# Faraday's law of induction **Faraday's law of induction** states that the electromotive force around a closed circuit equals minus the rate at which [[Magnetic_flux|magnetic flux]] through the circuit changes: `EMF = −N·dPhi/dt`, where Φ is the flux through one turn and N the number of turns.[^up2-ch13] It is the law that turns motion into [[Electric_current|electric current]], and almost every joule of electricity delivered to a grid anywhere in the world is generated by it. The crucial word is *changing*: a [[Magnetic_field|magnetic field]], however strong, induces nothing at all if it is held still. In the microsim below the reader pushes a bar magnet through a coil and sets three things: the magnet's speed, the number of turns N, and the direction of travel. Two traces run underneath. The flux Φ(t) is a single smooth bump, rising as the magnet approaches and falling as it recedes; the EMF trace is the slope of that bump, so it is zero at the moment the magnet sits at the centre of the coil — precisely where the flux is greatest — and peaks on either side of it. Stop the magnet anywhere and both the slope and the EMF go to zero while the flux stays large, which is the point the equation makes and intuition usually does not. Reverse the direction and the EMF changes sign, and the arrow drawn on the winding shows the induced current circulating so that its own field opposes the change that produced it — [[Lenz's_law|Lenz's law]], the minus sign made visible. The flux itself is computed from the closed form for a point magnetic dipole on the axis of a circular loop, `Phi = mu0·m·a^2/(2·(a^2 + z^2)^(3/2))`, so no field integration is needed at run time (ILLUSTRATIVE: a real bar magnet is not a point dipole, and the model is accurate only when the magnet is longer than a coil radius away). On the [[Physics]] flagship this article serves Part I — History, the *19th century* section (row P8). It sits between [[Hans_Christian_Ørsted|Ørsted]]'s discovery that a current moves a compass needle and [[Maxwell's_equations|Maxwell's equations]], which take Faraday's experimental rule, write it as a local relation between fields, and get light out of the result. ## History In July 1820 [[Hans_Christian_Ørsted|Hans Christian Ørsted]] reported that a wire carrying a current deflects a nearby compass needle, establishing that electricity and magnetism are connected at all.[^oersted1820] The obvious converse — that magnetism should produce electricity — was pursued for a decade without success, because experimenters looked for an effect of a steady magnet on a steady circuit, and there is none. [[Michael_Faraday|Michael Faraday]] found the effect on August 29, 1831, using an iron ring wound with two separate coils. Connecting a battery to the first coil produced a momentary deflection in a galvanometer attached to the second; disconnecting it produced a deflection the other way; while the battery stayed connected, nothing happened.[^faraday1832] Within weeks he had reproduced the effect by thrusting a bar magnet into a helix — the experiment the microsim reconstructs — and by October had built the first [[Electric_generator|generator]], a copper disc rotating between the poles of a magnet, which delivered a continuous current rather than a transient.[^faraday1832] Faraday described his results in terms of lines of force cut by a moving conductor, a picture he preferred to action at a distance and which [[James_Clerk_Maxwell|Maxwell]] later credited as the seed of field theory. Joseph Henry, working in Albany, had observed induction independently and arguably earlier, but published after Faraday; the unit of inductance carries his name.[^henry1832] The missing piece — which way the induced current flows — was supplied in 1834 by Emil Lenz, who stated that the current is always directed so as to oppose the change that caused it.[^lenz1834] Lenz's rule is not an independent law but a statement of [[Conservation_of_energy|energy conservation]]: were the induced current to reinforce the change, pushing the magnet in would generate a force pulling it further in, and the apparatus would supply energy from nothing. ## Flux rule The magnetic flux through a surface bounded by the circuit is the integral of the normal component of B over that surface, `Phi = ∫ B·dA`, measured in webers. Faraday's law in its circuit form says the induced electromotive force is `EMF = −N·dPhi/dt` for a coil of N identical turns.[^up2-ch13] Three separate things can change Φ: the strength of B, the area of the circuit, or the angle between them. The law does not care which. This is the equation the microsim computes. With the magnet treated as a point dipole of moment m on the axis of a loop of radius a, the flux through one turn at separation z has the closed form `Phi = mu0·m·a^2/(2·(a^2 + z^2)^(3/2))`, which follows from the reciprocity of mutual inductance and needs no numerical field integral.