The Flux Rule and Its Exceptions

Every generator engineer, every transformer designer, and every student of this chapter uses one formula more than any other: $\mathcal{E} = -\mathrm{d}\Phi/\mathrm{d}t$ . Watch the flux; the EMF follows. It works for coils in alternators spinning at 3600 rpm, for stationary windings around surging cores, for loops stretched, tilted, or dragged through field gradients. And yet Feynman, presenting exactly this rule, felt compelled to issue physics' strangest health warning: we know of no other place in physics where such a simple and accurate general principle requires for its real understanding an analysis in terms of two different phenomena. The flux rule is not one law. It is two — stitched together so seamlessly that the seam only shows when it tears. The two are genuinely different physics. Hold the loop still and change the field: a curling electric field appears (Faraday's Law of Induction) and pushes charges around the wire. Hold the field steady and move the loop: no induced electric field exists anywhere, but each charge is carried through $\mathbf{B}$ and feels the magnetic kick $q\mathbf{v}\times\mathbf{B}$ (Motional EMF). One mechanism lives in a Maxwell equation, the other in the Lorentz force law. That their contributions always sum to a single tidy time derivative is a theorem — proved here with the Leibniz transport theorem, and resting on premises that can fail. This article does both halves of the honest job: it proves the unification, term by term, and then it breaks it — on the Faraday disc, which generates a steady EMF while no flux changes anywhere, and on Feynman's rocking plates, where flux sweeps wildly and no EMF appears. The cure in every pathological case is the same: return to the force law, follow the actual charges.

Key takeaways

The flux rule $\mathcal{E} = -\mathrm{d}\Phi/\mathrm{d}t$ unifies two distinct mechanisms: transformer EMF (changing $\mathbf{B}$ , stationary loop — a Maxwell-equation effect) and motional EMF (moving conductor, steady $\mathbf{B}$ — a Lorentz-force effect).
The unification is a theorem, proved by the Leibniz transport theorem, and it needs $\nabla\cdot\mathbf{B} = 0$ : no monopoles, no flux rule.
The rule's hidden premise: a materially continuous circuit — one definite set of charges on one definite closed path. Sliding contacts, brushes, and reassigned conduction paths void the warranty.
Faraday disc: steady EMF $\tfrac12\omega BR^2$ , zero flux change — the rule under-predicts. Rocking plates: huge flux change, essentially zero EMF — the rule over-predicts. Both failures, opposite directions.
The fundamental statement never fails: $\mathcal{E} = \oint(\mathbf{E} + \mathbf{v}\times\mathbf{B})\cdot\mathrm{d}\boldsymbol\ell$ over the genuine charge path. When circuit identity is ambiguous, trust the force law.
KEY IDEA: the flux rule is a derived theorem, not a postulate — it equals the true EMF only for a well-defined material loop, and where the loop's identity breaks down, only $\oint(\mathbf{E}+\mathbf{v}\times\mathbf{B})\cdot\mathrm{d}\boldsymbol\ell$ can be trusted.

The core insight

The flux rule is a derived theorem, not a postulate. The fundamental statement is $\mathcal{E}=\oint(\mathbf{E}+\mathbf{v}\times\mathbf{B})\cdot\mathrm{d}\boldsymbol\ell$ ; it equals $-\mathrm{d}\Phi/\mathrm{d}t$ only for a well-defined material loop. When the loop's identity is ambiguous, trust the force law, not the flux.

