# Electromotive force

In electromagnetism and electronics, electromotive force (also electromotance, abbreviated emf, denoted ${\mathcal {E}}$ ) is the electrical action produced by a non-electrical source, measured in volts. Devices (known as transducers) provide an emf by converting other forms of energy into electrical energy, such as batteries (which convert chemical energy) or generators (which convert mechanical energy). Sometimes an analogy to water pressure is used to describe electromotive force. (The word "force" in this case is not used to mean forces of interaction between bodies).

In electromagnetic induction, emf can be defined around a closed loop of conductor as the electromagnetic work that would be done on an electric charge (an electron in this instance) if it travels once around the loop.

In the case of a two-terminal device (such as an electrochemical cell) which is modeled as a Thévenin's equivalent circuit, the equivalent emf can be measured as the open-circuit potential difference, or 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.

## Overview

Devices that can provide emf include electrochemical cells, thermoelectric devices, solar cells, photodiodes, electrical generators, 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 dW on that charge to move it to the high-potential terminal. The emf of the source is defined as the work dW done per charge dq. ${\textstyle {\mathcal {E}}={\frac {{\mathit {d}}W}{{\mathit {d}}q}}}$ .

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 was initially 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.

## Notation and units of measurement

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

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 charge, or W/Q. Like other measures of energy per charge, emf uses the SI unit volt, which is equivalent to a joule per coulomb.

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, A, and the positive terminal B. This leads to an electrostatic field ${\boldsymbol {E}}_{\mathrm {open\ circuit} }$ that points from B to A, whereas the emf of the source must be able drive current from A to B when connected to a circuit. This led Max Abraham  to introduce the concept of a nonelectrostatic field ${\boldsymbol {E}}'$ that exists only inside the source of emf. In the open-circuit case, ${\boldsymbol {E}}'=-{\boldsymbol {E}}_{\mathrm {open\ circuit} }$ , while when the source is connected to a circuit the electric field ${\boldsymbol {E}}$ inside the source changes but ${\boldsymbol {E}}'$ 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.  Mathematically:

${\mathcal {E}}_{\mathrm {source} }=\int _{A}^{B}{\boldsymbol {E}}'\cdot \mathrm {d} {\boldsymbol {\ell }}=-\int _{A}^{B}{\boldsymbol {E}}_{\mathrm {open\ circuit} }\cdot \mathrm {d} {\boldsymbol {\ell }}=V_{B}-V_{A}\ ,$ where ${\boldsymbol {E}}_{\mathrm {open\ circuit} }$ is the conservative electrostatic field created by the charge separation associated with the emf, $\mathrm {d} {\boldsymbol {\ell }}$ is an element of the path from terminal A to terminal B, '⋅' denotes the vector dot product, and $V$ is the electric scalar potential. This emf is the work done on a unit charge by the nonelectrostatic "battery" field ${\boldsymbol {E}}'$ when the charge moves from A to B.

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

${\mathcal {E}}_{C}=\oint _{C}{\boldsymbol {E}}\cdot \mathrm {d} {\boldsymbol {\ell }}=-{\frac {d\Phi _{C}}{dt}}=-{\frac {d}{dt}}\oint _{C}{\boldsymbol {A}}\cdot \mathrm {d} {\boldsymbol {\ell }}\ ,$ where ${\boldsymbol {E}}$ 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 time-varying flux $\Phi _{C}$ of magnetic magnetic field, and ${\boldsymbol {A}}$ 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  ). 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 is a conductor that carries current $I$ in the direction of integration around the loop, and the magnetic flux is due to that current, we have that $\Phi _{B}=LI$ , where $L$ 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' 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' is essentially the same as that in loop C, so we can write

${\mathcal {E}}_{C}={\mathcal {E}}_{C'}=-{\frac {d\Phi _{C'}}{dt}}=-L{\frac {dI}{dt}}=\oint _{C}{\boldsymbol {E}}\cdot \mathrm {d} {\boldsymbol {\ell }}=\int _{1}^{2}{\boldsymbol {E}}_{\mathrm {conductor} }\cdot \mathrm {d} {\boldsymbol {\ell }}-\int _{1}^{2}{\boldsymbol {E}}_{\mathrm {center\ line} }\cdot \mathrm {d} {\boldsymbol {\ell }}\ .$ For a good conductor, ${\boldsymbol {E}}_{\mathrm {conductor} }$ is negligible, so we have, to a good approximation,
$L{\frac {dI}{dt}}=\int _{1}^{2}{\boldsymbol {E}}_{\mathrm {center\ line} }\cdot \mathrm {d} {\boldsymbol {\ell }}=V_{1}-V_{2}\ ,$ where V is the electric scalar potential along the centerline between points 1 and 2. Thus, we can associate an effective "voltage drop" $L\ dI/dt$ with an inductor (even though our basic uunderstanding of induced emf is based on the vector potential rather than the scalar potential), and consider it as a load element in Kirchhoff's loop law, $\sum {\mathcal {E}}_{\mathrm {source} }=\sum _{\mathrm {load\ elements} }\mathrm {voltage\ drops} ,$ where now the induced emf is not considered to be a source emf.

This definition can be extended to arbitrary sources of emf and moving paths C:

{\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 $\oint _{C}\left[{\boldsymbol {E}}+{\boldsymbol {v}}\times {\boldsymbol {B}}\right]\cdot \mathrm {d} {\boldsymbol {\ell }}$ is often called a "motional emf".