[^dipole-flux] Differentiating and applying the chain rule gives the EMF directly in terms of the magnet's speed: `EMF = −N·v·dPhi/dz`. Because dΦ/dz vanishes at z = 0, the induced voltage is zero at the instant of maximum flux, and it is largest at z = ±a/2 (derived), where the flux is changing fastest. The numbers are small but easily measured. Take a magnet of moment 1 A·m², a coil of radius 25 mm and N = 200 turns. Peak flux through one turn is μ₀·m/(2·a) = 2.5×10⁻⁵ Wb, and at a speed of 0.5 m/s the peak EMF is about 86 mV (derived) — enough to drive a multimeter, not enough to light anything. Doubling the turns doubles the voltage, doubling the speed doubles it again, and holding the magnet still anywhere along the axis gives exactly zero. Those four facts are the whole content of the law, and they are the four things the sliders let the reader check. ### Motional emf When the flux changes because the conductor moves, the mechanism is the [[Lorentz_force|magnetic force]] on the charge carriers inside it. A carrier of charge q moving with velocity v through a field B feels a force q·v×B along the wire, which drives it toward one end and leaves the other positive. For a straight rod of length L moving perpendicular to a uniform field, the resulting EMF is `EMF = B·L·v`.[^up2-ch13] A 0.2 m rod crossing a 0.5 T field at 2 m/s develops 0.2 V (derived). Motional EMF is the operating principle of every rotating machine. In a generator the conductors are on a rotor and the flux linkage varies sinusoidally with angle, producing the [[Alternating_current|alternating]] output that a [[Steam_turbine|steam turbine]] or a [[Wind_turbine|wind turbine]] ultimately delivers to the [[Electrical_grid|grid]]; run the same machine backwards and it is an [[Electric_motor|electric motor]]. Faraday's rotating copper disc is the limiting case in which the "circuit" is a continuous sheet rather than a wire, and it is also the standard counterexample discussed below. ### Transformer emf The flux can also change with the circuit held rigidly still, because B itself varies in time. This is what happened in Faraday's iron ring, and it is the basis of the [[Transformer|transformer]]: a primary winding carrying alternating current produces a time-varying flux in a shared iron core, and the secondary sees that same flux through its own N₂ turns, so the voltages stand in the ratio of the turn counts.[^up2-ch13] Because the ratio can be made almost anything, power can be raised to hundreds of kilovolts for [[Electric_power_transmission|transmission]] and stepped down again at the point of use, which is why the flux rule is the reason grids are alternating-current systems at all. Transformer EMF needs no moving parts and no relative motion, and this is what makes the two cases feel like different phenomena even though one equation covers both. In the microsim the reader only ever produces the motional case; the transformer case would be a second instance of the same block with a stationary magnet whose strength is modulated instead. ### Direction of the induced current The minus sign in `EMF = −N·dPhi/dt` is [[Lenz's_law|Lenz's law]] in algebraic form: the induced current flows in the sense whose own [[Magnetic_field|magnetic field]] opposes the change in flux that produced it.[^lenz1834] Pushing a north pole toward a coil induces a current that makes the near face of the coil a north pole, repelling the magnet; pulling it away induces the opposite current, and the coil attracts it. Either way the magnet is resisted, and the mechanical [[Work_(physics)|work]] done against that resistance is exactly the electrical energy delivered to the circuit. In the sim the direction control makes this concrete: reversing the magnet's travel mirrors the EMF trace about the time axis without changing its shape. The same physics is the basis of eddy-current braking, in which a conductor moving through a field has currents induced in its bulk whose dissipation as [[Joule_heating|resistive heating]] removes kinetic energy without any contact, and of the damping a magnet feels when dropped down a copper tube. ## Maxwell–Faraday equation Faraday's rule refers to a circuit. The field-theoretic version refers only to points in space: a time-varying magnetic field is accompanied by a circulating electric field, whether or not any wire is present. In differential form this is `curl E = −∂B/∂t`, and in the equivalent integral form `∮ E·dl = −d/dt ∫ B·dA` taken over any fixed surface and its boundary.[^up2-ch16][^maxwell1865] The shift is not cosmetic. It says the [[Electric_field|electric field]] induced by changing magnetism is not conservative — its line integral around a closed loop is not zero — so it has no potential and the notion of "voltage between two points" becomes path-dependent inside an induction region. It also decouples the effect from matter entirely: the circulating E field exists in vacuum, and it is that fact, joined to [[Maxwell's_equations|Maxwell's]] displacement-current term, which allows a self-sustaining [[Electromagnetic_radiation|electromagnetic wave]] to propagate with no charges anywhere. The Maxwell–Faraday equation is the third of Maxwell's four, and it is the one in which Faraday's laboratory result survives intact into modern physics. Where the flux rule needs a circuit to be well defined, this form never does, which is why it — and not the flux rule — is taken as fundamental. ## Derivation of the flux rule from microscopic equations The flux through a moving, deforming circuit can change for two reasons at once, and the total time derivative separates them. Differentiating `Phi = ∫ B·dA` over a surface whose boundary moves with velocity v gives one term in ∂B/∂t, from the field changing where the circuit is, and a second term in the flux swept by the moving boundary, which reduces to a line integral of v×B around the loop. Each term comes from a different piece of physics. The first is supplied by the Maxwell–Faraday equation, which converts the surface integral of ∂B/∂t into the circulation of the induced electric field. The second is supplied by the [[Lorentz_force|Lorentz force]] law, which says a carrier moving with the wire experiences q·v×B and therefore an effective field v×B along it. Adding them reproduces `EMF = −dPhi/dt` exactly.