Where this lives today

The flux rule, used inside its premises, runs the grid: every transformer EMF and every alternator voltage on Earth is computed as $-N\,\mathrm{d}\Phi/\mathrm{d}t$ without a flicker of trouble, because windings are materially continuous loops. The exceptions run niches that are surprisingly alive. Homopolar generators — Faraday discs at industrial scale — powered the pulsed-energy community for decades (the Australian National University's 500 MJ machine drove railgun research), and the same physics in reverse makes the homopolar motor the world's simplest: a battery, a magnet, and a screw will spin on your desk. Liquid-metal versions are current technology: the DC conduction pumps that drive sodium through fast-breeder reactors are homopolar machines whose "disc" flows. The deeper legacy is diagnostic. Plasma physicists computing EMFs around channels whose current paths re-route mid-pulse, and electrical engineers analyzing commutator machines and brush noise, use exactly this article's recovery procedure: abandon the flux shortcut, integrate $\mathbf{E} + \mathbf{v}\times\mathbf{B}$ along the genuine charge path. And the two-mechanism seam Feynman flagged became one of physics' most productive irritations: the asymmetry between "magnet moves" and "loop moves" explanations is the opening paragraph of Einstein's 1905 relativity paper — the flux rule's hidden seam, pulled, unravelled into spacetime. The covariant resolution, where $\mathbf{E}$ and $\mathbf{B}$ become one tensor and the two stories become one story seen from two frames, is owned by the Relativistic electrodynamics chapter.

Origins · 1 minRichard Feynman and how this was discovered

Richard Feynman

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

Feynman's monument is quantum electrodynamics, but generations of physicists met him first through the Lectures on Physics — and Volume II, chapter 17, is where this article's honesty comes from. Presenting induction to Caltech sophomores in 1962, he refused to let the flux rule pass as a single law: he exhibited the disc and the rocking plates, showed the rule failing in both directions, and drew the moral that has been quoted ever since — we know of no other place in physics where such a simple and accurate general principle requires for its real understanding an analysis in terms of two different phenomena. The pedagogy was characteristic: Feynman treated apparent paradoxes as load-bearing, the places where a formula's premises become visible, and he built the Lectures around them (his disk-and-angular-momentum puzzle anchors this track's Conservation-laws chapter). A Nobel laureate (1965, with Schwinger and Tomonaga), a safecracker, a bongo player — and the sharpest warning label ever attached to $-\mathrm{d}\Phi/\mathrm{d}t$ .

History

1831

— Faraday builds the first homopolar generator: copper disc, horseshoe magnet, steady current — the exception precedes the rule

1834

— Lenz states the opposition law fixing the EMF's direction

1845

— Franz Neumann formalizes induction mathematically; the flux formulation takes its modern shape

1885

— Heaviside's vector recast of Maxwell separates the field law from the force law, making the two-mechanism structure visible

1905

— Einstein opens the relativity paper with the magnet-and-conductor asymmetry — the flux rule's seam becomes spacetime

1962

— Feynman's Lectures canonize the disc and rocking plates as the honest exceptions

The physical picture

Run the two stories side by side. Story A: a coil sits clamped to the bench while you shove a magnet through it. The magnet's changing field induces a swirling electric field throughout space — real, measurable, capable of accelerating charges in a vacuum (Induced Electric Fields maps it) — and that field drives the coil's electrons around. Story B: the magnet sits clamped while you drag the coil past it at the same relative speed. Now $\partial\mathbf{B}/\partial t = 0$ everywhere; there is no induced electric field at all. Instead, every electron in the moving wire is carried through the magnet's static field and feels the magnetic force $q\mathbf{v}\times\mathbf{B}$ along the wire. Different fields, different forces, different Maxwell-era physics — identical ammeter readings. That identity is no accident (it was one of the clues that led Einstein to relativity: his 1905 paper opens with precisely this asymmetry of explanation for a symmetric phenomenon), but within classical electrodynamics it must be proved, and the proof is the flux rule.

Figure

One dial, two engines: transformer EMF and motional EMF produce the same reading

The flux through a loop can change for exactly three reasons: the field varies in time, the loop moves bodily, or the loop deforms. The total derivative $\mathrm{d}\Phi/\mathrm{d}t$ swallows all three at once, and the theorem below shows the swallowing is legitimate — the field-variation piece is precisely the transformer EMF, the motion-and-deformation piece precisely the motional EMF, with nothing left over and nothing counted twice. But notice the phrase that has been doing quiet work: the loop. A circuit, for the theorem's purposes, is a definite set of identifiable charges threading a definite closed path, deforming continuously if it deforms at all. Brushes, sliding contacts, and switch-like geometry changes reassign the current to different material mid-story — the "loop" teleports, its flux jumps discontinuously for reasons that have nothing to do with induction, and the rule lies. Both of the classic exceptions below are exactly this failure, in opposite directions: a real EMF the flux misses, and a fake EMF the flux invents.

