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{{short description|Electrical action produced by a non-electrical source}}
{{distinguish|Electromagnetic field}}
{{Electromagnetism |Network}}

In [[electromagnetism]] and [[electronics]], '''electromotive force''' ('''emf''', or '''EMF''')<ref>{{Cite encyclopedia|title=EMF|url=https://www.ahdictionary.com/word/search.html?q=electromotive+force|year=1992|dictionary=American Heritage Dictionary of the English Language, 3rd ed.|publisher=Houghton Mifflin}}</ref><ref>{{Cite encyclopedia|title=EMF|url=https://www.oed.com/view/Entry/58919?rskey=PWMafP&result=1#eid5887886|dictionary=Oxford English Dictionary|url-access=subscription}}<!--template "cite OED" doesn't specify entry--></ref> or '''electromotance''', denoted <math>\mathcal{E}</math>, is an [[energy]] transfer to an [[electric circuit]] per unit of [[electric charge]], measured in [[volt]]s. Devices called electrical ''[[transducer]]s'' provide an emf<ref name="Tipler803" /> by [[Energy transformation|converting]] other forms of [[energy]] into [[electrical energy]].<ref name="Tipler803">{{cite book|last1=Tipler|first1=Paul A.|title=Physics|date=January 1976|publisher=Worth Publishers, Inc.|location=New York, NY|isbn=978-0-87901-041-6|page=[https://archive.org/details/physics00tipl/page/803 803]|url=https://archive.org/details/physics00tipl/page/803}}</ref> Other types of electrical equipment also produce an emf, such as [[Battery (electricity)|batteries]], which convert [[chemical energy]], and [[Electric generator|generator]]s, which convert [[mechanical energy]].<ref name="Stewart">{{cite book |last=Stewart |first=Joseph V. |title=Intermediate electromagnetic theory |publisher=World Scientific |year=2001 |isbn=978-981-02-4470-5 |publication-place=Singapore River Edge, NJ |page=389 |oclc=47127179}}</ref> This energy conversion is achieved by [[Force|physical forces]] applying [[Work (physics)|physical work]] on [[electric charge]]s. However, electromotive force itself is not a physical force,<ref>{{Cite book |last=Matthews |first=Michael R. |url=https://books.google.com/books?id=7bP0AwAAQBAJ&dq=electromotive+force+history&pg=PA142 |title=International Handbook of Research in History, Philosophy and Science Teaching |date=2014-07-03 |publisher=Springer |isbn=978-94-007-7654-8 |pages=142 |language=en |quote=[Volta] stated that a new type of "force" was acting upon the charges, separating them and keeping them separated, and he called this action the electromotive force, the name that is still applied.}}</ref> and [[ISO]]/[[International Electrotechnical Commission|IEC]] standards have deprecated the term in favor of '''source voltage''' or '''source tension''' instead (denoted {{nowrap|{{tmath|U_\mathrm{s} }}}}).<ref>{{Cite web |title=IEC 60050 - International Electrotechnical Vocabulary - Details for IEV number 131-12-22: "source voltage" |url=https://www.electropedia.org/iev/iev.nsf/display?openform&ievref=131-12-22 |access-date=2022-12-19 |website=www.electropedia.org}}</ref><ref name=80000-6:2022>{{cite web|title=IEC 80000-6:2022|url=https://www.iso.org/standard/77846.html|publisher=[[International Organization for Standardization]]|access-date=2022-12-19}}</ref>

An [[Hydraulic analogy|electronic–hydraulic analogy]] may view emf as the [[mechanical work]] done to water by a [[pump]], which results in a [[Volt#Water-flow analogy|pressure difference (analogous to voltage)]].<ref>{{cite journal |first=Irving|last=Langmuir |year=1916 |title=The Relation Between Contact Potentials and Electrochemical Action |url=https://books.google.com/books?id=OW0SAAAAYAAJ&q=%22electromotive%20force%20is%20that%22&pg=PA172 |journal=Transactions of the American Electrochemical Society |publisher=The Society |volume=29 |pages=175}}</ref>

In [[electromagnetic induction]], emf can be defined around a closed loop of a [[electrical conductor|conductor]] as the electromagnetic [[work (physics)|work]] that would be done on an [[Elementary charge|elementary electric charge]] (such as an [[electron]]) if it travels once around the loop.<ref name=Cook>{{cite book
 | title = The Theory of the Electromagnetic Field
 | first = David M.|last=Cook
 | publisher = Courier Dover
 | year = 2003
 | isbn = 978-0-486-42567-2
 | page = 157
 | url = https://books.google.com/books?id=bI-ZmZWeyhkC&pg=PA157
 }}</ref>

For two-[[Terminal (electronics)|terminal]] devices modeled as a [[Thévenin's theorem|Thévenin equivalent circuit]], an equivalent emf can be measured as the [[open-circuit voltage]] between the two terminals. This emf can drive an [[electric current]] if an external [[Electronic circuit|circuit]] is attached to the terminals, in which case the device becomes the [[voltage source]] of that circuit.

Although an emf gives rise to a voltage and can be measured as a voltage and may sometimes informally be called a "voltage", they are not the same phenomenon (see {{Section link||Distinction with potential difference}}).

==Overview==
Devices that can provide emf include [[electrochemical cell]]s, [[Thermoelectric effect|thermoelectric device]]s, [[solar cells]], [[photodiode]]s, [[electrical generator]]s, [[inductor]]s, [[electrical transformer|transformer]]s and even [[Van de Graaff generator]]s.<ref name="Lerner">{{cite book |first=Lawrence M. |last=Lerner |url=https://books.google.com/books?id=Nv5GAyAdijoC&pg=PA727 |title=Physics for scientists and engineers |publisher=Jones & Bartlett Publishers |year=1997 |isbn=978-0-7637-0460-5 |pages=724–727}}</ref><ref name=Tipler>{{cite book
 | title=Physics for Scientists and Engineers
 |first1=Paul A.|last1=Tipler |first2=Gene|last2=Mosca | url=https://books.google.com/books?id=BMVR37-8Jh0C&pg=PA850
 | page=850
 | isbn=978-1-4292-0124-7
 | year=2007
 | edition=6 
 | publisher=Macmillan
 }}</ref> In nature, emf is generated when [[magnetic field]] fluctuations occur through a surface. For example, the shifting of the [[Earth's magnetic field]] during a [[geomagnetic storm]] induces currents in an [[electrical grid]] as the lines of the magnetic field are shifted about and cut across the conductors.

