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# Electromotive force

Electromotive force (emf, $mathcal\left\{E\right\}$) is a term used to characterize electrical devices, such as voltaic cells, thermoelectric devices, electrical generators and transformers, and even resistors. For a given device, if an electric charge Q passes through that device, and gains an energy U, the net emf for that device is the energy gained per unit charge, or U/Q. This has units of volts, or newton meters per coulomb, and hence can be thought of as a voltage induced by the device in question. Since force has the unit of the newton, emf is a misnomer, but one that over time has resisted change.

In most circuits current is driven by a so-called "source of emf", which usually is a voltaic cell (or battery, which consists of voltaic cells in series and/or in parallel) or the power company. For a voltaic cell the source of emf is the chemical reactions that occur at each of the electrode-electrolyte interfaces, so that a voltaic cell can be thought of as two "surface pumps" of atomic dimension. The reactions at the electrode-electrolyte interfaces provide the "seat" of emf for the voltaic cell. For the power company, the source of emf is electromagnetic induction, which is more extended than an atomic size, but nevertheless is confined to the power generation building, usually many miles from the user.

## Sources and unit of measurement

Sources of electromotive force include electric generators (both alternating current and continuous current types), batteries, and thermocouples (in a heat gradient). Electromotive force is often denoted by $mathcal\left\{E\right\}$ or (script capital E).

Electromotive force is measured in volts (in the International System of Units equal in amount to a joule per coulomb of electric charge). 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).

## Terminology

The term electromotive force is due to Alessandro Volta (1745–1827), who invented the battery, or voltaic pile. "Electromotive force" originally referred to the 'force' with which positive and negative charges could be separated (i.e. moved, hence "electromotive"), and was also called "electromotive power" (although it is not a power in the modern sense). Maxwell's 1865 explanation of what are now called Maxwell's equations used the term "electromotive force" for what is now called the electric field strength.

## Formal definition of electromotive force

If the vector field f represents the force per unit charge on a charge carrier, the emf around a circuit C is
$mathcal\left\{E\right\}=oint_Cmathbf\left\{f\right\}cdot dmathbf\left\{l\right\}.$

This formal definition is not very helpful for a voltaic cell; there f due to the chemical reactions is either very large but not calculable (at the electrode-electrolyte interfaces), or zero (everywhere else). However, this definition is quite helpful for emfs generated by a time-dependent magnetic field (Faraday's Law of electromagnetic induction). Note that the electrostatic potential does not contribute to the net emf around a circuit (although it does contribute over parts of a circuit). Like the electric potential at a point and the voltage between two points, the emf around a loop is measured in volts.

The emf is sensitive to non-electrostatic forces, since the force f can include magnetic, chemical, mechanical, and gravitational components. In practice, the power sources for the non-electrostatic forces in a voltaic cell are the chemicals that react at the electrode-electrolyte interfaces; for the power company they are the moving rotors that produce a non-electrostatic field by Faraday's Law of electromagnetic induction; and for a thermoelectric device they are the heaters and coolers that maintain a temperature difference across the device. The EMF of a source (electromagnetic, chemical, thermal or otherwise) may be defined as the work done by an external agent, per unit charge, with sign reversed, in bringing a test charge once around a circuit that contains the source and no other source. Such a source is often described as a "seat" of EMF.

Another term for emf is electromotance.

## Electromotive force in thermodynamics

When multiplied by an amount of charge de the emf ℰ yields a thermodynamic work term ℰde that is used in the formulism for the change in Gibbs free energy when charge is passed in a battery:

dG = -SdT + VdP + ℰde.

The combination ℰ.e 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 (a measurable quantity) to the change in entropy when charge is passed isothermally and isobarically. The latter is closely related to the reaction entropy ΔrS of the electrochemical reaction that lends the battery its power.


left(frac{partial mathcal{E}}{partial T}right)_e= -left(frac{partial S}{partial e}right)_T

## Electromotive force and voltage difference

According to Maxwell, even a potential difference can have the same effect as an emf. Nevertheless, normal usage does not consider a voltage difference as a source of emf.

1. For a resistor the voltage difference across its ends serves as the sole source of emf.
2. For a voltaic cell the net emf is the sum of the chemical emf, which always tends to drive current so as to discharge the cell, and the voltage difference emf across its terminals. The combination of the two emfs can drive current in either direction, thus permitting both charge and discharge; in equilibrium, where there is zero current, these two emfs cancel.
3. For a circuit as a whole, such as one containing a resistor in series with a voltaic cell, voltage does not contribute to the overall emf, because the voltage difference on going around a circuit is zero. (See Kirchoff's Law)
4. For a circuit consisting of a capacitor that discharges through a resistor, the emf that drives current is solely due to the voltage difference across the resistor, and due to the capacitor.

