3.5.1.6 - Electromotive Force and Internal Resistance

Emf is energy supplied per coulomb

The electromotive force, or emf, of a source is the energy it supplies per unit charge. Despite the name it is not a mechanical force. It is measured in volts, equivalent to joules per coulomb:

A source with emf $1.8\,\mathrm{V}$ supplies $1.8\,\mathrm{J}$ per coulomb; transferring $5.0\,\mathrm{C}$ corresponds to $9.0\,\mathrm{J}$ supplied. In a chemical cell this comes from chemical energy.

Terminal pd is the potential difference between the external terminals. It is defined even when the circuit is open. When a cell is supplying current, terminal pd gives the energy delivered to the external circuit per coulomb; emf gives the total supplied per coulomb, including energy transferred inside the cell.

With an ideal voltmeter as the only connection, no current is drawn and its reading equals emf. A real high-resistance voltmeter draws a small current, so its reading only approximates emf. A nominal battery label also need not be an exact measurement of its present emf.

Internal resistance divides the supplied energy

A real source can be modelled as an ideal emf source in series with an internal resistance, $r$. This is an equivalent circuit: a chemical cell’s internal processes include ionic conduction and electrochemical effects, so they should not all be described as electrons colliding with a metal lattice.

[DIAGRAM: asset_name: 5.1.6 - Electromotive Force and Internal Resistance - Diagram 1; asset_slug: 5.1.6 - Electromotive Force and Internal Resistance - Diagram 1; caption: The ideal emf source and internal resistance lie inside the real-cell boundary. The same current passes through internal r and external R. The terminals span the complete source model.; description: The ideal emf source and internal resistance lie inside the real-cell boundary. The same current passes through internal r and external R. The terminals span the complete source model.]

For a discharging cell supplying an external resistance $R$, conservation of energy per charge gives

The internal pd $Ir$, sometimes called lost volts, represents energy transferred inside the cell per coulomb. Energy is transferred, not destroyed. For $I>0$ and $r>0$, terminal pd is below emf. These signs describe a cell delivering current; they must not be applied unchanged to charging a cell.

If the current doubles while $r$ stays constant, lost volts double. Internal heating power $I^2r$ instead increases fourfold. Multiplying the energy-per-charge balance by current gives a power balance:

This separates the source’s total power from its external output power.

Solving for a load or an unknown cell

The whole-circuit current is $I=\varepsilon/(R+r)$. Terminal pd can then be found from either $IR$ or $\varepsilon-Ir$; agreement checks the calculation. Do not put emf across $R$ alone when internal resistance matters.

For $\varepsilon=12\,\mathrm{V}$, $r=1.5\,\Omega$ and $R=4.5\,\Omega$, total resistance is $6.0\,\Omega$, so $I=2.0\,\mathrm{A}$. Lost volts are $3.0\,\mathrm{V}$ and terminal pd is $9.0\,\mathrm{V}$. External power is $18\,\mathrm{W}$; internal power is $6.0\,\mathrm{W}$, totaling $24\,\mathrm{W}=\varepsilon I$.

For a resistor network, first find the equivalent external resistance, then add $r$. Return to the external network using terminal pd to find its branch currents; internal resistance carries the total cell current.