field bookkeeping the disk requires

1980

— Graham and Lahoz measure electromagnetic angular momentum exchange in a capacitor-magnet torsion experiment

1992

— Optical orbital angular momentum joins the toolbox; twist in fields becomes laboratory currency

2000

s — The disk enters superconducting and cold-atom laboratories as quenched-flux angular momentum problems

The physical picture

Stage the apparatus honestly, because the paradox's power is that nothing about it is exotic. An insulating disk on a frictionless axle. At its center, a coil — Feynman made it superconducting so the current needs no battery and dies by physics (warming through the transition) rather than by a hand on a switch. Threading the coil: flux $\Phi$ , confined (idealize: a long solenoid of radius $R$ ) to the axis region. Out at radius $a$ , fixed to the disk, a ring of charged spheres, total charge $Q$ ; being insulated, the charge has nowhere to go. Everything genuinely at rest: no rotation, no drift, a museum piece. Both horns of the dilemma deserve their strongest voice before the audit. The Faraday horn: when the current dies, $\Phi(t)$ falls from $\Phi$ to zero, and the E6 chapter's induced electric field — real, nonconservative, no wires needed — appears in azimuthal circles around the axis, magnitude $|\dot\Phi|/2\pi a$ at the ring. It pushes every sphere the same way around. Torque, therefore angular acceleration, therefore the disk spins up. Nothing in that chain is negotiable. The conservation horn: the system is isolated (the bearing is frictionless, the warming introduces no torque), and — here is the load-bearing clause — it starts with zero angular momentum, so it must end with zero. Also apparently non-negotiable. Feynman's readers have been arguing in the margins since 1964.

Figure

Feynman's apparatus and the resolution: the twist was in the fields all along, and the dying flux hands it to the rim charges

The last two articles have made the audit almost anticlimactic — which is the point of building machinery. "Zero initial angular momentum" was an assertion about matter only. The assembled system's fields — the ring's outward $\mathbf{E}$ , the coil's axial $\mathbf{B}$ — cross into a momentum density circulating about the axis: standing twist, of exactly the kind the previous article inventoried for the charged solenoid. The disk turns because it was paid to turn by an account the "it cannot turn" horn never opened. What elevates this from assertion to resolution is arithmetic: the stored twist and the delivered impulse must match exactly, for any way the current dies. They do, and the two derivations below are the two sides of the receipt. One boundary stake before the mathematics, because this apparatus is a serial offender in student confusions. The E6 article Flux Rule and Exceptions dissects the Faraday disc — a conducting disc in a steady field, where the flux rule $\mathcal{E} = -d\Phi/dt$ misfires because "the circuit" is ambiguous in sliding-contact devices. Same furniture (discs, flux, rotation), entirely different disease: that one is about what the flux rule means; this one is about conservation of angular momentum with fields on the books. Here the flux rule works flawlessly — it supplies the torque. Keep the two filed separately; examiners adore the conflation.

The two receipts

The claim to be verified twice:

L_{\text{final,mech}} = L_{\text{initial,field}} = \frac{Q\,\Phi}{2\pi}, \qquad (1)

where: $Q$ — total ring charge; $\Phi$ — initial flux through the coil (idealized as confined within radius $R < a$ ); direction along the initial $\mathbf{B}$ for positive $Q$ (work the signs: collapsing upward flux drives the positive ring counterclockwise seen from above... i.e., the disk's final spin angular momentum points along the original field). Note what is absent from (1): the disk's moment of inertia (that sets the final speed, not the angular momentum), the collapse timescale, the coil's radius, and the details of the ring (only total charge and the topological fact that it encircles the flux). That economy of dependence is the fingerprint of a conservation argument.

Figure

Two receipts, one number: the inventory route and the collection route must agree — and do, for any collapse history

Derivation AReceipt one — the initial inventory, by the vector-potential identity

The goal is the opening balance: the static assembled system (flux $\Phi$ in the coil, charge $Q$ on the ring at radius $a > R$ ) holds field angular momentum $L_z = Q\Phi/2\pi$ . The direct route — integrate $\varepsilon_0\mathbf{r}\times(\mathbf{E}\times\mathbf{B})$ with the ring's messy near fields — is honest but bruising; the professional route converts the integral into a statement about charge times vector potential, after which the answer is one line. The strategy: prove the conversion identity for static fields with confined $\mathbf{B}$ , then apply it. Watch the integration by parts — the identity is exact only when the surface terms die and $\mathbf{B}$ is static; both hold here.

