The field that makes the other, and only while it is changing

A coil of wire and a magnet, sitting near each other, do nothing. There is a magnetic field through the coil, there are electrons in the wire, and there is no current. Nudge the magnet a centimetre and a current flows for as long as the nudge lasts, then stops.

That is the entire experiment Faraday reported in 1831, and it is worth pausing on how strange it should look. Every law of force met so far on this site produces an effect that lasts as long as its cause: a charge in an electric field feels a force the whole time it is there. Here the cause is present, unchanging, and produces nothing. What produces something is the change.

Flux, and why the law is written in terms of it #

The quantity that turns out to matter is the magnetic flux through the loop:

$$\Phi = \int \mathbf{B}\cdot\mathrm{d}\mathbf{A},$$

which for a flat loop in a uniform field is the field strength times the area times the cosine of the tilt. It is the same construction as the flux of an electric field through a closed surface, with one difference worth stating: here the surface is not closed. It is a cap bounded by the wire, and the flux through it is the number of field lines the loop encircles.

Flux appears here in a form the site met first as a closed-surface count, and the difference is worth marking: this surface is open, a cap bounded by the loop, and its boundary is what carries the electromotive force. That the answer does not depend on which cap is chosen is a theorem rather than an assumption, and it is what makes the flux rule a statement about a curve.

Faraday’s law is then

$$\mathcal{E} = -\frac{\mathrm{d}\Phi}{\mathrm{d}t}.$$

Three ways of changing $\Phi$ therefore all produce an emf, and they look like three different experiments: move the loop into or out of a field, hold the loop still and change the field, or leave both alone and rotate the loop so that the cosine changes. That the same number governs all three is not obvious, and it is the reason the law is written in terms of the flux rather than in terms of anything more concrete.

The middle section of that figure is the part worth dwelling on. A loop entirely inside a uniform field has an enormous amount of field passing through it and no emf whatever, because none of it is changing. The magnitude of $\mathbf{B}$ appears nowhere in the law.

Which way, and why it could not be otherwise #

The minus sign is Lenz’s law, and it says the induced current flows in the direction whose own magnetic field opposes the change that produced it. Pull a loop out of a field into the page and the current runs clockwise, making a field into the page inside itself — replacing, as far as it is able, what is being lost.

The sign is usually presented as an extra rule to be memorised alongside the law. It is better derived, because it follows from energy conservation in about four sentences and cannot be otherwise.

Suppose the sign were positive. Nudge the loop, and the induced current would produce a field reinforcing the change, which would increase the emf, which would increase the current, which would increase the reinforcement. The loop would accelerate itself out of the field, drawing unlimited energy from nowhere. Since it does not, the sign is negative — and the sign of Lenz’s law is a statement that energy is conserved rather than an independent fact about magnetism.

The mechanical consequence appears in the figure as the second arrow. The side of the loop still inside the field carries a current across a field, so it feels a force $\mathbf{F} = I\mathbf{L}\times\mathbf{B}$, and that force points backwards. Pulling the loop out requires work against it. The work done pulling equals the electrical energy dissipated in the wire, which is the accounting that makes a generator a machine for converting mechanical energy rather than for creating electrical energy. A generator with nothing plugged into it is easy to turn; the resistance a reader feels when the load is connected is Lenz’s law under their hand.

The field that has no potential #

The most disruptive consequence of induction is not the current. It is what a changing magnetic field does to the concept of voltage.

An emf around a loop means the electric field has a non-zero line integral round a closed path — a charge taken once round the circuit gains energy and comes back to where it started. That is precisely what the electrostatic field cannot do. The whole argument for replacing a field with a potential was that the work done between two points does not depend on the path, so a single number can be attached to each point of space.

A static charge arrangement has equipotentials, and round any closed loop in one the work comes to zero — which is what “having a potential” means. The induced electric field has no such picture, because its whole content is that the work round a closed loop is not zero. That is not a complication to be drawn more carefully; it is a field of a different kind, and no contour map can represent it.

An induced field has no such number. Its field lines close on themselves — they form loops with no beginning and no end, exactly like the magnetic field’s own lines — and the “voltage between two points” in a region of changing flux is not defined until a path is specified. Two voltmeters connected to the same two points on opposite sides of a transformer core read different values, and both are correct.

This is not a subtlety of no consequence; it is the reason that oscilloscope probes have their ground leads twisted, that the loop area of a circuit matters at high frequency, and that the phrase “ground” stops being a single node in any apparatus near a changing field. It is also a first sign that the electric and magnetic fields are not two separate things with a coincidental resemblance.