## In (electrochemical) thermodynamics

When multiplied by an amount of charge dQ the emf ℰ yields a thermodynamic work term ℰdQ that is used in the formalism for the change in Gibbs energy when charge is passed in a battery:

$dG=-S\,dT+V\,dP+{\mathcal {E}}\,dQ\ ,$ where G is the Gibbs 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 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:

$\left({\frac {\partial {\mathcal {E}}}{\partial T}}\right)_{Q}=-\left({\frac {\partial S}{\partial Q}}\right)_{T}$ If a mole of ions goes into solution (for example, in a Daniell cell, as discussed below) the charge through the external circuit is:

$\Delta Q=-n_{0}F_{0}\ ,$ where n0 is the number of electrons/ion, and F0 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:

$\Delta H=-n_{0}F_{0}\left({\mathcal {E}}-T{\frac {d{\mathcal {E}}}{dT}}\right)\ ,$ where ΔH is the enthalpy of reaction. The quantities on the right are all directly measurable. Assuming constant temperature and pressure:

$\Delta G=-n_{0}F_{0}{\mathcal {E}}$ which is used in the derivation of the Nernst equation.

## Potential difference

An electrical potential difference is sometimes called an emf. The points below illustrate the more formal usage, in terms of the distinction between emf and the voltage it generates:

1. 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 potential difference on going around a circuit is zero. (The ohmic IR voltage drop plus the applied electrical voltage sum to zero. See 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.
2. 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 Law)
3. 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".
4. 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.
5. Other devices that produce emf are fuel cells, thermocouples, and thermopiles.

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 and potential difference V, provided that the internal resistance r already has been measured:  = 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 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 Ohm and the one on the right is 200 Ohm, they are connected at the top and bottom at points A and B. The induced voltage, by Faraday's law is V, so the current I = V/(100+200). Therefore the voltage across the 100-Ohm resistor is 100 I and the voltage across the 200-Ohm resistor is 200 I, yet the two resistors are connected on both ends, but $V_{AB}$ measured with the voltmeter to the left of the solenoid is not the same as $V_{AB}$ measured with the voltmeter to the right of the solenoid. 

## Generation

### Chemical sources 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. and Transition state.

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

$\mathrm {Zn_{(s)}\rightarrow Zn_{(aq)}^{2+}+2e^{-}\ }$ The zinc sulfate is the electrolyte in that half cell. It is a solution which contains zinc cations $\mathrm {Zn} ^{2+}$ , and sulfate anions $\mathrm {SO} _{4}^{2-}$ 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:

$\mathrm {Cu_{(aq)}^{2+}+2e^{-}\rightarrow Cu_{(s)}\ }$ 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 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.

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. 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 = 1.0934 V, with a temperature coefficient of d/dT = −4.53×10−4 V/K.

#### Voltaic cells

Volta developed the voltaic cell about 1792, and presented his work March 20, 1800. Volta correctly identified the role of dissimilar electrodes in producing the voltage, but incorrectly dismissed any role for the electrolyte. 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." A typical symbolic convention in 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 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.

EMF Cell chemistry Common name
Anode Solvent, electrolyte Cathode
1.2 V Mischmetal (hydrogen absorbing) Water, potassium hydroxide Nickel nickel–metal hydride
1.5 V Zinc Water, ammonium or zinc chloride Carbon, manganese dioxide Zinc carbon
3.6 V to 3.7 V Graphite Organic solvent, Li salts LiCoO2 Lithium-ion
1.35 V Zinc Water, sodium or potassium hydroxide HgO Mercury cell

### Electromagnetic induction

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

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

An emf is induced in a coil or conductor whenever there is change in the flux 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. 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

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. 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 levels in the two solids when they are at charge neutrality, where the Fermi level (a name for the chemical potential of an electron system) describes the energy necessary to remove an electron from the body to some common point (such as ground). 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 levels 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. 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 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.

### Solar cell Solar cell voltage as a function of solar cell current delivered to a load for two light-induced currents IL; currents as a ratio with reverse saturation current I0. Compare with Fig. 1.4 in Nelson.

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 pairs 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 built-in potential, which arises from the contact potential between the two different materials in the junction.) The charge separation between positive holes and negative electrons across a p-n junction (a diode) yields a forward voltage, the photo voltage, between the illuminated diode terminals, 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 I0 = ISH + ID:

$I=I_{L}-I_{0}=I_{L}-I_{SH}-I_{D}$ Losses limit the current available to the external circuit. The light-induced charge separation eventually creates a current (called a forward current) ISH through the cell's junction in the direction opposite that the light is driving the current. In addition, the induced voltage tends to forward bias the junction. At high enough levels, this forward bias of the junction will cause a forward current, ID in the diode opposite that induced by the light. Consequently, the greatest current is obtained under short-circuit conditions, and is denoted as IL (for light-induced current) in the equivalent circuit. 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:

$I=I_{L}-I_{0}\left(e^{qV/(mkT)}-1\right)\ ,$ where I0 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). This relation is plotted in the figure using a fixed value m = 2. 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 equation as:

$V_{\text{oc}}=m\ {\frac {kT}{q}}\ \ln \left({\frac {I_{\text{L}}}{I_{0}}}+1\right)\ ,$ which is useful in indicating a logarithmic dependence of Voc upon the light-induced current. Typically, the open-circuit voltage is not more than about 0.5 V.

When driving a load, the photo voltage is variable. As shown in the figure, for a load resistance RL, the cell develops a voltage that is between the short-circuit value V = 0, I = IL and the open-circuit value Voc, I = 0, a value given by Ohm's law V = I RL, 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 (neglecting the parasitic resistances).

In contrast to the battery, at current levels delivered to the external circuit near IL, the solar cell acts more like a current generator rather than a voltage generator (near vertical part of the two illustrated curves) 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.