In electromagnetism and electronics, electromotive force (emf, or EMF) or electromotance, denoted E {\displaystyle {\mathcal {E}}} , is an energy transfer to an electric circuit per unit of electric charge, measured in volts. Devices called electrical transducers provide an emf by converting other forms of energy into electrical energy. Other types of electrical equipment also produce an emf, such as batteries, which convert chemical energy, and generators, which convert mechanical energy. This energy conversion is achieved by physical forces applying physical work on electric charges. However, electromotive force itself is not a physical force, and ISO/IEC standards have deprecated the term in favor of source voltage or source tension instead (denoted U s {\displaystyle U_{\mathrm {s} }} ). An electronic–hydraulic analogy may view emf as the mechanical work done to water by a pump, which results in a pressure difference (analogous to voltage). In electromagnetic induction, emf can be defined around a closed loop of a conductor as the electromagnetic work that would be done on an elementary electric charge (such as an electron) if it travels once around the loop. For two-terminal devices modeled as a 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 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 § Distinction with potential difference).
Overview Devices that can provide emf include electrochemical cells, thermoelectric devices, solar cells, photodiodes, electrical generators, inductors, transformers and even Van de Graaff generators. 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. 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 d W {\textstyle \mathrm {d} W} on that charge to move it to the high-potential terminal. The emf E {\textstyle {\mathcal {E}}} of the source is defined as the work d W {\textstyle \mathrm {d} W} done per charge d q {\displaystyle \mathrm {d} q} . E = d W d q {\displaystyle \textstyle {\mathcal {E}}={\frac {\mathrm {d} W}{\mathrm {d} q}}} . 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.
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). 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. 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 E {\displaystyle {\mathcal {E}}} (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 q {\displaystyle q} passing through that device gains an energy W {\displaystyle W} via work, the net emf for that device is the energy gained per unit charge: W q {\displaystyle \textstyle {\tfrac {W}{q}}} . Like other measures of energy per charge, emf uses the SI unit volt, which is equivalent to a joule (SI unit of energy) per coulomb (SI unit of 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).
Formal definitions 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 E o p e n c i r c u i t {\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 to introduce the concept of a nonelectrostatic field E ′ {\displaystyle {\boldsymbol {E}}'} that exists only inside the source of emf. In the open-circuit case, E ′ = − E o p e n c i r c u i t {\displaystyle {\boldsymbol {E}}'=-{\boldsymbol {E}}_{\mathrm {open\ circuit} }} , while when the source is connected to a circuit the electric field E {\displaystyle {\boldsymbol {E}}} inside the source changes but E ′ {\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. Expressed mathematically:
E s o u r c e = ∫ N P E ′ ⋅ d ℓ = − ∫ N P E o p e n c i r c u i t ⋅ d ℓ = V P − V N , {\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 E o p e n c i r c u i t {\displaystyle {\boldsymbol {E}}_{\mathrm {open\ circuit} }} is the conservative electrostatic field created by the charge separation associated with the emf, d ℓ {\displaystyle \mathrm {d} {\boldsymbol {\ell }}} is an element of the path from terminal N to terminal P, ⋅ (dot operator) denotes the vector dot product, and V {\displaystyle V} is the electric scalar potential. This emf is the work done on a unit charge by the source's nonelectrostatic field E ′ {\displaystyle {\boldsymbol {E}}'} when the charge moves from N to P. When the source is connected to a load, its emf is just
E s o u r c e = ∫ N P E ′ ⋅ d ℓ , {\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 E {\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 C {\displaystyle C} may be nonzero. Then, the "induced emf" (often called the "induced voltage") in the loop is:
E C = ∮ C E ⋅ d ℓ = − d Φ C d t = − d d t ∮ C A ⋅ d ℓ , {\displaystyle {\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 }}\ ,}
where E {\displaystyle {\boldsymbol {E}}} is the entire electric field, conservative and non-conservative, and the integral is around an arbitrary, but stationary, closed curve C {\displaystyle C} through which there is a time-varying magnetic flux Φ C {\displaystyle \Phi _{C}} , and A {\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, by Kirchhoff's voltage law, which is valid, as long as the circuit elements remain at rest and radiation is ignored). 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 C {\displaystyle C} is a conductor that carries current I {\displaystyle I} in the direction of integration around the loop, and the magnetic flux is due to that current, we have that Φ B = L I {\displaystyle \Phi _{B}=LI} , where L {\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 C ′ {\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 C ′ {\displaystyle C'} is essentially the same as that in loop C {\displaystyle C} :
E C = E C ′ = − d Φ C ′ d t = − L d I d t = ∮ C E ⋅ d ℓ = ∫ 1 2 E c o n d u c t o r ⋅ d ℓ − ∫ 1 2 E c e n t e r l i n e ⋅ d ℓ . {\displaystyle {\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 }}\ .}
For a good conductor, E c o n d u c t o r {\displaystyle {\boldsymbol {E}}_{\mathrm {conductor} }} is negligible, so we have, to a good approximation,
L d I d t = ∫ 1 2 E c e n t e r l i n e ⋅ d ℓ = V 1 − V 2 , {\displaystyle L{\frac {\mathrm {d} I}{\mathrm {d} t}}=\int _{1}^{2}{\boldsymbol {E}}_{\mathrm {center\ line} }\cdot \mathrm {d} {\boldsymbol {\ell }}=V_{1}-V_{2}\ ,}
where V {\displaystyle V} is the electric scalar potential along the centerline between points 1 and 2. Thus, we can associate an effective "voltage drop" L d I / d t {\displaystyle L\,\mathrm {d} I/\mathrm {d} t} 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,
∑ E s o u r c e = ∑ l o a d e l e m e n t s v o l t a g e d r o p s , {\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. This definition can be extended to arbitrary sources of emf and paths C {\displaystyle C} moving with velocity v {\displaystyle {\boldsymbol {v}}} through the electric field E {\displaystyle {\boldsymbol {E}}} and magnetic field B {\displaystyle {\boldsymbol {B}}} :
E = ∮ C [ E + v × B ] ⋅ d ℓ + 1 q ∮ C E f f e c t i v e c h e m i c a l f o r c e s ⋅ d ℓ + 1 q ∮ C E f f e c t i v e t h e r m a l f o r c e s ⋅ d ℓ , {\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 ∮ C [ E + v × B ] ⋅ d ℓ {\displaystyle \oint _{C}\left[{\boldsymbol {E}}+{\boldsymbol {v}}\times {\boldsymbol {B}}\right]\cdot \mathrm {d} {\boldsymbol {\ell }}}
is often called a "motional emf".