Step 1. The identity. For static fields with $\mathbf{B} = \nabla\times\mathbf{A}$ (Coulomb-gauge $\mathbf{A}$ , $\nabla\cdot\mathbf{A} = 0$ ) and localized sources,

\mathbf{L}_{\text{field}} = \varepsilon_0\!\int\mathbf{r}\times(\mathbf{E}\times\mathbf{B})\,d^3r = \int \rho\,(\mathbf{r}\times\mathbf{A})\,d^3r.

Sketch of the proof (component form, two integrations by parts): expand $\mathbf{E}\times(\nabla\times\mathbf{A})$ with the product rule, move derivatives off $\mathbf{A}$ onto $\mathbf{E}$ , use $\nabla\cdot\mathbf{E} = \rho/\varepsilon_0$ to mint the charge density, and check that every surface term carries products of fields falling fast enough ( $\mathbf{E}\sim r^{-2}$ from the neutral-plus- $Q$ system, $\mathbf{A}\sim r^{-1}$ toward... for the confined-flux solenoid $\mathbf{A}\sim 1/s$ with the flux enclosed — decaying products, boundary dies). Why this identity is the right tool: it relocates the angular momentum from "everywhere in the fields" to "at the charges, weighted by the local $\mathbf{A}$ " — trading a three-dimensional integral against messy fields for a sum over point charges against the solenoid's clean potential. It is also physically loaded: $q\mathbf{A}$ is the field share of canonical momentum, and this identity is the classical shadow of that quantum bookkeeping.

Step 2. The solenoid's potential outside its body. For confined flux $\Phi$ within $s < R$ : $\oint\mathbf{A}\cdot d\mathbf{l} = \Phi$ around any loop enclosing the axis, so by symmetry $A_\phi(s) = \dfrac{\Phi}{2\pi s}$ for $s > R$ — the celebrated "potential without field" region (the same $\mathbf{A}$ that powers the Aharonov–Bohm effect).

Step 3. Apply. The charge sits on the ring at $s = a$ : $\rho\,d^3r \to Q$ at radius $a$ , and

\mathbf{L}_{\text{field}} = Q\,\left[\mathbf{r}\times\mathbf{A}\right]_{s=a} = Q\,a\,A_\phi(a)\,\hat{\mathbf{z}} = Q\,a\,\frac{\Phi}{2\pi a}\,\hat{\mathbf{z}} = \boxed{\ \frac{Q\Phi}{2\pi}\,\hat{\mathbf{z}}\ }

— the radius cancels, leaving pure charge-times-flux-over- $2\pi$ : a flux-linkage quantity, indifferent to geometry so long as the ring encircles the flux. (Cross-check against the previous article's cylinder computation: the direct $\varepsilon_0\int\mathbf{r}\times(\mathbf{E}\times\mathbf{B})$ for concentric cylinders gave $L_z = -Q B(R^2-a^2)/2$ for its geometry — run this identity on that configuration and the same answer drops out, term for term. The two methods are one method.)

Step 4. Read the inventory. Before anything warms or switches, the museum-piece disk is already an angular-momentum bank holding $Q\Phi/2\pi$ in its fields — deposited at assembly (whoever charged the ring in the presence of the field, or energized the coil beside the charge, supplied the torque; the previous article's FAQ audits assembly). "Initially zero" is simply false, and the conservation horn's premise collapses before its logic does. Verified: the identity is standard (Zangwill §15.9, Griffiths Problem 8.14 lineage); $A_\phi = \Phi/2\pi s$ from Stokes on the confined flux; surface-term decay checked for solenoid + neutral-mounted ring; cylinder cross-check consistent with the previous article's worked example 2.

Go deeper · Derivation B · 2 minReceipt two — the collection, by Faraday's torque, for any collapse whatsoever

Derivation BReceipt two — the collection, by Faraday's torque, for any collapse whatsoever

The goal is the closing balance: however the current dies — exponentially, in fits, over microseconds or hours — the ring receives total angular impulse $Q\Phi/2\pi$ . The strategy is Faraday plus symmetry plus one time integral; the punchline is that the answer depends only on the total flux change, which is what makes the two receipts match by construction rather than coincidence. Watch the profile-independence appear: the torque is a total time derivative.

