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{{distinguish|Electromagnetic field}}
{{Electromagnetism |Network}}

'''Electromotive force''', abbreviated '''emf''' (denoted <math>\mathcal{E}</math> and measured in [[volts]]),<ref>emf. (1992). ''American Heritage Dictionary of the English Language'' 3rd ed. Boston:Houghton Mifflin.</ref> is the electrical action produced by a non-electrical source.<ref name="Stewart">{{cite book |last1=Stewart |first1=Joseph V. |title=Intermediate electromagnetic theory |date=2001 |publisher=World Scientific |page=389}}</ref> A device that converts other forms of energy into electrical energy (a "[[transducer]]"),<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> 
such as a [[Battery (electricity)|battery]] (converting [[chemical energy]]) or [[Electric generator|generator]] (converting mechanical energy),<ref name="Stewart"/> provides an emf as its output.<ref name="Tipler803"/> Sometimes an analogy to water "[[Volt#Water-flow_analogy|pressure]]" is used to describe electromotive force.<ref>
{{cite journal
 | journal = Transactions of the American Electrochemical Society
 | title = The Relation Between Contact Potentials and Electrochemical Action
 | author = Irving Langmuir
 | volume = 29
 | issue = 
 | publisher = The Society
 | pages = 175
 | year = 1916
 | url = https://books.google.com/books?id=OW0SAAAAYAAJ&pg=PA172&dq=%22electromotive+force+is+that%22&q=%22electromotive%20force%20is%20that%22
 }}</ref>  (The word "force" in this case is not used to mean [[force]] of interaction between bodies, as may be measured in [[Pound (force)|pounds]] or [[newton (unit)|newtons]].)

In [[electromagnetic induction]], emf can be defined around a closed loop of [[electrical conductor|conductor]] as the electromagnetic [[work (physics)|work]] that would be done on an  [[electric charge]] (an [[electron]] in this instance) if it travels once around the loop.<ref name=Cook>
{{cite book
 | title = The Theory of the Electromagnetic Field
 | author = David M. 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 a time-varying [[magnetic flux]] linking a loop, the electric potential scalar field is not defined due to a circulating electric vector field, but an emf nevertheless does work that can be measured as a virtual electric potential around the loop.<ref name=Lerner>
{{cite book
 | title=Physics for scientists and engineers
 | author=Lawrence M Lerner
 | url=https://books.google.com/books?id=Nv5GAyAdijoC&pg=PA727
 | publisher=Jones & Bartlett Publishers
 | isbn=978-0-7637-0460-5
 | pages=724–727
 | year=1997}}</ref>

In the case of a two-terminal device (such as an [[electrochemical cell]]) which is modeled as a [[Thévenin's theorem|Thévenin's equivalent circuit]], the equivalent emf can be measured as the open-circuit potential difference or "voltage" between the two terminals. This potential difference can drive an [[electric current]] if an external circuit is attached to the terminals.

==Overview==
Devices that can provide emf include [[electrochemical cell]]s, [[Thermoelectric effect|thermoelectric device]]s, [[solar cells]], [[photodiode]]s, [[electrical generator]]s, [[electrical transformer|transformer]]s and even [[Van de Graaff generator]]s.<ref name=Lerner/><ref name=Tipler>
{{cite book
 | title=Physics for Scientists and Engineers
 |author1=Paul A. Tipler |author2=Gene 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 whenever magnetic field fluctuations occur through a surface. The shifting of the [[Earth's magnetic field]] during a [[geomagnetic storm]] induces currents in the electrical grid as the lines of the magnetic field are shifted about and cut across the conductors.