Where the flux rule stops being a law #

Faraday’s law as written above — the “flux rule” — is a superb calculational tool and is not quite a fundamental statement, and the site’s habit is to say where a model stops rather than to imply it does not.

The difficulty is that two physically distinct things are being folded into one number. When the loop moves in a static field, the force on the charges is the magnetic force $q\mathbf{v}\times\mathbf{B}$ acting on carriers dragged sideways through the field. When the loop is stationary and the field changes, no charge is moving, so no magnetic force acts — what pushes the carriers is a genuine electric field created by the changing $\mathbf{B}$. Two different mechanisms, one formula, and no reason within the formula why they should agree.

They do agree, exactly, and the agreement is one of the strong hints that led to relativity: what one observer calls a magnetic force on a moving charge, another calls an electric force on a stationary one, and which observer is moving is a choice of frame rather than a fact about the apparatus.

The flux rule also fails outright in a small class of arrangements — circuits where the “loop” is ambiguous because part of it slides, such as a conducting disc rotating in a field with contacts at the rim and the axle. There the flux through any obvious surface does not change, and a current flows anyway. The force law $q(\mathbf{E} + \mathbf{v}\times\mathbf{B})$ handles it without complaint. The lesson is the one this site returns to: the flux rule is a theorem that follows from the field equations in most geometries, not the axiom it is usually taught as.

Every induced current makes a field of its own, and it is aimed to replace whatever flux is being lost. That is the geometrical content of Lenz’s law — not a sign convention to be memorised but a statement that the induced current’s field opposes the change, which is why the rule reads the way it does and why it could not read the other way without energy appearing from nowhere.

What it costs to make it useful #

An emf of a fifth of a volt from a loop moving at walking pace is not much. The two ways of making it large are both visible in the law.

Repeat the loop. $N$ turns of wire wound round the same area link the same flux $N$ times, and the emf multiplies by $N$. This is why a generator coil is a coil, and it is why the cheapest way to increase the output of an induction experiment is to wind more turns rather than to move faster.

Change faster. The rate of change is what appears, so alternating everything at fifty or sixty hertz gets an enormous improvement for free. This is the historical reason electricity distribution is alternating: not that alternating current is better to use — it is worse in several respects — but that transformers work only on changing flux, and transformers are the only cheap way to move power a long distance at high voltage and use it at low.

The transformer is the purest expression of the law on this page. Two coils on one iron core share a flux; the ratio of their voltages is the ratio of their turns, because both are $-\mathrm{d}\Phi/\mathrm{d}t$ multiplied by the number of times each links it. No moving parts, no contact, and an efficiency above 99 per cent in large units. It also cannot be made to work on direct current at all, which is a statement about the derivative rather than about engineering.

The unwanted version #

Every conductor exposed to a changing flux gets induced currents, whether or not anyone wanted them, and in a solid block those currents have no wire to follow. They circulate as eddies.

The energy they carry away comes from whatever is causing the change, and it ends up as heat. That is a loss in a transformer core — which is why cores are built from thin laminations, each insulated from its neighbours, so that the loops available to an eddy current are small and their resistance high. It is a brake in a train, where a magnet moved past a rail dissipates the train’s kinetic energy without touching anything. It is the reason a strong magnet dropped down a copper pipe falls slowly and smoothly: the current induced ahead of it and behind it always opposes its motion, and the terminal speed is set by the balance between gravity and the drag.

And it is the working principle of the induction hob, the metal detector and the smelting furnace, all of which deliver energy into a conductor without contact by presenting it with a flux that will not stop changing.

The loop nobody built #

The law does not require a loop to have been made on purpose, and the largest circuits on the planet are the ones nobody designed.

A magnetic storm rearranges the currents flowing in the Earth’s magnetosphere, and the field at the ground changes with them — by a few hundred nanotesla over a minute, in a severe event. That is a small field and a respectable rate, and a rate is what the law asks for. The flux threading the loop formed by a long transmission line, its two earthed substations, and the conducting ground between them is changing, so an emf appears along it: of order a volt per kilometre, and hundreds of volts across a long line.

What flows is effectively direct current, because the storm’s timescale is minutes and the grid’s is a fiftieth of a second. It enters through the earthed neutral of a transformer, and a transformer is the one device on this page that cannot tolerate it: a steady current in a winding pushes the core’s flux permanently off centre, so it saturates on one half of every cycle. A saturated core draws enormous magnetising current, dumps harmonics into the network, absorbs reactive power and heats.