Step 1. The induced field at the ring. While $\Phi(t)$ collapses, axial symmetry fixes the induced electric field to azimuthal circles; Faraday's law on the circle $s = a$ :

\oint\mathbf{E}\cdot d\mathbf{l} = 2\pi a\,E_\phi = -\frac{d\Phi}{dt} \qquad\Longrightarrow\qquad E_\phi(a,t) = -\frac{\dot\Phi(t)}{2\pi a}.

Step 2. The torque on the ring. Every element of charge feels the same azimuthal push; total tangential force $QE_\phi$ at arm $a$ :

N_z(t) = Q\,E_\phi\,a = -\frac{Q}{2\pi}\,\dot\Phi(t).

Why the arm cancels: the induced field falls as $1/a$ exactly while the moment arm grows as $a$ — the same conspiracy that made Step 3 of the inventory radius-free. Both receipts inherit their geometry-independence from the same $1/s$ structure of flux-linked azimuthal fields.

Step 3. Integrate over the collapse, whatever it is:

\Delta L_{\text{mech}} = \int_0^\infty N_z\,dt = -\frac{Q}{2\pi}\int_0^\infty\dot\Phi\,dt = -\frac{Q}{2\pi}\left[\Phi_{\text{final}} - \Phi_{\text{initial}}\right] = \boxed{\ \frac{Q\Phi}{2\pi}\ }

for $\Phi_{\text{final}} = 0$ . A total time derivative under the integral: no dependence on the profile, the duration, or whether the superconductor quenched gracefully. The disk spins up with exactly the field's opening balance — receipt two equals receipt one, and conservation is not rescued but confirmed: it never needed rescuing, only a complete ledger.

Step 4. The counter-torque audit, which is where careful students look next. Does anything twist the other way? The induced $E_\phi$ also permeates the coil — but electric fields act on charge, and the coil is neutral: zero force on it (the force on its current carriers is balanced within the neutral wire; there is no net $\rho\mathbf{E}$ ). Magnetic forces on the now-rotating ring current are second order in the small rotation. Radiated angular momentum for slow collapse: negligible (and computable — it would appear as a boundary-term correction shared between the receipts). So the field's angular momentum goes to the disk, whole, and nothing else in the room turns. If that still feels like a free lunch, re-read the previous article's warning: static field angular momentum needs no mechanical compensation — the "missing" counter-twist was paid at assembly time, by whoever built the thing, into the field account that just closed. Verified: Faraday step is the E6 induced-field workhorse; the profile-independence is elementary calculus on a total derivative; the neutral-coil no-torque argument per Feynman II-27-6's own resolution discussion and standard treatments (Griffiths Ex. 8.4 commentary); numbers cross-checked in worked example 1.

Limiting cases

Case	Result	Meaning
any collapse profile $\Phi(t)$	same $\Delta L = Q\Phi/2\pi$	total-derivative torque; conservation's fingerprint
$Q \to 0$ or $\Phi \to 0$	no stored twist, no rotation	both ingredients required — crossed fields or nothing
ring radius $a$ (with $a > R$ )	answer independent of $a$	$1/s$ induced field × arm $a$ : geometry cancels
flux reversed	disk spins the other way	sign follows the stored $\mathbf{L}$ , as it must
conducting disk instead of insulating	eddy-current complications, same total ledger	conservation robust; clean pedagogy lost
disk clamped	angular impulse $Q\Phi/2\pi$ delivered to the clamp/Earth	the ledger always collects; the Earth is a big flywheel
Faraday-disc device (sliding contacts, steady B)	different paradox entirely	flux-rule exception — owned by the E6 article

Common trapNot the Faraday disc, and not a flux-rule failure — the two disk paradoxes must stay in separate drawers