In a battery, the charge separation that gives rise to a potential difference ([[voltage]]) between the terminals is accomplished by [[chemical reaction]]s at the [[electrode]]s that convert chemical [[potential energy]] into electromagnetic potential energy.<ref name=Schaum>{{cite book
 |title=Schaum's outline of theory and problems of beginning physics II
 |first1=Alvin M.|last1=Halpern |first2=Erich|last2=Erlbach |url=https://books.google.com/books?id=vN2chIay624C&pg=PA138
 |page=138 |year=1998
 |publisher=McGraw-Hill Professional
 |isbn=978-0-07-025707-8
 }}</ref><ref>{{cite book
 | title = Physics the easy way
 | first = Robert L.|last=Lehrman
 | publisher = Barron's Educational Series
 | year = 1998
 | isbn = 978-0-7641-0236-3
 | page = [https://archive.org/details/physicseasyway00lehr_0/page/274 274]
 | url = https://archive.org/details/physicseasyway00lehr_0
 | url-access = registration
 | quote = emf  separated charge reaction potential.
 }}</ref> A [[Voltaic Cell|voltaic cell]] can be thought of as having a "charge pump" of atomic dimensions at each electrode, that is:
{{Blockquote
|text=A (chemical) source of emf can be thought of as a kind of ''charge pump'' that acts to move positive charges from a point of low potential through its interior to a point of high potential.&nbsp;... By chemical, mechanical or other means, the source of emf performs work <math display="inline">\mathrm{d}W</math> on that charge to move it to the high-potential terminal. The emf <math display="inline">\mathcal{E}</math> of the source is defined as the work <math display="inline">\mathrm{d}W</math> done per charge {{tmath|\mathrm{d}q}}. {{tmath|1= \textstyle \mathcal{E} = \frac{\mathrm{d}W}{\mathrm{d}q} }}.<ref name="Singh">{{cite book|chapter-url=https://books.google.com/books?id=oS_vSI-3yuwC|title=Basic Physics|first=Kongbam Chandramani|publisher=Prentice Hall India|year=2009|isbn=978-81-203-3708-4|page=152|chapter=§3.16 EMF of a source|last=Singh}}
</ref>
}}

In an electrical generator, a time-varying magnetic field inside the generator creates an [[electric field]] via [[electromagnetic induction]], which creates a potential difference between the generator terminals. Charge separation takes place within the generator because electrons flow away from one terminal toward the other, until, in the open-circuit case, an electric field is developed that makes further charge separation impossible. The emf is countered by the electrical voltage due to charge separation. If a [[Electrical load|load]] is attached, this voltage can drive a current. The general principle governing the emf in such electrical machines is [[Faraday's law of induction]].

==History==
In 1801, [[Alessandro Volta]] introduced the term "force motrice électrique" to describe the active agent of a battery (which he had invented around 1798).<ref>
{{cite journal
 |journal     = Annales de Chimie
 |publisher   = Chez Fuchs, Paris
 |year        = 1801
 |url         = https://gallica.bnf.fr/ark:/12148/bpt6k65721573
 |title       = De l'électricité dite galvanique
 |first      = Alessandro|last=Volta
}}</ref>
This is called the "electromotive force" in English.

Around 1830, [[Michael Faraday]] established that chemical reactions at each of two electrode–electrolyte interfaces provide the "seat of emf" for the voltaic cell. That is, these reactions drive the current and are not an endless source of energy as the earlier [[Superseded theories in science|obsolete theory]] thought.<ref name=cajori/> In the open-circuit case, charge separation continues until the electrical field from the separated charges is sufficient to arrest the reactions. Years earlier, [[Alessandro Volta]], who had measured a contact potential difference at the metal–metal (electrode–electrode) interface of his cells, held the incorrect opinion that contact alone (without taking into account a chemical reaction) was the origin of the emf. It is independent of size of the cell but depends on the nature of the electrolyte used.

== Notation and units of measurement ==
Electromotive force (emf) is typically denoted by the symbol <math>\mathcal E</math> (script "E"). It represents the energy provided by a source per unit electric charge. The standard unit of emf in the International System of Units (SI) is the '''volt (V)'''.

In a device without [[internal resistance]], if an electric charge <math>q</math> passing through that device gains an energy <math>W</math> via work, the net emf for that device is the energy gained per unit charge: {{tmath| \textstyle  \tfrac{W}{q} }}. Like other measures of energy per charge, emf uses the [[International System of Units|SI]] unit [[volt]], which is equivalent to a [[joule]] (SI unit of energy) per [[coulomb]] (SI unit of charge).<ref>
{{cite book
 | title = Basic Electricity
 | first = Van|last=Valkenburgh
 | publisher = Cengage Learning
 | year = 1995
 | isbn = 978-0-7906-1041-2
 | pages = 1–46
 | url = https://books.google.com/books?id=vmg1UKsTntAC&q=electromotive-force+joules-per-coulomb+volts+charge+energy&pg=PT67
 }}</ref>

Electromotive force in [[electrostatic unit]]s is the [[statvolt]] (in the [[centimeter gram second system of units]] equal in amount to an [[erg]] per electrostatic unit of charge).

==Formal definitions==
{{em|Inside}} a source of emf (such as a battery) that is open-circuited, a charge separation occurs between the negative terminal ''N'' and the positive terminal ''P''.
This leads to an [[electrostatic field]] <math>\boldsymbol{E}_\mathrm{open\ circuit}</math> that points from ''P'' to ''N'', whereas the emf of the source must be able to drive current from ''N'' to ''P'' when connected to a circuit.
This led [[Max Abraham]]<ref name=Abraham>
{{cite book
 | title = The Classical Theory of Electricity and Magnetism
 | first1 = M.|last1=Abraham|first2=R.|last2=Becker
 | publisher = Blackie & Son
 | year = 1932
 | pages = 116–122
 | url = https://pdfcoffee.com/abraham-max-amp-becker-richard-classical-theory-of-electricity-and-magnetism-1932-pdf-pdf-free.html
 }}</ref> to introduce the concept of a [[Nonelectrostatic electric fields|nonelectrostatic field]] <math>\boldsymbol{E}'</math> that exists only inside the source of emf.
In the open-circuit case, {{tmath|1= \boldsymbol{E}' = - \boldsymbol{E}_\mathrm{open\ circuit} }}, while when the source is connected to a circuit the electric field <math>\boldsymbol{E}</math> inside the source changes but <math>\boldsymbol{E}'</math> remains essentially the same.
In the open-circuit case, the [[Conservative field|conservative]] electrostatic field created by separation of charge exactly cancels the forces producing the emf.<ref name=Griffiths>
{{cite book
 | title = Introduction to Electrodynamics
 | first = David J|last=Griffiths
 | publisher = Pearson/Addison-Wesley
 | year = 1999
 | isbn = 978-0-13-805326-0
 | page = [https://archive.org/details/introductiontoel00grif_0/page/293 293]
 | edition = 3rd
 | url = https://archive.org/details/introductiontoel00grif_0/page/293
 }}</ref> 
Expressed mathematically:
<math display="block">\mathcal{E}_\mathrm{source}  = \int_{N}^{P} \boldsymbol{E}' \cdot \mathop{}\!\mathrm{d} \boldsymbol{ \ell }
= - \int_{N}^{P} \boldsymbol{E}_\mathrm{open\ circuit} \cdot \mathop{}\!\mathrm{d} \boldsymbol{ \ell }
=V_P - V_N \ ,</math>

where <math>\boldsymbol{E}_\mathrm{open\ circuit}</math> is the conservative electrostatic field created by the charge separation associated with the emf, <math>\mathrm{d}\boldsymbol{\ell}</math> is an element of the path from terminal ''N'' to terminal ''P'', {{code|⋅}} (dot operator) denotes the vector [[dot product]], and <math>V</math> is the electric scalar potential.<ref name="diode">
Only the electric field that results from charge separation caused by the emf is counted. While a solar cell has an electric field that results from a contact potential {{see below|{{slink||Contact potentials|Solar cell}}}}, this electric field component is not included in the integral. Only the electric field that results from charge separation caused by photon energy is included.
</ref>
This emf is the work done on a unit charge by the source's nonelectrostatic field <math>\boldsymbol{E}'</math> when the charge moves from ''N'' to ''P''.

