View source for Faraday's law of induction

{{For|applications and consequences of the law|Electromagnetic induction}}
{{Use American English|date=March 2019}}{{Short description|Basic law of electromagnetism of magnetic fields inducing a potential difference}}
{{electromagnetism|cTopic=Electrodynamics}}

'''Faraday's law of induction''' (briefly, '''Faraday's law''') is a basic law of [[electromagnetism]] predicting how a [[magnetic field]] will interact with an [[electric circuit]] to produce an [[electromotive force|electromotive force (EMF)]]—a phenomenon called [[electromagnetic induction]]. It is the fundamental operating principle of [[transformer]]s, [[inductor]]s, and many types of [[electricity|electrical]] [[electric motor|motors]], [[electrical generator|generators]] and [[solenoid]]s.<ref name="Sadiku386">{{cite book|last=Sadiku|first=M. N. O.|title=Elements of Electromagnetics|year=2007|page=386|publisher=Oxford University Press|edition=4th|location=New York & Oxford|url=https://books.google.com/books?id=w2ITHQAACAAJ&dq=ISBN0-19-530048-3|isbn=0-19-530048-3}}</ref><ref>{{cite web|date=1999-07-22|title=Applications of electromagnetic induction|url=http://physics.bu.edu/~duffy/py106/Electricgenerators.html|publisher=[[Boston University]]}}</ref>

The '''Maxwell–Faraday equation''' (listed as one of [[Maxwell's equations]]) describes the fact that a spatially varying (and also possibly time-varying, depending on how a magnetic field varies in time) electric field always accompanies a time-varying magnetic field, while Faraday's law states that there is EMF (electromotive force, defined as electromagnetic work done on a unit charge when it has traveled one round of a conductive loop) on the conductive loop when the magnetic flux through the surface enclosed by the loop varies in time.

Faraday's law had been discovered and one aspect of it (transformer EMF) was formulated as the Maxwell–Faraday equation later. The equation of Faraday's law can be derived by the Maxwell–Faraday equation (describing transformer EMF) and the [[Lorentz force]] (describing motional EMF). The integral form of the Maxwell–Faraday equation describes only the transformer EMF, while the equation of Faraday's law describes both the transformer EMF and the motional EMF.

==History==
[[File:Faraday emf experiment.svg|thumb|250px|A diagram of Faraday's iron ring apparatus. The changing magnetic flux of the left coil induces a current in the right coil.<ref name=Giancoli>{{cite book|last=Giancoli|first=Douglas C.|title=Physics: Principles with Applications|url=https://archive.org/details/physicsprinciple00gian|url-access=registration|year=1998|pages=[https://archive.org/details/physicsprinciple00gian/page/623 623–624]|edition=5th}}</ref>]]
[[File:Faraday disk generator.jpg|thumb|Faraday's disk, the first [[electric generator]], a type of [[homopolar generator]].]]

Electromagnetic induction was discovered independently by [[Michael Faraday]] in 1831 and [[Joseph Henry]] in 1832.<ref>{{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}}</ref> Faraday was the first to publish the results of his experiments.<ref>{{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=0-13-241326-4|page=255}}</ref><ref>{{cite web|url=http://www.nasonline.org/member-directory/deceased-members/20001467.html |title=Joseph Henry |accessdate=2016-12-30 |work=Member Directory, National Academy of Sciences}}</ref> In Faraday's first experimental demonstration of electromagnetic induction (August 29, 1831),<ref name="FaradayDay1999">{{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|accessdate=28 August 2011|date=1999-02-01|publisher=CRC Press|isbn=978-0-7503-0570-9|page=71}}</ref> he wrapped two wires around opposite sides of an iron ring ([[torus]]) (an arrangement similar to a modern [[toroidal transformer]]). Based on his assessment of recently discovered properties of electromagnets, he expected 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. He plugged one wire into a [[galvanometer]], and watched it as he connected the other wire to a battery. Indeed, he saw a transient current (which he called a "wave of electricity") when he connected the wire to the battery, and another when he disconnected it.<ref name=Williams>{{cite book|title=Michael Faraday|url=https://archive.org/details/michaelfaradaybi00will|url-access=registration|first=L. Pearce|last=Williams}}{{full citation needed|date=September 2018}}</ref>{{rp|182–183}} This induction was due to the change in [[magnetic flux]] that occurred when the battery was connected and disconnected.<ref name=Giancoli/> 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 ([[direct current|DC]]) current by rotating a copper disk near the bar magnet with a sliding electrical lead ("Faraday's disk").<ref name=Williams/>{{rp|191–195}}