The theorem: proving the unification

The fundamental definition — the quantity a voltmeter across a break in the loop actually reports — is the work per unit charge done by the total electromagnetic force around the circuit:

\mathcal{E} \equiv \oint_{C(t)} \big(\mathbf{E} + \mathbf{v}\times\mathbf{B}\big)\cdot \mathrm{d}\boldsymbol\ell \qquad (1)

where: $\mathbf{E}$ — the lab-frame electric field; $\mathbf{v}$ — the velocity of the material line element $\mathrm{d}\boldsymbol\ell$ (the wire's own motion, point by point); $C(t)$ — the closed material circuit, possibly moving and deforming. This definition never fails — it is the Lorentz force per unit charge, integrated along whatever path the charges genuinely follow. Everything else in this article is a statement about when the shortcut below may replace it. The shortcut is the flux rule:

\mathcal{E} = -\,\frac{\mathrm{d}\Phi}{\mathrm{d}t}, \qquad \Phi = \int_{S(t)} \mathbf{B}\cdot\mathrm{d}\mathbf{a} \qquad (2)

where: $S(t)$ — any surface spanning the loop; $\mathrm{d}\mathbf{a}$ — the oriented area element, sign tied to the direction of $\mathrm{d}\boldsymbol\ell$ by the right-hand rule (the bookkeeping Lenz's Law drills). Note what the derivative is: the total derivative of a quantity whose surface, boundary, and integrand all change together. Differentiating such an object is exactly what the Leibniz transport theorem is for, and applying it is the entire proof.

Derivation AThe flux rule from Maxwell plus Lorentz — the transport-theorem proof

The goal is the complete proof that $\oint(\mathbf{E}+\mathbf{v}\times\mathbf{B})\cdot\mathrm{d}\boldsymbol\ell = -\mathrm{d}\Phi/\mathrm{d}t$ for any moving, deforming, materially continuous loop — exhibiting exactly which physical law each term comes from, because the exceptions are diagnosed by watching those origins. The strategy: differentiate the flux with the Leibniz transport theorem (the kinematic identity for flux integrals over moving surfaces), then convert each resulting term with a law of physics — one with the no-monopole law, one with Faraday's law. Watch for the two inputs beyond calculus: $\nabla\cdot\mathbf{B} = 0$ and $\nabla\times\mathbf{E} = -\partial\mathbf{B}/\partial t$ . The rule is exactly as strong as those two laws plus the premise of a well-defined material boundary.

Step 1. State the kinematic engine. For any vector field $\mathbf{F}$ and any surface $S(t)$ whose points move with velocity $\mathbf{v}$ , the Leibniz transport theorem gives

\frac{\mathrm{d}}{\mathrm{d}t}\int_{S(t)}\mathbf{F}\cdot\mathrm{d}\mathbf{a} = \int_S \frac{\partial\mathbf{F}}{\partial t}\cdot\mathrm{d}\mathbf{a} + \int_S (\nabla\cdot\mathbf{F})\,\mathbf{v}\cdot\mathrm{d}\mathbf{a} - \oint_C (\mathbf{v}\times\mathbf{F})\cdot\mathrm{d}\boldsymbol\ell.

Why three terms: the flux can change because the field changes where the surface is (first term), because the moving surface engulfs sources of the field (second), or because the sweeping boundary adds and removes ribbon-shaped slivers of area (third). This is pure calculus — no physics yet.

Step 2. Set $\mathbf{F} = \mathbf{B}$ and kill the middle term with the first piece of physics: $\nabla\cdot\mathbf{B} = 0$ , the no-monopole law. A moving surface can never engulf magnetic charge, because there is none:

\frac{\mathrm{d}\Phi}{\mathrm{d}t} = \int_S \frac{\partial\mathbf{B}}{\partial t}\cdot\mathrm{d}\mathbf{a} - \oint_C (\mathbf{v}\times\mathbf{B})\cdot\mathrm{d}\boldsymbol\ell.