When the source is connected to a load, its emf is just
<math display="block">\mathcal{E}_\mathrm{source} = \int_{N}^{P} \boldsymbol{E}' \cdot \mathop{}\!\mathrm{d} \boldsymbol{ \ell}\ ,</math>
and no longer has a simple relation to the electric field <math>\boldsymbol{E}</math> inside it.

In the case of a closed path in the presence of a varying [[magnetic field]], the integral of the [[electric field]] around the (stationary) closed loop <math>C</math> may be nonzero.
Then, the "''induced emf''" (often called the "induced voltage") in the loop is:<ref>{{cite book
 | title = Beyond the mechanical universe: from electricity to modern physics
 |first1=Richard P.|last1=Olenick |first2=Tom M.|last2=Apostol |first3=David L.|last3=Goodstein 
 | publisher = Cambridge University Press
 | year = 1986
 | isbn = 978-0-521-30430-6
 | page = 245
 | url = https://books.google.com/books?id=Ht4T7C7AXZIC&q=define+electromotive-force+around-a-closed-path&pg=RA1-PA245
 }}</ref>
<math display="block">\mathcal{E}_C = \oint_{C} \boldsymbol{E} \cdot \mathrm{d} \boldsymbol{ \ell }
= - \frac{\mathrm{d}\Phi_C}{\mathrm{d}t}
= - \frac{\mathrm{d}}{\mathrm{d}t} \oint_{C} \boldsymbol{A} \cdot \mathrm{d} \boldsymbol{ \ell }\ ,
</math>

where <math>\boldsymbol{E}</math> is the entire electric field, conservative and non-conservative, and the integral is around an arbitrary, but stationary, closed curve <math>C</math> through which there is a time-varying [[magnetic flux]] {{tmath|\Phi_C}}, and <math>\boldsymbol{A}</math> is the [[vector potential]].
The electrostatic field does not contribute to the net emf around a circuit because the electrostatic portion of the electric field is [[Conservative force|conservative]] (i.e., the work done against the field around a closed path is zero, by [[Kirchhoff's voltage law]], which is valid, as long as the circuit elements remain at rest and radiation is ignored).<ref name="McDonald">
{{cite web 
|last1=McDonald |first1=Kirk T. 
|title=Voltage Drop, Potential Difference and EMF 
|url=http://kirkmcd.princeton.edu/examples/volt.pdf 
|website=Physics Examples 
|publisher=Princeton University
|date=2012
}}</ref>
That is, the "induced emf" (like the emf of a battery connected to a load) is not a "voltage" in the sense of a difference in the electric scalar potential.

If the loop <math>C</math> is a conductor that carries current <math>I</math> in the direction of integration around the loop, and the magnetic flux is due to that current, we have that {{tmath|1= \Phi_B = L I}}, where <math>L</math> is the self inductance of the loop.
If in addition, the loop includes a coil that extends from point 1 to 2, such that the magnetic flux is largely localized to that region, it is customary to speak of that region as an [[inductor]], and to consider that its emf is localized to that region.
Then, we can consider a different loop <math>C'</math> that consists of the coiled conductor from 1 to 2, and an imaginary line down the center of the coil from 2 back to 1.  
The magnetic flux, and emf, in loop <math>C'</math> is essentially the same as that in loop {{tmath|C}}:
<math display="block">\mathcal{E}_C = \mathcal{E}_{C'}
= - \frac{\mathrm{d}\Phi_{C'}}{\mathrm{d}t}
= - L \frac{\mathrm{d}I}{\mathrm{d}t}
= \oint_C \boldsymbol{E} \cdot \mathrm{d} \boldsymbol{ \ell }
= \int_1^2 \boldsymbol{E}_\mathrm{conductor} \cdot \mathrm{d} \boldsymbol{ \ell }
- \int_1^2 \boldsymbol{E}_\mathrm{center\ line} \cdot \mathrm{d} \boldsymbol{ \ell }\ .
</math>

For a good conductor, <math>\boldsymbol{E}_\mathrm{conductor}</math> is negligible, so we have, to a good approximation,
<math display="block">L \frac{\mathrm d I}{\mathrm dt}
= \int_1^2 \boldsymbol{E}_\mathrm{center\ line} \cdot \mathrm{d} \boldsymbol{ \ell } 
= V_1 - V_2\ ,
</math>
where <math>V</math> is the electric scalar potential along the centerline between points 1 and 2.

Thus, we can associate an effective "voltage drop" <math>L\mathop{}\!\mathrm{d}I /\mathrm{d}t</math> with an inductor (even though our basic understanding of induced emf is based on the vector potential rather than the scalar potential), and consider it as a load element in Kirchhoff's voltage law,
<math display="block"> \sum \mathcal{E}_\mathrm{source} = \sum_\mathrm{load\ elements} \mathrm{voltage\ drops},
</math>

where now the induced emf is not considered to be a source emf.<ref name="FeynmanCook2">{{cite book
 | title = The Feynman Lectures on Physics, Vol. II, chap. 22
 | first1 = R.P.|last1=Feynman|first2=R.B.|last2=Leighton|first3=M.|last3=Sands
 | publisher = Addison Wesley
 | year = 1964
 | url = https://www.feynmanlectures.caltech.edu/II_22.html }}</ref>

This definition can be extended to arbitrary sources of emf and paths <math>C</math> moving with velocity <math>\boldsymbol{v}</math> through the electric field <math>\boldsymbol{E}</math> and magnetic field {{tmath|\boldsymbol{B} }}:<ref name=Cook2>{{cite book
 | title = The Theory of the Electromagnetic Field
 | first = David M.|last=Cook
 | publisher = Courier Dover
 | year = 2003
 | isbn = 978-0-486-42567-2
 | page = 158
 | url = https://books.google.com/books?id=bI-ZmZWeyhkC&pg=PA158
 }}</ref>
<math display="block">\begin{align}
\mathcal{E} &= \oint_{C} \left[\boldsymbol{E}  + \boldsymbol{v} \times \boldsymbol{B} \right] \cdot \mathrm{d} \boldsymbol{ \ell } \\
&\qquad+\frac{1}{q}\oint_{C}\mathrm {Effective \ chemical \ forces \ \cdot} \ \mathrm{d} \boldsymbol{ \ell } \\
&\qquad\qquad+\frac{1}{q}\oint_{C}\mathrm { Effective \ thermal \ forces\ \cdot}\  \mathrm{d} \boldsymbol{ \ell } \ ,
\end{align} </math>

which is a conceptual equation mainly, because the determination of the "effective forces" is difficult.
The term <math> \oint_{C} \left[\boldsymbol{E}  + \boldsymbol{v} \times \boldsymbol{B} \right] \cdot \mathrm{d} \boldsymbol{ \ell } </math>
is often called a "motional emf".

==In (electrochemical) thermodynamics==
When multiplied by an amount of charge <math>\mathrm{d}Q</math> the emf <math>\mathcal{E}</math> yields a thermodynamic work term <math>\mathcal{E}\mathop{}\!\mathrm{d}Q</math> that is used in the formalism for the change in [[Gibbs free energy|Gibbs energy]] when charge is passed in a battery:
<math display="block">\mathrm{d}G = -S\mathop{}\!\mathrm{d}T + V\mathop{}\!\mathrm{d}P + \mathcal{E}\mathop{}\!\mathrm{d}Q\ , </math>
where <math>G</math> is the Gibbs free energy, <math>S</math> is the [[entropy]], <math>V</math> is the system volume, <math>P</math> is its pressure, and <math>T</math> is its [[absolute temperature]].