[[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.<ref name=Williams/>{{rp|510}} An exception was [[James Clerk Maxwell]], who in 1861–62 used Faraday's ideas as the basis of his quantitative electromagnetic theory.<ref name=Williams/>{{rp|510}}<ref>{{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}}</ref><ref name="IEEUK">{{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}}</ref> 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 [[#Two phenomena|motional EMF]]. Heaviside's version (see [[#Maxwell–Faraday equation|Maxwell–Faraday equation below]]) is the form recognized today in the group of equations known as [[Maxwell's equations]].

[[Lenz's law]], formulated by [[Emil Lenz]] in 1834,<ref>{{cite journal|last=Lenz |first=Emil |date=1834 |url=http://gallica.bnf.fr/ark:/12148/bpt6k151161/f499.image.r=lenz.langEN |title=Ueber<!--[sic]--> 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 }}<br>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}}</ref> 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).
[[Image:Induction experiment.png|thumb|300px|Faraday's experiment showing induction between coils of wire: The liquid battery ''(right)'' provides a current which flows through the small coil ''(A)'', creating a magnetic field. When the coils are stationary, no current is induced. But when the small coil is moved in or out of the large coil ''(B)'', the magnetic flux through the large coil changes, inducing a current which is detected by the galvanometer ''(G)''.<ref>{{cite book|url=https://books.google.com/books?id=JzBAAAAAYAAJ&pg=PA285 |last=Poyser |first=Arthur William |date=1892 |title=Magnetism and Electricity: A manual for students in advanced classes |location=London and New York |publisher=Longmans, Green, & Co. |at=Fig. 248, p. 245 |accessdate=2009-08-06}}</ref>]]

==Faraday's law==

The most widespread version of Faraday's law states:
{{Quotation|The electromotive force around a closed path is equal to the negative of the time rate of change of the [[magnetic flux]] enclosed by the path.<ref name="Jordan & Balmain (1968)">{{cite book| last=Jordan|first= Edward |last2=Balmain|first2=Keith G.| title = Electromagnetic Waves and Radiating Systems | edition = 2nd| page = 100| publisher = Prentice-Hall| date = 1968|quote=Faraday's Law, which states that the electromotive force around a closed path is equal to the negative of the time rate of change of magnetic flux enclosed by the path.}}</ref><ref name="Hayt (1989)">{{cite book| last = Hayt| first = William| title = Engineering Electromagnetics| edition = 5th| page = 312| isbn = 0-07-027406-1| publisher = McGraw-Hill| date = 1989| quote = The magnetic flux is that flux which passes through any and every surface whose perimeter is the closed path.| url = https://archive.org/details/engineeringelect5thhayt}}</ref>}}
The closed path here is, in fact, conductive.

===Mathematical statement===
[[Image:Surface integral illustration.svg|right|thumb|The definition of surface integral relies on splitting the surface {{math|Σ}} into small surface elements. Each element is associated with a vector {{math|d'''A'''}} of magnitude equal to the area of the element and with direction normal to the element and pointing "outward" (with respect to the orientation of the surface).]]
For a loop of wire in a [[magnetic field]], the [[magnetic flux]] {{math|Φ<sub>''B''</sub>}} is defined for any [[Surface (mathematics)|surface]] {{math|Σ}} whose [[Boundary (topology)|boundary]] is the given loop. Since the wire loop may be moving, we write {{math|Σ(''t'')}} for the surface. The magnetic flux is the [[surface integral]]:

:<math> \Phi_B = \iint\limits_{\Sigma(t)} \mathbf{B}(t) \cdot \mathrm{d} \mathbf{A}\, , </math>

where {{math|d'''A'''}} is an element of surface area of the moving surface {{math|Σ(''t'')}}, {{math|'''B'''}} is the magnetic field, and {{math|'''B'''·d'''A'''}} is a [[dot product|vector dot product]] representing the element of flux through {{math|d'''A'''}}. In more visual terms, the magnetic flux through the wire loop is proportional to the number of [[field line|magnetic flux lines]] that pass through the loop.

