formula of electromotive force

Electromotive force

Electromotive force Definition

It is a unit measurement of energy that causes current to flow through a circuit. It is generally abbreviated as EMF. An electric current arises only when there is a difference in potential, called voltage. EMF can either be generated by an electrochemical cell or a magnetic field. The energy per unit charge is normally denoted by the symbol ε. An electric potential difference results when the positive and negative charges are separated. Attachment of a circuit to the source of EMF causes a current flow. There is a sudden dip in the voltage due to internal resistance that develops within the EMF source.

Electromotive force Measurement

EMF is normally measured in volts in SI units, or statvolt in electrostatic units. If an electric charge Q passes through a device with a null internal resistance, there is always a gain in energy W. Thus, the resultant EMF is considered to be W/Q, meaning energy gained per unit charge. In a similar manner, the source of EMF performs work dW on that charge to move it to the positive/high potential terminal. Therefore, we get ε= dW/dq.

Electromotive force Formula

In an open circuit EMF source, the conservative electrostatic field generated after the separation of charge counters the forces that produce the EMF. This is usually termed as “back electromotive force”. The value of the EMF is equal to the integral of the electric field aligned with an internal path between two terminals A and B of an EMF source, but with an opposite sign. Hence, the following formula is obtained:

formula of electromotive force


ECS= the conservative electrostatic field created by the charge separation

dl= the length of path between terminal A and B

·= the vector dot product

The above equation is only a function of the electrostatic electric field and does not involve any non-conservative component of electric field due to Faraday’s law of induction, which states that a time varying magnetic field will induce an electric current.

In a closed circuit, the integral of the electric field is positive due to a varying magnetic field. The net electric potential called induced EMF is highly applicable in a closed loop. The equation obtained for the induced EMF around a closed path C is:

induced EMF equation

Here, E refers to the electric field comprising of both conservative and non-conservative components. The integral used in this equation is an arbitrary but stationary closed curve C with a varying magnetic field. Therefore, the work is actually done against the magnetic field in a closed circuit, which equals to zero.

Electromotive force in Thermodynamics

The EMF on multiplication with an amount of charge dZ, gives εdZ as a function of Gibbs free energy (G), which is the enthalpy (H) of any system minus the product of the temperature (T) and entropy (S). Mathematically it can be represented as:

dG= -SdT + VdP + εdZ ,

Where V is the volume of the system, P refers to pressure and T is the absolute temperature.

At constant pressure, the above equation gives rise to a Maxwell relation between the change in open cell voltage and temperature T with respect to the change in entropy S. However, in the latter case the charge is passed under constant temperature and pressure. Thus, the Maxwell relation is:maxwell relation

In case, a mole of ions enters into the solution then the resulting charge passing through the external circuit is negative due to the discharge of the cell.

∆Z= -n0F0


n0= the total number of electrons or ions

F0= Faraday constant

The thermodynamic properties of the cell under constant pressure and volume share a strong relationship with its EMF, giving rise to:

∆H= -n0F0 (ε-Tdε/dT),

Here, ∆H represents the heat of reaction.

Electromotive force Generation

Some of the commonly used devices that can provide EMF are:

Electrochemical cell

It is capable of producing a potential difference between electrodes using chemical reactions. Molecules constitute a group of atoms that are held tightly by chemical bonds. When the atoms dissociate into electrons and protons there is a separation of charge. During reconfiguration of the atoms, reduction-oxidation or redox reactions occur. In a typical cell, one electrode gains electrons from the solute, and the other loses electrons. For example, a Daniel cell comprises of a zinc anode, which dissolves into a zinc sulfate solution and leaves behind its electrons in the electrode as a part of the oxidation reaction.

Zn(s) →Zn2+ + 2e

On the other hand, the copper ions in a copper sulfate electrolyte extract electrons from the cathode by reduction reaction:

Cu2+ + 2e → Cu(s)

The potential difference formed between the cathode and electrode is the EMF of the cell.

Voltaic/Galvanic cell

It is an electric battery that generates EMF by an irreversible conversion of chemical to electrical energy. This type of cell cannot be recharged and is rendered useless after a while. Here, non-identical electrodes in the form of metals are used for generating voltage. A salt bridge connects the anode and cathode for maintaining the electrical potential difference. A thermodynamic equilibrium can only be achieved when one of the metal assume a higher electrical potential than the other.

Solar/Photovoltaic cell

The device is known to directly convert solar energy into electrical energy. Generally in a semiconductor, photons-containing light can induce electron–hole pairs. The electric field around the p-n junction remains in a state of thermal equilibrium and facilitate the process of charge separation. Subsequently, an electric potential is generated between the positive holes and negative electrons, giving rise to photo voltage that drives current through any attached load. For an illuminated diode, the current-voltage relation is given by:

I= IL –I0 (eqv/ (mkT) – 1)


I= the current delivered to the load

I0= the reverse flow of current

KT/q= the thermal voltage

On rearranging the previous equation, we get the voltage for an open circuit:

VOC= m kT/q ln (I0/I0 + 1)

The other sources of EMF include:

  • Electrical generators

  • Electrical transformer

  • Thermoelectric devices

  • Van de Graaff generators

  • Earth