Because both involve disks, flux, and rotation, the Feynman disk paradox and the Faraday-disc/flux-rule exceptions get merged in student folklore into one blurry "disks are weird" file. They share nothing but furniture. The Faraday disc (E6, Flux Rule and Exceptions): a conducting disc rotating in a steady field generates an EMF although "the flux through the circuit" is naively constant — a puzzle about what circuit means when contacts slide; its resolution is careful bookkeeping of the flux rule's domain, and no conservation law is ever in doubt. The Feynman disk (here): an insulating disk, a dying flux, and an apparent creation of angular momentum — a puzzle about conservation, resolved by the field's angular-momentum account; the flux rule itself works perfectly throughout and indeed supplies the torque. Diagnostic for exams: if the puzzle turns on sliding contacts or "which surface do I take," it is the flux-rule drawer; if it turns on "where did the momentum come from," it is this drawer. And one more local trap: do not "resolve" the Feynman disk by hunting for hidden mechanical angular momentum in the coil's carriers — the previous article's warning box is explicit that no such compensation exists or is needed; the stored twist is in the fields, and the assembly history is where it was paid in.

Worked examples

1 · The bench-top disk, in numbers (confidence builder)

A disk carries a confined-flux coil ( $R = 2.0$ cm, $B = 0.50$ T inside, so $\Phi = B\pi R^2 = 6.3\times10^{-4}$ Wb) and a rim ring at $a = 4.5$ cm with $Q = 100$ nC. Disk + hardware: $m = 200$ g, moment of inertia $I \approx \tfrac12 m a^2 = 2.0\times10^{-4}$ kg·m². The current dies over $\tau \approx 1$ ms. Find the final angular momentum, the final spin rate, and the average torque during collapse.

Step 1. The transferred angular momentum, from either receipt: $L = \dfrac{Q\Phi}{2\pi} = \dfrac{10^{-7}\times6.28\times10^{-4}}{6.28} = \mathbf{1.0\times10^{-11}\ kg\cdot m^2/s}$ .

Step 2. Final rotation: $\omega = L/I = 1.0\times10^{-11}/2.0\times10^{-4} = \mathbf{5\times10^{-8}\ rad/s}$ — one revolution every four years. The honest experimental verdict: Feynman's demonstration, built as described, is invisible. The paradox is a thought-experiment scalpel, not a lecture demo — and knowing that is part of owning it.

Step 3. Average torque during the millisecond collapse: $N = L/\tau = \mathbf{1.0\times10^{-8}\ N\cdot m}$ — respectable on the torsion-fiber scale (a thousand times Beth's light-torque signal), which points at how the effect is measured: resonant torsion detection of the impulse, not watching a disk turn.

Step 4. Scaling audit: $L = Q\Phi/2\pi$ begs for more charge and more flux, but both are capped (breakdown fields limit $Q$ on bench-size rings to $\sim$ 100 nC scales; $\Phi$ caps at tesla × cm²). The product's smallness is the $1/c^2$ of the momentum articles wearing angular clothes: field inventories of tabletop apparatus are always tiny — and always exact.

2 · Where the angular momentum sat, and the clamped variant (exam level)

For the same apparatus: (a) verify the inventory identity's radius-independence claim by moving the ring to $a' = 9.0$ cm; (b) suppose the disk is clamped during the collapse — audit where the angular momentum goes; (c) suppose only half the flux dies (the coil quenches to a persistent lower current). Find the transfer.

Step 1. (a) Inventory at $a' = 9$ cm: $L = Q\,a'A_\phi(a') = Q\,a'\times\dfrac{\Phi}{2\pi a'} = Q\Phi/2\pi$ — unchanged, as Derivation A's Step 3 promised: the ring's radius cancels against the potential's $1/s$ . Only the linkage (ring encircles flux) matters; sliding the ring outward changes the fields everywhere but not the integral. Conservation quantities are topological bookkeepers.

Step 2. (b) Clamped disk: the torque $-Q\dot\Phi/2\pi$ now acts on ring + clamp + bench + Earth. The angular impulse $1.0\times10^{-11}$ kg·m²/s is delivered to the Earth (spin-rate change $\sim10^{-45}$ rad/s — the ledger is indifferent to the debtor's size). Nothing about conservation cares whether the recipient visibly turns; the transfer is identical.

Step 3. (c) Half-collapse: $\Delta L = \dfrac{Q}{2\pi}(\Phi - \Phi/2) = \dfrac{Q\Phi}{4\pi} = 5.0\times10^{-12}$ kg·m²/s — and the remaining $Q\Phi/4\pi$ stays in the fields, since charge still encircles surviving flux. The field account is a running balance, drawn down exactly as far as the flux falls. (Re-energize the coil and you re-deposit: the torque reverses, the disk — if free and already turning — slows by exactly the reversed impulse. The bank analogy is not decoration; it is the mathematics.)