When 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 [[electromotive force|EMF]], {{mathcal|E}}, defined as the energy available from a unit charge that has travelled once around the wire loop.<ref name=Feynman/><ref name=Griffiths2>{{cite book|last=Griffiths|first=David J.|title=Introduction to Electrodynamics|url=https://archive.org/details/introductiontoel00grif_0|edition=3rd|pages=301–303|publisher=Prentice Hall|year=1999|location=Upper Saddle River, NJ|isbn=0-13-805326-X}}</ref><ref>{{cite book|last1=Tipler|last2=Mosca|title=Physics for Scientists and Engineers|page=795|url=https://books.google.com/books?id=R2Nuh3Ux1AwC&pg=PA795}}</ref> (Note that different textbooks may give different definitions. The set of equations used throughout the text was chosen to be compatible with the special relativity theory.) Equivalently, it is the voltage that would be measured by cutting the wire to create an [[Electric circuit|open circuit]], and attaching a [[voltmeter]] to the leads.

Faraday's law states that the EMF is also given by the [[time derivative|rate of change]] of the magnetic flux:
:<math>\mathcal{E} = -\frac{\mathrm{d}\Phi_B}{\mathrm{d}t}, </math>
where <math>\mathcal{E}</math> is the [[electromotive force]] (EMF) and {{math|Φ<sub>''B''</sub>}} is the [[magnetic flux]].

The direction of the electromotive force is given by [[Lenz's law]].

The laws of induction of electric currents in mathematical form was established by [[Franz Ernst Neumann]] in 1845.<ref>{{cite journal|first=Franz Ernst|last=Neumann |title=Allgemeine Gesetze der inducirten elektrischen Ströme|journal=Annalen der Physik|volume=143|number=1|pages=31–44|year=1846|doi=10.1002/andp.18461430103|url=https://isidore.co/misc/Physics%20papers%20and%20books/Zotero/storage/3UM3CRQ2/18461430103_ftp.pdf|bibcode=1846AnP...143...31N}}</ref>

Faraday's law contains the information about the relationships between both the magnitudes and the directions of its variables. However, the relationships between the directions are not explicit; they are hidden in the mathematical formula.
[[File:Salu's left-hand rule (magnetic induction).png|thumb|Salu's left-hand rule (magnetic induction)|A Left Hand Rule for Faraday's Law. The sign of {{math|ΔΦ<sub>''B''</sub>}}, the change in flux, is found based on the relationship between the magnetic field {{math|'''B'''}}, the area of the loop {{math|''A''}}, and the normal n to that area, as represented by the fingers of the left hand. If {{math|ΔΦ<sub>''B''</sub>}} is positive, the direction of the EMF is the same as that of the curved fingers (yellow arrowheads). If {{math|ΔΦ<sub>''B''</sub>}} is negative, the direction of the EMF is against the arrowheads.<ref name=Salu2014/>]]
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:<ref name="Salu2014">{{cite journal|year=2014|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|bibcode=2014PhTea..52...48S}}</ref><ref>{{cite web|url=http://Physicsforarchitects.com/bypassing-lenzs-rule|title=A Left Hand Rule for Faraday's Law|website=www.PhysicsForArchitects.com/bypassing-lenzs-rule|last1=Salu|first1=Yehuda|accessdate=30 July 2017}}</ref>
* 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<sub>''B''</sub>}}, the change in flux. Determine the initial and final fluxes (whose difference is {{math|ΔΦ<sub>''B''</sub>}}) with respect to the normal {{math|'''n'''}}, as indicated by the stretched thumb.
* If the change in flux, {{math|ΔΦ<sub>''B''</sub>}}, is positive, the curved fingers show the direction of the electromotive force (yellow arrowheads).
* If {{math|ΔΦ<sub>''B''</sub>}} 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 [[inductor|coil of wire]], composed of {{mvar|N}} identical turns, each with the same {{math|Φ<sub>''B''</sub>}}, Faraday's law of induction states that<ref>{{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}}</ref><ref>{{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|accessdate=2011-08-29}}</ref>