The combination <math>(\mathcal{E}, Q)</math> is an example of a [[conjugate variables (thermodynamics)|conjugate pair of variables]]. At constant pressure the above relationship produces a [[Maxwell relation]] that links the change in open cell voltage with temperature <math>T</math> (a measurable quantity) to the change in entropy <math>S</math> when charge is passed [[isothermally]] and [[isobarically]]. The latter is closely related to the reaction [[entropy]] of the electrochemical reaction that lends the battery its power. This Maxwell relation is:<ref name=Finn/>
<math display="block">
\left(\frac{\partial \mathcal{E}}{\partial T}\right)_Q =
-\left(\frac{\partial S}{\partial Q}\right)_T
</math>

If a mole of ions goes into solution (for example, in a Daniell cell, as discussed below) the charge through the external circuit is:
<math display="block"> \Delta Q = -n_0 F_0 ,</math>

where <math> n_0 </math> is the number of electrons/ion, and <math> F_0 </math> is the [[Faraday constant]] and the minus sign indicates discharge of the cell. Assuming constant pressure and volume, the thermodynamic properties of the cell are related strictly to the behavior of its emf by:<ref name=Finn/>

<math display="block">\Delta H = -n_0 F_0 \left( \mathcal{E} - T \frac {\mathrm{d}\mathcal{E}}{\mathrm{d}T}\right) \, , </math>

where <math> \Delta H </math> is the [[standard enthalpy of reaction|enthalpy of reaction]]. The quantities on the right are all directly measurable. Assuming constant temperature and pressure:

<math display="block">\Delta G = -n_0 F_0\mathcal{E}</math>

which is used in the derivation of the [[Nernst equation]].

== Distinction with potential difference ==
Although an [[Voltage|electrical potential difference (voltage)]] is sometimes called an emf,<ref name=Fogiel>
{{cite book |title=Basic Electricity
 |page=76
 |url=https://books.google.com/books?id=_DapslzANfwC&pg=PA76
 |first= M.|last=Fogiel
 |isbn=978-0-87891-420-3
 |year=2002
 |publisher= Research & Education Association
 }}</ref><ref name="Halliday">
{{cite book
 |title=Fundamentals of Physics
 |edition=6th
 |first1=David|last1=Halliday |first2=Robert|last2=Resnick |first3=Jearl|last3=Walker 
 |url=https://books.google.com/books?id=VXlEQlznCO0C&pg=PA638
 |page=638
 |isbn= 978-0-471-75801-3
 |publisher=Wiley
 |year=2008
 }}</ref><ref name=Freeman>
{{cite book
 |title=Fundamentals of Telecommunications
 |first=Roger L| last=Freeman 
 |url=https://books.google.com/books?id=6_yQ-dEGc5wC&pg=PA576
 |page=576 |isbn=978-0-471-71045-5
 |publisher=Wiley
 |year=2005
 |edition=2nd
 }}</ref><ref name=Croft>
{{cite book
 |title=Practical Electricity
 |year=1917
 |first=Terrell|last=Croft
 |page=[https://archive.org/details/practicalelectr01crofgoog/page/n551 533]
 |publisher=McGraw-Hill
 |url=https://archive.org/details/practicalelectr01crofgoog
 }}</ref><ref name=Loeb>
{{cite book
 |title=Fundamentals of Electricity and Magnetism
 |first=Leonard B.|last=Loeb
 |page =86
 |url=https://books.google.com/books?id=zw-3icfx9qAC&pg=PA86
 |isbn=978-1-4067-0733-5
 |publisher=Read Books
 |edition=Reprint of Wiley 1947 3rd
 |year=2007
 }}</ref> they are formally distinct concepts:

* Potential difference is a more general term that includes emf.
* Emf is the cause of a potential difference. 
* In a circuit of a voltage source and a resistor, the sum of the source's applied voltage plus the ohmic voltage drop through the resistor is zero. But the resistor provides no emf, only the voltage source does:
** For a circuit using a battery source, the emf is due solely to the chemical forces in the battery.
** For a circuit using an electric generator, the emf is due solely to a time-varying magnetic forces within the generator.
* Both a 1 volt emf and a 1 volt potential difference correspond to 1 joule per coulomb of charge.

In the case of an open circuit, the electric charge that has been separated by the mechanism generating the emf creates an electric field opposing the separation mechanism. For example, the chemical reaction in a voltaic cell stops when the opposing electric field at each electrode is strong enough to arrest the reactions. A larger opposing field can reverse the reactions in what are called ''reversible'' cells.<ref name=Peters>
{{cite book
 |title=Concise Chemical Thermodynamics
 |first1=J. R. W.|last1=Warn |first2=A. P. H.|last2=Peters |page=123
 |url=https://books.google.com/books?id=oCTRVcJ1mqYC&pg=PA123
 |isbn=978-0-7487-4445-9
 |year=1996
 |edition=2
 |publisher=CRC Press
 }}</ref><ref>
{{cite book
 |title=Thermodynamics for Chemists
 |first=Samuel|last=Glasstone
 |url=https://books.google.com/books?id=oW5XqmTSXyEC&pg=RA1-PA301
 |page=301
 |isbn=978-1-4067-7322-4
 |publisher=Read Books
 |year=2007
 |edition= Reprint of D. Van Nostrand Co (1964)
 }}</ref>

The electric charge that has been separated creates an electric [[potential difference]] that can (in many cases) be measured with a [[voltmeter]] between the terminals of the device, when not connected to a load. The magnitude of the emf for the battery (or other source) is the value of this open-circuit voltage. 
When the battery is charging or discharging, the emf itself cannot be measured directly using the external voltage because some voltage is lost inside the source.<ref name="Halliday"/>
It can, however, be inferred from a measurement of the current <math>I</math> and potential difference {{tmath|V}}, provided that the internal resistance <math>R</math> already has been measured:
<math display="block">\mathcal{E} = V_\text{load} + IR \, .</math>

"Potential difference" is not the same as "induced emf" (often called "induced voltage"). 
The potential difference (difference in the electric scalar potential) between two points A and B is independent of the path we take from ''A'' to ''B''.  
If a voltmeter always measured the potential difference between ''A'' and ''B'', then the position of the voltmeter would make no difference. 
However, it is quite possible for the measurement by a voltmeter between points ''A'' and ''B'' to depend on the position of the voltmeter, if a time-dependent magnetic field is present. 
For example, consider an infinitely long [[solenoid]] using an [[Alternating current|AC current]] to generate a varying flux in the interior of the solenoid. 
Outside the solenoid we have two resistors connected in a ring around the solenoid. 
The resistor on the left is {{val|100|u=Ω}} and the one on the right is {{val|200|u=Ω}}, they are connected at the top and bottom at points ''A'' and ''B''. 
The induced voltage, by Faraday's law is {{tmath|V}}, so the current {{tmath|1= I = V/(100+200)}}. Therefore, the voltage across the {{val|100|u=Ω}} resistor is <math>100 \ I</math> and the voltage across the {{val|200|u=Ω}} resistor is {{tmath|200 \ I}}, yet the two resistors are connected on both ends, but <math>V_{AB}</math> measured with the voltmeter to the left of the solenoid is not the same as <math>V_{AB}</math> measured with the voltmeter to the right of the solenoid.<ref name=Shadowitz>{{cite book
 | title = The Electromagnetic Field
 | first = Albert|last=Shadowitz
 | publisher = McGraw-Hill Book Company
 | year = 1975
 | isbn = 0-07-056368-3
 | pages = 396–398
 | edition = 1st
 }}</ref>
<ref name=McDonald2>
{{cite web 
|last1=McDonald |first1=Kirk T. 
|title=Lewin's Circuit Paradox 
|url=http://kirkmcd.princeton.edu/examples/lewin.pdf 
|website=Physics Examples 
|publisher=Princeton University
|date=2010
 }}</ref>

