Faraday's law of induction#magnetic flow meter

In 1820, Hans Christian Ørsted demonstrated that an electric current produces a magnetic field, showing that a compass needle could be deflected by a nearby current-carrying wire. This discovery prompted scientists to ask whether the reverse was also true—whether a magnetic field could generate an electric current.{{Sfn|Darrigol|2000|pp=3-6, 32}}

Initial experiments revealed that a static magnetic field had no effect on a nearby circuit: simply placing a magnet near a wire loop produced no current.{{Sfn|Darrigol|2000|pp=3-6, 32}} The breakthrough came in 1831, when Michael Faraday demonstrated that a changing magnetic field could indeed induce an electric current in a circuit. Independently, Joseph Henry made similar observations in 1832,{{Cite web |title=A Brief History of Electromagnetism |url=http://web.hep.uiuc.edu/home/serrede/P435/Lecture_Notes/A_Brief_History_of_Electromagnetism.pdf}} though Faraday was the first to publish his findings.{{cite book |last=Ulaby |first=Fawwaz |url=https://www.amazon.com/exec/obidos/tg/detail/-/0132413264/ref=ord_cart_shr?%5Fencoding=UTF8&m=ATVPDKIKX0DER&v=glance |title=Fundamentals of applied electromagnetics |publisher=Pearson:Prentice Hall |year=2007 |isbn=978-0-13-241326-8 |edition=5th |page=255}}{{cite web |title=Joseph Henry |url=http://www.nasonline.org/member-directory/deceased-members/20001467.html |access-date=2016-12-30 |work=Member Directory, National Academy of Sciences}}

File:Faraday emf experiment.svg

Faraday's notebook on August 29, 1831{{Cite web |last=Faraday |first=Michael |date=1831-08-29 |title=Faraday's notebooks: Electromagnetic Induction |url=https://www.rigb.org/docs/faraday_notebooks__induction_0.pdf |url-status=dead |archive-url=https://web.archive.org/web/20210830003053/https://www.rigb.org/docs/faraday_notebooks__induction_0.pdf |archive-date=2021-08-30 |access-date= |website=The Royal Institution of Great Britain}} describes an experimental demonstration of induction.{{cite book|last1=Faraday|first1=Michael|last2=Day|first2=P.|title=The philosopher's tree: a selection of Michael Faraday's writings|url=https://books.google.com/books?id=ur6iKVmzYhcC&pg=PA71|access-date=28 August 2011|date=1999-02-01|publisher=CRC Press|isbn=978-0-7503-0570-9|page=71}} He wrapped two coils of wire around opposite sides of an iron ring, forming a primitive toroidal transformer. When he connected one coil to a battery, he observed a brief deflection in a galvanometer attached to the second coil. He concluded that a changing current in the first coil created a changing magnetic field in the ring, which in turn induced a current in the second coil. He described this as a "wave of electricity" propagated through the iron.{{cite book |last=Williams |first=L. Pearce |url=https://archive.org/details/michaelfaradaybi00will |title=Michael Faraday |publisher=New York, Basic Books |year=1965 |url-access=registration}}{{full citation needed|date=September 2018}}{{rp|182–183}}

Over the following months, Faraday discovered other manifestations of electromagnetic induction. He observed transient currents when a bar magnet was rapidly moved into or out of a coil of wire. He also built a device, now known as Faraday's disk or homopolar generator, that produced a steady (DC) current by rotating a copper disk in the presence of a stationary magnet, using a sliding electrical contact.{{rp|191–195}}

File:Faraday disk generator.jpg, a type of homopolar generator|left]]

Faraday explained these phenomena using the concept of lines of force. However, his theoretical ideas were met with skepticism, as they were not formulated mathematically.{{rp|510}} James Clerk Maxwell later gave Faraday's insights mathematical expression, incorporating them into his broader electromagnetic theory in the early 1860s.{{rp|510}}{{cite book|last=Clerk Maxwell |first=James |date=1904 |title=A Treatise on Electricity and Magnetism |volume=2 |edition=3rd |publisher=Oxford University Press |pages=178–179, 189}}{{cite web|url=http://www.theiet.org/resources/library/archives/biographies/faraday.cfm |title=Archives Biographies: Michael Faraday |publisher=The Institution of Engineering and Technology}}

In Maxwell's papers, the time-varying aspect of electromagnetic induction is expressed as a differential equation which Oliver Heaviside referred to as Faraday's law even though it is different from the original version of Faraday's law, and does not describe motional emf. Heaviside's version is the form recognized today in the group of equations known as Maxwell's equations.

Lenz's law, formulated by Emil Lenz in 1834,{{cite journal|last=Lenz |first=Emil |date=1834 |url=http://gallica.bnf.fr/ark:/12148/bpt6k151161/f499.image.r=lenz.langEN |title=Ueber die Bestimmung der Richtung der durch elektodynamische Vertheilung erregten galvanischen Ströme |journal=Annalen der Physik und Chemie |volume=107 |issue=31 |pages=483–494|bibcode=1834AnP...107..483L |doi=10.1002/andp.18341073103 }}
A partial translation of the paper is available in {{cite book|last=Magie |first=W. M. |date=1963 |title=A Source Book in Physics |publisher=Harvard Press |location=Cambridge, MA |pages=511–513}} describes "flux through the circuit", and gives the direction of the induced emf and current resulting from electromagnetic induction (elaborated upon in the examples below).