:<math> \mathcal{E} = -N \frac{\mathrm{d}\Phi_B}{\mathrm{d}t} </math>

where {{mvar|N}} is the number of turns of wire and {{math|Φ<sub>''B''</sub>}} is the magnetic flux through a single loop.

===Maxwell–Faraday equation===
{{anchor|Faraday equation}}
[[Image:Stokes' Theorem.svg|thumb|right|An illustration of the Kelvin–Stokes theorem with surface {{math|'''Σ'''}}, its boundary {{math|∂'''Σ'''}}, and orientation {{math|'''n'''}} set by the [[right-hand rule]].]]

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

{{Equation box 1
|indent =:
|equation = <math>\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}</math>
|cellpadding
|border
|border colour = #50C878
|background colour = #ECFCF4}}

(in [[SI units]]) where {{math|∇ ×}} is the [[Curl (mathematics)|curl]] [[linear operator|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}}.

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]],<ref name=Harrington>{{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&pg=PA57&dq=%22faraday%27s+law+of+induction%22}}</ref> thereby reproducing Faraday's law:

{{Equation box 1
|indent =:
|equation = <math> \oint_{\partial \Sigma} \mathbf{E} \cdot \mathrm{d}\mathbf{l} = - \int_\Sigma \frac{\partial \mathbf{B}}{\partial t} \cdot \mathrm{d}\mathbf{A} </math>
|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|'''E'''}} is the electric field, {{math|'''B'''}} is the [[magnetic field]].
:{{math|d'''l'''}} is an [[infinitesimal]] vector element of the contour {{math|'''∂Σ'''}},
:{{math|d'''A'''}} is an infinitesimal vector element of surface {{math|'''Σ'''}}. If its direction is [[orthogonal]] to that surface patch, the magnitude is the area of an infinitesimal patch of surface.

Both {{math|d'''l'''}} and {{math|d'''A'''}} 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|d'''l'''}} 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 integral around {{math|∂'''Σ'''}} is called a path integral or [[line integral]].

Notice that a nonzero [[Line integral|path integral]] for {{math|'''E'''}} is different from the behavior of the electric field generated by 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. See [[gradient theorem]].

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:
:<math> \oint_{\partial \Sigma} \mathbf{E} \cdot \mathrm{d}\mathbf{l} = - \frac{\mathrm{d}}{\mathrm{d}t} \int_{\Sigma} \mathbf{B} \cdot \mathrm{d}\mathbf{A}. </math>
The [[surface integral]] at the right-hand side is the explicit expression for the [[magnetic flux]] {{math|Φ<sub>''B''</sub>}} through {{math|'''Σ'''}}.

The electric vector field induced by a changing magnetic flux, the [[solenoidal vector field|solenoidal component]] of the overall electric field, can be approximated in the non-relativistic limit by the following [[volume integral]] equation:<ref name=Griffiths4>Note, Griffiths uses τ for volume elements.{{cite book|last=Griffiths|first=David J.|title=Introduction to Electrodynamics|edition=4th|pages=321|publisher=Pearson India Education Services|year=2015|isbn=978-93-325-5044-5}}</ref>
:<math> \mathbf E_s (\mathbf r) \approx -\frac{1}{4\pi}\iiint_V \ \frac{(\frac{\partial \mathbf{B}} {\partial t}\,dV)\times\mathbf{\hat r'}}{|\mathbf r'|^2} </math>