Why this matters more than it looks: if monopoles existed, a loop sweeping past one would register a flux change with no corresponding force on its charges — the flux rule would fail for ordinary honest circuits. The rule silently certifies Gauss's law for magnetism every time it works.

Step 3. Convert the field-change term with the second piece of physics. Faraday's law in differential form, $\nabla\times\mathbf{E} = -\partial\mathbf{B}/\partial t$ , plus Stokes' theorem:

\int_S \frac{\partial\mathbf{B}}{\partial t}\cdot\mathrm{d}\mathbf{a} = -\int_S (\nabla\times\mathbf{E})\cdot\mathrm{d}\mathbf{a} = -\oint_C \mathbf{E}\cdot\mathrm{d}\boldsymbol\ell.

This is the transformer EMF wearing its field-theory clothes: the curling induced electric field, integrated around the loop.

Step 4. Assemble and negate:

\frac{\mathrm{d}\Phi}{\mathrm{d}t} = -\oint_C \mathbf{E}\cdot\mathrm{d}\boldsymbol\ell - \oint_C (\mathbf{v}\times\mathbf{B})\cdot\mathrm{d}\boldsymbol\ell = -\,\mathcal{E}.

The transformer term came from Faraday's law; the motional term came from the boundary sweep in the transport theorem — two independent laws delivering the two halves of one derivative. The premise used silently throughout: $C(t)$ is a single continuous curve whose points have well-defined velocities $\mathbf{v}$ — a material loop. Every exception in this article is a violation of that premise, never of Maxwell.

The two clean limits deserve their own display. For a stationary loop, only Step 3's term survives:

\mathcal{E}_{\text{trans}} = -\int_S \frac{\partial\mathbf{B}}{\partial t}\cdot\mathrm{d}\mathbf{a} \qquad (3)

where: everything is evaluated on a fixed surface — this is Faraday's law in integral form, the physics of every transformer, and the regime Induced Electric Fields explores. For a rigid loop moving through a static field, only the sweep term survives:

\mathcal{E}_{\text{mot}} = \oint_C (\mathbf{v}\times\mathbf{B})\cdot\mathrm{d}\boldsymbol\ell \qquad (4)

where: $\mathbf{v}$ is the wire's velocity — the physics of Motional EMF, with no electric field in sight. A rotating generator coil uses both at once in general; the flux rule's genius is that you never need to know the split to get the total. Its danger is that you can forget a split exists.

The exceptions: where the bookkeeping breaks

Exception 1 — the Faraday disc: real EMF, no flux change

Faraday's own 1831 generator is the canonical counterexample. A conducting disc spins about its axis in a uniform axial field $\mathbf{B}$ ; brushes touch the axle and the rim; a steady DC current flows through the external circuit. Sketch any loop you like through the apparatus: the geometry is stationary, $\mathbf{B}$ is constant, and the flux through every fixed loop is rigorously constant. The flux rule predicts zero. The ammeter disagrees.

Figure

The Faraday disc: brushes at axle and rim harvest a steady EMF while no flux changes anywhere

Go deeper · Derivation B · 2 minThe disc audited — the force law delivers what the flux rule cannot see

Derivation BThe disc audited — the force law delivers what the flux rule cannot see

The goal is to compute the homopolar EMF honestly from the force law, and then to locate — precisely, not rhetorically — why the flux rule returns zero for the same device. The strategy: follow the actual charges (co-rotating with the disc), integrate the magnetic force per unit charge along the actual conduction path (axle to rim), then attempt the flux-rule calculation and watch its premise fail. Watch the brushes: they are where circuit identity dies.

Step 1. Identify the moving charges and their velocity field. Charges locked in the disc co-rotate: at radius $r$ , $\mathbf{v} = \omega r\,\hat{\boldsymbol\phi}$ . Since $\mathbf{B} = B\hat{\mathbf{z}}$ is static, $\partial\mathbf{B}/\partial t = 0$ and there is no induced electric field anywhere — whatever drives the current must be magnetic.

