# Self-⁠Induction and Eddy Currents

Laplace’s Mini World · Plate 65 · Matter, Light & Sound · Matter

> Change a magnetic field and a current appears in a conductor, running against the change. It slows a magnet in a copper pipe, brakes trains, and heats only the pan on a hob.

- Interactive model: https://world.lapla.si/en/eddy/
- Embed: https://world.lapla.si/embed/en/eddy/
- 中文: https://world.lapla.si/zh/eddy/
- Updated: 2026-10-30

## 01 A Slow Fall Through Copper

A demonstration published in 2006: a vertical copper pipe 1.7 m long and a neodymium magnet of 6 g, about 1.3 cm across. Dropped in at the top, the magnet comes out of the bottom 22.9 s later; a non-⁠magnetic object falls the same height in under a second.

Copper is not magnetic, and the magnet does not cling to it. Lay the magnet still on copper and there is no force at all. The force appears only while the magnet moves: then currents run round the pipe’s wall, and their field holds the magnet back.

The fall soon settles to a steady speed. All the potential energy the magnet loses on the way becomes heat in the pipe.

*The 2006 demonstration · a 1.7 m copper pipe · one magnet: 22.9 s · free fall: 0.59 s (computed here)*

## 02 Only While It Changes

On 29 August 1831 Michael Faraday wound two coils on a ring of soft iron 6 inches (about 15 cm) across, one to a battery and one to a galvanometer.

The moment the battery was connected, the galvanometer’s needle kicked; while the current flowed, the needle went back to zero; when the battery was disconnected, it kicked the other way. A steady current from the battery gave no deflection.

Today: when the magnetic flux through a coil changes, an emf appears in the coil, ε = −N dΦ/dt. The faster the change, the larger the emf; no change, nothing.

*29 August 1831 · a soft iron ring 6 inches across · ε = −N dΦ/dt*

## 03 Always Against

In November 1833 Emil Lenz read a rule to the Academy of Sciences in St Petersburg, published the next year: when a conductor moves near a magnet or a current, the current produced in it runs in the direction that would push the conductor against that motion.

Today it is usually put this way: the induced current’s field opposes the change in flux. Where the flux grows, it weakens it; where it shrinks, it makes up for it. It opposes the change, not the field itself.

In the pipe, the flux through the wall below the magnet is growing, and the current there pushes the magnet up; above it the flux is shrinking, and the current runs the other way and pulls it back. Pushed from the front and pulled from behind, the magnet slows.

*Lenz, 1834 · pushed in front, pulled behind · two loops running opposite ways*

## 04 A Coil Against Itself

In 1832 Joseph Henry found that a battery joined by a wire a foot long gave no spark when the circuit was broken, but a wire thirty or forty feet long gave a vivid spark, brighter still when coiled.

When the current in a coil changes, its own flux changes with it, and that induces an emf in the coil itself, against the change:

ε = −L dI/dt

This is self-⁠induction; L is the inductance, measured in henries. Again it opposes the change in current, not the current.

So when a coil is switched on, the current does not jump to its final value but climbs, at a pace set by L/R. A teaching coil of 900 turns, 24 mH and 6 Ω has L/R = 4 ms; on a closed iron core its inductance is far larger and the climb far slower.

*ε = −L dI/dt · 900 turns · 24 mH · 6 Ω · L/R = 4 ms (computed here)*

## 05 The Spark at the Break

In 1835 Faraday reported that with an electromagnet’s coil joined to a single pair of plates, there was hardly a spark on making contact but a large, bright one on breaking it, and a shock to hands holding the ends; a bar of soft iron in the coil raised the effect greatly, a bar of copper changed nothing. He called the current that appeared at the break the “extra current”.

A coil holds energy in its field, ½LI². Open the switch and the current must fall to zero almost at once: dI/dt is huge, and so is the self-⁠induced emf. The energy has nowhere to go but a spark across the opening switch, or a neon lamp that the low-⁠voltage battery could never light.