==Proof==
The four [[Maxwell's equations]] (including the Maxwell–Faraday equation), along with Lorentz force law, are a sufficient foundation to derive ''everything'' in [[classical electromagnetism]].<ref name=Feynman/><ref name=Griffiths2/> Therefore, it is possible to "prove" Faraday's law starting with these equations.<ref name=Davison>{{Cite journal | last1 = Davison | first1 = M. E. | title = A Simple Proof that the Lorentz Force, Law Implied Faraday's Law of Induction, when '''B''' is Time Independent | doi = 10.1119/1.1987339 | journal = American Journal of Physics | volume = 41 | issue = 5 | page = 713| year = 1973 | pmid =  | pmc = |bibcode = 1973AmJPh..41..713D }}</ref><ref name=Krey>{{cite book|title=Basic Theoretical Physics: A Concise Overview|last1=Krey|last2=Owen|page=155|url=https://books.google.com/books?id=xZ_QelBmkxYC&pg=PA155}}</ref>

The starting point is the time-derivative of flux through an arbitrary surface {{math|Σ}} (that can move or be deformed) in space:

:<math>\frac{\mathrm{d}\Phi_B}{\mathrm{d}t} = \frac{\mathrm{d}}{\mathrm{d}t}\int_{\Sigma(t)} \mathbf{B}(t)\cdot \mathrm{d}\mathbf{A}</math>

(by definition). This total time derivative can be evaluated and simplified with the help of the Maxwell–Faraday equation and some vector identities; the details are in the box below:
{| class="wikitable mw-collapsible mw-collapsed"
|+
|-
|Consider the time-derivative of magnetic flux through a closed boundary (loop) that can move or be deformed. The area bounded by the loop is denoted as {{math|Σ(''t'')}}), then the time-derivative can be expressed as

:<math>\frac{\mathrm{d}\Phi_B}{\mathrm{d}t} = \frac{\mathrm{d}}{\mathrm{d}t}\int_{\Sigma(t)} \mathbf{B}(t)\cdot \mathrm{d}\mathbf{A}</math>

The integral can change over time for two reasons: The integrand can change, or the integration region can change. These add linearly, therefore:

:<math>\left. \frac{\mathrm{d}\Phi_B}{\mathrm{d}t}\right|_{t=t_0} = \left( \int_{\Sigma(t_0)} \left. \frac{\partial\mathbf{B}}{\partial t}\right|_{t=t_0} \cdot \mathrm{d}\mathbf{A}\right) + \left( \frac{\mathrm{d}}{\mathrm{d}t} \int_{\Sigma(t)} \mathbf{B}(t_0) \cdot \mathrm{d}\mathbf{A} \right)</math>

where {{math|''t''<sub>0</sub>}} is any given fixed time. We will show that the first term on the right-hand side corresponds to transformer EMF, the second to motional EMF (from the magnetic Lorentz force on charge carriers due to the motion or deformation of the conducting loop in the magnetic field). The first term on the right-hand side can be rewritten using the integral form of the Maxwell–Faraday equation:

:<math> \int_{\Sigma(t_0)} \left. \frac{\partial \mathbf{B}}{\partial t}\right|_{t=t_0} \cdot \mathrm{d}\mathbf{A} = - \oint_{\partial \Sigma(t_0)} \mathbf{E}(t_0) \cdot \mathrm{d}\mathbf{l} </math>

Next, we analyze the second term on the right-hand side:

:<math>\frac{\mathrm{d}}{\mathrm{d}t} \int_{\Sigma(t)} \mathbf{B}(t_0) \cdot \mathrm{d}\mathbf{A}</math>