If a source of emf is not connected to an external resistor, then an electric current cannot flow through that resistor (Ohm's Law). In this case, between the terminals of the source there must exist a true electric field that produces a voltage difference that exactly cancels the emf of the source.

The source of this true electric field is the electric charge that has been separated by the mechanism generating the emf . For example, the chemical reaction in a voltaic cell stops when the electric field across each electrode is strong enough to stop the reactions at each electrode.

This electric field between the terminals of the battery creates an electric potential difference that can be measured with a voltmeter. The polarity of this measured potential difference is always opposite to that of the generated emf. The value 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 $mathcal\left\{E\right\}$ itself cannot be measured directly. It can, however, be inferred from a measurement of the current I and voltage difference V, provided that the internal resistance has already been measured: I=($mathcal\left\{E\right\}$ -V)/r.

One of Volta's great contributions to science was to recognize that a voltaic cell has two sources of emf, the chemical reactions at each electrode. He showed that they provide distinct emfs $mathcal\left\{E\right\}_\left\{1\right\}$ and $mathcal\left\{E\right\}_\left\{2\right\}$ that oppose one another, so that two identical electrodes give no net emf, but that two different electrodes give a net emf of $mathcal\left\{E\right\}_\left\{1\right\} - mathcal\left\{E\right\}_\left\{2\right\}$, which we assume is positive. A schematic of this circuit would have a long electrode 1 and a short electrode 2, to indicate that electrode 1 dominates. Volta's law about opposing electrode emfs means that, given ten electrodes (e.g. zinc and nine other materials), which can be used to produce 45 types of voltaic cells (10*9/2), only nine relative measurements (e.g. copper and each of the nine others) are needed to get all 45 possible emfs that these ten electrodes can produce.

## Electromotive force generation

Besides voltaic cells, other devices that produce chemical emfs from electrochemical reactions are fuel cells. Radiant and thermal energy (e.g., a solar cell or a thermocouple) can also produce emfs. Some other sources of emf include thermocouples, thermopiles, and photodiodes.

Dissimilar metals in contact also produce what is known as a contact electromotive force or contact potential (eg., the volta effect). However, this is a truly electrostatic effect, and does not affect the overall emf of a circuit.

The principle of electromagnetic induction, noted above, states that a time-dependent magnetic field can produce 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. The electromotive force generated by motion is often referred to as '''motional electromotive force

For a given circuit, the electromagnetically induced electric field is determined purely by the geometry and the rate of change of the magnetic flux through the circuit, by Faraday's law of induction. However, the accompanying electrostatic field does depend on the details of the circuit, since the emf across a resistor will have contributions from both the electromagnetic and electrostatic fields, and their detailed form will depend on the value and shape of the resistor.

If an electric circuit has self-inductance L, and carries current i, then by Faraday's Law

$mathcal\left\{E\right\} = -L \left\{ di over dt \right\}.$

Given this emf and the resistance of the circuit, the instantaneous current can be computed with Ohm's Law, for example, or more generally by solving the differential equations that arise out of Kirchhoff's laws.

### Classification of induced emfs

An emf is induced in a coil or conductor whenever there is change in the flux linkages. 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. When the change in flux linkage arises from a change in the magnetic field around the stationary conductor, the emf is dynamically induced.

## Electromotive force of cells

The electromotive force produced by primary and secondary 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.

Emf Cell chemistry
1.2 V Nickel Metal-Hydride
1.5 V Zinc-carbon
3.6 V to 3.7 V Lithium-Ion

## References

General

• Griffiths, David (1999). Introduction to Electrodynamics. 3e, Prentice-Hall. ISBN 0-13-805326-X.
• Saslow, Wayne M. (2002). Electricity, Magnetism, and Light. Toronto: Thomson Learning. Ch. 8 (especially pp. 302-315) discusses emfs for voltaic cells. Ch.12 discusses electromagnetically induced emfs; especially see example 12.13 and Fig.12.16(b), where a distributed electromagnetic emf is discussed and shown in a circuit.Citations

Ohm's Law (PDF in German)

• 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.
• 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.
• "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.
• Nabendu S. Choudhury, "Electromotive force measurements on cells involving [beta]-alumina solid electrolyte". NASA technical note, D-7322.
• 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 //r88
• 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 //r88
• 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.

## External articles

• Doug Gingrich, "Physics lecture notes, electronics", Direct Current Circuits, Electromotive Force (EMF) University of Alberta, Department of Physics, 1999.
• Advanced Physics lecture notes, "Electromagnetism", Faraday’s Law—Electromagnetic Induction. Electromotive Force". Semiconductor Physics Group, Department of Physics, University of Cambridge, 2006. (PDF)