The laws of induction of electric currents in mathematical form were established by Franz Ernst Neumann in 1845.{{cite journal |last=Neumann |first=Franz Ernst |year=1846 |title=Allgemeine Gesetze der inducirten elektrischen Ströme |url=https://isidore.co/misc/Physics%20papers%20and%20books/Zotero/storage/3UM3CRQ2/18461430103_ftp.pdf |journal=Annalen der Physik |volume=143 |pages=31–44 |bibcode=1846AnP...143...31N |doi=10.1002/andp.18461430103 |archive-url=https://web.archive.org/web/20200312012028/https://isidore.co/misc/Physics%20papers%20and%20books/Zotero/storage/3UM3CRQ2/18461430103_ftp.pdf |archive-date=12 March 2020 |number=1}}{{Non-primary source needed|date=June 2025}}

According to Albert Einstein, much of the groundwork and discovery of his special relativity theory was presented by this law of induction by Faraday in 1834.{{cite news |last=Siegel |first=Ethan |authorlink=Ethan Siegel |title=Relativity Wasn't Einstein's Miracle; It Was Waiting In Plain Sight For 71 Years |url=https://www.forbes.com/sites/startswithabang/2019/03/01/relativity-wasnt-einsteins-miracle-it-was-waiting-in-plain-sight-for-71-years/ |date=1 March 2019 |work=Forbes |url-status=live |archiveurl=https://archive.today/20230703120353/https://www.forbes.com/sites/startswithabang/2019/03/01/relativity-wasnt-einsteins-miracle-it-was-waiting-in-plain-sight-for-71-years/?sh=5d837e5b644c |archivedate=3 July 2023 |accessdate=3 July 2023 }}{{cite news |last=Siegel |first=Ethan |authorlink=Ethan Siegel |title=71 years earlier, this scientist beat Einstein to relativity - Michael Faraday's 1834 law of induction was the key experiment behind the eventual discovery of relativity. Einstein admitted it himself. |url=https://bigthink.com/starts-with-a-bang/scientist-beat-einstein-relativity/ |date=28 June 2023 |work=Big Think |url-status=live |archiveurl=https://archive.today/20230628183501/https://bigthink.com/starts-with-a-bang/scientist-beat-einstein-relativity/ |archivedate=28 June 2023 |accessdate=3 July 2023 }}

{{clear}}

File:Electromagnetic_induction_-_solenoid_to_loop_-_animation.gif

Faraday's law of induction, also known as the flux rule, flux law, and Faraday{{endash}}Lenz law,{{cite book | last=Fujimoto | first=Minoru | title=Physics of Classical Electromagnetism | publisher=Springer Science & Business Media | publication-place=New York | date=2007-09-06 | isbn=978-0-387-68018-7 | page=105}} states that the electromotive force (emf) around a closed circuit is equal to the negative rate of change of the magnetic flux through the circuit. This rule holds for any circuit made of thin wire and accounts for changes in flux due to variations in the magnetic field, movement of the circuit, or deformation of its shape.{{cite book | last1=Landau| first1=Lev Davidovich | last2=Lifshitz | first2=Evgeniĭ Mikhaĭlovich | last3=Pitaevskiĭ | first3=Lev Petrovich | title=Electrodynamics of Continuous Media | publisher=Pergamon press | publication-place=Oxford | date=1984 | isbn=0-08-030276-9 | page=219}} The direction of the induced emf is given by Lenz's law, which states that the induced current will flow in such a way that its magnetic field opposes the change in the original magnetic flux.{{sfn|Griffiths|2023|p=319}}

Mathematically, in SI units, the law is expressed as

$$\backslash mathcal\{E\}\; =\; -\backslash frac\{\backslash mathrm\{d\}\backslash Phi\_B\}\{\backslash mathrm\{d\}t\},$$

where $\backslash mathcal\{E\}$ is the electromotive force (emf) and {{math|Φ_B}} is the magnetic flux throught the circuit. The magnetic flux is defined as the surface integral of the magnetic field {{math|B}} over a time-dependent surface {{math|Σ(t)}}, whose boundary is the wire loop:

$$\backslash Phi\_B\; =\; \backslash iint\_\{\backslash Sigma(t)\}\; \backslash mathbf\{B\}(t)\; \backslash cdot\; \backslash mathrm\{d\}\; \backslash mathbf\{A\}\backslash ,\; ,$$

where {{math|dA}} is an infinitesimal area vector normal to the surface. The dot product {{math|B · dA}} represents the flux through the differential area element.

In more visual terms, the magnetic flux is proportional to the number of magnetic field lines passing through the loop.