That is how a geomagnetic storm brought down the Hydro-Québec grid in 1989, in about ninety seconds, leaving six million people without power. The 1859 event was larger and there was almost no grid to affect — what there was, was the telegraph network, and operators reported shocks from their equipment and found they could send messages with the batteries disconnected, running on the emf the sky was providing.

Pipelines have the same problem in a different form, since a buried steel pipe is another long conductor with earthed ends, and the induced current corrodes it.

The emf Faraday could not find #

There is one more natural circuit worth mentioning, because Faraday looked for it himself and failed.

Seawater conducts, and the Earth’s magnetic field has a large vertical component in northern latitudes. So water flowing horizontally is a conductor moving through a field, and the charges in it are pushed sideways — an emf across the flow, of the order of a volt across a strait a few tens of kilometres wide.

In January 1832, five months after the ring experiment, Faraday hung copper plates from Waterloo Bridge into the Thames and connected them to a galvanometer, expecting to measure the river’s flow. He got nothing he could trust. The signal was there; what defeated him was that two copper plates in river water form a battery of their own, whose voltage drifts by far more than the effect being sought.

The measurement works, and it is made routinely now. A disused telegraph cable across a strait, with its ends earthed, reports a voltage proportional to the volume of water crossing it — so a cable laid for one purpose becomes a flow meter for a whole channel, continuously, with no instrument in the water at all. The electrode problem is solved by measuring changes rather than absolute values, and by using electrode materials whose drift is slow.

It is a good closing illustration of the page’s own theme. The physics was right in 1832 and the experiment failed on a systematic error that had nothing to do with induction, which is the ordinary way experiments fail.

What a single frame cannot show #

Every figure on this page is one instant, and the quantity the page is about is a rate of change. That gap is worth naming, because it is the reason the second figure exists at all.

A snapshot of the loop in the field carries no information about whether an emf is present. The same picture — a loop half in, half out, in a field of a stated strength — describes a loop being pulled out at speed, a loop being pushed in, and a loop sitting motionless with no current at all. Nothing visible in the arrangement distinguishes them. The velocity arrow in the scene figure is drawn precisely because the geometry cannot supply what the law needs, and the current direction shown follows from the arrow rather than from the field.

That is a stronger statement than it looks. Almost every other figure on this site can be checked against its own contents: the rainbow angle is a property of the drawn rays, and the Carnot efficiency is a property of the drawn loop. An induced emf is a property of a sequence, so the only honest way to draw it is to draw the quantity against time and read the slope — which is what the second figure does, and why its lower panel is computed by differencing the upper one rather than by evaluating a formula.

The same limitation runs through the physics and not merely the pictures. A field configuration does not determine what is happening; a field history does. This is the first place on the site where that is true, and it is where the subject stops being describable by a single map of space.

The ten years, and the experiment nobody was looking for #

Ørsted had shown in 1820 that a current deflects a compass needle: electricity makes magnetism. The converse question — whether magnetism makes electricity — was then obvious, was pursued by several people including Faraday himself, and resisted for a decade.

The reason it resisted is the content of this page. Everyone was doing the experiment with steady currents and stationary magnets, because a steady cause was expected to produce a steady effect. A powerful magnet placed beside the most sensitive galvanometer available gives nothing, and gives nothing however long the experimenter waits, and the natural conclusion is that the effect does not exist.

Faraday’s decisive apparatus in August 1831 was an iron ring with two separate coils wound on it, one connected to a battery through a switch and the other to a galvanometer. There is no moving part in it. What he saw was a kick of the needle when the switch was closed, nothing while the current flowed steadily, and a kick the other way when the switch was opened. The effect was at the switch, and the ten-year search had been failing because the interesting moments were the ones nobody was watching.

Two things about that are worth carrying. The first is that the ring was the first transformer, built before there was a theory of what it did. The second is a methodological one: an effect proportional to a derivative is invisible to an experiment designed around steady states, and no amount of improving the sensitivity of the galvanometer would have found it. What was needed was a different question.

Where the ladder goes next #

The rungs from here: the motional emf derived from the force law alone, and its exact agreement with the flux rule; self-inductance, where a circuit’s own changing current opposes itself, and the oscillation that appears once that is put beside a capacitor; mutual inductance and the transformer treated properly; the differential form, $\nabla\times\mathbf{E} = -\partial\mathbf{B}/\partial t$, which is the local statement of the same law; the displacement current, which is the missing symmetric term and the one that makes light possible; and the homopolar generator, which is where the flux rule visibly fails.

The idea to carry forward is the one in the middle of the flux graph. A large field through a loop produces nothing whatever. It is the rate at which the flux changes that appears in the law — and a physics in which effects are caused by rates of change rather than by quantities is a different kind of physics from the one that came before it.