A relay coil switched off can throw up hundreds or even thousands of volts across its ends, which is why a diode is often fitted across it. A car’s ignition coil puts the effect to work: breaking a 12 V current in the primary gives about 40,⁠000 V in the secondary, enough to fire the spark plug.

*½LI² · relay coils: hundreds to thousands of volts · ignition coil: 12 V → about 40,⁠000 V*

## 06 An Invisible Brake

In 1824 Arago found that a copper disc turning under a compass needle drags the needle round with it; at rest relative to each other, the two do nothing at all. In 1831 Faraday explained it with induced currents.

Currents are induced in a solid piece of metal too. They run round in loops inside it, like eddies in water, hence eddy currents, and their field opposes the motion in the same way.

In 1855 Léon Foucault set the bronze rotor of his gyroscope between the poles of a strong electromagnet; a crank spun it at 150 to 200 turns a second. With the magnet switched on the rotor stopped within a few seconds, as if an invisible brake had been applied. He kept cranking, and the rotor warmed from the room’s 16 °C to 34 °C, hot to the hand: the work of the crank had become heat.

*Arago 1824 · Faraday 1831 · Foucault 1855: 16 °C → 34 °C*

## 07 Cutting Their Path

Eddy currents need a closed path to flow; cut the path and they weaken. A solid copper plate swinging between an electromagnet’s poles soon stops; a plate cut with a row of comb-⁠like slits swings for much longer. This is Waltenhofen’s pendulum, listed in Max Kohl’s instrument catalogue of 1900.

The copper tube on the board with a slit down its length works the same way: the rings of its wall are cut, the current cannot run all the way round, and a magnet falls faster than in the whole pipe. The acrylic tube does not conduct, and in it the magnet simply falls freely.

An aluminium ring placed over an iron-⁠cored coil fed with alternating current is thrown into the air; a ring with a slot cut in it does not jump.

*Comb slits · a slit tube · a cut ring: the currents cannot close*

## 08 Cores in Thin Sheets

In the cores of transformers and motors the field changes dozens of times a second, and iron conducts, so eddy currents would waste power as heat. That is why a core is not one solid block but a stack of thin steel sheets insulated from each other.

In each sheet the eddy currents can only run in small loops. The loss goes as the square of the sheet’s thickness: a tenth of the thickness, a hundredth of the loss. Sheets on the market go down to about 0.3 mm.

The steel is also alloyed with silicon, which raises its resistance and weakens the eddy currents; but more silicon makes the steel less magnetic, so it stays below about 3.5 %. The U-⁠shaped core under the coil on the board is stacked from sheets too.

*Loss ∝ thickness² · sheets down to about 0.3 mm · silicon below about 3.5 %*

## 09 The Rail as a Brake Disc

During the Expo 2000 world fair in Hannover, Germany’s ICE 3 high-⁠speed trains began carrying passengers. They were the first production trains in Europe to use eddy-⁠current rail brakes in service; the brakes were made by Knorr-⁠Bremse.

On both bogies of each trailer car hang two electromagnets 1,⁠540 mm long, each with eight coils, their poles alternately north and south along their length. Normally they are held up; to brake, they are lowered to 7 mm above the rail head. The magnet never touches the rail: the head is like a strip of metal dragged past the poles, and eddy currents are induced in it.

Each unit gives around 21 kN, a whole train about 170 kN at 200 km/h; each kilonewton takes about 4 kW to excite, about 680 kW for the train (computed here). The train’s kinetic energy ends up as heat in the rails.

*ICE 3 · since 2000 · a 7 mm gap · about 21 kN a unit · about 170 kN a train*

## 10 Weaker as It Slows

Eddy currents come from relative motion. At low speed the braking force is proportional to speed; faster still, it peaks at a critical speed and then falls away.

When an ICE 3 brakes from 300 km/h, the eddy-⁠current brakes do most of the work above about 230 km/h; they stay in use down to 50 km/h, and then the disc brakes bring the train to a stop. They cannot serve as a parking brake. They heat the rails, so in Germany they are used for service braking only on lines that can take it (rails kept at or below 75 °C, and slab track), and only for emergencies on lines with wooden sleepers.