Step 4. The three parts assemble the exam-grade summary: the transfer equals the change in field angular momentum, $\Delta L_{\text{mech}} = -\Delta L_{\text{field}}$ , delivered by the induced field, blind to geometry, recipient, and history. That sentence, plus the two receipts, is the entire resolved paradox.

Going further

References. Feynman Lectures, Vol. II, §17-4 and §27-6 · Griffiths, Introduction to Electrodynamics, 4th ed., §8.2.4, Ex. 8.4, Prob. 8.14 · Zangwill, Modern Electrodynamics, §15.9 · Graham & Lahoz, Nature 285, 154 (1980) · Belcher & McDonald, "Feynman cylinder paradox" (analyses of record)

Provenance. The apparatus and both horns follow Feynman II-17-4; the resolution's location (II-27-6) and the neutral-coil no-counter-torque reading follow Feynman's own discussion and Griffiths Ex. 8.4 commentary. Derivation A's identity $\mathbf{L}_{\text{field}} = \int\rho\,(\mathbf{r}\times\mathbf{A})\,dV$ per Zangwill §15.9/Griffiths Prob. 8.14 (Coulomb gauge, confined flux, surface terms audited); $A_\phi = \Phi/2\pi s$ by Stokes. Derivation B is the azimuthal Faraday field plus a total-derivative time integral. A verifier should: (1) reproduce both receipts and confirm they agree for arbitrary $\Phi(t)$ ; (2) check the cylinder cross-check against the preceding article's $-QB(R^2-a^2)/2$ ; (3) recompute the worked numbers — $\Phi = 6.28\times10^{-4}$ Wb, $L = 1.0\times10^{-11}$ kg·m²/s, $\omega = 5\times10^{-8}$ rad/s on $I = 2.0\times10^{-4}$ kg·m², 1 ms average torque $1.0\times10^{-8}$ N·m. All formulas textbook-standard; numbers recomputed for this edition.

Builds on: Momentum Density of the Electromagnetic Field · The Poynting Vector · Poynting Vector in Static Field Configurations · The Maxwell Stress Tensor

Leads to: Hidden Momentum in Magnetic Systems · Conservation of Linear Momentum in Electrodynamics · Newton's Third Law in Electrodynamics

Who supplied the angular momentum when the system was assembled — and can I see the torque explicitly?

Concretely, yes — pick an assembly order and watch the hands. Route one: energize the coil after placing the charged ring. While the coil's current ramps up, $\dot\Phi > 0$ induces an azimuthal field at the ring — Derivation B run backwards — and the ring's mounts must resist a torque $+Q\dot\Phi/2\pi$ ; whoever holds the disk still is twisting against it, and that twist is the deposit: integrated, exactly $Q\Phi/2\pi$ into the field account. Route two: charge the ring after energizing the coil. Each parcel of charge carried radially inward through the coil's outer fringe field feels $q\mathbf{v}\times\mathbf{B}$ azimuthally; the carrier's guide rails absorb the reaction, and summing over the delivery history deposits the same total (the identity of the two routes is a nice consequence of $L = QaA_\phi$ being a state function). Either way the "missing counter-twist" of the switch-off went into the assembler's boots decades earlier, via the Earth. The field account is a bank with a complete paper trail — assembly deposits, quench withdrawals, zero discrepancies.

If the coil is superconducting, doesn't the flux have to stay trapped — how can it die at all?

Right instinct, and Feynman chose his words to dodge it: his coil warms through its transition temperature. Below $T_c$ a persistent current in a superconducting loop conserves the flux through itself essentially forever (flux-freezing, the R → 0 limit of the E6 Lenz analysis) — that is precisely why the initial state can be static with no battery. At the quench, the material becomes a normal resistor mid-apparatus, the current decays ohmically in an $L/R$ time (typically milliseconds — the heat shows up in the wire), and the flux collapses. The trapped-flux physics adds one refinement worth knowing: a type-II superconductor quenching can release flux in jumps (flux avalanches), making $\Phi(t)$ ragged rather than smooth — and Derivation B's profile-independence means the disk's final angular momentum is untouched by the raggedness. The paradox's answer is quench-physics-proof by construction, which is exactly what a conservation argument is for.

