Electromotive force and internal resistance

A supply transfers energy to charges in a circuit. Its electromotive force, or emf, is the energy transferred by the supply per unit charge:

Here, $$\varepsilon$$ is the emf, $$E$$ is the energy transferred and $$Q$$ is the charge. Emf is measured in volts: one volt is one joule per coulomb. Despite its name, emf is not a force.

Internal resistance and terminal potential difference

A real supply has internal resistance, $$r$$. When the supply delivers a current $$I$$, some energy is dissipated within it. The energy dissipated per unit charge corresponds to an internal potential difference of $$Ir$$.

The terminal potential difference, $$V$$, is the energy transferred per unit charge to the external circuit. The energy supplied per unit charge is therefore shared between the external circuit and the supply’s internal resistance:

When no current flows, $$I = 0$$, so there is no internal potential difference and the terminal potential difference equals the emf. When the supply delivers a current, $$Ir$$ is positive, so the terminal potential difference is lower than the emf.

Electromotive force and external resistance

If the external circuit has an equivalent resistance $$R$$, its potential difference is $$V = IR$$. Substituting this into the emf relationship gives:

Thus the current depends on the emf and the combined external and internal resistances. These relationships can be rearranged to find an unknown quantity when the relevant others are known.

For example, suppose a supply has an emf of $$6\ \text{V}$$ and an internal resistance of $$1\ \Omega$$, and the external resistance is $$5\ \Omega$$. The current is:

The terminal potential difference is $$V = IR = 5\ \text{V}$$. The remaining $$1\ \text{V}$$ is the internal potential difference, $$Ir$$: of the $$6\ \text{J}$$ supplied per coulomb, $$5\ \text{J}$$ is transferred to the external circuit and $$1\ \text{J}$$ is dissipated within the supply.