The magnetic brakes of roller coasters and drop towers use permanent magnets: a metal fin on the car passes between two rows of magnets. In a power cut the magnets are still there and the brake still works; but with no relative motion there is no force, so the exit speed is never zero.

*Low speed: ∝ speed · greatest at the critical speed · the ICE 3 uses it down to 50 km/h*

## 11 A Hob That Heats the Pan

An induction hob first rectifies the 230 V, 50 Hz mains to direct current, then switches it into an alternating current at 20 to 75 kHz and feeds it to a coil under the glass. Each cooking zone takes about 1 to 3.6 kW. The first commercial induction range was made by Westinghouse in 1972.

The coil is a flat spiral of litz wire: many fine insulated strands twisted together, each thinner than the skin depth at that frequency. Under it lie a few ferrite bars, which guide the flux up towards the pan and keep it from the electronics below; under them is an aluminium shield.

The alternating field passes through the glass and induces eddy currents in the pan’s base, so the heat is made in the base itself: mostly the eddy currents’ Joule heat, with hysteresis losses too in a ferromagnetic pan.

*230 V · 50 Hz → 20⁠–⁠75 kHz · 1⁠–⁠3.6 kW a zone · Westinghouse, 1972*

## 12 A Thin Layer at the Bottom

The higher the frequency, the more the eddy currents crowd into the layer nearest the coil. The depth δ = 1/√(π f σ μ₀ μᵣ) is called the skin depth: the better the conductor and the more magnetic it is, the shallower it gets.

By this page’s arithmetic, at 25 kHz it is about 0.42 mm in copper; in ferritic stainless steel, with about 35 times copper’s resistivity and μᵣ taken as 100 to 1,⁠000, it is only about 0.08 to 0.25 mm. Squeezed into so thin a layer, the currents meet a resistance about 60 to 190 times copper’s.

That is why a stainless pan made for induction often has a capsule base: an outer layer of magnetic 18/0 stainless, an aluminium core to spread the heat, and an 18/10 stainless body. Aluminium and copper pans have too little resistance for an ordinary hob; they need an “all-⁠metal” hob working at higher frequencies.

*δ = 1/√(π f σ μ₀ μᵣ) · at 25 kHz: copper 0.42 mm · ferritic stainless 0.08⁠–⁠0.25 mm (computed here)*

## 13 The Glass Is Warmed by the Pan

The top is a glass-⁠ceramic. It does not conduct, so the alternating field passes through it without inducing currents in it. It gets hot only because a hot pan sits on it.

The glass-⁠ceramic barely expands with heat and withstands sudden temperature changes of up to 700 °C; it is also a poor conductor of heat, about 1.6 W/(m·K), so heat does not spread far sideways through it.

In 2020, 66 % of the 233 million hobs in use in Europe were electric, and induction made up 41 % of those.

*Glass-⁠ceramic · about 4 mm · withstands sudden changes of 700 °C · about 1.6 W/(m·K)*

## 14 The Hob Taken Apart

The hob comes apart in six pieces: the pan, the glass-⁠ceramic top, the coil with its ferrite bars, the aluminium shield, the circuit board (switch, capacitors, fan) and the case.

Rest the pointer on a piece and it is lifted out on its own and named; otherwise all six are named beside the hob.

*Pan · top · coil · shield · board · case*

## 15 Drop a Magnet

This is the page’s model: the 1.7 m copper pipe of 2006, inner radius 7.85 mm, with magnets of the paper’s size and the field it measured at their faces. The lever sets the wall’s thickness; drop one magnet, or two or three stuck together.

Each press of “Drop the magnet” starts a fresh fall from the top, played four times faster than life, and the chart draws the magnet’s height; the pale line is the path it will take. With a 1.9 mm wall, the time measured in 2006 is shown beside it. The model leaves out the currents’ own field and the pipe’s ends, and differs from the measurements by up to a little over a fifth.