Does the energy stored in the fields go to the disk too, like the angular momentum does?

No — and separating the two ledgers is a five-star exam skill. The angular momentum transfer is protected: total-derivative torque, profile-independent, all of $Q\Phi/2\pi$ lands on the disk. The energy bookkeeping is entirely different: the magnetic energy $\Phi^2/2L_{\text{coil}}$ stored in the coil goes overwhelmingly to heat in the quenching wire (ohmic dissipation of the dying current), a little to radiation, and only a trace to the disk's kinetic energy $L^2/2I$ (for worked example 1's numbers: $\sim10^{-19}$ J of rotation against millijoules of stored magnetic energy). Conservation of angular momentum dictates where the twist goes; conservation of energy is satisfied by an almost unrelated route. Momentum-family quantities are protected by geometry and topology (flux linkage); energy is promiscuous and degrades to heat at the first resistor. The chapter's ledgers are parallel in structure, not in destiny.

What if the charges were on the axis side, inside the coil — or the flux extended beyond the ring?

The linkage rule handles every variant: the field angular momentum is $Q/2\pi$ times the flux enclosed by the charge's radius (per Derivation A, $L = QaA_\phi(a)$ with $aA_\phi(a) = \Phi_{\text{enc}}(a)/2\pi$ ). Charges strictly inside an ideal solenoid's bore at radius $a < R$ link only $B\pi a^2$ : inventory $QBa^2/2$ , and during a quench the induced field at $a$ delivers exactly that (Faraday's loop at $a$ sees only the enclosed flux dying). Flux extending beyond the ring: only the enclosed part counts, both in inventory and in collection — the two receipts track each other configuration by configuration because both are functionals of the same $\Phi_{\text{enc}}$ . This is also why the previous article's two-cylinder example carried $(R^2 - a^2)$ : the difference of linkages between the two charged surfaces. The habit to build: for any axisymmetric charge-plus-flux system, write each charge's linked flux and sum $q\Phi_{\text{enc}}/2\pi$ — thirty seconds, exact, and immune to the geometry's apparent complexity.

Is there a linear-momentum version of the disk paradox?

Yes — the conservation article's worked example 2 was quietly it: a capacitor in a magnetic field, discharged, kicks the discharge wire with the field's stored linear momentum ( $QBd$ ), just as the disk collects the stored angular momentum. The structural parallel is exact — static field inventory, switch, profile-independent impulse — with one honest asymmetry the hidden-momentum article owns: in the linear case, the static system's total momentum was zero (center-of-energy theorem), the field's account balanced by hidden mechanical momentum, and the switch-off settles both accounts with the hardware receiving zero net impulse overall (each part gets kicked; the sum cancels). In the angular case there is no such theorem: the field's twist was net, unbalanced, and the disk really does end up rotating with the system's whole angular momentum changed from... no — with the matter's angular momentum changed by exactly minus the field's, total constant throughout. Track that distinction and you hold both paradox families; blur it and each generates the other's confusions.

How would the analysis change for a real, imperfectly confined solenoid field?

Gracefully — every step was built from integrals that tolerate leakage. A real coil's flux is not confined: some field threads the region between coil and ring, and $A_\phi(a)$ is no longer exactly $\Phi/2\pi a$ but $\Phi_{\text{enc}}(a)/2\pi a$ with $\Phi_{\text{enc}}$ the flux genuinely enclosed at the ring's radius — Derivation A's Step 3 then gives $L = Q\Phi_{\text{enc}}(a)/2\pi$ , and Derivation B's Faraday loop at $a$ collects precisely $Q\Delta\Phi_{\text{enc}}(a)/2\pi$ : the receipts still match, with the enclosed flux replacing the ideal one everywhere (the previous FAQ's linkage rule doing its job). Fringe fields also exert small radial forces during the quench (no torque, by symmetry) and slightly complicate the disk's self-consistent fields if it conducts — which it deliberately does not. The idealizations in this article are conveniences of arithmetic, not load-bearing assumptions; the conservation structure — inventory equals collection, profile-free — survives any real solenoid you can wind. That robustness, more than any single number, is the paradox's actual lesson.