In the case of a battery, the charge separation that gives rise to a voltage difference between the terminals is accomplished by chemical reactions at the electrodes 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
 |author1=Alvin M. Halpern |author2=Erich 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
 | author = Robert L. Lehrman
 | publisher = Barron's Educational Series
 | year = 1998
 | isbn = 978-0-7641-0236-3
 | page = 274
 | url = https://books.google.com/books?id=wMhCxOsPNE8C&pg=PA274&dq=emf++separated+charge+reaction+potential
 }}</ref> A voltaic cell can be thought of as having a "charge pump" of atomic dimensions at each electrode, that is:<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|author=Singh}}
</ref>

A source of emf can be thought of as a kind of ''charge pump'' that acts to move positive charge from a point of low potential through its interior to a point of high potential. … By chemical, mechanical or other means, the source of emf performs work ''dW'' on that charge to move it to the high potential terminal. The emf ''ℰ'' of the source is defined as the work ''dW'' done per charge ''dq'': {{nowrap|''ℰ'' {{=}} ''dW/dq''}}.


In the case of an electrical generator, a time-varying magnetic field inside the generator creates an electric field via [[electromagnetic induction]], which in turn creates a voltage difference between the generator terminals. Charge separation takes place within the generator, with electrons flowing away from one terminal and toward the other, until, in the open-circuit case, sufficient electric field builds up to make further charge separation impossible. Again, the emf is countered by the electrical voltage due to charge separation. If a 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==
Around 1830, [[Michael Faraday]] established that the reactions at each of the 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 was initially 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, had held the incorrect opinion that contact alone (without taking into account a chemical reaction) was the origin of the emf.

== Notation and units of measurement ==
Electromotive force is often denoted by <math>\mathcal{E}</math> or ''ℰ'' (script capital E, Unicode U+2130).

In a device without internal resistance, if an [[electric charge]] ''Q'' passes through that device, and gains an [[energy]] ''W'', the net emf for that device is the energy gained per unit [[electric charge|charge]], or ''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]] per [[coulomb]].<ref>
{{cite book
 | title = Basic Electricity
 | author = Van Valkenburgh
 | publisher = Cengage Learning
 | year = 1995
 | isbn = 978-0-7906-1041-2
 | pages = 1–46
 | url = https://books.google.com/books?id=vmg1UKsTntAC&pg=PT67&dq=electromotive-force+joules-per-coulomb+volts+charge+energy
 }}</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 [[electric charge|charge]]).

==Formal definitions==
''Inside'' a source of emf that is open-circuited, the conservative electrostatic field created by separation of charge exactly cancels the forces producing the emf. Thus, the emf has the same value but opposite sign as the integral of the [[electric field]] aligned with an internal path between two terminals ''A'' and ''B'' of a source of emf in open-circuit condition (the path is taken from the negative terminal to the positive terminal to yield a positive emf, indicating work done on the electrons moving in the circuit).<ref name=Griffiths>
{{cite book
 | title = Introduction to Electrodynamics
 | author = David J 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> 
Mathematically:

:<math>\mathcal{E} = -\int_{A}^{B} \boldsymbol{E}_\mathrm{cs} \cdot \mathrm{d} \boldsymbol{ \ell } \ ,</math>

where '''''E'''''<sub>cs</sub> is the conservative electrostatic field created by the charge separation associated with the emf, d'''ℓ''' is an element of the path from terminal ''A'' to terminal ''B'', and ‘·’ denotes the vector [[dot product]].<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 [[Electromotive force#Contact potentials|contact potentials]] and [[Electromotive force#Solar cell|solar cells]]), 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 equation applies only to locations ''A'' and ''B'' that are terminals, and does not apply to paths between points ''A'' and ''B'' with portions outside the source of emf. This equation involves the electrostatic electric field due to charge separation '''''E'''''<sub>cs</sub> and does not involve (for example) any non-conservative component of electric field due to Faraday's law of induction.

In the case of a closed path in the presence of a varying [[magnetic field]], the integral of the [[electric field]] around a closed loop may be nonzero; one common application of the concept of emf, known as "''induced emf''" is the voltage induced in such a loop.<ref>{{cite book
 | title = Beyond the mechanical universe: from electricity to modern physics
 |author1=Richard P. Olenick |author2=Tom M. Apostol |author3=David L. Goodstein | publisher = Cambridge University Press
 | year = 1986
 | isbn = 978-0-521-30430-6
 | page = 245
 | url = https://books.google.com/books?id=Ht4T7C7AXZIC&pg=RA1-PA245&dq=define+electromotive-force+around-a-closed-path
 }}</ref>  The "''induced emf''" around a stationary closed path ''C'' is:

:<math>\mathcal{E}=\oint_{C} \boldsymbol{E} \cdot \mathrm{d} \boldsymbol{ \ell } \ ,</math>

where '''''E''''' is the entire electric field, conservative and non-conservative, and the integral is around an arbitrary but stationary closed curve ''C'' through which there is a varying magnetic field. 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, see [[Kirchhoff's circuit laws#Kirchhoff's voltage law (KVL)|Kirchhoff's voltage law]], which is valid, as long as the circuit elements remain at rest and radiation is ignored<ref>{{cite web |last1=McDonald |first1=Kirk T. |title=Voltage Drop, Potential Difference and EMF |url=http://physics.princeton.edu/~mcdonald/examples/volt.pdf |website=Physics Examples |publisher=Princeton University |page=1, fn. 3 |date=2012}}</ref>).

This definition can be extended to arbitrary sources of emf and moving paths ''C'':<ref name=Cook2>{{cite book
 | title = The Theory of the Electromagnetic Field
 | author = David M. 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>\mathcal{E}=\oint_{C} \left[\boldsymbol{E}  + \boldsymbol{v} \times \boldsymbol{B} \right] \cdot \mathrm{d} \boldsymbol{ \ell } \ </math>
:::<math> +\frac{1}{q}\oint_{C}\mathrm {Effective \ chemical \ forces \ \cdot} \ \mathrm{d} \boldsymbol{ \ell } \ </math>
::::<math> +\frac{1}{q}\oint_{C}\mathrm { Effective \ thermal \ forces\ \cdot}\  \mathrm{d} \boldsymbol{ \ell } \ ,</math>

which is a conceptual equation mainly, because the determination of the "effective forces" is difficult.

==In (electrochemical) thermodynamics==
When multiplied by an amount of charge ''dQ'' the emf ℰ yields a thermodynamic work term ℰ''dQ'' that is used in the formalism for the change in [[Gibbs free energy|Gibbs energy]] when charge is passed in a battery:

:: <math>dG = -SdT + VdP + \mathcal{E}dQ\ , </math>

where ''G'' is the Gibb's free energy, ''S'' is the [[entropy]], ''V'' is the system volume, ''P'' is its pressure and ''T'' is its [[absolute temperature]].

The combination  ( ℰ, ''Q'' ) 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 ''T'' (a measurable quantity) to the change in entropy ''S'' 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>
\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> \Delta Q = -n_0F_0 \ , </math>

where ''n''<sub>0</sub> is the number of electrons/ion, and ''F''<sub>0</sub> 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>\Delta H = -n_0 F_0 \left( \mathcal{E} - T \frac {d\mathcal{E}}{dT}\right) \ , </math>

where Δ''H'' is the [[standard enthalpy of reaction|enthalpy of reaction]]. The quantities on the right are all directly measurable.

== Voltage difference ==
An electrical voltage difference 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
 |author= M. 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
 |author1=David Halliday |author2=Robert Resnick |author3=Jearl 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
 |author=Roger L 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
 |author=Terrell Croft
 |page=533
 |publisher=McGraw-Hill
 |url=https://books.google.com/books?id=zuZMAAAAMAAJ&pg=PA533
 }}</ref><ref name=Loeb>
{{cite book
 |title=Fundamentals of Electricity and Magnetism
 |author=Leonard B 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> The points below illustrate the more formal usage, in terms of the distinction between emf and the voltage it generates:

# For a circuit as a whole, such as one containing a resistor in series with a voltaic cell, electrical voltage does not contribute to the overall emf, because the voltage difference on going around a circuit is zero. (The ohmic ''IR'' voltage drop plus the applied electrical voltage sum to zero. See [[Kirchhoff's circuit laws#Kirchhoff's voltage law (KVL)|Kirchhoff's voltage law]]). The emf is due solely to the chemistry in the battery that causes charge separation, which in turn creates an electrical voltage that drives the current.
# For a circuit consisting of an electrical generator that drives current through a resistor, the emf is due solely to a time-varying magnetic field within the generator that generates an electrical voltage that in turn drives the current. (The ohmic ''IR'' drop plus the applied electrical voltage again is zero. See [[Kirchhoff's circuit laws|Kirchhoff's Law]])
# 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 example illustrates the origin of the term "transformer emf".
#A [[photodiode]] or [[solar cell]] may be considered as a source of emf, similar to a battery, resulting in an electrical voltage generated by charge separation driven by light rather than chemical reaction.<ref name=Nelson_page7>
{{cite book |title=The Physics of Solar Cells
 |author=Jenny Nelson
 |url=https://books.google.com/books?id=s5NN34HLWO8C&pg=PA6
 |isbn=978-1-86094-349-2
 |year=2003
 |publisher=Imperial College Press
 |page=7
 }}</ref>
#Other devices that produce emf are [[fuel cell]]s,  [[thermocouple]]s, and [[thermopiles]].<ref>John S. Rigden, (editor in chief), ''Macmillan encyclopedia of physics''. New York : Macmillan, 1996.</ref>

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
 |author1=J. R. W. Warn |author2=A. P. H. 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
 |author=Samuel 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 be measured with a [[voltmeter]] between the terminals of the device. 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 ''I'' and voltage difference ''V'', provided that the internal resistance ''r'' already has been measured: ''ℰ''&nbsp;=&nbsp;''V''&nbsp;+&nbsp;''Ir''.

==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 |author =Nikolaus 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&printsec=frontcover#PPA374,M1}}

</ref> and [[Transition state]].]]

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

The question of how batteries ([[galvanic cell]]s) 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
 | author = Florian Cajori
 | publisher = The Macmillan Company
 | year = 1899
 | pages = 218–219
 | url = https://books.google.com/books?id=ICASAAAAYAAJ&pg=PA219&dq=%22seat+of%22+%22electromotive+force%22
 }}</ref>

Atoms in molecules or solids are held together by [[chemical bond|chemical bonding]], which stabilizes the molecule or solid (i.e. 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="Schmidt-Rohr 18"> Schmidt-Rohr, K. (2018). "How Batteries Store and Release Energy: Explaining Basic Electrochemistry" ‘’J. Chem. Educ.’’ '''95''': 1801-1810. https://doi.org/10.1021/acs.jchemed.8b00479</ref> <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 |author=Christian 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.<ref name="Schmidt-Rohr 18"/> 

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 (''s'' = solid electrode; ''aq'' = aqueous solution):

:<math>\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 <math>\mathrm{Zn}_{} ^{2+}</math>, 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> \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 |author=BE 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 (213 kJ per 65.4 g of zinc) can be attributed mostly due to the 207 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.<ref name="Schmidt-Rohr 18"/>

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://books.google.com/books?id=ZVgOLCXNoMoC&pg=PA267 |page=267 |author=R. J. D. Tilley |isbn=978-0-470-85275-0 |year=2004 |publisher=Wiley}}</ref> The so-called "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 298 K (room temperature), the emf ''ℰ'' = 1.0934 V, with a temperature coefficient of d''ℰ''/d''T''&nbsp;= −4.53×10<sup>−4</sup> V/K.<ref name= Finn>

{{cite book |title=Thermal Physics |author=Colin B P 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 |author=Paul Fleury 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
 |author      = Helge Kragh
 |url-status     = dead
 |archiveurl  = https://web.archive.org/web/20090320064922/http://ppp.unipv.it/Collana/Pages/Libri/Saggi/NuovaVoltiana_PDF/sei.pdf
 |archivedate = 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 |author=Linnaus 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

