Electromotive force

Devices that can provide emf include electrochemical cells, thermoelectric devices, solar cells, photodiodes, electrical generators, transformers and even Van de Graaff generators.[8][9] 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 reactions at the electrodes that convert chemical potential energy into electromagnetic potential energy.[10][11] A voltaic cell can be thought of as having a "charge pump" of atomic dimensions at each electrode, that is:

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. … By chemical, mechanical or other means, the source of emf performs work ${\textstyle {\mathit {d}}W}$ on that charge to move it to the high-potential terminal. The emf ${\textstyle {\mathcal {E}}}$ of the source is defined as the work ${\textstyle {\mathit {d}}W}$ done per charge ${\textstyle dq}$. ${\textstyle {\mathcal {E}}={\frac {{\mathit {d}}W}{{\mathit {d}}q}}}$.[12]

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 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.

In 1801, Alessandro Volta introduced the term "force motrice électrique" to describe the active agent of a battery (which he had invented around 1798).[13] 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 obsolete theory thought.[14] 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.

Electromotive force is often denoted by ${\displaystyle {\mathcal {E}}}$ or ℰ.

In a device without internal resistance, if an electric charge ${\displaystyle q}$ passing through that device gains an energy ${\displaystyle W}$ via work, the net emf for that device is the energy gained per unit charge: ${\displaystyle W/q\,.}$ Like other measures of energy per charge, emf uses the SI unit volt, which is equivalent to a joule per coulomb.[15]

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

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 ${\displaystyle {\boldsymbol {E}}_{\mathrm {open\ circuit} }}$ 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[16] to introduce the concept of a nonelectrostatic field ${\displaystyle {\boldsymbol {E}}'}$ that exists only inside the source of emf. In the open-circuit case, ${\displaystyle {\boldsymbol {E}}'=-{\boldsymbol {E}}_{\mathrm {open\ circuit} }}$, while when the source is connected to a circuit the electric field ${\displaystyle {\boldsymbol {E}}}$ inside the source changes but ${\displaystyle {\boldsymbol {E}}'}$ remains essentially the same. In the open-circuit case, the conservative electrostatic field created by separation of charge exactly cancels the forces producing the emf.[17] Mathematically:

$${\displaystyle {\mathcal {E}}_{\mathrm {source} }=\int _{N}^{P}{\boldsymbol {E}}'\cdot \mathrm {d} {\boldsymbol {\ell }}=-\int _{N}^{P}{\boldsymbol {E}}_{\mathrm {open\ circuit} }\cdot \mathrm {d} {\boldsymbol {\ell }}=V_{P}-V_{N}\ ,}$$

where ${\displaystyle {\boldsymbol {E}}_{\mathrm {open\ circuit} }}$ is the conservative electrostatic field created by the charge separation associated with the emf, ${\displaystyle \mathrm {d} {\boldsymbol {\ell }}}$ is an element of the path from terminal N to terminal P, '${\displaystyle \cdot }$' denotes the vector dot product, and ${\displaystyle V}$ is the electric scalar potential.[18] This emf is the work done on a unit charge by the source's nonelectrostatic field ${\displaystyle {\boldsymbol {E}}'}$ when the charge moves from N to P.

When the source is connected to a load, its emf is just ${\displaystyle {\mathcal {E}}_{\mathrm {source} }=\int _{N}^{P}{\boldsymbol {E}}'\cdot \mathrm {d} {\boldsymbol {\ell }}\ ,}$ and no longer has a simple relation to the electric field ${\displaystyle {\boldsymbol {E}}}$ 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 ${\displaystyle C}$ may be nonzero. Then, the "induced emf" (often called the "induced voltage") in the loop is:[19]

$${\displaystyle {\mathcal {E}}_{C}=\oint _{C}{\boldsymbol {E}}\cdot \mathrm {d} {\boldsymbol {\ell }}=-{\frac {d\Phi _{C}}{dt}}=-{\frac {d}{dt}}\oint _{C}{\boldsymbol {A}}\cdot \mathrm {d} {\boldsymbol {\ell }}\ ,}$$