==Generation==

===Chemical sources===
{{Main|Electrochemical cell}}

[[File:Reaction path.JPG|thumb|380px|A typical reaction path requires the initial reactants to cross an energy barrier, enter an intermediate state and finally emerge in a lower energy configuration. If charge separation is involved, this energy difference can result in an emf. See Bergmann et al.<ref name=Bergmann>{{cite book |title=Constituents of Matter: Atoms, Molecules, Nuclei, and Particles |first=Nikolaus|last=Risch |chapter=Molecules - bonds and reactions |editor=L Bergmann |display-editors=etal |isbn=978-0-8493-1202-1  |year=2002 |publisher=CRC Press |chapter-url=https://books.google.com/books?id=mGj1y1WYflMC}}</ref> and [[Transition state]].]]

[[Image:Galvanic cell labeled.svg|thumb|380px|[[Galvanic cell]] using a [[salt bridge]]]]

The question of how batteries (galvanic cells) generate an emf occupied scientists for most of the 19th century. The "seat of the electromotive force" was eventually determined in 1889 by [[Walther Nernst]]<ref>{{cite journal |last1=Nernst |first1=Walter |title=Die elektromotorische Wirksamkeit der Ionen |journal=[[Z. Phys. Chem.]] |date=1889 |volume=4 |page=129}}</ref> to be primarily at the interfaces between the [[electrode]]s and the [[electrolyte]].<ref name=cajori>{{cite book
 | title = A History of Physics in Its Elementary Branches: Including the Evolution of Physical Laboratories
 | first = Florian|last=Cajori
 | publisher = The Macmillan Company
 | year = 1899
 | pages = [https://archive.org/details/ahistoryphysics03cajogoog/page/n232 218]–219
 | url = https://archive.org/details/ahistoryphysics03cajogoog
 | quote = seat of electromotive force.
 }}</ref>

Atoms in molecules or solids are held together by [[chemical bond]]ing, which stabilizes the molecule or solid (i.e. [[Minimum total potential energy principle|reduces its energy]]). When molecules or solids of relatively high energy are brought together, a spontaneous chemical reaction can occur that rearranges the bonding and reduces the (free) energy of the system.<ref name=reconfigure>
The brave reader can find an extensive discussion for organic electrochemistry in {{cite book |title=Organic electrochemistry |edition=4 |year=2000 |publisher=CRC Press |isbn=978-0-8247-0430-8 |editor1=Henning Lund |editor2=Ole Hammerich |chapter-url=https://books.google.com/books?id=tBxxZclgKyMC&pg=PA23 |first=Christian| last=Amatore |chapter=Basic concepts}}
</ref> In batteries, coupled half-reactions, often involving metals and their ions, occur in tandem, with a gain of electrons (termed "reduction") by one conductive electrode and loss of electrons (termed "oxidation") by another (reduction-oxidation or [[redox|redox reactions]]). The spontaneous overall reaction can only occur if electrons move through an external wire between the electrodes. The electrical energy given off is the free energy lost by the chemical reaction system.

As an example, a [[Daniell cell]] consists of a zinc anode (an electron collector) that is oxidized as it dissolves into a zinc sulfate solution. The dissolving zinc leaving behind its electrons in the electrode according to the oxidation reaction (the subscripts {{math|(''s'')}} stand for solid electrode; {{math|(''aq'')}} stands for aqueous solution):
<math display="block">\mathrm{Zn_{(s)} \rightarrow Zn^{2+}_{(aq)} + 2  e ^- \ } </math>

The zinc sulfate is the [[electrolyte]] in that half cell. It is a solution which contains zinc cations {{tmath|\mathrm{Zn}^{2+} }}, and sulfate anions <math>\mathrm{SO}_4^{2-} </math> with charges that balance to zero.

In the other half cell, the copper cations in a copper sulfate electrolyte move to the copper cathode to which they attach themselves as they adopt electrons from the copper electrode by the reduction reaction:
<math display="block"> \mathrm{Cu^{2+}_{(aq)} + 2 e^- \rightarrow Cu_{(s)}\ } </math>

which leaves a deficit of electrons on the copper cathode. The difference of excess electrons on the anode and deficit of electrons on the cathode creates an electrical potential between the two electrodes. (A detailed discussion of the microscopic process of electron transfer between an electrode and the ions in an electrolyte may be found in Conway.)<ref name=Conway>
{{cite book |title=Electrochemical supercapacitors |first=BE|last=Conway |chapter=Energy factors in relation to electrode potential |page=37 |chapter-url=https://books.google.com/books?id=8yvzlr9TqI0C&pg=PA37 |isbn=978-0-306-45736-4 |year=1999 |publisher=Springer}}
</ref> The electrical energy released by this reaction ({{val|213|u=kJ}} per {{val|65.4|u=g}} of zinc) can be attributed mostly due to the {{val|207|u=kJ}} weaker bonding (smaller magnitude of the cohesive energy) of zinc, which has filled 3d- and 4s-orbitals, compared to copper, which has an unfilled orbital available for bonding.

If the cathode and anode are connected by an external conductor, electrons pass through that external circuit (light bulb in figure), while ions pass through the [[salt bridge]] to maintain charge balance until the anode and cathode reach electrical equilibrium of zero volts as chemical equilibrium is reached in the cell. In the process the zinc anode is dissolved while the copper electrode is plated with copper.<ref name= Tilley>{{cite book |title=Understanding Solids |url=https://archive.org/details/understandingsol0000till |url-access=registration |page=[https://archive.org/details/understandingsol0000till/page/267 267] |first=R. J. D.|last=Tilley |isbn=978-0-470-85275-0 |year=2004 |publisher=Wiley}}</ref> The salt bridge has to close the electrical circuit while preventing the copper ions from moving to the zinc electrode and being reduced there without generating an external current. It is not made of salt but of material able to wick [[cations and anions]] (a dissociated salt) into the solutions. The flow of positively charged cations along the bridge is equivalent to the same number of negative charges flowing in the opposite direction.

If the light bulb is removed (open circuit) the emf between the electrodes is opposed by the electric field due to the charge separation, and the reactions stop.