In (electrochemical) thermodynamics When multiplied by an amount of charge d Q {\displaystyle \mathrm {d} Q} the emf E {\displaystyle {\mathcal {E}}} yields a thermodynamic work term E d Q {\displaystyle {\mathcal {E}}\,\mathrm {d} Q} that is used in the formalism for the change in Gibbs energy when charge is passed in a battery:
d G = − S d T + V d P + E d Q , {\displaystyle \mathrm {d} G=-S\,\mathrm {d} T+V\,\mathrm {d} P+{\mathcal {E}}\,\mathrm {d} Q\ ,}
where G {\displaystyle G} is the Gibbs free energy, S {\displaystyle S} is the entropy, V {\displaystyle V} is the system volume, P {\displaystyle P} is its pressure, and T {\displaystyle T} is its absolute temperature. The combination ( E , Q ) {\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 T {\displaystyle T} (a measurable quantity) to the change in entropy S {\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:
( ∂ E ∂ T ) Q = − ( ∂ S ∂ Q ) T {\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:
Δ Q = − n 0 F 0 , {\displaystyle \Delta Q=-n_{0}F_{0},}
where n 0 {\displaystyle n_{0}} is the number of electrons/ion, and F 0 {\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:
Δ H = − n 0 F 0 ( E − T d E d T ) , {\displaystyle \Delta H=-n_{0}F_{0}\left({\mathcal {E}}-T{\frac {\mathrm {d} {\mathcal {E}}}{\mathrm {d} T}}\right)\,,}
where Δ H {\displaystyle \Delta H} is the enthalpy of reaction. The quantities on the right are all directly measurable. Assuming constant temperature and pressure:
Δ G = − n 0 F 0 E {\displaystyle \Delta G=-n_{0}F_{0}{\mathcal {E}}}
which is used in the derivation of the Nernst equation.
Distinction with potential difference Although an electrical potential difference (voltage) is sometimes called an emf, 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. 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. It can, however, be inferred from a measurement of the current I {\displaystyle I} and potential difference V {\displaystyle V} , provided that the internal resistance R {\displaystyle R} already has been measured:
E = V load + I R . {\displaystyle {\mathcal {E}}=V_{\text{load}}+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 V {\displaystyle V} , so the current I = V / ( 100 + 200 ) {\displaystyle I=V/(100+200)} . Therefore, the voltage across the 100 Ω resistor is 100 I {\displaystyle 100\ I} and the voltage across the 200 Ω resistor is 200 I {\displaystyle 200\ I} , yet the two resistors are connected on both ends, but V A B {\displaystyle V_{AB}} measured with the voltmeter to the left of the solenoid is not the same as V A B {\displaystyle V_{AB}} measured with the voltmeter to the right of the solenoid.
Generation
Chemical sources
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 to be primarily at the interfaces between the electrodes and the electrolyte. Atoms in molecules or solids are held together by 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. 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 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 (s) stand for solid electrode; (aq) stands for aqueous solution):
Z n ( s ) → Z n ( a q ) 2 + + 2 e − {\displaystyle \mathrm {Zn_{(s)}\rightarrow Zn_{(aq)}^{2+}+2e^{-}\ } }
The zinc sulfate is the electrolyte in that half cell. It is a solution which contains zinc cations Z n 2 + {\displaystyle \mathrm {Zn} ^{2+}} , and sulfate anions S O 4 2 − {\displaystyle \mathrm {SO} _{4}^{2-}} 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:
C u ( a q ) 2 + + 2 e − → C u ( s ) {\displaystyle \mathrm {Cu_{(aq)}^{2+}+2e^{-}\rightarrow Cu_{(s)}\ } }
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.) The electrical energy released by this reaction (213 kJ per