where ${\displaystyle {\boldsymbol {E}}}$ is the entire electric field, conservative and non-conservative, and the integral is around an arbitrary, but stationary, closed curve ${\displaystyle C}$ through which there is a time-varying magnetic flux ${\displaystyle \Phi _{C}}$, and ${\displaystyle {\boldsymbol {A}}}$ 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 (i.e., the work done against the field around a closed path is zero, see Kirchhoff's voltage law, which is valid, as long as the circuit elements remain at rest and radiation is ignored[20]). 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 ${\displaystyle C}$ is a conductor that carries current ${\displaystyle I}$ in the direction of integration around the loop, and the magnetic flux is due to that current, we have that ${\displaystyle \Phi _{B}=LI}$, where ${\displaystyle L}$ 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 ${\displaystyle C'}$ 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 ${\displaystyle C'}$ is essentially the same as that in loop ${\displaystyle C}$:

$${\displaystyle {\mathcal {E}}_{C}={\mathcal {E}}_{C'}=-{\frac {d\Phi _{C'}}{dt}}=-L{\frac {dI}{dt}}=\oint _{C}{\boldsymbol {E}}\cdot \mathrm {d} {\boldsymbol {\ell }}=\int _{1}^{2}{\boldsymbol {E}}_{\mathrm {conductor} }\cdot \mathrm {d} {\boldsymbol {\ell }}-\int _{1}^{2}{\boldsymbol {E}}_{\mathrm {center\ line} }\cdot \mathrm {d} {\boldsymbol {\ell }}\ .}$$

For a good conductor, ${\displaystyle {\boldsymbol {E}}_{\mathrm {conductor} }}$ is negligible, so we have, to a good approximation,

$${\displaystyle L{\frac {dI}{dt}}=\int _{1}^{2}{\boldsymbol {E}}_{\mathrm {center\ line} }\cdot \mathrm {d} {\boldsymbol {\ell }}=V_{1}-V_{2}\ ,}$$

where ${\displaystyle V}$ is the electric scalar potential along the centerline between points 1 and 2.

Thus, we can associate an effective "voltage drop" ${\displaystyle L\ dI/dt}$ 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,

$${\displaystyle \sum {\mathcal {E}}_{\mathrm {source} }=\sum _{\mathrm {load\ elements} }\mathrm {voltage\ drops} ,}$$

where now the induced emf is not considered to be a source emf.[21]

This definition can be extended to arbitrary sources of emf and paths ${\displaystyle C}$ moving with velocity ${\displaystyle {\boldsymbol {v}}}$ through the electric field ${\displaystyle {\boldsymbol {E}}}$ and magnetic field ${\displaystyle {\boldsymbol {B}}}$:[22]

$${\displaystyle {\begin{aligned}{\mathcal {E}}&=\oint _{C}\left[{\boldsymbol {E}}+{\boldsymbol {v}}\times {\boldsymbol {B}}\right]\cdot \mathrm {d} {\boldsymbol {\ell }}\\&\qquad +{\frac {1}{q}}\oint _{C}\mathrm {Effective\ chemical\ forces\ \cdot } \ \mathrm {d} {\boldsymbol {\ell }}\\&\qquad \qquad +{\frac {1}{q}}\oint _{C}\mathrm {Effective\ thermal\ forces\ \cdot } \ \mathrm {d} {\boldsymbol {\ell }}\ ,\end{aligned}}}$$

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

When multiplied by an amount of charge ${\displaystyle dQ}$ the emf ${\displaystyle {\mathcal {E}}}$ yields a thermodynamic work term ${\displaystyle {\mathcal {E}}\,dQ}$ that is used in the formalism for the change in Gibbs energy when charge is passed in a battery:

${\displaystyle dG=-S\,dT+V\,dP+{\mathcal {E}}\,dQ\ ,}$

where ${\displaystyle G}$ is the Gibbs free energy, ${\displaystyle S}$ is the entropy, ${\displaystyle V}$ is the system volume, ${\displaystyle P}$ is its pressure and ${\displaystyle T}$ is its absolute temperature.

The combination ${\displaystyle ({\mathcal {E}},Q)}$ is an example of a conjugate pair of variables. At constant pressure the above relationship produces a Maxwell relation that links the change in open cell voltage with temperature ${\displaystyle T}$ (a measurable quantity) to the change in entropy ${\displaystyle S}$ 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:[23]

${\displaystyle \left({\frac {\partial {\mathcal {E}}}{\partial T}}\right)_{Q}=-\left({\frac {\partial S}{\partial Q}}\right)_{T}}$

If a mole of ions goes into solution (for example, in a Daniell cell, as discussed below) the charge through the external circuit is:

${\displaystyle \Delta Q=-n_{0}F_{0}\ ,}$

where ${\displaystyle n_{0}}$ is the number of electrons/ion, and ${\displaystyle F_{0}}$ 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:[23]

${\displaystyle \Delta H=-n_{0}F_{0}\left({\mathcal {E}}-T{\frac {d{\mathcal {E}}}{dT}}\right)\ ,}$

where ${\displaystyle \Delta H}$ is the enthalpy of reaction. The quantities on the right are all directly measurable. Assuming constant temperature and pressure:

${\displaystyle \Delta G=-n_{0}F_{0}{\mathcal {E}}}$

which is used in the derivation of the Nernst equation.