Step 2. Compute the driving force per unit charge:

\mathbf{v}\times\mathbf{B} = (\omega r\,\hat{\boldsymbol\phi})\times(B\,\hat{\mathbf{z}}) = \omega r B\,\hat{\mathbf{r}},

purely radial, pushing positive charge from axle toward rim — the disc is a battery whose "chemistry" is rotation.

Step 3. Integrate along the genuine conduction path, a radius from axle brush to rim brush:

\mathcal{E} = \int_0^R \omega r B\,\mathrm{d}r = \tfrac{1}{2}\,\omega B R^2.

Steady, DC, and real: this is the EMF the external circuit sees, and homopolar generators built on it have delivered mega-ampere currents.

Step 4. Now attempt the flux rule and watch the premise fail. To evaluate $-\mathrm{d}\Phi/\mathrm{d}t$ you must nominate a closed material loop. Try the natural one: external wire, rim brush, a radius of the disc, axle brush. But which radius? The material radius you nominated at $t = 0$ rotates away from the rim brush immediately — the current does not follow it. The path through the disc is defined by the brushes, not by any persisting set of charges; disc material under each brush is continuously replaced. No materially continuous $C(t)$ carries the current, so " $\Phi$ through the circuit" refers to nothing definite. Nominate instead any honest fixed loop: its flux is constant, its flux-rule EMF zero. The rule has not been violated — it has been misapplied, outside its premise. (If you insist on a co-rotating material loop, its swept flux does change — but that loop leaves the brushes and carries no current; the bookkeeping and the physics have parted company.) As a consistency check with worked example 2's numbers ( $R = 0.10$ m, $B = 0.50$ T, 3000 rpm): $\mathcal{E} = 0.5 \times 314.16 \times 0.50 \times 0.010 = 0.785$ V. The absence of any induced electric field follows directly from $\partial\mathbf{B}/\partial t = 0$ together with $\nabla\times\mathbf{E} = 0$ in magnetostatics.

Exception 2 — Feynman's rocking plates: flux change, no EMF

The disc showed a real EMF the flux rule misses. Feynman's second counterexample is the mirror image: a huge $\mathrm{d}\Phi/\mathrm{d}t$ and no EMF. Two slightly curved conducting plates rock against each other in a uniform field, touching at a contact point that rolls rapidly from one edge toward the other as they rock. Draw the circuit through the two plates and the external wires: as the contact point sweeps, the enclosed area — and with it the flux — changes dramatically. The flux rule predicts a healthy EMF. The galvanometer reads essentially nothing. The diagnosis is the same premise, violated the other way. No charge anywhere moves with the contact point: the point of bookkeeping races across the plates, but the material barely moves at all ( $\mathbf{v} \approx 0$ everywhere, so $\oint(\mathbf{v}\times\mathbf{B})\cdot\mathrm{d}\boldsymbol\ell \approx 0$ ), and $\mathbf{B}$ is static (so $\oint\mathbf{E}\cdot\mathrm{d}\boldsymbol\ell = 0$ ). The force law honestly reports zero. The "flux change" was an artifact of redrawing the circuit through new material at each instant — a switching operation, not a deformation. The same trap, in engineering dress: any circuit whose geometry changes by reassigning conductors (commutators, brushes, make-before-break switches, plasma current channels re-routing) can show flux jumps that induce nothing. The two exceptions bracket the honest rule:

situation	$-\mathrm{d}\Phi/\mathrm{d}t$ says	force law says (and nature agrees)	premise violated
Faraday disc	$0$	$\tfrac12\omega BR^2$	current path defined by brushes, not by persisting material
rocking plates	large	$\approx 0$	"circuit" redrawn through new material as contact rolls
bar on rails	$B\ell v$	$B\ell v$	none — the loop deforms continuously; both methods agree