When the flux changes, an emf is induced around the loop. This emf corresponds to the energy per unit charge required to move it once around the loop.{{sfn|Feynman|Leighton|Sands|2006|loc=Ch. 17}}{{sfn|Griffiths|2023|pp=304–306}}{{cite book |last1=Tipler|last2=Mosca |title=Physics for Scientists and Engineers |year=2004|page=795|publisher=Macmillan |isbn=9780716708100 |url=https://books.google.com/books?id=R2Nuh3Ux1AwC&pg=PA795}} In a simple circuit with resistance $R$, an emf $\backslash mathcal\; E$ gives rise to a current $I$ according to the Ohm's law $\backslash mathcal\; E\; =\; IR$.{{sfn|Zangwill|2013|pp=462-464}} Equivalently, if the loop is broken to form an open circuit and a voltmeter is connected across the terminals, the emf is equal to the voltage measured across the open ends.{{cite book | last1=Paul | first1=Clayton R. | last2=Scully | first2=Robert C. | last3=Steffka | first3=Mark A. | title=Introduction to Electromagnetic Compatibility | publisher=John Wiley & Sons | date=2022-11-01 | isbn=978-1-119-40434-7 | page=703}}

For a tightly wound coil of wire, composed of {{mvar|N}} identical turns, the same magnetic field lines cross the surface {{mvar|N}} times. In this case, Faraday's law of induction states that{{cite book |last1=Whelan |first1=P. M. |title=Essential Principles of Physics |last2=Hodgeson |first2=M. J. |date=1978 |publisher=John Murray |isbn=0-7195-3382-1 |edition=2nd}}{{cite web |last=Nave |first=Carl R. |title=Faraday's Law |url=http://hyperphysics.phy-astr.gsu.edu/hbase/electric/farlaw.html |access-date=2011-08-29 |work=HyperPhysics |publisher=Georgia State University}}

$$\backslash mathcal\{E\}\; =\; -N\; \backslash frac\{\backslash mathrm\{d\}\backslash Phi\_B\}\{\backslash mathrm\{d\}t\}$$

where {{mvar|N}} is the number of turns of wire and {{math|Φ_B}} is the magnetic flux through a single loop. The product {{math|NΦ_B}} is known as linked flux.{{Cite web |title=121-11-77: "linked flux" |url=https://www.electropedia.org/iev/iev.nsf/display?openform&ievref=121-11-77 |access-date=2025-06-20 |website=IEC 60050 - International Electrotechnical Vocabulary}}

The flux can change either because the loop moves or deforms over time, or because the field itself varies in time. These two possibilities correspond to the two mechanisms described by the flux rule:{{sfn|Griffiths|2023|pp=298-319}}

• Motional emf: The circuit moves through a static but non-uniform magnetic field.
• Transformer emf: The circuit remains stationary while the magnetic field changes over time.

File:Lorentz_force_on_moving_conductor.svg

The basic mechanism behind motional emf is illustrated by a conducting rod moving through a magnetic field that is perpendicular to both the rod and its direction of motion. Due to movement in magnetic field, the mobile electrons of the conductor experience the magnetic component ({{math|qv × B}}) of the Lorentz force that drives them along the length of the rod. This leads to a separation of charge between the two ends of the rod. In the steady state, the electric field from the accumulated charge balances the magnetic force.{{sfn|Griffiths|2023|p=307}}

File:Faraday's_law_of_induction.svg

If the rod is part of a closed conducting loop moving through a nonuniform magnetic field, the same effect can drive a current around the circuit. For instance, suppose the magnetic field is confined to a limited region of space, and the loop initially lies outside this region. As it moves into the field, the area of the loop that encloses magnetic flux increases, and an emf is induced. From the Lorentz force perspective, this is because the field exerts a magnetic force on charge carriers in the parts of the loop entering the region. Once the entire loop lies in a uniform magnetic field and continues at constant speed, the total enclosed flux remains constant, and the emf vanishes. In this situation, magnetic forces on opposite sides of the loop cancel out.

A complementary case is transformer emf, which occurs when the conducting loop remains stationary but the magnetic flux through it changes due to a time-varying magnetic field. This can happen in two ways: either the source of the magnetic field moves, altering the field distribution through the fixed loop, or the strength of the magnetic field changes over time at a fixed location, as in the case of a powered electromagnet.

In either situation, no magnetic force acts on the charges, and the emf is entirely due to the electric component ({{math|qE}}) of the Lorentz force. According to the Maxwell–Faraday equation, a time-varying magnetic field produces a circulating electric field, which drives current in the loop. This phenomenon underlies the operation of electrical machines such as synchronous generators.{{sfn|Sadiku|2018|pp=424-427}} The electric field induced in this way is non-conservative, meaning its line integral around a closed loop is not zero.{{sfn|Purcell|Morin|2013|p=259}}{{sfn|Sadiku|2018|pp=424-425}}{{sfn|Griffiths|2023|p=317}}

File:Salu's left-hand rule (magnetic induction).png |volume=52 |issue=1 |pages=48 |bibcode=2014PhTea..52...48S |doi=10.1119/1.4849156}} [https://www.youtube.com/watch?v=ipUD9VcAd9o Video Explanation]|alt=]]

It is possible to find out the direction of the electromotive force (emf) directly from Faraday's law, without invoking Lenz's law. A left hand rule helps doing that, as follows:{{cite web |last1=Salu |first1=Yehuda |date=17 January 2017 |title=Bypassing Lenz's Rule - A Left Hand Rule for Faraday's Law |url=http://Physicsforarchitects.com/bypassing-lenzs-rule |archive-url=https://web.archive.org/web/20200507170609/http://physicsforarchitects.com/bypassing-lenzs-rule |archive-date=7 May 2020 |access-date=30 July 2017 |website=www.PhysicsForArchitects.com}}