Although an electrical potential difference (voltage) is sometimes called an emf,[24][25][26][27][28] however they are formally distinct concepts:

• Emf is the cause of a potential difference. Potential difference in turn is a cause of current flow.
• Potential difference itself is not the cause of an emf.
  • Consider Kirchhoff's voltage law, which says the sum of potential differences going through any loop in a circuit is zero. For 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 chemistry in the battery that causes charge separation, which generates a potential difference.
    • For a circuit using an electric generator, the emf is due solely to a time-varying magnetic field within the generator that causes charge separation, which generates a potential difference.
• Both a 1 volt emf and a 1 volt potential difference correspond to 1 joule per coulomb of charge. However:
  • a 1 volt emf means that the source supplies an energy of 1 joule to each coulomb of charge passing through.
  • a 1 volt potential difference between two points on a circuit means that each coulomb of charge will need to either:
    • gain 1 joule of energy to move up that potential difference,
    • or give up 1 joule of energy to move down that potential difference.[29]

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.[30][31]

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.[25] It can, however, be inferred from a measurement of the current ${\displaystyle I}$ and potential difference ${\displaystyle V}$, provided that the internal resistance ${\displaystyle R}$ already has been measured: ${\displaystyle {\mathcal {E}}=V+IR\ .}$

"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 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 100 Ω and the one on the right is 200 Ω, they are connected at the top and bottom at points A and B. The induced voltage, by Faraday's law is ${\displaystyle V}$, so the current ${\displaystyle I=V/(100+200).}$ Therefore the voltage across the 100 Ω resistor is ${\displaystyle 100\ I}$ and the voltage across the 200 Ω resistor is ${\displaystyle 200\ I}$, yet the two resistors are connected on both ends, but ${\displaystyle V_{AB}}$ measured with the voltmeter to the left of the solenoid is not the same as ${\displaystyle V_{AB}}$ measured with the voltmeter to the right of the solenoid.[32] [33]

• Counter-electromotive force
• Electric battery
• Electrochemical cell
• Electrolytic cell
• Galvanic cell
• Voltaic pile

• George F. Barker, "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", Electromotive force. Macmillan and co., 1884.
• Charles Albert Perkins, "Outlines of Electricity and Magnetism", Measurement of Electromotive Force. Henry Holt and co., 1896.
• John Livingston Rutgers Morgan, "The Elements of Physical Chemistry", 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 07-3935
• 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 20-20413
• Hazel Rossotti, "Chemical applications of potentiometry". London, Princeton, N.J., Van Nostrand, 1969. ISBN 0-442-07048-9 LCCN 69-11985
• Nabendu S. Choudhury, 1973. "Electromotive force measurements on cells involving beta-alumina solid electrolyte". NASA technical note, D-7322.
• John O'M. Bockris; Amulya K. N. Reddy (1973). "Electrodics". Modern Electrochemistry: An Introduction to an Interdisciplinary Area (2 ed.). Springer. ISBN 978-0-306-25002-6.
• Roberts, Dana (1983). "How batteries work: A gravitational analog". Am. J. Phys. 51 (9): 829. Bibcode:1983AmJPh..51..829R. doi:10.1119/1.13128.
• 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.
• Norio Sato (1998). "Semiconductor photoelectrodes". Electrochemistry at metal and semiconductor electrodes (2nd ed.). Elsevier. p. 326 ff. ISBN 978-0-444-82806-4.
• Hai, Pham Nam; Ohya, Shinobu; Tanaka, Masaaki; Barnes, Stewart E.; Maekawa, Sadamichi (2009-03-08). "Electromotive force and huge magnetoresistance in magnetic tunnel junctions". Nature. 458 (7237): 489–92. Bibcode:2009Natur.458..489H. doi:10.1038/nature07879. PMID 19270681. S2CID 4320209.