Limiting cases

Case	Result	Meaning
stationary loop	$\mathcal{E} = -\int_S (\partial\mathbf{B}/\partial t)\cdot\mathrm{d}\mathbf{a}$	pure transformer EMF — Faraday's integral law, exact
rigid loop, static field	$\mathcal{E} = \oint(\mathbf{v}\times\mathbf{B})\cdot\mathrm{d}\boldsymbol\ell$	pure motional EMF — Lorentz force, exact
moving and deforming material loop	$\mathcal{E} = -\mathrm{d}\Phi/\mathrm{d}t$ , total derivative	both mechanisms, one derivative — the theorem at full strength
monopoles existing	extra term $\int(\nabla\cdot\mathbf{B})\mathbf{v}\cdot\mathrm{d}\mathbf{a}$	the rule would break for ordinary loops; its success is evidence for $\nabla\cdot\mathbf{B}=0$
sliding contacts / brushes	rule unreliable in both directions	disc: EMF without flux change; plates: flux change without EMF
ambiguous circuit, any geometry	$\mathcal{E} = \oint(\mathbf{E}+\mathbf{v}\times\mathbf{B})\cdot\mathrm{d}\boldsymbol\ell$	the fundamental definition — never fails, merely requires knowing the real charge path

Common trapThe flux rule is not more fundamental than the forces it summarizes

The failure mode is treating $\mathcal{E} = -\mathrm{d}\Phi/\mathrm{d}t$ as a law of nature and hunting for a loop that makes it work. It is a theorem about materially continuous circuits, and when you cannot point to the specific charges and the specific closed path they traverse — brushes, commutators, rolling contacts, re-routed channels — the "flux through the circuit" is not a physical quantity at all. Both pathologies occur: real EMFs at constant flux (Faraday disc) and flux sweeps that induce nothing (rocking plates). Neither is a paradox and neither touches Maxwell's equations, which have no exceptions; each is a bookkeeping construction applied outside its premises. The recovery procedure is mechanical: identify the actual conduction path, assign each element its actual material velocity, and integrate $\mathbf{E} + \mathbf{v}\times\mathbf{B}$ along it.

Worked examples

1 · The alternator coil, audited both ways (confidence builder)

A generator coil of $N = 100$ turns, each of area $A = 0.010\ \mathrm{m^2}$ (a $10\ \mathrm{cm} \times 10\ \mathrm{cm}$ square), rotates at $f = 50\ \mathrm{Hz}$ in a uniform field $B = 0.20\ \mathrm{T}$ . Find the peak EMF by the flux rule, then confirm it by the force law.

Step 1. Flux-rule route. The flux linkage is $N\Phi = NBA\cos\omega t$ with $\omega = 2\pi f = 314\ \mathrm{rad/s}$ , so

\mathcal{E} = -N\frac{\mathrm{d}\Phi}{\mathrm{d}t} = NBA\omega\sin\omega t, \qquad \mathcal{E}_{\rm peak} = NBA\omega = 100 \times 0.20 \times 0.010 \times 314 = \mathbf{63\ V}.

Step 2. Force-law route, at the instant the coil plane contains $\mathbf{B}$ (peak EMF). Only the two sides parallel to the rotation axis (length $a = 0.10\ \mathrm{m}$ ) contribute; each moves at $v = \omega(b/2) = 314 \times 0.05 = 15.7\ \mathrm{m/s}$ perpendicular to $\mathbf{B}$ , and at this instant $\mathbf{v}\times\mathbf{B}$ points along the wire with full strength $vB$ in opposite senses on the two sides — which is what drives current around the loop rather than cancelling.

Step 3. Sum: per turn, $\mathcal{E}_1 = 2 \times vBa = 2 \times 15.7 \times 0.20 \times 0.10 = 0.63\ \mathrm{V}$ ; for $N = 100$ turns, $\mathcal{E} = \mathbf{63\ V}$ ✓. The flux rule's one-line answer and the term-by-term Lorentz audit agree exactly — as Derivation A guarantees they must, because this rotating loop is materially continuous. (RMS value for the electrician: $63/\sqrt{2} = 44\ \mathrm{V}$ .)