• Align the curved fingers of the left hand with the loop (yellow line).
• Stretch your thumb. The stretched thumb indicates the direction of {{math|n}} (brown), the normal to the area enclosed by the loop.
• Find the sign of {{math|ΔΦ_B}}, the change in flux. Determine the initial and final fluxes (whose difference is {{math|ΔΦ_B}}) with respect to the normal {{math|n}}, as indicated by the stretched thumb.
• If the change in flux, {{math|ΔΦ_B}}, is positive, the curved fingers show the direction of the electromotive force (yellow arrowheads).
• If {{math|ΔΦ_B}} is negative, the direction of the electromotive force is opposite to the direction of the curved fingers (opposite to the yellow arrowheads).

{{anchor|Faraday equation}}

Image:Stokes' Theorem.svg]]

The Maxwell–Faraday equation is one of the four Maxwell's equations, and therefore plays a fundamental role in the theory of classical electromagnetism. It states that a time-varying magnetic field always accompanies a spatially varying, non-conservative electric field. In differential form and in SI units, it reads:

{{Equation box 1

|indent =:

|equation = $\backslash nabla\; \backslash times\; \backslash mathbf\{E\}\; =\; -\backslash frac\{\backslash partial\; \backslash mathbf\{B\}\}\; \{\backslash partial\; t\}$

|cellpadding

|border

|border colour = #50C878

|background colour = #ECFCF4}}

where {{math|∇ ×}} is the curl operator, {{math|E(r, t)}} is the electric field and {{math|B(r, t)}} is the magnetic field. These fields can generally be functions of position {{math|r}} and time {{mvar|t}}.{{cite book

|last = Griffiths

|first = David J.

|title = Introduction to Electrodynamics

|publisher = Cambridge University Press

|series = 4

|year = 2017

|isbn = 978-1-108-42041-9

|url = https://www.worldcat.org/oclc/965197645

|edition = Fourth

| oclc = 965197645

}}

It can also be written in an integral form by the Kelvin–Stokes theorem:{{cite book|first=Roger F.| last=Harrington|title=Introduction to electromagnetic engineering |year=2003 |page=56 |publisher=Dover Publications |location=Mineola, NY |isbn=0-486-43241-6 |url=https://books.google.com/books?id=ZlC2EV8zvX8C&q=%22faraday%27s+law+of+induction%22&pg=PA57}}

{{Equation box 1

|indent =:

|equation = $\backslash oint\_\{\backslash partial\; \backslash Sigma\}\; \backslash mathbf\{E\}\; \backslash cdot\; \backslash mathrm\{d\}\backslash mathbf\{l\}\; =\; -\; \backslash int\_\backslash Sigma\; \backslash frac\{\backslash partial\; \backslash mathbf\{B\}\}\{\backslash partial\; t\}\; \backslash cdot\; \backslash mathrm\{d\}\backslash mathbf\{A\}$

|cellpadding

|border

|border colour = #50C878

|background colour = #ECFCF4}}

where, as indicated in the figure, {{math|Σ}} is a surface bounded by the closed loop {{math|∂Σ}} and {{math|dl}} is an infinitesimal vector element along that loop. The vector area element {{math|dA}} is perpendicular to the surface and oriented according to the right-hand rule: when the thumb points in the direction of the {{math|dA}}, the curled fingers indicate the direction of {{math|dl}} along the boundary.

The left-hand side of the equation represents the circulation of the electric field around the loop {{math|∂Σ}}.{{Cite web| last=Feynman|first=Richard P. |title=The Feynman Lectures on Physics Vol. II |url=https://feynmanlectures.caltech.edu/II_toc.html|access-date=2020-11-07 |website=feynmanlectures.caltech.edu}}{{Rp|ch3}} For static electric fields, the circulation is zero, since the field can be expressed as the gradient of a scalar potential. In contrast, a time-varying magnetic field produces a non-conservative electric field with nonzero circulation. When such field acts on a conducting loop, it drives a current around the loop.

If the surface {{math|Σ}} is not changing in time, the right-hand side equation becomes the time-derivative of the magnetic flux {{math|Φ_B}} through the surface:

$$\backslash oint\_\{\backslash partial\; \backslash Sigma\}\; \backslash mathbf\{E\}\; \backslash cdot\; \backslash mathrm\{d\}\backslash mathbf\{l\}\; =\; -\; \backslash frac\{\backslash mathrm\{d\}\}\{\backslash mathrm\{d\}t\}\; \backslash int\_\{\backslash Sigma\}\; \backslash mathbf\{B\}\; \backslash cdot\; \backslash mathrm\{d\}\backslash mathbf\{A\}\; =\; -\backslash frac\{\backslash mathrm\{d\}\backslash Phi\_B\}\{\backslash mathrm\{d\}t\}.$$If the left-hand side is identified with the work per unit charge done by the electric field on charges in a fixed conducting loop, this equation reproduces the flux rule in the special case of a stationary circuit.