[[File:Derivation of Faraday Equation Wikipedia 20181127 - 4.png|alt=|thumb|300x300px|The area swept out by a vector element {{math|d'''l'''}} of a loop {{math|∂'''Σ'''}} in time {{math|d''t''}} when it has moved with velocity {{math|'''v'''<sub>'''l'''</sub>}} .]]The proof of this is a little more difficult than the first term; more details and alternate approaches for the proof can be found in the references.<ref name="Davison" /><ref name="Krey" /><ref name=":0">{{cite book|title=Theoretische Elektrotechnik|last=Simonyi|first=K.|date=1973|publisher=VEB Deutscher Verlag der Wissenschaften|edition=5th|location=Berlin|at=eq.&nbsp;20, p.&nbsp;47}}</ref> As the loop moves and/or deforms, it sweeps out a surface (see the right figure). As a small part of the loop {{math|d'''l'''}} moves with velocity {{math|'''v'''<sub>'''l'''</sub>}} over a short time {{math|d''t''}}, it sweeps out an area which vector is {{math|d'''A'''<sub>'''sweep'''</sub> {{=}} '''v'''<sub>'''l'''</sub> d''t'' × d'''l'''}} (note that this vector is toward out from the display in the right figure). Therefore, the change of the magnetic flux through the loop due to the deformation or movement of the loop over the time {{math|d''t''}} is<math>\mathbf{d}\Phi_B=\int \mathbf{B} \cdot \mathbf{dA}_{sweep}=\int \mathbf{B} \cdot (\mathbf{v}_{\mathbf{l}} \mathrm{d}t \times \mathrm{d}\mathbf{l}) = -\int \mathrm{d}t \mathrm{d}\mathbf{l} \cdot (\mathbf{v}_{\mathbf{l}}\times\mathbf{B})</math>

Here, [[Vector calculus identities|identities of triple scalar products]] are used. Therefore,

:<math>\frac{\mathrm{d}}{\mathrm{d}t} \int_{\Sigma(t)} \mathbf{B}(t_0) \cdot \mathrm{d}\mathbf{A} = -\oint_{\partial \Sigma(t_0)} (\mathbf{v}_{\mathbf{l}}(t_0)\times \mathbf{B}(t_0))\cdot \mathrm{d}\mathbf{l}</math>

where {{math|'''v'''<sub>'''l'''</sub>}} is the velocity of a part of the loop {{math|∂'''Σ'''}}.

Putting these together results in,

:<math>\left. \frac{\mathrm{d}\Phi_B}{\mathrm{d}t}\right|_{t=t_0} = \left(- \oint_{\partial \Sigma(t_0)} \mathbf{E}(t_0) \cdot \mathrm{d}\mathbf{l}\right) + \left(- \oint_{\partial \Sigma(t_0)} \bigl(\mathbf{v}_{\mathbf{l}}(t_0)\times \mathbf{B}(t_0)\bigr)\cdot \mathrm{d}\mathbf{l} \right)</math>
:<math>\left. \frac{\mathrm{d}\Phi_B}{\mathrm{d}t}\right|_{t=t_0} = - \oint_{\partial \Sigma(t_0)} \bigl( \mathbf{E}(t_0) + \mathbf{v}_{\mathbf{l}}(t_0)\times \mathbf{B}(t_0) \bigr) \cdot \mathrm{d}\mathbf{l}.</math>
|}

The result is:

:<math>\frac{\mathrm{d}\Phi_B}{\mathrm{d}t} = - \oint_{\partial \Sigma} \left( \mathbf{E} + \mathbf{v}_{\mathbf{l}} \times \mathbf{B} \right) \cdot \mathrm{d}\mathbf{l}.</math>
where {{math|∂Σ}} is the boundary (loop) of the surface {{math|Σ}}, and {{math|'''v'''<sub>'''l'''</sub>}} is the velocity of a part of the boundary.

In the case of a conductive loop, EMF (Electromotive Force) is the electromagnetic work done on a unit charge when it has traveled around the loop once, and this work is done by the [[Lorentz force law|Lorentz force]].  Therefore, EMF is expressed as

:<math>\mathcal{E} = \oint \left(\mathbf{E} + \mathbf{v}\times\mathbf{B}\right) \cdot \mathrm{d}\mathbf{l}</math>

where <math>\mathcal{E}</math> is EMF and {{math|'''v'''}} is the unit charge velocity.