For this particular cell chemistry, at {{val|298|u=K}} (room temperature), the emf <math>\mathcal{E}</math> is {{val|1.0934|u=V}}, with a temperature coefficient of {{tmath| \mathrm{d}\mathcal{E}/\mathrm{d}T }}&nbsp;= {{val|−4.53|e=-4|u=V|up=K}}.<ref name= Finn>{{cite book |title=Thermal Physics |first=Colin B P|last=Finn |page=163 |url=https://books.google.com/books?id=BTMPThGxXQ0C&pg=PA162 |isbn=978-0-7487-4379-7 |year=1992 |publisher=CRC Press}}</ref>

====Voltaic cells====
Volta developed the voltaic cell about 1792, and presented his work March 20, 1800.<ref name=Mottelay>{{cite book |title=Bibliographical History of Electricity and Magnetism |first=Paul Fleury|last=Mottelay |page=247 |url=https://books.google.com/books?id=9vzti90Q8i0C&pg=RA1-PA247 |isbn=978-1-4437-2844-7 |publisher=Read Books |year=2008 |edition=Reprint of 1892}}</ref> Volta correctly identified the role of dissimilar electrodes in producing the voltage, but incorrectly dismissed any role for the electrolyte.<ref name=Kragh>{{cite journal
 |journal     = Nuova Voltiana:Studies on Volta and His Times
 |publisher   = Università degli studi di Pavia
 |year        = 2000
 |url         = http://ppp.unipv.it/Collana/Pages/Libri/Saggi/NuovaVoltiana_PDF/sei.pdf
 |title       = Confusion and Controversy: Nineteenth-century theories of the voltaic pile
 |first = Helge|last=Kragh
 |url-status     = dead
 |archive-url  = https://web.archive.org/web/20090320064922/http://ppp.unipv.it/Collana/Pages/Libri/Saggi/NuovaVoltiana_PDF/sei.pdf
 |archive-date = 2009-03-20
}}</ref> Volta ordered the metals in a 'tension series', "that is to say in an order such that any one in the list becomes positive when in contact with any one that succeeds, but negative by contact with any one that precedes it."<ref name=Cumming>{{cite book |title=An Introduction to the Theory of Electricity |first=Linnaus|last=Cumming |url=https://books.google.com/books?id=Nrb8723u4WEC&pg=PA118 |page=118 |isbn=978-0-559-20742-6 |publisher=BiblioBazaar |year=2008 |edition=Reprint of 1885}}</ref> A typical symbolic convention in a schematic of this circuit ( –<big>|</big>'''<small>|</small>'''– ) would have a long electrode 1 and a short electrode 2, to indicate that electrode 1 dominates.  Volta's law about opposing electrode emfs implies that, given ten electrodes (for example, zinc and nine other materials), 45 unique combinations of voltaic cells (10 × 9/2) can be created.

====Typical values====
The electromotive force produced by primary (single-use) and secondary (rechargeable) cells is usually of the order of a few volts. The figures quoted below are nominal, because emf varies according to the size of the load and the state of exhaustion of the cell.

{| class="wikitable sortable plainrowheaders"
|+ EMF and cell chemistries of several common voltaic cells
|-
! scope="col" rowspan="2" | Common name
! scope="col" rowspan="2" | EMF (volts)
! scope="col" colspan="3" | Cell chemistry
|-
! scope="col" | Anode
! scope="col" | Solvent, electrolyte
! scope="col" | Cathode 
|-
! scope="row" | [[Nickel–cadmium battery|nickel-cadmium]]
| 1.2 || Cadmium || Water, potassium hydroxide || {{chem2|NiO(OH)}}
|-
! scope="row" | [[Nickel–metal hydride battery|nickel–metal hydride]]
| 1.2 || [[Mischmetal]] (hydrogen absorbing) || Water, potassium hydroxide || Nickel
|-
! scope="row" | [[zinc–carbon battery|Zinc carbon]]
| 1.5 || Zinc || Water, ammonium or zinc chloride || Carbon, manganese dioxide
|-
! scope="row" | [[Lead–acid battery|Lead–acid]]
| 2.1 || Lead || Water, sulfuric acid || Lead dioxide
|-
! scope="row" | [[lithium-ion battery|Lithium-ion]]
| 3.6–3.7 || Graphite || Organic solvent, Li salts || {{chem2|LiCoO2}}
|-
! scope="row" | [[Mercury cell]]
| 1.35 || Zinc || Water, sodium or potassium hydroxide || {{chem2|HgO}}
|}

==== Other chemical sources ====
Other chemical sources include [[fuel cell]]s.

===Electromagnetic induction===
{{Main|Faraday's law of induction}}

Electromagnetic induction is the production of a circulating electric field by a time-dependent magnetic field. A time-dependent magnetic field can be produced either by motion of a magnet relative to a circuit, by motion of a circuit relative to another circuit (at least one of these must be carrying an electric current), or by changing the electric current in a fixed circuit. The effect on the circuit itself, of changing the electric current, is known as self-induction; the effect on another circuit is known as [[mutual induction]].

For a given circuit, the electromagnetically induced emf is determined purely by the rate of change of the magnetic flux through the circuit according to [[Faraday's law of induction]].

An emf is induced in a coil or conductor whenever there is change in the [[flux linkage]]s. Depending on the way in which the changes are brought about, there are two types: When the conductor is moved in a stationary magnetic field to procure a change in the flux linkage, the emf is ''statically induced''. The electromotive force generated by motion is often referred to as ''motional emf''. When the change in flux linkage arises from a change in the magnetic field around the stationary conductor, the emf is ''dynamically induced.'' The electromotive force generated by a time-varying magnetic field  is often referred to as ''transformer emf''.

===Contact potentials===
{{See also|Volta potential|Electrochemical potential}}
When solids of two different materials are in contact, [[thermodynamic equilibrium]] requires that one of the solids assume a higher electrical potential than the other. This is called the ''contact potential''.<ref name=Trigg>{{cite book |title=Landmark experiments in twentieth century physics |first=George L.|last=Trigg |page=138 ''ff'' |url=https://books.google.com/books?id=YOQ9fi5yQ4sC&pg=PA138 |isbn=978-0-486-28526-9 |year=1995 |publisher=Courier Dover |edition=Reprint of Crane, Russak & Co 1975}}</ref> Dissimilar metals in contact produce what is known also as a contact electromotive force or [[Galvani potential]]. The magnitude of this potential difference is often expressed as a difference in [[Fermi level]]s in the two solids when they are at charge neutrality, where the Fermi level (a name for the [[chemical potential]] of an electron system<ref name=Rockett>{{cite book |title=Materials science of semiconductors |first=Angus|last=Rockett |chapter=Diffusion and drift of carriers |page=74 ''ff'' |chapter-url=https://books.google.com/books?id=n5zMiMfw6ZUC&pg=PA74 |isbn=978-0-387-25653-5 |year=2007 |publisher=Springer Science |location=New York, NY}}</ref><ref name=Kittel>{{cite book |title=Elementary Statistical Physics |first=Charles|last=Kittel |chapter-url=https://books.google.com/books?id=5sd9SAoRjgQC&pg=PA67 |chapter= Chemical potential in external fields |page=67 |isbn=978-0-486-43514-5 |publisher=Courier Dover |year=2004 |edition=Reprint of Wiley 1958}}
</ref>) describes the energy necessary to remove an electron from the body to some common point (such as ground).<ref name=Hanson>{{cite book |title=Fundamentals of Nanoelectronics |first=George W.|last=Hanson |page=100 |url=https://books.google.com/books?id=L7AUi7ltCksC&pg=PA100 |isbn=978-0-13-195708-4 |year=2007 |publisher=Prentice Hall}}</ref> If there is an energy advantage in taking an electron from one body to the other, such a transfer will occur. The transfer causes a charge separation, with one body gaining electrons and the other losing electrons. This charge transfer causes a potential difference between the bodies, which partly cancels the potential originating from the contact, and eventually equilibrium is reached. At thermodynamic equilibrium, the [[Fermi level]]s are equal (the electron removal energy is identical) and there is now a built-in electrostatic potential between the bodies.
The original difference in Fermi levels, before contact, is referred to as the emf.<ref name=Sato>{{cite book |title=Electrochemistry at metal and semiconductor electrodes |first=Norio|last=Sato |page=110 ''ff'' |chapter-url=https://books.google.com/books?id=olQzaXNgM74C&pg=PA110 |isbn=978-0-444-82806-4 |year=1998 |publisher=Elsevier |edition=2nd |chapter= Semiconductor photoelectrodes}}</ref>
The contact potential cannot drive steady current through a load attached to its terminals because that current would involve a charge transfer. No mechanism exists to continue such transfer and, hence, maintain a current, once equilibrium is attained.