In the non-relativistic limit, the solenoidal component of the induced electric field can be approximated by the volume integral{{Rp|321}}

$$\backslash mathbf\; E\_s\; (\backslash mathbf\; r,t)\; \backslash approx\; -\backslash frac\{1\}\{4\backslash pi\}\backslash iiint\_V\; \backslash \; \backslash frac\{\backslash left(\backslash frac\{\backslash partial\; \backslash mathbf\{B\}(\backslash mathbf\{r\}\text{'},t)\}\{\backslash partial\; t\}\; \backslash right)\; \backslash times\; \backslash left(\backslash mathbf\{r\}-\backslash mathbf\{r\}\text{'}\; \backslash right)\; \}\{|\backslash mathbf\; \{r\}\; -\; \backslash mathbf\{r\}\text{'}|^3\}\; d^3\backslash mathbf\{r\text{'}\}$$This expression shows how changes in the magnetic field across space contribute to the induced electric field at a given point, with each contribution weighted by the inverse square of the distance.

The four Maxwell's equations, together with the Lorentz force law, form a complete foundation for classical electromagnetism.{{cite book|last=Griffiths|first=David J. | title=Introduction to Electrodynamics | url=https://archive.org/details/introductiontoel00grif_0/page/301 | edition=3rd |pages=[https://archive.org/details/introductiontoel00grif_0/page/301 301–303] | publisher=Prentice Hall| year=1999 | location=Upper Saddle River, NJ | isbn=0-13-805326-X}} From these, Faraday's law can be derived directly.{{Cite journal |last1=Davison |first1=M. E. |year=1973 |title=A Simple Proof that the Lorentz Force, Law Implied Faraday's Law of Induction, when B is Time Independent |journal=American Journal of Physics |volume=41 |issue=5 |page=713 |bibcode=1973AmJPh..41..713D |doi=10.1119/1.1987339}}{{cite book |last1=Krey |url=https://books.google.com/books?id=xZ_QelBmkxYC&pg=PA155 |title=Basic Theoretical Physics: A Concise Overview |last2=Owen |date=14 August 2007 |publisher=Springer |isbn=9783540368052 |page=155}}{{cite book|title=Theoretische Elektrotechnik|last=Simonyi|first=K.|date=1973|publisher=VEB Deutscher Verlag der Wissenschaften|edition=5th|location=Berlin|at=eq. 20, p. 47}}

The derivation begins by considering the time derivative of the magnetic flux through a surface {{math|Σ(t)}} that may vary with time:

$$\backslash frac\{\backslash mathrm\{d\}\backslash Phi\_B\}\{\backslash mathrm\{d\}t\}\; =\; \backslash frac\{\backslash mathrm\{d\}\}\{\backslash mathrm\{d\}t\}\; \backslash int\_\{\backslash Sigma(t)\}\; \backslash mathbf\{B\}(t)\; \backslash cdot\; \backslash mathrm\{d\}\backslash mathbf\{A\}.$$

The magnetic flux can change for two reasons: the magnetic field itself may vary with time, and the surface may move or change shape, enclosing a different region of space. Both effects are captured by the three-dimensional version of the Leibniz integral rule, sometimes referred to as the "flux theorem":{{sfn|Zangwill|2013|p=10}}{{sfn|Zangwill|2013|pp=462–464}}

$$\backslash frac\{\backslash mathrm\{d\}\}\{\backslash mathrm\{d\}t\}\; \backslash int\_\{\backslash Sigma(t)\}\; \backslash mathbf\{B\}\; \backslash cdot\; \backslash mathrm\{d\}\backslash mathbf\{A\}\; =\; \backslash int\_\{\backslash Sigma(t)\}\; \backslash left(\backslash frac\{\backslash partial\; \backslash mathbf\{B\}\}\{\backslash partial\; t\}\; +\; (\backslash nabla\backslash cdot\backslash mathbf\; B)\backslash mathbf\{v\}\_c\; \backslash right)\backslash cdot\; \backslash mathrm\{d\}\backslash mathbf\{A\}\; -\; \backslash oint\_\{\backslash partial\; \backslash Sigma(t)\}\; (\backslash mathbf\{v\}\_c\; \backslash times\; \backslash mathbf\{B\})\; \backslash cdot\; \backslash mathrm\{d\}\backslash mathbf\{l\}$$

Here, {{math|∂Σ(t)}} is the moving boundary of the surface and $\backslash mathbf\{v\}\_c$ is the local velocity of the boundary at each point. By Gauss's law for magnetism ($\backslash nabla\backslash cdot\backslash mathbf\; B\; =\; 0$), the second term under the area integral vanishes. Applying the Maxwell–Faraday equation to the remaining term,

$$\backslash int\_\{\backslash Sigma(t)\}\; \backslash frac\{\backslash partial\; \backslash mathbf\{B\}\}\{\backslash partial\; t\}\; \backslash cdot\; \backslash mathrm\{d\}\backslash mathbf\{A\}\; =\; -\; \backslash oint\_\{\backslash partial\; \backslash Sigma(t)\}\; \backslash mathbf\{E\}\; \backslash cdot\; \backslash mathrm\{d\}\backslash mathbf\{l\},$$

and combining the two line integrals gives

$$\backslash frac\{\backslash mathrm\{d\}\backslash Phi\_B\}\{\backslash mathrm\{d\}t\}\; =\; -\; \backslash oint\_\{\backslash partial\; \backslash Sigma(t)\}\; \backslash left(\; \backslash mathbf\{E\}\; +\; \backslash mathbf\{v\}\_c\; \backslash times\; \backslash mathbf\{B\}\; \backslash right)\; \backslash cdot\; \backslash mathrm\{d\}\backslash mathbf\{l\}.$$