! rowspan="2" | EMF
! colspan="3" | Cell chemistry
! rowspan="2" | Common name
|-
! Anode
! Solvent, electrolyte
! Cathode 
|-
| 1.2&nbsp;V || Cadmium || Water, potassium hydroxide || NiO(OH) || [[Nickel–cadmium battery|nickel-cadmium]]
|-
| 1.2&nbsp;V || [[Mischmetal]] (hydrogen absorbing) || Water, potassium hydroxide || Nickel|| [[Nickel–metal hydride battery|nickel–metal hydride]]
|-
| 1.5&nbsp;V || Zinc || Water, ammonium or zinc chloride || Carbon,  manganese dioxide|| [[zinc-carbon battery|Zinc carbon]]
|-
| 2.1&nbsp;V || Lead || Water, sulfuric acid || Lead dioxide || [[Lead–acid battery|Lead–acid]]
|-
| 3.6&nbsp;V to 3.7&nbsp;V || Graphite || Organic solvent, Li salts || LiCoO<sub>2</sub> || [[lithium-ion battery|Lithium-ion]]
|-
| 1.35&nbsp;V || Zinc || Water, sodium or potassium hydroxide || HgO || [[Mercury cell]]
|-
|}

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

The principle of [[electromagnetic induction]], noted above, states that a time-dependent magnetic field produces a circulating electric 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 a current), or by changing the current in a fixed circuit. The effect on the circuit itself, of changing the 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 |author=George L. 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 |author=Angus 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 |author=Charles 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 |author=George W. 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 |author=Norio 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 circuit laws|Kirchhoff's law of voltages]] 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 ''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 |author=Richard S. 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 |author=Donald A. Neamen |url=https://books.google.com/books?id=9oEifMuMAVsC&pg=PA240 |page=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 equivalent circuit of a solar cell; parasitic resistances are ignored in the discussion of the text.]]

[[File:Solar cell characterisitcs.JPG|thumb|250px |Solar cell voltage as a function of solar cell current delivered to a load for two light-induced currents ''I''<sub>L</sub>; currents as a ratio with reverse saturation current ''I''<sub>0</sub>. Compare with Fig. 1.4 in Nelson.<ref>{{cite book
 |author=Jenny Nelson
 |title=The physics of solar cells
 |url=https://books.google.com/books?id=s5NN34HLWO8C&pg=PA8
 |page=8
 |isbn=978-1-86094-349-2
 |year=2003
 |publisher=Imperial College Press
 }}</ref>]]

Operation of a solar cell can be understood from the equivalent circuit at right. Light, of sufficient energy (greater than the [[bandgap]] of the material), creates mobile [[electron–hole pair]]s in a semiconductor. Charge separation occurs because of a pre-existing electric field associated with the [[p-n junction]] in thermal equilibrium. (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 a [[p-n junction]] (a [[diode]]) yields a ''forward voltage'', the ''photo voltage'', between the illuminated diode terminals,<ref name=Dhir>
{{cite book
 | title=Electronic Components and Materials: Principles, Manufacture and Maintenance
 | author=S M Dhir
 | chapter-url=https://books.google.com/books?id=sGbwj4J76tEC&pg=PA283
 | chapter=§3.1 Solar cells |publisher=Tata McGraw-Hill
 | year=2000
 | isbn=978-0-07-463082-2
 }}</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. 

The current available to the external circuit is limited by internal losses I<sub>0</sub>=I<sub>SH</sub> + I<sub>
D</sub>:

<math> I = I_L - I_0 = I_L - I_{SH} - I_D</math>

Losses limit the current available to the external circuit. The light-induced charge separation eventually creates a current (called a forward current) I<sub>SH</sub> through the cell's junction in the  direction opposite that the light is driving the current. In addition, the induced voltage tends to [[p-n junction#Forward bias|forward bias]] the junction. At high enough levels, this forward bias of the junction will cause a forward current, I<sub>
D</sub> in the diode opposite that induced by the light. Consequently, the greatest current is obtained under short-circuit conditions, and is denoted as ''I''<sub>L</sub> (for light-induced current) in the equivalent circuit.<ref name=Lorenzo>{{cite book
 |title=Solar Electricity: Engineering of photovoltaic systems
 |editor=Eduardo Lorenzo
 |author= Gerardo L. 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 is:

:<math>I = I_L -I_0 \left( e^{qV/(mkT)} - 1 \right) \ , </math>

where ''I''<sub>0</sub> is the reverse saturation current.  Where the two parameters that depend on the solar cell construction and to some degree upon the voltage itself are ''m'', the ideality factor,  and ''kT/q'' the [[thermal voltage]] (about 0.026 V at room temperature).<ref name= Lorenzo/> This relation is plotted in the figure using a fixed value ''m''&nbsp;= 2.<ref name =params>

In practice, at low voltages ''m''&nbsp;→ 2, whereas at high voltages ''m''&nbsp;→ 1. See Araújo, ''op. cit.'' {{ISBN|84-86505-55-0}}. [https://books.google.com/books?id=lYc53xZyxZQC&pg=PA72 page 72]

</ref> Under open-circuit conditions (that is, as ''I'' = 0), the open-circuit voltage is the voltage at which forward bias of the junction is enough that the forward current completely balances the photocurrent. Solving the above for the voltage V and designating it the open-circuit voltage of the [[I–V curve|''I–V'' equation]] as:

:<math>V_\text{oc} = m\ \frac{kT}{q}\ \ln \left( \frac{I_\text{L}}{I_0}+1 \right) \ , </math>

which is useful in indicating a logarithmic dependence of ''V''<sub>oc</sub> upon the light-induced current. Typically, the open-circuit voltage is not more than about 0.5 V.<ref name=Northrop>
{{cite book
 | title=Introduction to Instrumentation and Measurements
 | author=Robert B. 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>

When driving a load, the photo voltage is variable.  As shown in the figure, for a load resistance ''R''<sub>L</sub>, the cell develops a voltage that is between the short-circuit value ''V''&nbsp;= 0, ''I''&nbsp;= ''I''<sub>L</sub> and the open-circuit value ''V''<sub>oc</sub>, ''I''&nbsp;= 0, a value given by [[Ohm's law]] ''V&nbsp;= I R''<sub>L</sub>, where the current ''I'' is the difference between the short-circuit current and current due to forward bias of the junction, as indicated by the equivalent circuit<ref>
{{cite book
 |author=Jenny Nelson
 |title=The physics of solar cells
 |url=https://books.google.com/books?id=s5NN34HLWO8C&pg=PA8
 |page=6
 |isbn=978-1-86094-349-2
 |year=2003
 |publisher=Imperial College Press
 }}</ref> (neglecting the [[Parasitic element (electrical networks)|parasitic resistances]]).<ref>
{{cite book
 |author=Jenny Nelson
 |title=The physics of solar cells
 |url=https://books.google.com/books?id=s5NN34HLWO8C&pg=PA8
 |page=13
 |isbn=978-1-86094-349-2
 |year=2003
 |publisher=Imperial College Press
 }}</ref>

In contrast to the battery, at current levels delivered to the external circuit near ''I''<sub>L</sub>, the solar cell acts more like a ''current generator'' rather than a voltage generator (near vertical part of the two illustrated curves)<ref name=Nelson_page7/>
The current drawn is nearly fixed over a range of load voltages, to one electron per converted [[photon]]. The [[quantum efficiency]], or probability of getting an electron of photocurrent per incident photon, depends not only upon the solar cell itself, but upon the spectrum of the light.

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

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

==Further reading==
* George F. Barker, "[https://books.google.com/books?vid=0uowlltB5bCx84xQrO&id=zXAQ97d7YiIC&pg=PA649&lpg=PA650 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?vid=0pkd5YYtaGRtjR6Oes&id=WxeFSg38JLQC&pg=PA41&dq= Electromotive force]. Macmillan and co., 1884.
* Charles Albert Perkins, "Outlines of Electricity and Magnetism", [https://books.google.com/books?vid=OCLC02316583&id=jd1vvFcD-MAC&pg=PA158&vq=&dq= Measurement of Electromotive Force]. Henry Holt and co., 1896.
* John Livingston Rutgers Morgan, "The Elements of Physical Chemistry", [https://books.google.com/books?vid=OCLC10759094&id=ovjORAvJZZkC&pg=PA235&dq= 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]]|url=http://www.nature.com/nature/journal/vaop/ncurrent/abs/nature07879.html|accessdate=2009-03-10| 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 }}


{{Authority control}}

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