One might inquire why the contact potential does not appear in [[Kirchhoff's voltage law]] as one contribution to the sum of potential drops. The customary answer is that any circuit involves not only a particular diode or junction, but also all the contact potentials due to wiring and so forth around the entire circuit. The sum of {{em|all}} the contact potentials is zero, and so they may be ignored in Kirchhoff's law.<ref name=Quimby>{{cite book |title=Photonics and lasers |first=Richard S.|last=Quimby |page=176 |url=https://books.google.com/books?id=82f-gIvtC7wC&pg=PA176 |isbn=978-0-471-71974-8 |publisher=Wiley |year=2006}}</ref><ref name=Neamen>{{cite book |title=Semiconductor physics and devices |first=Donald A.|last=Neamen |url=https://archive.org/details/semiconductorphy00neam |url-access=registration |page=[https://archive.org/details/semiconductorphy00neam/page/240 240] |year=2002 |isbn=978-0-07-232107-4 |publisher=McGraw-Hill Professional |edition=3rd}}</ref>

===Solar cell===
{{Main|Theory of solar cells}}

[[Image:Solar cell equivalent circuit.svg|thumb|250px |The [[Theory of solar cells#Equivalent circuit of a solar cell|equivalent circuit of a solar cell]], ignoring parasitic resistances.]]

Operation of a [[solar cell]] can be understood from [[Theory of solar cells#Equivalent circuit of a solar cell|its equivalent circuit]]. [[Photon]]s with energy greater than the [[bandgap]] of the [[semiconductor]] create mobile [[electron–hole pair]]s. Charge separation occurs because of a pre-existing electric field associated with the [[p-n junction]]. This electric field is created from a [[p–n junction#Equilibrium (zero bias)|built-in potential]], which arises from the [[Volta potential|contact potential]] between the two different materials in the junction. The charge separation between positive [[Electron hole|hole]]s and negative [[electron]]s across the [[p–n diode]] yields a ''[[forward voltage]]'', the ''photo voltage'', between the illuminated diode terminals,<ref name="Dhir">{{cite book |first=S. M. |last=Dhir |title=Electronic Components and Materials: Principles, Manufacture & Maintenance |date=2000 |orig-year=1999 |publisher=[[Tata McGraw-Hill Publishing Company Limited]] |location=India |edition=2007 fifth reprint |isbn=0-07-463082-2 |page=283 |chapter=§3.1 Solar cells |url=https://books.google.com/books?id=sGbwj4J76tEC |chapter-url=https://books.google.com/books?id=sGbwj4J76tEC&pg=PA283}}</ref> which drives current through any attached load. ''Photo voltage'' is sometimes referred to as the ''photo emf'', distinguishing between the effect and the cause.

==== Solar cell current–voltage relationship ====
Two internal current losses <math>I_\mathrm{SH} + I_\mathrm{D}</math> limit the total current <math>I</math> available to the external circuit. The light-induced charge separation eventually creates a forward current <math> I_\mathrm{SH}</math> through the cell's internal resistance <math>R_\mathrm{SH}</math> in the direction opposite the light-induced current {{tmath|I_\mathrm{L} }}. In addition, the induced voltage tends to [[p-n junction#Forward bias|forward bias]] the junction, which at high enough voltages will cause a recombination current <math> I_\mathrm{D}</math> in the diode opposite the light-induced current.

When the output is short-circuited, the output voltage is zeroed, and so the voltage across the diode is smallest. Thus, short-circuiting results in the smallest <math>I_\mathrm{SH} + I_\mathrm{D}</math> losses and consequently the maximum output current, which for a high-quality solar cell is approximately equal to the light-induced current {{tmath| I_\mathrm{L} }}.<ref name="Lorenzo">{{cite book
 |title=Solar Electricity: Engineering of photovoltaic systems
 |editor=Eduardo Lorenzo
 |first= Gerardo L.|last=Araújo
 |chapter-url=https://books.google.com/books?id=lYc53xZyxZQC&pg=PA74
 |chapter=§2.5.1 Short-circuit current and open-circuit voltage
 |isbn=978-84-86505-55-4
 |year=1994
 |page=74
 |publisher=Progenza for Universidad Politechnica Madrid
 }}</ref> Approximately this same current is obtained for forward voltages up to the point where the diode conduction becomes significant.

The current delivered by the illuminated diode to the external circuit can be simplified (based on certain assumptions) to:
<math display="block">I = I_\mathrm{L} -I_0 \left( e^{\frac{V}{mV_\mathrm{T}}} - 1 \right) \ . </math>

<math>I_0</math> is the [[reverse saturation current]]. Two parameters that depend on the solar cell construction and to some degree upon the voltage itself are the [[ideality factor]] {{tmath|m}} and the [[thermal voltage]] {{tmath|1= V_\mathrm{T} = \tfrac{k T}{q} }}, which is about {{val|26|u=millivolts}} at [[room temperature]].<ref name= Lorenzo/>

==== Solar cell photo emf ====

[[File:Solar cell characterisitcs.JPG|thumb|250px |Solar cell output voltage for two light-induced currents {{math|''I''{{sub|L}}}} expressed as a ratio to the reverse saturation current {{math|''I''{{sub|0}}}}<ref>{{cite book |first=Jenny|last=Nelson |url=https://books.google.com/books?id=s5NN34HLWO8C&pg=PA8 |title=The physics of solar cells |publisher=Imperial College Press |year=2003 |isbn=978-1-86094-349-2 |page=8}}</ref> and using a fixed ideality factor {{mvar|m}} of 2.<ref name="params">In practice, at low voltages {{mvar|m}}&nbsp;→ 2, whereas at high voltages {{mvar|m}}&nbsp;→ 1. See Araújo, ''op. cit.'' {{ISBN|84-86505-55-0}}. [https://books.google.com/books?id=lYc53xZyxZQC&pg=PA72 page 72]</ref> Their emf is the voltage at their y-axis intercept.]]

Solving the illuminated diode's above simplified [[Current-voltage relationship|current–voltage relationship]] for output voltage yields:
<math display="block">V = mV_\mathrm{T} \ln \left( \frac{I_\mathrm{L} - I}{I_0}+1 \right) \ , </math>
which is plotted against <math>I / I_0 </math> in the figure.