[^up2-ch13] The flux rule is therefore a theorem rather than a postulate, and — as Richard Feynman emphasised in his lectures — it is a single simple statement covering two logically independent effects, a coincidence with no obvious reason behind it in the non-relativistic formulation.[^feynman2] The relativistic account below removes the coincidence. ## Limitations of the flux rule Because it is a derived statement about a material circuit, the flux rule fails whenever "the circuit" is not well defined. The standard counterexample is Faraday's own homopolar disc: a conducting disc spinning in an axial field with brushes at the axle and the rim. A current flows, and an EMF is measurable, yet no reasonable choice of circuit has a flux through it that is changing — the geometry and the field are both static. The microscopic account handles the case without difficulty, because the v×B force on carriers inside the rotating metal is perfectly well defined.[^up2-ch13] Related failures arise with sliding contacts, with circuits whose topology changes as a switch closes, and with conductors in which the current path is set by the geometry of the material rather than by a wire; Feynman's lectures collect several such cases under the heading of exceptions to the flux rule.[^feynman2] In all of them the diagnosis is the same: the flux rule is a bookkeeping device for a loop of identifiable material, and it stops applying when that loop stops being identifiable. Nothing goes wrong with the Maxwell–Faraday equation or the Lorentz force in any of these cases, which is the practical argument for treating those two as the physics and the flux rule as a convenience. ## Flux rule and relativity The split between motional and transformer EMF depends on who is watching. A magnet moved toward a stationary coil and a coil moved toward a stationary magnet are the same relative motion, and produce the same current, but the first is described as a changing B at the wire and the second as a v×B force on the carriers. [[Albert_Einstein|Einstein]] opened his 1905 paper on the electrodynamics of moving bodies with exactly this asymmetry, noting that the observable phenomenon depends only on the relative motion while the customary explanation draws a sharp distinction between the two cases.[^einstein1905] [[Special_relativity|Special relativity]] dissolves the distinction. The electric and magnetic fields are components of a single object that mix under a change of frame: what one observer calls a pure [[Magnetic_field|magnetic field]] another, moving relative to the first, sees as a combination of magnetic and [[Electric_field|electric]] fields.[^einstein1905] The v×B term of one frame is literally the induced E field of another, and the "coincidence" noted in the previous section becomes a requirement of [[Lorentz_transformation|Lorentz]] covariance. For the sim this is more than a footnote. The reader can push the magnet toward the coil or, conceptually, the coil toward the magnet; the traces are identical, and the equation is indifferent. That indifference is not an accident of the model — it is the experimental fact that motivated relativity in the first place. ## See also - [[Electromagnetic_induction]] - [[Lenz's_law]] - [[Magnetic_flux]] - [[Maxwell's_equations]] - [[Transformer]] - [[Electric_generator]] - [[Lorentz_force]] - [[Michael_Faraday]] ## Notes Two conventions matter for reading the equations above. First, *electromotive force* is not a force: it is the work done per unit charge around the loop, measured in volts, and the name is an eighteenth-century survival. Second, the sign of the EMF depends on an arbitrary choice of which way round the boundary is traversed and which way the surface normal points; the two choices are linked by a right-hand rule, and Lenz's law is the statement that, once they are made consistently, the sign is negative. Reversing both conventions leaves the physics unchanged. Full citations for every claim are collected under References. ## References [^up2-ch13]: Sanny, Jeff; Ling, Samuel, et al. (2016). *University Physics Volume 2*. OpenStax, CC BY. Chapter 13, "Electromagnetic Induction", pp. 557–602 (page to pin) — Faraday's law in circuit form, motional EMF `EMF = B·L·v`, the transformer case, Lenz's law, eddy currents and the homopolar-disc counterexample. Portal Book 078, https://open.umn.edu/opentextbooks/textbooks/university-physics-volume-2. The text extraction of Portal Book 078 lost nearly every displayed equation; the forms quoted on this page are the