In a macroscopic view, for charges on a segment of the loop, {{math|'''v'''}} consists of two components in average; one is the velocity of the charge along the segment {{math|'''v'''<sub>'''t'''</sub>}}, and the other is the velocity of the segment {{math|'''v'''<sub>'''l'''</sub>}} (the loop is deformed or moved). {{math|'''v'''<sub>'''t'''</sub>}} does not contribute to the work done on the charge since the direction of {{math|'''v'''<sub>'''t'''</sub>}} is same to the direction of <math>d\mathbf{l}</math>. Mathematically,

<math>(\mathbf{v}\times B)\cdot d\mathbf{l}=((\mathbf{v}_t + \mathbf{v}_l)\times B)\cdot d\mathbf{l}=(\mathbf{v}_t\times B+\mathbf{v}_l\times B)\cdot d\mathbf{l}=(\mathbf{v}_l\times B)\cdot d\mathbf{l}</math>

since <math>(\mathbf{v}_t\times B)</math> is perpendicular to <math>d\mathbf{l}</math> as <math>\mathbf{v}_t</math> and <math>d\mathbf{l}</math> are along the same direction. Now we can see that, for the conductive loop, EMF is same to the time-derivative of the magnetic flux through the loop except for the sign on it. Therefore, we now reach the equation of Faraday's law (for the conductive loop) as

:<math>\frac{\mathrm{d}\Phi_B}{\mathrm{d}t} = -\mathcal{E}</math>

where <math>\mathcal{E} = \oint \left(\mathbf{E} + \mathbf{v}_l\times\mathbf{B}\right) \cdot \mathrm{d}\mathbf{l}</math>. With breaking this integral, <math>\oint\mathbf{E}\cdot\mathrm{d}\mathbf{l}</math> is for the transformer EMF (due to a time-varying magnetic field) and <math>\oint \left(\mathbf{v}_l\times\mathbf{B}\right) \cdot \mathrm{d}\mathbf{l}</math> is for the motional EMF (due to the magnetic Lorentz force on charges by the motion or deformation of the loop in the magnetic field).

==EMF for non-thin-wire circuits==

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 and the reason is not just from the obvious reason that EMF is undefined in empty space when no conductor is present. 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.<ref name=Stewart>{{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.}}</ref> The two examples illustrated below show that one often obtains incorrect results when the motion of {{math|∂Σ}} is divorced from the motion of the material.<ref name=Feynman/>

<gallery widths="300">
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'' Vol. II page 17-3.
</gallery>

One can analyze examples like these by taking care that the path {{math|∂Σ}} moves with the same velocity as the material.<ref name=Stewart/> Alternatively, one can always correctly calculate the EMF by combining Lorentz force law with the Maxwell–Faraday equation:<ref name=Feynman/><ref name=HughesYoung>{{cite book|title=The Electromagnetodynamics of Fluid|first1=W. F.|last1=Hughes|first2=F. J.|last2=Young|publisher=John Wiley|date=1965|at=Eq. (2.6–13) p. 53}}</ref>
:<math>\mathcal{E} = \int_{\partial \Sigma} (\mathbf{E} + \mathbf{v}_m \times \mathbf{B}) \cdot \mathrm{d}\mathbf{l} = -\int_\Sigma \frac{\partial \mathbf{B}}{\partial t} \cdot \mathrm{d}\Sigma + \oint_{\partial \Sigma} (\mathbf{v}_m\times\mathbf{B}) \cdot \mathrm{d}\mathbf{l}</math>
where "it is very important to notice that (1) {{math|['''v'''<sub>''m''</sub>]}} is the velocity of the conductor ... not the velocity of the path element {{math|d'''l'''}} and (2) in general, the partial derivative with respect to time cannot be moved outside the integral since the area is a function of time."<ref name=HughesYoung/>

==Faraday's law and relativity==
{{Further|Moving magnet and conductor problem}}

===Two phenomena===
Faraday's law is a single equation describing two different phenomena: the ''motional EMF'' generated by a magnetic force on a moving wire (see the [[Lorentz force#Force on a current-carrying wire|Lorentz force]]), and the ''transformer EMF'' generated by an electric force due to a changing magnetic field (described by the [[#Maxwell–Faraday equation|Maxwell–Faraday equation]]).

