Jump to content

Faraday's law of induction: Difference between revisions

From Wikipedia, the free encyclopedia
Browse history interactively
← Previous editNext edit →
Content deleted Content added
VisualWikitext
No edit summary
←Blanked the page
Tag: Blanking
Line 1: Line 1:
{{For|applications and consequences of the law|Electromagnetic induction}}
{{electromagnetism|cTopic=Electrodynamics}}
HI GUYS HERE IS FARADAY LAW Any change in the magnetic environment of a coil of wire will cause a voltage (emf) to be "induced" in the coil. No matter how the change is produced, the voltage will be generated. The change could be produced by changing the magnetic field strength, moving a magnet toward or away from the coil, moving the coil into or out of the magnetic field, rotating the coil relative to the magnet, etc.What is electromagnetic induction?
Electromagnetic induction is the process by which a current can be induced to flow due to a changing magnetic field.
In our article on the magnetic force we looked at the force experienced by moving charges in a magnetic field. The force on a current-carrying wire due to the electrons which move within it when a magnetic field is present is a classic example. This process also works in reverse. Either moving a wire through a magnetic field or (equivalently) changing the strength of the magnetic field over time can cause a current to flow.
How is this described?
There are two key laws that describe electromagnetic induction:
Faraday's law, due to 19ᵗʰ century physicist Michael Faraday. This relates the rate of change of magnetic flux through a loop to the magnitude of the electro-motive force \mathcal{E}E induced in the loop. The relationship is
\mathcal{E} = \frac{\mathrm{d}\Phi}{\mathrm{d}t}E=
dt
dΦ
​
The electromotive force or EMF refers to the potential difference across the unloaded loop (i.e. when the resistance in the circuit is high). In practice it is often sufficient to think of EMF as voltage since both voltage and EMF are measured using the same unit, the volt. [Explain]
Lenz's law is a consequence of conservation of energy applied to electromagnetic induction. It was formulated by Heinrich Lenz in 1833. While Faraday's law tells us the magnitude of the EMF produced, Lenz's law tells us the direction that current will flow. It states that the direction is always such that it will oppose the change in flux which produced it. This means that any magnetic field produced by an induced current will be in the opposite direction to the change in the original field.
Lenz's law is typically incorporated into Faraday's law with a minus sign, the inclusion of which allows the same coordinate system to be used for both the flux and EMF. The result is sometimes called the Faraday-Lenz law,
\mathcal{E} = -\frac{\mathrm{d}\Phi}{\mathrm{d}t}E=−
dt
dΦ
​
In practice we often deal with magnetic induction in multiple coils of wire each of which contribute the same EMF. For this reason an additional term NNN representing the number of turns is often included, i.e.
\mathcal{E} = -N \frac{\mathrm{d}\Phi}{\mathrm{d}t}E=−N
dt
dΦ
​
What is the connection between Faraday's law of induction and the magnetic force?
While the full theoretical underpinning of Faraday's law is quite complex, a conceptual understanding of the direct connection to the magnetic force on a charged particle is relatively straightforward.
'''Faraday's law of induction''' 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''' is a generalization of Faraday's law, and is listed as one of [[Maxwell's equations]].

==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|year=1998|pages=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|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 |page=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 |page=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 |page=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==

===Qualitative statement===
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.}}</ref>}}
This version of Faraday's law strictly holds only when the closed circuit is a loop of infinitely thin wire,<ref name=Feynman/> and is invalid in other circumstances as discussed [[#"Counterexamples" to Faraday's law|below]]. A different version, the Maxwell–Faraday equation (discussed [[#Maxwell–Faraday equation|below]]), is valid in all circumstances.

===Quantitative===
[[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).]]
Faraday's law of induction makes use of the [[magnetic flux]] {{math|Φ<sub>''B''</sub>}} through a hypothetical surface {{math|Σ}} whose boundary is a wire loop. Since the wire loop may be moving, we write {{math|Σ(''t'')}} for the surface. The magnetic flux is defined by a [[surface integral]]:

:<math> \Phi_B = \iint\limits_{\Sigma(t)} \mathbf{B}(\mathbf{r}, 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]] (also called "magnetic flux density"), and {{math|'''B'''·d'''A'''}} is a [[dot product|vector dot product]] (the infinitesimal amount of magnetic flux through the infinitesimal area element {{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://www.amazon.com/gp/reader/013805326X/ref=sib_dp_pt/104-2951702-6987112#reader-link|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 {{mathcal|E}} 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===
[[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 is a modification and generalisation of Faraday's law that states that a time-varying magnetic field will always accompany a spatially 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>

{{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'''}}.