This is an exact result, derived from Maxwell's equations and vector calculus.{{sfn|Zangwill|2013|pp=462–464}}

However, the quantity inside the integral is not the full Lorentz force per unit charge, because the velocity $\backslash mathbf\{v\}\_c$ represents the motion of loop boundary, not the actual velocity of the charge carriers. To recover the physical electromotive force, we must distinguish between these velocities. Let us choose the integration path to coincide with the physical circuit. The velocity of a charge carrier in the conductor is then given by

:$\backslash mathbf\; v(\backslash mathbf\; r,\; t)\; =\; \backslash mathbf\; v\_c(\backslash mathbf\; r,\; t)\; +\; \backslash mathbf\; v\_d(\backslash mathbf\; r,\; t)$,

where $\backslash mathbf\; v\_c$ is the velocity of the conductor (the ions in the material), and $\backslash mathbf\; v\_d$ is the drift velocity of the electrons relative to the material. This decomposition assumes nonrelativistic (Galilean) addition of velocities.{{sfn|Zangwill|2013|pp=462–464}}

The emf $\backslash mathcal\{E\}$ associated with the Lorentz force is defined as

$$\backslash mathcal\{E\}\; =\; \backslash oint\_\{\backslash partial\; \backslash Sigma(t)\}\; \backslash left(\backslash mathbf\{E\}\; +\; \backslash mathbf\{v\}\backslash times\backslash mathbf\{B\}\backslash right)\; \backslash cdot\; \backslash mathrm\{d\}\backslash mathbf\{l\}.$$

Substituting the expression for the carrier velocity and the above result yields:{{Equation box 1|cellpadding|border|indent=:|equation=$\backslash mathcal\{E\}\; =\; -\backslash frac\{\backslash mathrm\{d\}\backslash Phi\_B\}\{\backslash mathrm\{d\}t\}\; +\; \backslash oint\_\{\backslash partial\; \backslash Sigma(t)\}\; \backslash left(\; \backslash mathbf\{v\}\_d\backslash times\backslash mathbf\{B\}\backslash right)\; \backslash cdot\; \backslash mathrm\{d\}\backslash mathbf\{l\}.$|border colour=#50C878|background colour=#ECFCF4}}

Equivalently, this can be expressed as

:$\backslash mathcal\{E\}\; =\; -\backslash int\_\{\backslash partial\; \backslash Sigma(t)\}\; \backslash frac\{\backslash partial\; \backslash mathbf\; B\}\{\backslash partial\; t\}\backslash cdot\{\backslash rm\; d\}\backslash mathbf\; A\; +\; \backslash oint\_\{\backslash partial\; \backslash Sigma(t)\}\; \backslash left(\; \backslash mathbf\{v\}\backslash times\backslash mathbf\{B\}\backslash right)\; \backslash cdot\; \backslash mathrm\{d\}\backslash mathbf\{l\},$

where the first term is the "transformer emf" due to a time-varying magnetic field, and the second term is the "motional emf" due to the magnetic Lorentz force by the motion of the charges in the magnetic field.{{sfn|Zangwill|2013|pp=462–464}}

In circuits made of thin, one-dimensional wires, the drift velocity is aligned with the wire, and hence with the integration element $\{\backslash rm\; d\}\backslash mathbf\; l$. In that case, the cross product $\backslash mathbf\; v\_d\backslash times\backslash mathbf\; B$ is perpendicular to $\{\backslash rm\; d\}\backslash mathbf\; l$, and the term proportional to the drift velocity vanishes. This recovers the standard form of Faraday's law:

$$\backslash mathcal\{E\}\; =\; -\backslash frac\{\backslash mathrm\{d\}\backslash Phi\_B\}\{\backslash mathrm\{d\}t\}$$

In this case, the emf can also be expressed as a sum{{sfn|Zangwill|2013|pp=462–464}}

$$\backslash mathcal\{E\}\; =\; -\backslash int\_\{\backslash Sigma(t)\}\; \backslash frac\{\backslash partial\; \backslash mathbf\{B\}\}\{\backslash partial\; t\}\; \backslash cdot\; \backslash mathrm\{d\}\backslash mathbf\{A\}\; +\; \backslash oint\_\{\backslash partial\; \backslash Sigma(t)\}\; \backslash left(\backslash mathbf\{v\}\_c\backslash times\backslash mathbf\{B\}\backslash right)\; \backslash cdot\; \backslash mathrm\{d\}\backslash mathbf\{l\}.$$In conductors that are not thin wires, the drift velocity term $\backslash oint\_\{\backslash partial\; \backslash Sigma(t)\}\; \backslash left(\; \backslash mathbf\{v\}\_d\backslash times\backslash mathbf\{B\}\backslash right)\; \backslash cdot\; \backslash mathrm\{d\}\backslash mathbf\{l\}$ may not vanish exactly. However, electrons typically drift at speeds of the order of 10^-4 m/s, and the contribution is often negligible compared to other effects.{{Cite web |last1=Ling |first1=Samuel J. |last2=Moebs |first2=William |last3=Sanny |first3=Jeff |date=2016-10-06 |title=9.2 Model of Conduction in Metals - University Physics Volume 2 {{!}} OpenStax |url=https://openstax.org/books/university-physics-volume-2/pages/9-2-model-of-conduction-in-metals |access-date=2025-06-19 |website=openstax.org |language=English}} A notable exception is the Hall effect, where magnetic flux term $\backslash mathrm\{d\}\backslash Phi\_B/\backslash mathrm\{d\}t$ vanishes, and the observed Hall voltage arises entirely from the drift velocity term.{{sfn|Zangwill|2013|pp=462–464}}