[[James Clerk Maxwell]] drew attention to this fact in his 1861 paper ''[[On Physical Lines of Force]]''.<ref>{{cite journal|author-link = James Clerk Maxwell|last=Clerk Maxwell|first= James|journal = [[Philosophical Magazine]]|doi = 10.1080/1478643100365918|pages = 11–23|publisher = [[Taylor & Francis]]|title = On physical lines of force|volume = 90|year = 1861}}</ref> 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.<ref name=Griffiths1>{{cite book|last=Griffiths|first=David J.|title=Introduction to Electrodynamics|url=https://archive.org/details/introductiontoel00grif_0|edition=3rd|pages=301–3|publisher=Prentice Hall|year=1999|location=Upper Saddle River, NJ|isbn=0-13-805326-X}}<br>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".</ref> As Richard Feynman states:

{{Quotation|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''' {{=}} −∂<sub>''t''</sub>'''B'''}} 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]]''<ref name=Feynman>{{cite book|author-link=Richard Feynman|last1=Feynman|first1=R. P.|editor1-last=Leighton|editor1-first=R. B.|editor2-last=Sands|editor2-first=M. L.|title=The Feynman Lectures on Physics|year=2006|at=Vol. II, p. 17-2|publisher=Pearson/Addison-Wesley|location=San Francisco|url=https://books.google.com/books?id=zUt7AAAACAAJ&dq=intitle:Feynman+intitle:Lectures+intitle:on+intitle:Physics|isbn=0-8053-9049-9}}<br>"The flux rule" is the terminology that Feynman uses to refer to the law relating magnetic flux to EMF.</ref>}}

===Einstein's view===
Reflection on this apparent dichotomy was one of the principal paths that led [[Einstein]] to develop [[special relativity]]:
{{Quotation|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]]''<ref>{{cite web|first=Albert|last=Einstein|authorlink=Albert Einstein|url=http://www.fourmilab.ch/etexts/einstein/specrel/specrel.pdf|title=On the Electrodynamics of Moving Bodies}}</ref>}}

==See also==
{{Wikipedia books|Maxwell's equations}}
{{columns-list|colwidth=22em|
* [[Eddy current]]
* [[Inductance]]
* [[Maxwell's equations]]
* [[Crosstalk]]
* [[Faraday paradox]]
}}

==References==
{{Reflist|30em}}

==Further reading==
* {{cite book|url=https://books.google.com/books?id=vAsJAAAAIAAJ&printsec=frontcover&dq=intitle:a+intitle:treatise+intitle:on+intitle:electricity+intitle:an+intitle:magnetism&cad=0_1#v=onepage&q&f=false |last= Clerk Maxwell |first=James |authorlink=James Clerk Maxwell |date=1881 |title=A treatise on electricity and magnetism, Vol. II |at=ch.&nbsp;III, sec.&nbsp;530, p.&nbsp;178|location=Oxford|publisher=Clarendon Press|ISBN=0-486-60637-6}}

==External links==
* [https://web.archive.org/web/20080725021957/http://www.magnet.fsu.edu/education/tutorials/java/electromagneticinduction/index.html A simple interactive Java tutorial on electromagnetic induction] National High Magnetic Field Laboratory
* [https://web.archive.org/web/20080530092914/http://www.physics.smu.edu/~vega/em1304/lectures/lect13/lect13_f03.ppt R. Vega ''Induction: Faraday's law and Lenz's law'' – Highly animated lecture]
* [http://hyperphysics.phy-astr.gsu.edu/HBASE/hframe.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 java simulation on motional EMF]

{{Michael Faraday}}

{{DEFAULTSORT:Electromagnetic Induction}}
[[Category:Electrodynamics]]
[[Category:Michael Faraday]]
[[Category:Maxwell's equations]]