Faraday's law of induction#Quantitative

Electromagnetic induction was discovered independently by Michael Faraday in 1831 and Joseph Henry in 1832.{{Cite web|url=http://web.hep.uiuc.edu/home/serrede/P435/Lecture_Notes/A_Brief_History_of_Electromagnetism.pdf|title=A Brief History of Electromagnetism}} Faraday was the first to publish the results of his experiments.{{cite book|last=Ulaby|first=Fawwaz|title=Fundamentals of applied electromagnetics|edition=5th|year=2007|url=https://www.amazon.com/exec/obidos/tg/detail/-/0132413264/ref=ord_cart_shr?%5Fencoding=UTF8&m=ATVPDKIKX0DER&v=glance|publisher=Pearson:Prentice Hall|isbn=978-0-13-241326-8|page=255}}{{cite web|url=http://www.nasonline.org/member-directory/deceased-members/20001467.html |title=Joseph Henry |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 electromagnetic 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}} that wraps two wires around opposite sides of an iron ring (like a modern toroidal transformer). His assessment of newly-discovered properties of electromagnets suggested that when current started to flow in one wire, a sort of wave would travel through the ring and cause some electrical effect on the opposite side. Indeed, a galvanometer's needle measured a transient current (which he called a "wave of electricity") on the right side's wire when he connected or disconnected the left side's wire to a battery.{{cite book|title=Michael Faraday|url=https://archive.org/details/michaelfaradaybi00will|url-access=registration|first=L. Pearce|last=Williams|year=1965|publisher=New York, Basic Books}}{{full citation needed|date=September 2018}}{{rp|182–183}} This induction was due to the change in magnetic flux that occurred when the battery was connected and disconnected. His notebook entry also noted that fewer wraps for the battery side resulted in a greater disturbance of the galvanometer's needle.

Within two months, Faraday had found several other manifestations of electromagnetic induction. For example, he saw transient currents when he quickly slid a bar magnet in and out of a coil of wires, and he generated a steady (DC) current by rotating a copper disk near the bar magnet with a sliding electrical lead ("Faraday's disk").{{rp|191–195}}

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

Michael Faraday explained electromagnetic induction using a concept he called lines of force. However, scientists at the time widely rejected his theoretical ideas, mainly because they were not formulated mathematically.{{rp|510}} An exception was James Clerk Maxwell, who in 1861–62 used Faraday's ideas as the basis of his quantitative electromagnetic theory.{{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 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 | last=Landau| first=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.{{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|page=319}}

Mathematically, in SI units, the law is written 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. It is defined as the surface integral of the magnetic field over a 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 element of area vector of the moving surface {{math|Σ(t)}}, directed normal to the surface, and {{math|B}} is the magnetic field. The dot product {{math|B · dA}} represents the element of flux through {{math|dA}}. In more visual terms, the magnetic flux through the wire loop is proportional to the number of magnetic field lines that pass through the loop.

Image:Surface integral illustration.svgWhen the flux changes—because {{math|B}} changes, or because the wire loop is moved or deformed, or both—Faraday's law of induction says that the wire loop acquires an emf, defined as the energy available from a unit charge that has traveled once around the wire loop.{{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|ch17}}{{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}}{{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}} (Although some sources state the definition differently, this expression was chosen for compatibility with the equations of special relativity.{{Cn|date=June 2025}}) Equivalently, it is the voltage that would be measured by cutting the wire to create an open circuit, and attaching a voltmeter to the leads.{{Cn|date=June 2025|reason=The voltmeter closes the circuit. Does the voltage depend on the shape of the circuit which open circuit and the voltmeter form?}}File:Salu's left-hand rule (magnetic induction).png

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 journal|year=2014 |url=https://www.researchgate.net/publication/262986189 |title=A Left Hand Rule for Faraday's Law | journal=The Physics Teacher | volume=52|pages=48 |doi=10.1119/1.4849156 |author=Yehuda Salu| issue=1 |bibcode=2014PhTea..52...48S}} [https://www.youtube.com/watch?v=ipUD9VcAd9o Video Explanation]{{cite web |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 |title=Bypassing Lenz's Rule - A Left Hand Rule for Faraday's Law |website=www.PhysicsForArchitects.com |last1=Salu|first1=Yehuda |date=17 January 2017 |access-date=30 July 2017}}

• 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).

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| title=Essential Principles of Physics| first1=P. M.|last1=Whelan|first2=M. J.|last2=Hodgeson|edition=2nd|date=1978|publisher=John Murray|isbn=0-7195-3382-1}}{{cite web|last=Nave|first=Carl R. | title=Faraday's Law | url=http://hyperphysics.phy-astr.gsu.edu/hbase/electric/farlaw.html | work=HyperPhysics |publisher=Georgia State University |access-date=2011-08-29}}

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

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}}

{{anchor|Faraday equation}}

Image:Stokes' Theorem.svg]]

The Maxwell–Faraday equation states that a time-varying magnetic field always accompanies a spatially varying (also possibly time-varying), non-conservative electric field, and vice versa. The Maxwell–Faraday equation is

{{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}}

(in SI units) where {{math|∇ ×}} is the curl operator and again {{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

}}

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 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 contour {{math|∂Σ}}, {{math|dl}} is an infinitesimal vector element of the contour {{math|∂Σ}}, and {{math|dA}} is an infinitesimal vector element of surface {{math|Σ}}. Its direction is orthogonal to that surface patch, the magnitude is the area of an infinitesimal patch of surface.

