Editing Electromagnetic induction (section)

==Theory==

===Faraday's law of induction and Lenz's law===
{{main|Faraday's law of induction}}
[[File:Solenoid-1.png|right|thumb|upright=1.3|A solenoid]]
[[File:VFPt_Solenoid_correct2.svg|right|thumb|upright=1.3|The longitudinal cross section of a solenoid with a constant electrical current running through it. The magnetic field lines are indicated, with their direction shown by arrows. The magnetic flux corresponds to the 'density of field lines'. The magnetic flux is thus densest in the middle of the solenoid, and weakest outside of it.]]

Faraday's law of induction makes use of the [[magnetic flux]] Φ<sub>B</sub> through a region of space enclosed by a wire loop. The magnetic flux is defined by a [[surface integral]]:<ref>
{{cite book
 |last=Good |first=R. H.
 |year=1999
 |title=Classical Electromagnetism
 |page=107
 |publisher=[[Saunders College Publishing]]
 |isbn=0-03-022353-9
}}</ref>
<math display="block"> \Phi_\mathrm{B} = \int_{\Sigma} \mathbf{B} \cdot d \mathbf{A}\, , </math>
where ''d'''''A''' is an element of the surface Σ enclosed by the wire loop, '''B''' is the magnetic field. The [[dot product]] '''B'''·''d'''''A''' corresponds to an infinitesimal amount of magnetic flux. In more visual terms, the magnetic flux through the wire loop is proportional to the number of [[field line|magnetic field lines]] that pass through the loop.

When the flux through the surface changes, [[Faraday's law of induction]] says that the wire loop acquires an [[electromotive force]] (emf).{{refn|group=note|The EMF is the voltage that would be measured by cutting the wire to create an [[Open-circuit voltage|open circuit]], and attaching a [[voltmeter]] to the leads. Mathematically, <math>\mathcal{E}</math> is defined as the energy available from a unit charge that has traveled once around the wire loop.<ref name=Feynman>
{{cite book
 |last1=Feynman |first1=R. P.
 |last2=Leighton |first2=R. B.
 |last3=Sands |first3=M. L.
 |year=2006
 |title=The Feynman Lectures on Physics, Volume 2
 |url=https://feynmanlectures.caltech.edu/II_17.html#Ch17-S2
 |page=17{{hyphen}}2
 |publisher=[[Pearson Education|Pearson]]/[[Addison-Wesley]]
 |isbn=0-8053-9049-9 
}}</ref><ref name=Griffiths2>
{{cite book
 |last=Griffiths
 |first=D. J.
 |year=1999
 |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]]
 |isbn=0-13-805326-X
 }}</ref><ref>
{{cite book
 |last1=Tipler |first1=P. A.
 |last2=Mosca |first2=G.
 |year=2003
 |title=Physics for Scientists and Engineers
 |page=795
 |edition=5th
 |url=https://books.google.com/books?id=R2Nuh3Ux1AwC&pg=PA795
 |publisher=[[W.H. Freeman]]
 |isbn=978-0716708100
}}</ref>}} The most widespread version of this law states that the induced electromotive force in any closed circuit is equal to the [[time derivative|rate of change]] of the [[magnetic flux]] enclosed by the circuit:<ref name="Jordan & Balmain (1968)">
{{cite book
 |last1=Jordan |first1=E.
 |last2=Balmain |first2=K. G.
 |year=1968
 |title=Electromagnetic Waves and Radiating Systems
 |url=https://archive.org/details/electromagneticw00jord_157 |url-access=limited |page=[https://archive.org/details/electromagneticw00jord_157/page/n113 100]
 |edition=2nd
 |publisher=[[Prentice-Hall]]
|isbn=978-0132499958
 }}</ref><ref name="Hayt (1989)">
{{cite book
 |last=Hayt
 |first=W.
 |year=1989
 |title=Engineering Electromagnetics
 |page=[https://archive.org/details/engineeringelect5thhayt/page/312 312]
 |edition=5th
 |publisher=[[McGraw-Hill]]
 |isbn=0-07-027406-1
 |url=https://archive.org/details/engineeringelect5thhayt/page/312
 }}</ref>
<math display="block">\mathcal{E} = -\frac{d\Phi_\mathrm{B}}{dt} \, , </math>
where <math>\mathcal{E}</math> is the emf and Φ<sub>B</sub> is the [[magnetic flux]]. The direction of the electromotive force is given by [[Lenz's law]] which states that an induced current will flow in the direction that will oppose the change which produced it.<ref>
{{cite book
 |last=Schmitt |first=R.
 |year=2002
 |title=Electromagnetics Explained
 |publisher=Newnes |url=https://archive.org/details/electromagnetics0000schm
 |url-access=registration |page=[https://archive.org/details/electromagnetics0000schm/page/75 75]
 |isbn=978-0750674034
}}</ref> This is due to the negative sign in the previous equation. To increase the generated emf, a common approach is to exploit [[flux linkage]] by creating a tightly wound [[inductor|coil of wire]], composed of ''N'' identical turns, each with the same magnetic flux going through them. The resulting emf is then ''N'' times that of one single wire.<ref>
{{cite book
 |last1=Whelan |first1=P. M.
 |last2=Hodgeson |first2=M. J.
 |year=1978
 |title=Essential Principles of Physics
 |edition=2nd
 |publisher=[[John Murray (publishing house)|John Murray]]
 |isbn=0-7195-3382-1
}}</ref><ref>
{{cite web
 |last=Nave |first=C. 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
}}</ref>
<math display="block"> \mathcal{E} = -N \frac{d\Phi_\mathrm{B}}{dt} </math>