2 · Designing with the exception: a homopolar generator (exam level)

A copper disc of radius $R = 0.10\ \mathrm{m}$ spins at $3000\ \mathrm{rpm}$ in an axial field $B = 0.50\ \mathrm{T}$ . Brushes at axle and rim feed a load; the total circuit resistance is $R_{\rm ckt} = 10\ \mathrm{m\Omega}$ . Find the EMF, the current, the electrical power, and the torque the motor driving the disc must supply.

Step 1. Angular speed: $\omega = 3000 \times 2\pi/60 = 314\ \mathrm{rad/s}$ . The homopolar EMF (Derivation B — the flux rule is useless here, by design of the problem):

\mathcal{E} = \tfrac12\,\omega B R^2 = 0.5 \times 314 \times 0.50 \times (0.10)^2 = \mathbf{0.79\ V}.

Low voltage is the signature of the device — the "battery" is only one disc-radius deep.

Step 2. Current: $I = \mathcal{E}/R_{\rm ckt} = 0.785/0.010 = \mathbf{79\ A}$ . Homopolar machines are low-voltage, high-current by construction, which is why their historical niches were electrolysis and welding, and why railgun power supplies used them.

Step 3. Electrical power: $P = \mathcal{E}I = 0.785 \times 78.5 = \mathbf{62\ W}$ .

Step 4. The mechanical audit — Lenz's law never sleeps, even where the flux rule fails. The radial current $I$ crossing the axial field feels an azimuthal force opposing the rotation; the required driving torque is

\tau = \frac{P}{\omega} = \frac{61.7}{314} = \mathbf{0.20\ N\,m},

and one can verify this directly: the retarding torque on the radial current path is $\tau = \int_0^R I B r\,\mathrm{d}r = \tfrac12 I B R^2 = 0.5 \times 78.5 \times 0.50 \times 0.01 = 0.20\ \mathrm{N\,m}$ ✓. The energy books close through the force law alone: the exception breaks the flux bookkeeping, never the energy ledger.

Going further

References. Griffiths, Introduction to Electrodynamics, 4th ed., §7.2.1–7.2.3 (the flux rule and its exceptions) · Jackson, Classical Electrodynamics, 3rd ed., §5.15 (Faraday's law, moving circuits) · Purcell & Morin, Electricity and Magnetism, 3rd ed., Ch. 7 · Zangwill, Modern Electrodynamics, §14.3 (motional EMF and the flux rule) · Feynman Lectures on Physics, Vol. II, §17-2 (exceptions to the flux rule). All results are analytic; no empirical constants are quoted. Key numerical checks: the sliding-bar EMF is $vBL$ and the homopolar-disc EMF is $\tfrac12\omega BR^2$ , both obtained by direct integration of $(\mathbf{v}\times\mathbf{B})$ . Worked numbers — alternator ( $N = 100$ , $A = 0.010$ m², $B = 0.20$ T, $f = 50$ Hz): $\mathcal{E}_{\rm peak} = NBA\omega = 63$ V, RMS 44 V, force-law check $2NvBa = 63$ V with $v = 15.7$ m/s; homopolar disc ( $R = 0.10$ m, $B = 0.50$ T, 3000 rpm, $R_{\rm ckt} = 10$ mΩ): $\mathcal{E} = 0.79$ V, $I = 79$ A, $P = 62$ W, $\tau = P/\omega = \tfrac12 IBR^2 = 0.20$ N·m.

Builds on: Faraday's Law of Induction · Motional EMF · Electromotive Force · Maxwell's Equations in Vacuum

Leads to: Lenz's Law · Self-Inductance · Induced Electric Fields · The Displacement Current

If the flux rule can fail, why is it taught as if it were a law?

Because inside its premises it is exact, and its premises cover nearly everything: any circuit made of wires that move or deform continuously — every coil, transformer winding, generator loop, and antenna in ordinary practice. The failures require a specific pathology (the conduction path being reassigned to new material, as at brushes or rolling contacts), which most courses never encounter. The honest formulation costs one sentence: for a materially continuous circuit, $\mathcal{E} = -\mathrm{d}\Phi/\mathrm{d}t$ . This article's job is to make you the person who knows what that sentence is guarding against.

Is the Faraday disc evidence that Maxwell's equations are incomplete?