{{See also|Faraday paradox|Hering's Paradox}}

It is tempting to generalize Faraday's law to state: If {{math|∂Σ}} is any arbitrary closed loop in space whatsoever, then the total time derivative of magnetic flux through {{math|Σ}} equals the emf around {{math|∂Σ}}. This statement, however, is not always true. As noted in the previous section, Faraday's law is not guaranteed to work unless the velocity of the abstract curve {{math|∂Σ}} matches the actual velocity of the material conducting the electricity.{{cite book |title=Intermediate Electromagnetic Theory |first1=Joseph V. |last1=Stewart |page=396 |quote=This example of Faraday's Law [the homopolar generator] makes it very clear that in the case of extended bodies care must be taken that the boundary used to determine the flux must not be stationary but must be moving with respect to the body.}} If the conductor is not an infinitely thin wire, one may also have take into account the velocity of charges with respect to the material.{{sfn|Zangwill|2013|pp=462–464}} The two examples illustrated below show that one often obtains incorrect results when Faraday's law is applied too broadly.

File:Faraday's disc.PNG|Faraday's homopolar generator. The disc rotates with angular rate {{mvar|ω}}, sweeping the conducting radius circularly in the static magnetic field {{math|B}} (which direction is along the disk surface normal). The magnetic Lorentz force {{math|v × B}} drives a current along the conducting radius to the conducting rim, and from there the circuit completes through the lower brush and the axle supporting the disc. This device generates an emf and a current, although the shape of the "circuit" is constant and thus the flux through the circuit does not change with time.

File:FaradaysLawWithPlates.gif|A wire (solid red lines) connects to two touching metal plates (silver) to form a circuit. The whole system sits in a uniform magnetic field, normal to the page. If the abstract path {{math|∂Σ}} follows the primary path of current flow (marked in red), then the magnetic flux through this path changes dramatically as the plates are rotated, yet the emf is almost zero. After Feynman Lectures on Physics{{Rp|ch17}}|alt=A wire (solid red lines) connects to two touching metal plates (silver) to form a circuit. The whole system sits in a uniform magnetic field, normal to the page. If the abstract path ∂Σ follows the primary path of current flow (marked in red), then the magnetic flux through this path changes dramatically as the plates are rotated, yet the emf is almost zero. After Feynman Lectures on Physics: ch17

One can analyze examples like these by taking care that the path {{math|∂Σ}} moves with the same velocity as the material. The electromotive force can always be correctly calculated by combining the Lorentz force law with the Maxwell–Faraday equation:{{Rp|ch17}}

$$\backslash mathcal\{E\}\; =\; \backslash int\_\{\backslash partial\; \backslash Sigma\}\; (\backslash mathbf\{E\}\; +\; \backslash mathbf\{v\}\; \backslash times\; \backslash mathbf\{B\})\; \backslash cdot\; \backslash mathrm\{d\}\backslash mathbf\{l\}\; =\; -\backslash int\_\{\backslash partial\; \backslash Sigma(t)\}\; \backslash frac\{\backslash partial\; \backslash mathbf\; B\}\{\backslash partial\; t\}\backslash cdot\{\backslash rm\; d\}\backslash mathbf\; A\; +\; \backslash oint\_\{\backslash partial\; \backslash Sigma(t)\}\; \backslash left(\; \backslash mathbf\{v\}\backslash times\backslash mathbf\{B\}\backslash right)\; \backslash cdot\; \backslash mathrm\{d\}\backslash mathbf\{l\},$$

where {{math|v}} is the velocity of the conductor in the frame of reference in which {{math|B}} in described. The time derivative cannot in general be moved outside the integral since the position or shape of the loop may be a function of time.{{cite book |last1=Hughes |first1=W. F. |title=The Electromagnetodynamics of Fluid |last2=Young |first2=F. J. |date=1965 |publisher=John Wiley |at=Eq. (2.6–13) p. 53}}

{{Further|Moving magnet and conductor problem}}

Historically, the two distinct mechanisms encompassed by the flux rule, motional emf and transformer emf, posed a conceptual challenge.{{Cite web |title=The Feynman Lectures on Physics Vol. II Ch. 17: The Laws of Induction |url=https://www.feynmanlectures.caltech.edu/II_17.html |website=www.feynmanlectures.caltech.edu}} James Clerk Maxwell already recognized that electromagnetic induction could arise through different physical processes, even though the induced emf obeyed a single mathematical expression. In his 1861 paper On Physical Lines of Force,{{cite journal |last=Clerk Maxwell |first=James |author-link=James Clerk Maxwell |year=1861 |title=On physical lines of force |journal=Philosophical Magazine |publisher=Taylor & Francis |volume=90 |pages=11–23 |doi=10.1080/14786431003659180 |s2cid=135524562}} he gave separate physical explanations for each case.