*The page’s model · a 1.7 m copper pipe · one magnet of 6 g · measured times for the 1.9 mm wall*

**Try this**: Thicken the copper tube’s wall from 1 mm to 3 mm: how many more seconds does one magnet take to fall the 1.7 m? And two or three stuck together?

## What the model leaves out

The four exhibits are not at one scale: the hob and pan are a little larger than life; the copper tubes are much shorter than the 2006 pipe and a little wider; the rail and brake magnet are about 1:5, and the magnet has six poles (each of the ICE 3’s has eight coils). The hob and pan have their front right quarter sawn away, and the copper tube has a window in its front. The blue loops are a drawing: real eddy currents are invisible, and the loops are drawn where they flow, in the pan’s magnetic layer, the pipe’s wall and the rail’s head; the loops in the pan are drawn in the sawn-⁠away quarter and stand for the currents all round its base. The hob’s current reverses tens of thousands of times a second; the drawing shows it standing still. The hob is no maker’s design, and its parts are placed by eye. The lab only computes: it leaves out the currents’ own field, the pipe’s ends and air drag.

## A few common misreadings

Lenz wrote in 1834 that the current’s force opposes the motion; “the induced current opposes the change in flux” is the later wording. It opposes the change, not the field: when the flux is shrinking, the induced field points the same way as the original. Copper is not attracted by a magnet; the magnet is slow in the pipe because it moves, and at rest there is no force. Foucault used the bronze rotor of his gyroscope in 1855, not the copper disc often described, and Arago saw the effect first, in 1824. An induction hob does not heat with microwaves, and the glass is not the source of heat: the heat is made in the pan’s base and the glass is warmed by the pan. An eddy-⁠current brake cannot hold a train still: the ICE 3’s is used only down to 50 km/h, and a roller coaster’s magnetic brake never brings the exit speed to zero; nor is the force always greater at higher speed: past the critical speed it falls. And eddy currents are not only waste: hobs and the brakes of trains and roller coasters put them to work.

## Common questions

### Why does a magnet fall so slowly down a copper pipe?

The force appears only while the magnet moves: then currents run round the pipe’s wall, and their field holds the magnet back.

See: [01 A Slow Fall Through Copper](https://world.lapla.si/en/eddy/#drop)

### What does Lenz’s law say?

Today it is usually put this way: the induced current’s field opposes the change in flux.

See: [03 Always Against](https://world.lapla.si/en/eddy/#lenz)

### Why is a diode fitted across a relay coil?

A relay coil switched off can throw up hundreds or even thousands of volts across its ends, which is why a diode is often fitted across it.

See: [05 The Spark at the Break](https://world.lapla.si/en/eddy/#spark)

### Does a magnetic brake work in a power cut, and can it stop a car?

In a power cut the magnets are still there and the brake still works; but with no relative motion there is no force, so the exit speed is never zero.

See: [10 Weaker as It Slows](https://world.lapla.si/en/eddy/#slow)

### How does an induction hob heat a pan?

The alternating field passes through the glass and induces eddy currents in the pan’s base, so the heat is made in the base itself: mostly the eddy currents’ Joule heat, with hysteresis losses too in a ferromagnetic pan.

See: [11 A Hob That Heats the Pan](https://world.lapla.si/en/eddy/#hob)