Generating an emf through a variation of the magnetic flux through the surface of a wire loop can be achieved in several ways:
# the magnetic field '''B''' changes (e.g. an alternating magnetic field, or moving a wire loop towards a bar magnet where the B field is stronger),
# the wire loop is deformed and the surface Σ changes,
# the orientation of the surface ''d'''''A''' changes (e.g. spinning a wire loop into a fixed magnetic field),
# any combination of the above

===Maxwell–Faraday equation===
{{See also|Faraday's law of induction#Maxwell–Faraday equation}}
In general, the relation between the emf <math> \mathcal{E}</math> in a wire loop encircling a surface &Sigma;, and the electric field '''E''' in the wire is given by
<math display="block"> \mathcal{E} = \oint_{\partial \Sigma} \mathbf{E} \cdot d\boldsymbol{\ell} </math>
where ''d'''''ℓ''' is an element of contour of the surface Σ, combining this with the definition of flux
<math display="block"> \Phi_\mathrm{B} = \int_{\Sigma} \mathbf{B} \cdot d \mathbf{A}\, , </math>
we can write the integral form of the Maxwell–Faraday equation
<math display="block"> \oint_{\partial \Sigma} \mathbf{E} \cdot d\boldsymbol{\ell} = -\frac{d}{d t} { \int_{\Sigma} \mathbf{B} \cdot d\mathbf{A}} </math>

It is one of the four [[Maxwell's equations]], and therefore plays a fundamental role in the theory of [[classical electromagnetism]].

===Faraday's law and relativity===
Faraday's law describes 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'' that is generated by an electric force due to a changing magnetic field (due to the differential form of the [[#Maxwell–Faraday equation|Maxwell–Faraday equation]]). [[James Clerk Maxwell]] drew attention to the separate physical phenomena in 1861.<ref>
{{cite journal
 |last=Maxwell |first=J. C.
 |year=1861
 |title=On physical lines of force
 |journal = [[Philosophical Magazine]]
 |volume=90 |issue=139
 |pages=11–23
 |doi=10.1080/14786446108643033
 |doi-access=free
 }}</ref><ref name=Griffiths1>
{{cite book
 |last=Griffiths
 |first=D. J.
 |year=1999
 |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]]
 |isbn=0-13-805326-X
 }} Note that the law relating flux to EMF, which this article calls "Faraday's law", is referred to by Griffiths as the "universal flux rule". He uses the term "Faraday's law" to refer to what this article calls the "Maxwell–Faraday equation".</ref> This is believed to be a unique example in physics of where such a fundamental law is invoked to explain two such different phenomena.<ref name=Feynman2>"The flux rule" is the terminology that Feynman uses to refer to the law relating magnetic flux to EMF. {{cite book
 |last1=Feynman
 |first1=R. P.
 |last2=Leighton
 |first2=R. B.
 |last3=Sands
 |first3=M. L.
 |year=2006
 |title=The Feynman Lectures on Physics, Volume II
 |page=17{{hyphen}}2
 |publisher=[[Pearson Education|Pearson]]/[[Addison-Wesley]]
 |url=https://books.google.com/books?id=zUt7AAAACAAJ&q=intitle:Feynman+intitle:Lectures+intitle:on+intitle:Physics
 |isbn=0-8053-9049-9
 }}{{Dead link|date=October 2023 |bot=InternetArchiveBot |fix-attempted=yes }}</ref>

[[Albert Einstein]] noticed that the two situations both corresponded to a relative movement between a conductor and a magnet, and the outcome was unaffected by which one was moving. This was one of the principal paths that led him to develop [[special relativity]].<ref>
{{cite journal
 |last=Einstein |first=A.
 |year=1905
 |title=Zur Elektrodynamik bewegter Körper
 |journal=[[Annalen der Physik]]
 |volume=17 |issue=10 |pages=891–921
 |bibcode=1905AnP...322..891E
 |doi=10.1002/andp.19053221004
 |url=http://sedici.unlp.edu.ar/bitstream/handle/10915/2786/Documento_completo__.pdf?sequence=1
 |doi-access=free
 }}<br />
:Translated in {{cite book
 |last=Einstein |first=A.
 |others=Jeffery, G.B.; Perret, W. (transl.)
 |year=1923
 |chapter=On the Electrodynamics of Moving Bodies
 |chapter-url=http://www.fourmilab.ch/etexts/einstein/specrel/specrel.pdf
 |title=The Principle of Relativity
 |publisher=[[Methuen and Company]]
 |location=London
}}</ref>