The rule that is two laws wearing one coat

A copper disc spins in a steady magnetic field. Brushes touch its axle and its rim, wires run from them to a voltmeter, and the voltmeter reads. The machine is Faraday’s, built in 1831, and it is the oldest electrical generator there is. The flux through every surface its circuit bounds stays exactly the same throughout. It also produces its voltage with no changing flux anywhere in it, which makes it the standard counterexample to the law it was built to demonstrate.

The usual response is that the disc is a special case, or a degenerate one, or that the circuit is ambiguous. It is none of those things. It is an ordinary machine, and what it shows is that the flux rule is a theorem with hypotheses rather than a law.

Two mechanisms with one formula #

Consider the standard demonstration first. A rectangular loop is pulled sideways out of a region of uniform field.

What acts on the carriers here is the magnetic part of the Lorentz force. A charge in the moving bar has a velocity, the field is perpendicular to it, and $q\mathbf{v}\times\mathbf{B}$ pushes it along the bar. This is a magnetic force, it acts only because the charge is moving, and it has nothing whatever to do with electric fields.

The magnetic force is always perpendicular to the velocity, so it does no work and cannot change a particle’s energy — charges of three different energies in the same field go round three different circles at the same rate, and none of them speeds up. In a conductor being dragged through a field that same perpendicular force pushes carriers along the wire, and the work that ends up in the circuit is done by whoever is doing the dragging. The field is a router, not a source.

Now take the same loop, hold it perfectly still, and change the field instead. The carriers are not moving, so no magnetic force acts on them at all. What pushes them is a genuine electric field, produced by the changing magnetic one, and its existence is one of Maxwell’s four equations rather than a consequence of anything mechanical:

$$\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}.$$

Two different forces, acting on different things, for different reasons. And both are computed by the same formula, $\text{emf} = -\mathrm{d}\Phi/\mathrm{d}t$, which mentions neither of them.

There is a further asymmetry between the two that is easy to miss. The magnetic force does no work — it is always perpendicular to the velocity, so it can redirect a carrier but never speed it up — and yet a moving-bar generator delivers power to a circuit. The resolution is that the force on a carrier is perpendicular to the carrier’s total velocity, which has two parts: the sideways drag of the bar and the along-the-bar drift the force itself produces. The component of the magnetic force along the bar does positive work on the carrier; the component of the same force opposing the bar’s motion does exactly as much negative work on whoever is pulling the bar. The magnetic force is a broker. It moves energy from the puller to the circuit and keeps none of it, which is what “does no work” means when the accounting is done over the whole system rather than one particle.

Nothing analogous is needed in the transformer case, because the induced electric field does work in the ordinary way, and the energy comes from whatever is changing the magnetic field.

Both at once, computed separately #

Nothing prevents both mechanisms from acting together, and separating them requires computing each one.

The two terms cross zero at different times, have different shapes and are individually meaningless as “the emf”. The identity

$$-\frac{\mathrm{d}}{\mathrm{d}t}\int \mathbf{B}\cdot\mathrm{d}\mathbf{A} = -\int \frac{\partial\mathbf{B}}{\partial t}\cdot\mathrm{d}\mathbf{A} + \oint (\mathbf{v}\times\mathbf{B})\cdot\mathrm{d}\boldsymbol\ell$$

is a piece of vector calculus about a surface whose boundary is moving, and it is why the flux rule works. It is a statement about geometry, not about physics: the physics is on the right-hand side, in two pieces, and the left-hand side is a convenient way of adding them up.

Plot the flux through that loop as it leaves the field region, and its derivative beneath, and the distinction the law rests on is drawn in one picture. A large flux produces nothing; a changing one produces an emf; and the emf is largest where the flux is changing fastest, which is not where the flux is largest. That is the entire content of the law, and it is what makes induction a subject about rates rather than about quantities.

Where the folding comes apart #

The identity above requires the circuit to be a definite curve made of definite material — a curve that can be followed from one instant to the next, so that “the surface it bounds” and “the velocity of each element of it” mean something. A disc with sliding contacts has no such curve.

There is no surface, at any angle, through any part of the apparatus, whose flux changes while the disc turns. The field is constant in time; the geometry of the wires, the brushes and the disc is constant in time; and the disc is a solid of revolution, so turning it is indistinguishable from not turning it as far as any surface is concerned. The flux rule gives zero and the machine gives most of a volt.

The force law has no difficulty. Ask what force acts on a carrier at radius $r$ in the disc, get $qv B = q\omega r B$, integrate from axle to rim, and the answer is $\tfrac12 B\omega R^2$ — which is what the voltmeter reads, at every rotation rate, in every field, for every disc radius. Nothing has to be said about circuits at all.