The solar cell's ''photo emf'' <math>\mathcal{E}_\text{photo}</math> has the same value as the open-circuit voltage {{tmath|V_\mathrm{oc} }}, which is determined by zeroing the output current {{tmath|I}}:
<math display="block">\mathcal{E}_\text{photo} = V_\mathrm{oc} = mV_\mathrm{T} \ln \left( \frac{I_\mathrm{L}}{I_0}+1 \right) \ . </math>

It has a [[logarithm]]ic dependence on the light-induced current <math>I_\mathrm{L}</math> and is where the junction's forward bias voltage is just enough that the forward current completely balances the light-induced current. For silicon junctions, it is typically not much more than {{val|0.5|u=volts}}.<ref name="Northrop">{{cite book
 | title=Introduction to Instrumentation and Measurements
 | first=Robert B.|last=Northrop
 | page=176
 | chapter=§6.3.2 Photovoltaic Cells
 | chapter-url= https://books.google.com/books?id=mcpcfpQfxB4C&pg=PA176
 | isbn=978-0-8493-7898-0 |year=2005 |publisher=CRC Press
 }}</ref> While for high-quality silicon panels it can exceed {{val|0.7|u=volts}} in direct sunlight.<ref>{{cite web| url = https://www.pveducation.org/pvcdrom/solar-cell-operation/open-circuit-voltage#:~:text=Silicon%20solar%20cells%20on%20high,circuit%20voltages%20around%20690%20mV. | title = Open-Circuit Voltage}}</ref>

When driving a resistive load, the output voltage can be determined using [[Ohm's law]] and will lie between the short-circuit value of zero volts and the open-circuit voltage {{tmath|V_\mathrm{oc} }}.<ref>{{cite book |first=Jenny|last=Nelson |url=https://books.google.com/books?id=s5NN34HLWO8C&pg=PA8 |title=The physics of solar cells |publisher=Imperial College Press |year=2003 |isbn=978-1-86094-349-2 |page=6}}</ref> When that resistance is small enough such that <math>I \approx I_\mathrm{L}</math> (the near-vertical part of the two illustrated curves), the solar cell acts more like a ''current generator'' rather than a voltage generator,<ref name="Nelson_page7">
{{cite book |first=Jenny|last=Nelson |url=https://books.google.com/books?id=s5NN34HLWO8C&pg=PA8 |title=The physics of solar cells |publisher=Imperial College Press |year=2003 |isbn=978-1-86094-349-2 |page=7}}</ref> since the current drawn is nearly fixed over a range of output voltages. This contrasts with batteries, which act more like voltage generators.

=== Other sources that generate emf ===

* A [[transformer]] coupling two circuits may be considered a source of emf for one of the circuits, just as if it were caused by an electrical generator; this is the origin of the term "transformer emf".
* For converting [[sound waves]] into voltage [[signal]]s:
** a [[microphone]] generates an emf from a moving [[Diaphragm (acoustics)|diaphragm]].
** a [[Pickup (music technology)#Magnetic pickups|magnetic pickup]] generates an emf from a varying magnetic field produced by an instrument.
** a [[piezoelectric sensor]] generates an emf from strain on a [[piezoelectric crystal]].
* Devices that use temperature to produce emfs include [[thermocouple]]s and [[thermopiles]].<ref>{{cite book|editor=John S. Rigden|title=Macmillan encyclopedia of physics|location=New York|publisher=Macmillan|year=1996}}</ref>
* Any electrical [[Transducer#Applications|transducer]] which converts a physical energy into electrical energy.

==See also==
*[[Counter-electromotive force]]
*[[Electric battery]]
*[[Electrochemical cell]]
*[[Electrolytic cell]]
*[[Galvanic cell]]
*[[Magnetomotive force]]
*[[Voltaic pile]]

==References==
{{reflist|30em}}

==Further reading==
* George F. Barker, "[https://books.google.com/books?id=zXAQ97d7YiIC&pg=PA649 On the measurement of electromotive force]". Proceedings of the American Philosophical Society Held at Philadelphia for Promoting Useful Knowledge, American Philosophical Society. January 19, 1883.
* Andrew Gray, "Absolute Measurements in Electricity and Magnetism", [https://books.google.com/books?id=WxeFSg38JLQC&pg=PA41 Electromotive force]. Macmillan and co., 1884.
* Charles Albert Perkins, "Outlines of Electricity and Magnetism", [https://archive.org/details/outlineselectri00perkgoog/page/n169 <!-- pg=158 --> Measurement of Electromotive Force]. Henry Holt and co., 1896.
* John Livingston Rutgers Morgan, "The Elements of Physical Chemistry", [https://archive.org/details/elementsphysica01morggoog/page/n246 <!-- pg=235 --> Electromotive force].  J. Wiley, 1899.
*  "Abhandlungen zur Thermodynamik, von H. Helmholtz. Hrsg. von Max Planck". (Tr. "Papers to thermodynamics, on H. Helmholtz. Hrsg. by Max Planck".) Leipzig, W. Engelmann,  Of Ostwald classical author of the accurate sciences series. New consequence. No. 124, 1902.
* Theodore William Richards and Gustavus Edward Behr, jr., "The electromotive force of iron under varying conditions, and the effect of occluded hydrogen". Carnegie Institution of Washington publication series, 1906. {{LCCN |07003935}}
*  Henry S. Carhart,  "Thermo-electromotive force in electric cells, the thermo-electromotive force between a metal and a solution of one of its salts".  New York, D. Van Nostrand company, 1920. {{LCCN|20020413}}
* [[Hazel Rossotti]], "Chemical applications of potentiometry".  London, Princeton, N.J., Van Nostrand, 1969. {{ISBN|0-442-07048-9}} {{LCCN |69011985}}
* Nabendu S. Choudhury, 1973. [https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19730015362.pdf "Electromotive force measurements on cells involving beta-alumina solid electrolyte"]. NASA technical note, D-7322.
*{{cite book |title=Modern Electrochemistry: An Introduction to an Interdisciplinary Area |author1=John O'M. Bockris |author2=Amulya K. N. Reddy |chapter-url=https://books.google.com/books?id=5OGsg_v_7yoC&pg=PA647 |chapter=Electrodics |isbn=978-0-306-25002-6 |year=1973 |edition=2 |publisher=Springer}}
*{{cite journal |author=Roberts, Dana |title= How batteries work: A gravitational analog |journal= Am. J. Phys.|volume= 51 |issue= 9 |page= 829 |year=1983 |doi=10.1119/1.13128|bibcode = 1983AmJPh..51..829R }}
* G. W. Burns, et al., "Temperature-electromotive force reference functions and tables for the letter-designated thermocouple types based on the ITS-90". Gaithersburg, MD : U.S. Dept. of Commerce, National Institute of Standards and Technology, Washington, Supt. of Docs., U.S. G.P.O., 1993.
* {{cite book |author=Norio Sato |title=Electrochemistry at metal and semiconductor electrodes |page=326 ''ff'' |chapter-url=https://books.google.com/books?id=olQzaXNgM74C&pg=PA328 |isbn=978-0-444-82806-4 |year=1998 |publisher=Elsevier |edition=2nd |chapter= Semiconductor photoelectrodes}}
* {{cite journal|date=2009-03-08|title=Electromotive force and huge magnetoresistance in magnetic tunnel junctions|journal=[[Nature (journal)|Nature]]| doi= 10.1038/nature07879|last1=Hai|first1=Pham Nam|last2=Ohya|first2=Shinobu|last3=Tanaka|first3=Masaaki|last4=Barnes|first4=Stewart E.|last5=Maekawa|first5=Sadamichi|volume=458|pages=489–92|pmid=19270681|issue=7237|bibcode = 2009Natur.458..489H |s2cid=4320209}}

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{{DEFAULTSORT:Electromotive Force}}
[[Category:Electromagnetism]]
[[Category:Electrodynamics]]
[[Category:Voltage]]