## Sources

1. [M. Faraday: Experimental Researches in Electricity, vol. 1 (First Series, Phil. Trans. 1832; Ninth Series, 1835)](https://www.gutenberg.org/cache/epub/14986/pg14986.txt)
2. [The Royal Institution: Michael Faraday’s ring-⁠coil apparatus (29 August 1831)](https://rigb.org/node/143)
3. [APS News (2001): Faraday and Electromagnetism](https://www.aps.org/apsnews/2001/08/faraday-electromagnetism)
4. [E. Lenz (1834): Ueber die Bestimmung der Richtung der durch elektrodynamische Vertheilung erregten galvanischen Ströme, Ann. Phys. Chem. 31, 483](https://archive.org/details/sim_annalen-der-physik_1834_31_31)
5. [W. F. Magie (1935): A Source Book in Physics (Lenz and Henry in English, and their lives)](https://archive.org/details/magie-a-source-book-in-physics)
6. [J. Henry (1832): On the Production of Currents and Sparks of Electricity from Magnetism, Am. J. Sci. 22, 403](https://archive.org/details/sim_american-journal-of-science_1832-07_22)
7. [L. Foucault (1855): De la chaleur produite par l’influence de l’aimant sur les corps en mouvement, Comptes rendus 41, 450](https://archive.org/details/comptesrendusheb41acad)
8. [MIT 8.02 course notes, ch. 10: Faraday’s Law of Induction (Lenz’s law, eddy currents, laminations)](https://web.mit.edu/8.02t/www/802TEAL3D/visualizations/coursenotes/modules/guide10.pdf)
9. [MIT 8.02 course notes, ch. 11: Inductance and Magnetic Energy (self-⁠inductance, ½LI², L/R)](https://web.mit.edu/8.02t/www/802TEAL3D/visualizations/coursenotes/modules/guide11.pdf)
10. [HyperPhysics: Faraday’s law, Lenz’s law, the jumping ring](http://hyperphysics.phy-astr.gsu.edu/hbase/electric/farlaw.html)
11. [HyperPhysics: Automobile Ignition (12 V to about 40,⁠000 V)](http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/ignition.html)
12. [Y. Levin, F. L. da Silveira and F. B. Rizzato (2006): Electromagnetic braking: A simple quantitative model, Am. J. Phys. 74, 815 (the 1.7 m pipe, 22.9 s)](https://doi.org/10.1119/1.2203645)
13. [M. H. Partovi and E. J. Morris (2006): Electrodynamics of a Magnet Moving through a Conducting Pipe, Can. J. Phys. 84, 253](https://doi.org/10.1139/P06-065)
14. [C. Weigel et al. (2015): Predicting the Influence of Plate Geometry on the Eddy Current Pendulum, arXiv:1506.06401](https://arxiv.org/abs/1506.06401)
15. [T. B. Greenslade (Kenyon College): The Eddy Current Pendulum (Max Kohl’s 1900 catalogue)](https://physics.kenyon.edu/EarlyApparatus/Electricity/Eddy_Current_Pendulum/Eddy_Current_Pendulum.html)
16. [G. Moritz (2010): Eddy currents in accelerator magnets, CERN-⁠2010-⁠004 (laminations, silicon steel, about 0.3 mm)](https://arxiv.org/abs/1103.1800)
17. [J. Barranger (1965): Hysteresis and Eddy-⁠Current Losses of a Transformer Lamination…, NASA Technical Note](https://jontalle.web.engr.illinois.edu/Public/Allen/Noori-pdf/Barranger.65.pdf)
18. [Railway Gazette International (2000): ICE3 pioneers commercial application of eddy-⁠current rail brakes](https://www.railwaygazette.com/passenger/2000/09/01/ice3-pioneers-commercial-application-of-eddy-current-rail-brakes/)
19. [Railway Gazette International (2008): Eddy-⁠current braking: a long road to success](https://www.railwaygazette.com/business/2008/06/02/eddy-current-braking-a-long-road-to-success/)
20. [H2W Technologies (2019): Failsafe Linear Magnetic Brakes, Tech Briefs](https://www.techbriefs.com/component/content/article/34113-failsafe-linear-magnetic-brakes)
21. [US patent 6,⁠523,⁠650 (Magnetar): Permanent magnet eddy brake with flux-⁠steering poles](https://patents.google.com/patent/US6523650B1/en)
22. [US patent application 2004/0262103: a permanent-⁠magnet eddy-⁠current brake that brakes in a power cut](https://patents.google.com/patent/US20040262103A1/en)
23. [C. Chen et al. (2019): Characteristic Analysis of the Peak Braking Force and the Critical Speed of Eddy Current Braking in a High-⁠Speed Maglev, Energies 12, 2622](https://doi.org/10.3390/en12132622)
24. [TE Connectivity: Coil Suppression Can Reduce Relay Life (13C3264)](https://www.te.com/en/products/relays-and-contactors/relays/intersection/coil-suppression-relay-life.html)
25. [PHYWE: coil, 900 turns (06512-⁠01: 24 mH, 6 Ω)](https://phywe.com/equipment-accessories/electrical-magnetic-components/coils-iron-cores/coil-900-turns_976)
26. [P. Guillén et al. (2025): Induction Heating Appliances: Fifty Years of Technological Success, IEEE Industrial Electronics Magazine 19(3), 51](https://doi.org/10.1109/MIE.2025.3591256)
27. [Infineon: AN2014-⁠01 (induction cooking: frequency and coil)](https://www.infineon.com/dgdl/Infineon-AN2014_01_Reverse_Conducting_IGBT-ApplicationNotes-v03_01-EN.pdf)
28. [SCHOTT: CERAN glass-⁠ceramic, technical details](https://www.schott.com/en-gb/products/ceran/technical-details)
29. [Fissler: induction pots (18/10 body, 18/0 base capsule, aluminium core)](https://us.fissler.com/collections/induction-pots)
30. [ATI: technical data sheet, 430 ferritic stainless (resistivity 60.0 μΩ·cm)](https://www.atimaterials.com/Products/Documents/datasheets/stainless-specialty-steel/ferritic/ati_430_tds_en.pdf)
31. [US National Bureau of Standards (1966): Copper Wire Tables, Handbook 100 (the International Annealed Copper Standard)](https://nvlpubs.nist.gov/nistpubs/Legacy/hb/nbshandbook100.pdf)
32. [NIST: CODATA 2022 fundamental physical constants (μ₀, standard gravity)](https://physics.nist.gov/cgi-bin/cuu/Value?mu0)