One more feature of the disc deserves recording, because it is the thing that makes people reach for a special explanation. Where, in the machine, is the seat of the emf? In a battery it is the chemistry at the electrodes; in the moving-bar generator it is the bar. In the disc it is everywhere in the disc at once, distributed continuously along every radius, with each element of radius contributing $\omega r B\,\mathrm{d}r$. There is no localised source and no pair of terminals across which something happens — the terminals merely tap a potential difference that is being maintained throughout the metal. That is unfamiliar, and being unfamiliar is not the same as being anomalous.

The statement that is never wrong #

It is worth writing down what replaces the flux rule, because it is shorter than the rule and has no hypotheses in it.

An emf is the work done per unit charge in carrying a charge once round the circuit, and the force per unit charge on a carrier is the whole Lorentz force divided by the charge. So

$$\mathcal{E} = \oint \left(\mathbf{E} + \mathbf{v}\times\mathbf{B}\right)\cdot\mathrm{d}\boldsymbol\ell,$$

with $\mathbf{v}$ the velocity of the material at each point of the path and $\mathbf{E}$ whatever electric field is there. That expression is correct for the loop leaving the field region, for the stationary loop in a swinging field, for both at once, for the disc, for a circuit that is being rerouted, and for cases nobody has thought of, because it is the force law integrated along a path and nothing else.

The flux rule is what it becomes when the path can be followed through time. Where it can, the two agree exactly and the flux version is far easier to evaluate — a single number read off a geometry, rather than a line integral of a cross product. Where it cannot, the flux version has no meaning to be right or wrong about, and the integral above is still an integral.

That ordering is the useful thing to carry. A general statement that is harder to use, and a special one that is easier and hides its own conditions, is an extremely common arrangement, and it is almost always the easier one that gets called the law.

The torque, which turns out to be the emf again #

The disc has a second measurement in it that settles any remaining doubt about where its energy comes from, and it takes two lines.

Close the circuit and a current $I$ flows radially through the disc. Every element of that radial current sits in the same field $B$, so it feels a force $BI\,\mathrm{d}r$ sideways, opposing the rotation, at a lever arm $r$. Integrating from axle to rim,

$$\tau = \int_0^R B I r\,\mathrm{d}r = \tfrac{1}{2}BIR^2,$$

and the mechanical power needed to keep the disc turning is $\tau\omega = \tfrac12 B\omega R^2 I$. The bracket is the emf computed at the start of this essay, so the mechanical power in equals the electrical power out, exactly, with no approximation and no appeal to conservation as a principle. The same integral, read once as a force per unit charge and once as a torque, gives both sides of the ledger.

There is nothing surprising in the result and something worth noticing in the route. A flux argument would have reached it too, in a machine where a flux argument applies; here there was none available, and the accounting closed anyway. Whenever a rule fails and the underlying force law is still there, everything the rule was used for is still computable — it is only less convenient.

What the rule of thumb was really doing #

The phrase that usually accompanies induction is that a conductor “cuts lines of force”. It is a useful mnemonic and it is not a statement about anything.

The number of field lines drawn is a choice. Twice as many would be the same field, and nothing in the equations identifies a line at one instant with a line at the next — a field is a value at each point, and a line is a curve somebody chose to draw through those values. “How many were cut” is therefore a question about the drawing rather than about the apparatus, and the choice was made by whoever drew it.

The rule works in ordinary geometries because counting cut lines is a way of computing the flux change, and the flux rule is right in ordinary geometries. It fails in exactly the same place the flux rule fails, and for the same reason: applied to the disc, it invites the question of whether the lines turn with the magnet, which has no answer, and different answers to it give different predictions for a machine whose behaviour is not in doubt.

That last point is worth being explicit about. Spin the disc and keep the magnet still, and the machine generates. Spin the magnet and keep the disc still, and it does not. Spin both together, and it does. If the lines were carried round by the magnet, the second and third cases would have to be the other way round. The force law gets all three right without ever asking where a line is, because it only ever asks what field exists at the place where a charge is and how fast that charge is moving.

Where the two mechanisms turn out to be one #

The exact agreement between the two terms is not a coincidence, and the reason is the deepest thing on this page.

Work the same wire out in two frames and the two mechanisms turn out to be one. In the laboratory frame the wire is neutral and the force on a moving charge is magnetic. In the charge’s own frame it is at rest and cannot feel a magnetic force at all — and the wire is not neutral there, because the two lattices of charge are contracted by different amounts, so the force is electric. Same force, same magnitude, two names for it depending on who is looking.

That is the subject of its own essay, and its consequence here is direct. Whether an emf is “motional” or “transformer” depends on which frame the problem is set up in, because whether a conductor is moving is a statement about a frame. A single physical situation splits into the two terms differently for different observers, while the total — which is what a voltmeter reads — comes out the same. Two mechanisms that transform into each other are not really two mechanisms; they are one piece of physics whose description depends on a choice.

A magnet falling through a copper tube is the rule working perfectly with no wire in it at all. Every ring of the tube is a circuit, the flux through each one changes as the magnet passes, and the retarding force is the sum over all of them. That case is worth putting beside the disc, because it shows what actually breaks the rule: not the absence of a circuit, but the absence of an identifiable one — a copper tube has a well-defined set of rings, and a rotating disc sliding against a brush does not.