Both {{math|dl}} and {{math|dA}} have a sign ambiguity; to get the correct sign, the right-hand rule is used, as explained in the article Kelvin–Stokes theorem. For a planar surface {{math|Σ}}, a positive path element {{math|dl}} of curve {{math|∂Σ}} is defined by the right-hand rule as one that points with the fingers of the right hand when the thumb points in the direction of the normal {{math|n}} to the surface {{math|Σ}}.

The line integral around {{math|∂Σ}} is called circulation.{{Rp|ch3}} A nonzero circulation of {{math|E}} is different from the behavior of the electric field generated by static charges. A charge-generated {{math|E}}-field can be expressed as the gradient of a scalar field that is a solution to Poisson's equation, and has a zero path integral.

The integral equation is true for any path {{math|∂Σ}} through space, and any surface {{math|Σ}} for which that path is a boundary.

If the surface {{math|Σ}} is not changing in time, the equation can be rewritten:

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

The surface integral at the right-hand side is the explicit expression for the magnetic flux {{math|Φ_B}} through {{math|Σ}}.

The electric vector field induced by a changing magnetic flux, the solenoidal component of the overall electric field, can be approximated in the non-relativistic limit by the volume integral equation{{Rp|321}}

$$\backslash mathbf\; E\_s\; (\backslash mathbf\; r,t)\; \backslash approx\; -\backslash frac\{1\}\{4\backslash pi\}\backslash iiint\_V\; \backslash \; \backslash frac\{\backslash left(\backslash frac\{\backslash partial\; \backslash mathbf\{B\}(\backslash mathbf\{r\}\text{'},t)\}\{\backslash partial\; t\}\; \backslash right)\; \backslash times\; \backslash left(\backslash mathbf\{r\}-\backslash mathbf\{r\}\text{'}\; \backslash right)\; \}\{|\backslash mathbf\; \{r\}\; -\; \backslash mathbf\{r\}\text{'}|^3\}\; d^3\backslash mathbf\{r\text{'}\}$$

The four Maxwell's equations, together with the Lorentz force law, form a complete foundation for classical electromagnetism. 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":{{cite book |last=Zangwill |first=Andrew |title=Modern Electrodynamics |date=2013 |publisher=Cambridge University Press |isbn=978-0-521-89697-9 |publication-place=Cambridge |pages=10, 462–464}} Proof of the theorem is found on page 10.

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

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.

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.

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.

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

$$\backslash mathcal\{E\}\; =\; -\backslash int\_\{\backslash Sigma(t)\}\; \backslash frac\{\backslash partial\; \backslash mathbf\{B\}\}\{\backslash partial\; t\}\; \backslash cdot\; \backslash mathrm\{d\}\backslash mathbf\{A\}\; +\; \backslash oint\_\{\backslash partial\; \backslash Sigma(t)\}\; \backslash left(\backslash mathbf\{v\}\_c\backslash times\backslash mathbf\{B\}\backslash right)\; \backslash cdot\; \backslash mathrm\{d\}\backslash mathbf\{l\}.$$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 |last=Ling |first=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.

{{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. 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}}

{{rewrite section|date=June 2025}}

Faraday's law is a single equation describing two different phenomena: the motional emf generated by a magnetic force on a moving wire, and the transformer emf generated by an electric force due to a changing magnetic field (described by the Maxwell–Faraday equation).

James Clerk Maxwell drew attention to this fact in his 1861 paper On Physical Lines of Force.{{cite journal|author-link = James Clerk Maxwell|last=Clerk Maxwell|first= James|journal = Philosophical Magazine|doi = 10.1080/14786431003659180 |pages = 11–23|publisher = Taylor & Francis|title = On physical lines of force|volume = 90|year = 1861|s2cid=135524562}} In the latter half of Part II of that paper, Maxwell gives a separate physical explanation for each of the two phenomena.

A reference to these two aspects of electromagnetic induction is made in some modern textbooks.{{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/301 301–3]|publisher=Prentice Hall|year=1999|location=Upper Saddle River, NJ|isbn=0-13-805326-X}}
Note that the law relating flux to emf, which this article calls "Faraday's law", is referred to in Griffiths' terminology as the "universal flux rule". Griffiths uses the term "Faraday's law" to refer to what this article calls the "Maxwell–Faraday equation". So in fact, in the textbook, Griffiths' statement is about the "universal flux rule". As Richard Feynman states:

{{Blockquote|So the "flux rule" that the emf in a circuit is equal to the rate of change of the magnetic flux through the circuit applies whether the flux changes because the field changes or because the circuit moves (or both) ...