No — and this is the crucial logical point. Maxwell's equations plus the Lorentz force predict the disc's EMF perfectly: $\oint(\mathbf{v}\times\mathbf{B})\cdot\mathrm{d}\boldsymbol\ell = \tfrac12\omega BR^2$ , confirmed by every homopolar machine ever built. What fails is a derived bookkeeping theorem applied outside its premises. The distinction matters because "exceptions to the flux rule" is sometimes miscited in fringe literature as evidence for new electromagnetic physics. It is the opposite: the exceptions are predictions of standard theory, testifying that the force law, not the flux shortcut, is fundamental.

How do the two mechanisms split for a generator coil that both rotates and sits in an AC field?

Compute either way, but not half-and-half by guesswork. The clean procedure: the total EMF is $-\mathrm{d}\Phi/\mathrm{d}t$ with $\Phi(t) = \int_{S(t)}\mathbf{B}(t)\cdot\mathrm{d}\mathbf{a}$ , the total derivative capturing both the explicit time dependence of $\mathbf{B}$ (transformer part) and the motion of $S(t)$ (motional part). If you want the split itself, evaluate $-\int(\partial\mathbf{B}/\partial t)\cdot\mathrm{d}\mathbf{a}$ on the instantaneous surface for the transformer piece and $\oint(\mathbf{v}\times\mathbf{B})\cdot\mathrm{d}\boldsymbol\ell$ for the motional piece; Derivation A guarantees they sum to the total. The split is frame-dependent — which is Einstein's 1905 point — but the sum is not.

Where exactly does the rocking-plates flux change go, if it induces nothing?

Nowhere — it was never a physical flux change experienced by charges. The flux through a mathematical loop you keep redrawing is just a number attached to your drawings; nature integrates forces along the paths of real charges. In the rocking plates the material velocities are tiny (gentle rocking) and $\mathbf{B}$ is static, so both terms of $\oint(\mathbf{E}+\mathbf{v}\times\mathbf{B})\cdot\mathrm{d}\boldsymbol\ell$ are tiny; the galvanometer agrees. The rapid area change comes entirely from the contact point migrating — and a geometric point is not a charge carrier. Compare the disc, where the failure runs the other way: there the charges genuinely move ( $\omega r$ can be large) and the EMF is real, while the drawable flux is constant.

Do commutators in ordinary DC motors suffer from these exceptions?

Yes, and machine designers know it. A commutator deliberately reassigns winding connections every half turn — exactly the "circuit identity" surgery this article warns about. For computing the average torque and back-EMF, flux-linkage methods survive because the reassignment is periodic and accounted segment by segment; but brush arcing, commutation spikes, and the detailed EMF during the switching interval require the force-law/charge-path analysis. The practical engineering rule mirrors the physics rule: flux methods between switching events, honest bookkeeping across them.

Why does the disc EMF go as R squared rather than R?

Because both the lever arm and the speed grow with radius. The force per unit charge at radius $r$ is $\omega r B$ — linear in $r$ — and the EMF is its line integral from axle to rim: $\int_0^R \omega r B\,\mathrm{d}r = \tfrac12\omega B R^2$ . Doubling the disc radius quadruples the voltage, which is why practical homopolar machines push disc size and rim speed to their mechanical limits and still deliver only volts. The same integral read backwards explains the motor version: a radial current in an axial field exerts torque $\tfrac12 IBR^2$ , quadratic for the same reason.

Does relativity make one of the two mechanisms fundamental and the other derived?

It makes them the same mechanism seen from different frames. In the loop's rest frame, the approaching magnet's field is time-dependent and the driving force is electric; in the magnet's rest frame, the field is static and the force is magnetic, $q\mathbf{v}\times\mathbf{B}$ . The Lorentz transformation converts one description into the other exactly — $\mathbf{E}$ and $\mathbf{B}$ are facets of one field-strength tensor, and "transformer versus motional" is a statement about your reference frame, not about nature. The covariant machinery belongs to the Relativistic electrodynamics chapter; the flux rule's seam is where that chapter's necessity first shows.