==Proof of Faraday's law==
The four [[Maxwell's equations]] (including the Maxwell–Faraday equation), along with the [[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, possibly moving surface in space {{math|Σ}}:
:<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, [[Gauss's law for magnetism]], and some vector calculus; the details are in the box below:
:{| class="toccolours collapsible collapsed" width="80%" style="text-align:left"
!Click [show] (right) to see the detailed evaluation and simplification of the time-derivative of flux.
|-
|Consider the time-derivative of flux through a possibly moving loop, with area {{math|Σ(''t'')}}:
:<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 (see above). 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>
[[Image:Faraday Area.PNG|thumbnail|300px|Area swept out by vector element {{math|d'''l'''}} of curve {{math|∂'''Σ'''}} in time {{math|d''t''}} when moving with velocity {{math|'''v'''}}.]]
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>
This is the most difficult part of the proof; more details and alternate approaches can be found in references.<ref name=Davison/><ref name=Krey/><ref>{{cite book|first=K.|last=Simonyi|title=Theoretische Elektrotechnik|edition=5th|publisher=VEB Deutscher Verlag der Wissenschaften|location=Berlin|date=1973|at=eq.&nbsp;20, p.&nbsp;47}}</ref> As the loop moves and/or deforms, it sweeps out a surface (see figure on right). The magnetic flux through this swept-out surface corresponds to the magnetic flux that is either entering or exiting the loop, and therefore this is the magnetic flux that contributes to the time-derivative. (This step implicitly uses [[Gauss's law for magnetism]]: Since the flux lines have no beginning or end, they can only get into the loop by getting cut through by the wire.) As a small part of the loop {{math|d'''l'''}} moves with velocity {{math|'''v'''<sub>'''l'''</sub>}} for a short time {{math|d''t''}}, it sweeps out a vector area vector {{math|d'''A''' {{=}} '''v'''<sub>'''l'''</sub> d''t'' × d'''l'''}}. Therefore, the change in magnetic flux through the loop here is
:<math>\mathbf{B} \cdot (\mathbf{v}_{\mathbf{l}} \, \mathrm{d}t \times \mathrm{d}\mathbf{l}) = -\mathrm{d}t \, \mathrm{d}\mathbf{l} \cdot (\mathbf{v}_{\mathbf{l}}\times\mathbf{B})</math>
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 the curve {{math|∂'''Σ'''}}.

Putting these together,
:<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 of the surface {{math|Σ}}, and {{math|'''v'''<sub>'''l'''</sub>}} is the velocity of that boundary.

While this equation is true for any arbitrary moving surface {{math|Σ}} in space, it can be simplified further in the special case that {{math|∂Σ}} is a loop of wire. In this case, we can relate the right-hand side to the EMF. Specifically, EMF is defined as the energy available per unit charge that travels once around the loop. Therefore, by the [[Lorentz force law]],
:<math>\mathcal{E} = \oint \left(\mathbf{E} + \mathbf{v}_m\times\mathbf{B}\right) \cdot \mathrm{d}\mathbf{l}</math>
where {{mathcal|E}} is EMF and {{math|'''v'''<sub>''m''</sub>}} is the material velocity, i.e. the velocity of the atoms that make up the circuit. If {{math|∂Σ}} is a loop of wire, then {{math|'''v'''<sub>''m''</sub> {{=}} '''v'''<sub>'''l'''</sub>}}, and hence:
:<math>\frac{\mathrm{d}\Phi_B}{\mathrm{d}t} = -\mathcal{E}</math>

==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 not just for 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="300px">
Image: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'''}}. The magnetic Lorentz force {{math|'''v''' × '''B'''}} drives the 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.
Image: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 the [[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 [[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 (due to 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://www.amazon.com/gp/reader/013805326X/ref=sib_dp_pt/104-2951702-6987112#reader-link|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==
* [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]

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

Revision as of 04:36, 20 November 2018

Retrieved from "https://en.wikipedia.org/w/index.php?title=Faraday%27s_law_of_induction&oldid=869756008"