Yet in our explanation of the rule we have used two completely distinct laws for the two cases – {{math|v × B}} for "circuit moves" and {{math|∇ × E {{=}} −∂_tB}} for "field changes".

We know of no other place in physics where such a simple and accurate general principle requires for its real understanding an analysis in terms of two different phenomena.|Richard P. Feynman, The Feynman Lectures on Physics{{Cite web|url=https://www.feynmanlectures.caltech.edu/II_17.html|title=The Feynman Lectures on Physics Vol. II Ch. 17: The Laws of Induction|website=www.feynmanlectures.caltech.edu}}}}{{Dubious|reason=Feynman Lectures sometimes uses intentionally misleading statements to set up later revelations. This may be one of those times. Also see talk section "Two phenomena"|date=June 2023}}

In the general case, explanation of the motional emf appearance by action of the magnetic force on the charges in the moving wire or in the circuit changing its area is unsatisfactory. As a matter of fact, the charges in the wire or in the circuit could be completely absent, will then the electromagnetic induction effect disappear in this case? This situation is analyzed in the article, in which, when writing the integral equations of the electromagnetic field in a four-dimensional covariant form, in the Faraday’s law the total time derivative of the magnetic flux through the circuit appears instead of the partial time derivative.{{cite journal | last1 = Fedosin | first1 = Sergey G. | title = On the Covariant Representation of Integral Equations of the Electromagnetic Field | journal = Progress in Electromagnetics Research C | volume = 96 | pages = 109–122| year = 2019 | url = https://rdcu.be/ccV9o| doi = 10.2528/PIERC19062902|arxiv=1911.11138 | s2cid = 208095922 }} Thus, electromagnetic induction appears either when the magnetic field changes over time or when the area of the circuit changes. From the physical point of view, it is better to speak not about the induction emf, but about the induced electric field strength $\backslash mathbf\; E\; =\; -\; \backslash nabla\; \backslash mathcal\{E\}\; -\; \backslash frac\{\; \backslash partial\; \backslash mathbf\; A\}\{\; \backslash partial\; t\}$, that occurs in the circuit when the magnetic flux changes. In this case, the contribution to $\backslash mathbf\; E$ from the change in the magnetic field is made through the term $-\; \backslash frac\{\; \backslash partial\; \backslash mathbf\; A\}\{\; \backslash partial\; t\}$ , where $\backslash mathbf\; A$ is the vector potential. If the circuit area is changing in case of the constant magnetic field, then some part of the circuit is inevitably moving, and the electric field $\backslash mathbf\; E$ emerges in this part of the circuit in the comoving reference frame K’ as a result of the Lorentz transformation of the magnetic field $\backslash mathbf\; B$, present in the stationary reference frame K, which passes through the circuit. The presence of the field $\backslash mathbf\; E$ in K’ is considered as a result of the induction effect in the moving circuit, regardless of whether the charges are present in the circuit or not. In the conducting circuit, the field $\backslash mathbf\; E$ causes motion of the charges. In the reference frame K, it looks like appearance of emf of the induction $\backslash mathcal\{E\}$, the gradient of which in the form of $-\; \backslash nabla\; \backslash mathcal\{E\}$, taken along the circuit, seems to generate the field $\backslash mathbf\; E$.

Reflection on this apparent dichotomy was one of the principal paths that led Albert Einstein to develop special relativity:

{{Blockquote|It is known that Maxwell's electrodynamics—as usually understood at the present time—when applied to moving bodies, leads to asymmetries which do not appear to be inherent in the phenomena. Take, for example, the reciprocal electrodynamic action of a magnet and a conductor.

The observable phenomenon here depends only on the relative motion of the conductor and the magnet, whereas the customary view draws a sharp distinction between the two cases in which either the one or the other of these bodies is in motion. For if the magnet is in motion and the conductor at rest, there arises in the neighbourhood of the magnet an electric field with a certain definite energy, producing a current at the places where parts of the conductor are situated.

But if the magnet is stationary and the conductor in motion, no electric field arises in the neighbourhood of the magnet. In the conductor, however, we find an electromotive force, to which in itself there is no corresponding energy, but which gives rise—assuming equality of relative motion in the two cases discussed—to electric currents of the same path and intensity as those produced by the electric forces in the former case.

Examples of this sort, together with unsuccessful attempts to discover any motion of the earth relative to the "light medium," suggest that the phenomena of electrodynamics as well as of mechanics possess no properties corresponding to the idea of absolute rest.

| Albert Einstein, On the Electrodynamics of Moving Bodies{{cite web|first=Albert|last=Einstein|author-link=Albert Einstein|url=http://www.fourmilab.ch/etexts/einstein/specrel/specrel.pdf|title=On the Electrodynamics of Moving Bodies}}}}

{{columns-list|colwidth=22em|

• Eddy current
• Inductance
• Maxwell's equations
• Crosstalk
• Faraday paradox
• Felici's law

}}{{Clear}}

{{Reflist|30em}}

• {{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