## Data

The falling magnet follows the demonstration of Y. Levin, F. L. da Silveira and F. B. Rizzato (Am. J. Phys. 74, 815, 2006): their pipe, magnets and measured times; the page’s model sums the heat in thin rings of the pipe’s wall for a uniformly magnetised cylinder (Maxwell’s mutual inductance of coaxial loops; docs/plates/eddy/computed-⁠eddy.py), with CODATA 2022 constants. Skin depths use the IACS resistivity of copper (NBS Handbook 100) and ATI’s data for 430 ferritic stainless steel; the coil’s time constant uses PHYWE’s 900-⁠turn coil. History from Faraday’s Experimental Researches in Electricity, the Royal Institution, APS News, Lenz (Ann. Phys. 31, 1834) in W. F. Magie’s Source Book in Physics (1935), Henry (Am. J. Sci. 22, 1832) and Foucault (Comptes rendus 41, 1855); physics from MIT’s 8.02 course notes and HyperPhysics; laminations from J. Barranger (NASA, 1965) and G. Moritz (CERN-⁠2010-⁠004); the ICE 3 brake from Railway Gazette International (2000, 2008); magnetic brakes from H2W Technologies, Magnetar’s and Rosner’s patents and C. Chen et al. (Energies, 2019); the hob from P. Guillén et al. (IEEE Industrial Electronics Magazine, 2025), Infineon, SCHOTT and Fissler; relay coils from TE Connectivity. The four exhibits are this page’s own models, not any maker’s design, and not at one scale. Material presets, parts and the softbox studio of the site’s model kit are ported from AirsupHQ/machine-⁠skill (MIT, © 2026 Air Supply Technologies Inc.).

## Licence CC BY-NC 4.0

The text, pictures and models on this page are licensed CC BY-NC 4.0: free to use and adapt for teaching and other non-commercial purposes, with credit to world.lapla.si. Third-party material listed above keeps its own licence.

https://creativecommons.org/licenses/by-nc/4.0/deed.en

## Cite this plate

Laplace’s Mini World, Plate 65, “Self-⁠Induction and Eddy Currents”, https://world.lapla.si/en/eddy/, CC BY-NC 4.0, updated 2026-10-30.