The other counterexample: a circuit that changes its mind #

The disc is the famous failure and it is not the only one. The second kind is easier to build and harder to argue about, because it involves no rotation and no symmetry.

Take two conducting plates touching along a line, with a wire from each running to a meter, and a steady field through them. Rock the plates so that the line of contact moves across from one side to the other. The circuit before and the circuit after both enclose an area, and between the two instants the enclosed area jumps — the surface bounded by “the circuit” has changed by a finite amount in no time at all. The flux rule, differentiated, predicts an enormous emf at that instant. The meter reads nothing.

What went wrong is not the arithmetic but the identification. There is no continuous family of material curves connecting the before-circuit to the after-circuit, so $\mathrm{d}\Phi/\mathrm{d}t$ is not the derivative of anything a piece of metal is doing. The line integral above has no such trouble: at every instant it is taken round whatever conductor exists, every carrier is at rest, $\mathbf{E}$ is zero because nothing is changing, and the answer is zero at every instant including the awkward one.

The two failures are the same failure. In the disc there is no way to say which material points make up the circuit; in the rocking plates there is, but it changes discontinuously. Both violate the one hypothesis the theorem needs, and both are ordinary objects rather than pathologies.

Why anybody builds one #

The homopolar machine’s peculiarity is also its use, which is worth knowing before dismissing it as a demonstration piece.

It is the only rotating generator with no commutator and no alternation: the output is genuinely direct current, steady rather than rectified, with no ripple to filter. And it is a single-turn machine, so it produces a very low voltage at a very high current — the opposite of what a wound machine does. Where that combination is wanted, nothing else will do: pulsed-power supplies storing energy in a spinning rotor and dumping tens of thousands of amperes, electrolytic plant, and experiments needing large steady currents.

The engineering difficulty is entirely in the sliding contacts, which have to carry the whole output. Brushes that would be adequate at a few amperes are hopeless at tens of thousands, and the usual answer is a liquid metal — a channel of sodium–potassium alloy or mercury between rotor and stator, which makes electrical contact over a large area with almost no friction. That the hardest part of the machine is the part with no physics in it is a familiar pattern and worth stating: the counterexample that unsettles a law is often a device whose real problems are elsewhere entirely.

Where the model stops #

The flux rule’s hypotheses are rarely stated, and this essay is about them. They are that the circuit is a closed curve of identifiable material points moving with a well-defined velocity field, and that the surface spanning it deforms continuously with the curve. A sliding contact violates the first. A circuit that changes topology — a switch closing, a wire breaking — violates the second, and produces a flux change with no emf at all, which is the second classical counterexample and is drawn in no figure here because the interesting instant lasts no time.

The disc is treated as a rigid conductor with free carriers. At the rotation rates a real homopolar generator runs at, the disc is also under enormous stress, and the same rotation that generates the voltage would tear the disc apart if it were made much larger — which is why these machines are used for very high currents at low voltage rather than scaled up.

And nothing here is time-varying fast enough to radiate. Every field is quasi-static, the retardation is ignored, and the emf is computed as though the field at each point were the field the sources have right now. That is excellent for a machine turning at 3000 rpm and wrong for anything whose size is comparable with a wavelength at its operating frequency.

What the pictures cannot show #

The disc figures draw a force per unit charge and a flux, and neither is a picture of the machine. What is missing is the current path — through the disc radially, out at the brushes, round the external circuit and back — and the fact that the radial current in the disc is itself in the field, so it experiences a force opposing the rotation. That force is how the machine takes mechanical work in, and it is the reason the disc is hard to turn when the circuit is closed and easy when it is open.

The terms figure draws two voltages against time and cannot show where in the apparatus each one is acting. The transformer term is distributed round three sides of the loop; the motional term is entirely in the bar. What couples two circuits is the first term alone, which is why a transformer has no moving parts. A reading of the two curves as “two contributions to the same thing” is right about the arithmetic and hides the fact that they are forces on different pieces of metal.

Where the ladder goes next #

The induction ladder began with the field that makes the other, and only while it is changing and continued with the magnet that falls slowly, which applied the law to a solid with no wire in it. This rung asks what the law actually is. The rungs after it: self-inductance, where a circuit’s own changing current opposes its change and the flux rule is applied to a circuit whose flux it produces itself; the skin effect, where the induced currents confine themselves to a depth that falls as the square root of frequency; and the differential form used in earnest, where the four equations are solved together and the flux rule never appears because it is never needed.

The habit worth carrying away is about what a formula’s success proves. Two different mechanisms that agree numerically in every ordinary case will be taught as one, and the teaching will be undetectable until a case turns up where they disagree. The disc is that case for induction, and the general lesson is to look for the arrangement in which a rule’s hypotheses fail rather than for another case in which it works.