In 1905, Albert Einstein highlighted this asymmetry in classical electrodynamics in his paper On the Electrodynamics of Moving Bodies.{{cite web |last=Einstein |first=Albert |author-link=Albert Einstein |title=On the Electrodynamics of Moving Bodies |url=http://www.fourmilab.ch/etexts/einstein/specrel/specrel.pdf}} He pointed out that the physical outcome, such as the induced current, depends only on relative motion between the conductor and the magnet, yet classical theory provided different explanations depending on which object was considered to be in motion. This inconsistency suggested the absence of a preferred frame and helped motivate the development of special relativity.{{sfn|Griffiths|2023|pp=316-318}}

In modern terms, electric and magnetic fields are understood as components of a single electromagnetic field tensor. Under a change of inertial frame, the two fields transform into one another.{{cite book |author=Tai L. Chow |url=https://books.google.com/books?id=dpnpMhw1zo8C&pg=PA153 |title=Electromagnetic theory |publisher=Jones and Bartlett |year=2006 |isbn=0-7637-3827-1 |location=Sudbury, Massachusetts |page=395}}

{{columns-list|colwidth=22em|

• Eddy current
• Inductance
• Crosstalk
• Felici's law

}}{{Clear}}

{{Reflist|30em}}

• {{cite book | last=Darrigol | first=Olivier | title=Electrodynamics from Ampère to Einstein | publisher=Clarendon Press | publication-place=Oxford ; New York | date=2000 | isbn=0-19-850594-9}}
• {{cite book |first1 = Richard Phillips |last1 = Feynman |author-link = Richard Feynman |first2 = Robert B. |last2 = Leighton | first3 = Matthew L. |last3 = Sands |title = The Feynman lectures on physics |publisher = Pearson / Addison-Wesley | year = 2006 |isbn = 0-8053-9047-2 |volume=2|url=https://www.feynmanlectures.caltech.edu/II_toc.html}}
• {{cite book | last=Griffiths | first=David J. | title=Introduction to Electrodynamics | publisher=Cambridge University Press | date=2023 | isbn=978-1-009-39773-5 | doi=10.1017/9781009397735 | url=https://www.cambridge.org/highereducation/product/9781009397735/book}}
• {{cite book | last1=Purcell | first1=Edward M. | last2=Morin | first2=David J. | title=Electricity and Magnetism | publisher=Cambridge University Press | date=2013 | isbn=978-1-139-01297-3 | doi=10.1017/cbo9781139012973 | url=https://www.cambridge.org/core/product/identifier/9781139012973/type/book}}
• {{cite book | first=Matthew N. O. |last=Sadiku | title=Elements of electromagnetics | year= 2018 | edition=7th | publisher=Oxford University Press | location=New York/Oxford | isbn = 978-0-19-069861-4 |lccn=2017046497 | url=https://lccn.loc.gov/2017046497}}
• {{cite book | last=Zangwill | first=Andrew | title=Modern Electrodynamics | publisher=Cambridge University Press | date=2013 | isbn=978-0-521-89697-9 | doi=10.1017/CBO9781139034777}}

• {{cite book|url=https://archive.org/details/atreatiseonelec04maxwgoog |quote=a treatise on electricity and magnetism. |last= Clerk Maxwell |first=James |author-link=James Clerk Maxwell |date=1881 |title=A treatise on electricity and magnetism, Vol. II |at=ch. III, sec. 530, p. 178|location=Oxford|publisher=Clarendon Press|isbn=0-486-60637-6}}

• {{Commons category-inline}}
• [https://nationalmaglab.org/magnet-academy/watch-play/interactive-tutorials/electromagnetic-induction/ A simple interactive tutorial on electromagnetic induction] (click and drag magnet back and forth) National High Magnetic Field Laboratory
• [https://web.archive.org/web/20080530092914/http://www.physics.smu.edu/~vega/em1304/lectures/lect13/lect13_f03.ppt Roberto Vega. Induction: Faraday's law and Lenz's law – Highly animated lecture, with sound effects], [https://web.archive.org/web/20081228092644/http://www.physics.smu.edu/vega/em1304/p1304.html Electricity and Magnetism course page]
• [http://hyperphysics.phy-astr.gsu.edu/hbase/emcon.html Notes from Physics and Astronomy HyperPhysics at Georgia State University]
• [https://web.archive.org/web/20120617020014/http://usna.edu/Users/physics/tank/Public/FaradaysLaw.pdf Tankersley and Mosca: Introducing Faraday's law]
• [http://www.phy.hk/wiki/englishhtm/Induction.htm A free simulation on motional emf]

{{Michael Faraday}}

{{DEFAULTSORT:Faraday's law of electromagnetic induction}}

Faraday's law of electromagnetic induction

Category:Michael Faraday

Category:Maxwell's equations