standard, unambiguous ones and should be verified against the PDF pages before the page numbers are frozen. [^up2-ch16]: *University Physics Volume 2* (2016), Chapter 16, "Electromagnetic Waves", pp. 669–710 (page to pin), for the Maxwell–Faraday equation in the set of four and its role in wave propagation. Portal Book 078. Same extraction caveat as above: the differential and integral forms quoted here are standard forms supplied, not transcriptions. [^dipole-flux]: The on-axis closed form `Phi = mu0·m·a^2/(2·(a^2 + z^2)^(3/2))` for the flux of a point dipole of moment m through a coaxial circular loop of radius a follows from the reciprocity of mutual inductance applied to the on-axis field of a current loop, `B = mu0·I·a^2/(2·(a^2 + z^2)^(3/2))`, given in Portal Book 078, Chapter 12, "Sources of Magnetic Fields", pp. 513–556 (page to pin). It is the form implemented as `wt-em.induction.dipoleFluxThroughLoop`. The peak flux 2.5×10⁻⁵ Wb, the extremum of dΦ/dz at z = ±a/2, the 86 mV peak EMF for m = 1 A·m², a = 25 mm, N = 200 and v = 0.5 m/s, and the 0.2 V motional example are all derived here from these expressions and are not printed in the book. [^oersted1820]: Ørsted, Hans Christian (1820). *Experimenta circa effectum conflictus electrici in acum magneticam*. Copenhagen. The pamphlet announcing that a current-carrying wire deflects a magnetic needle, circulated to European scientific societies in July 1820. [^faraday1832]: Faraday, Michael (1832). "Experimental Researches in Electricity." *Philosophical Transactions of the Royal Society of London* 122: 125–162. The first series, reporting the induction-ring experiment of August 29, 1831, the induction of a current by a magnet thrust into a helix, and the rotating copper disc. [^henry1832]: Henry, Joseph (1832). "On the Production of Currents and Sparks of Electricity from Magnetism." *The American Journal of Science and Arts*, vol. 22 (page range to pin). Henry's independent observation of induction and self-induction, published after Faraday's announcement. [^lenz1834]: Lenz, Emil (1834). "Ueber die Bestimmung der Richtung der durch elektrodynamische Vertheilung erregten galvanischen Ströme." *Annalen der Physik und Chemie* (volume and page range to pin). The rule in modern statement, with worked cases, is in Portal Book 078, Chapter 13, pp. 557–602 (page to pin). [^maxwell1865]: Maxwell, James Clerk (1865). "A Dynamical Theory of the Electromagnetic Field." *Philosophical Transactions of the Royal Society of London* 155: 459–512. [^feynman2]: Feynman, Richard P.; Leighton, Robert B.; Sands, Matthew (1964). *The Feynman Lectures on Physics*, Volume II, Chapter 17, "The Laws of Induction" (section and page to pin) — the observation that the flux rule joins two physically distinct effects under one statement, and the collected exceptions to it, including the homopolar disc and rocking-plate contacts. [^einstein1905]: Einstein, Albert (1905). "Zur Elektrodynamik bewegter Körper." *Annalen der Physik* 322 (10): 891–921. The opening paragraph takes the magnet-and-conductor asymmetry as its motivating example; §6 gives the transformation of the electric and magnetic fields between frames. ## Further reading - Sanny, Jeff; Ling, Samuel, et al., *University Physics Volume 2* (2016), Chapter 13 "Electromagnetic Induction" and Chapter 16 "Electromagnetic Waves" — the standard undergraduate treatment of the flux rule, motional and transformer EMF, Lenz's law and the Maxwell–Faraday equation. Portal Book 078, CC BY. - Sanny, Ling, et al., *University Physics Volume 2* (2016), Chapter 12 "Sources of Magnetic Fields", pp. 513–556 — the on-axis loop field from which the sim's closed-form flux is obtained by reciprocity. Portal Book 078. - Faraday, Michael, *Experimental Researches in Electricity* (from 1832) — the primary record, readable without mathematics because Faraday used none. - Maxwell, James Clerk, "A Dynamical Theory of the Electromagnetic Field" (1865) — Faraday's lines of force turned into field equations. ## External links - [*University Physics Volume 2*](https://openstax.org/books/university-physics-volume-2) — OpenStax, CC BY; Chapter 13 is the induction chapter this page follows (Portal Book 078) - Further archival and museum sources are listed in the Wikipedia pair's *External links*; none is reproduced here until its URL has been checked. <!-- MATTERSIM:BEGIN g33 — Matter & Energy Cluster microsim (framework build, specs/sims/Faraday's_law_of_induction.json); do not hand-edit inside --> **Microsim — three.js (Wikitube framework):** *Faraday's law of induction* <div class="wt-sim" data-src="https://wikitube-3d-microsims.netlify.app/matter/Faraday's_law_of_induction.html" data-title="Faraday's law of induction"></div> *Built from `MICROSIM_GUIDE/specs/sims/Faraday's_law_of_induction.json`; part of the [[PORTAL_Matter|Matter portal]] spine (section sims and See-also variants).* <!-- MATTERSIM:END --> ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Faraday's_law_of_induction) : [Wikitube](https://en.wikitube.io/wiki/Faraday's_law_of_induction) · pinned revision [1367211175](https://en.wikipedia.org/w/index.php?oldid=1367211175) · 2026-09-11 ## Previous hub tags Hubs: `Life_Physics`. 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