EMF and internal resistanceAQA A-Level Physics: Revision notes

The emf ($\varepsilon$) of a source is the energy transferred to each coulomb of charge passing through it: $\varepsilon = E/Q$, measured in volts. It is the total energy supplied per unit charge.

The terminal potential difference ($V$) is the pd across the terminals of the source when it is delivering current. Charge also does work pushing through the internal resistance $r$ of the source, so some energy is transferred inside the source itself and the terminal pd is less than the emf.

With no current, there are no lost volts and $V = \varepsilon$.

For a cell of emf $\varepsilon$ and internal resistance $r$ connected to an external resistor $R$:

$\varepsilon = I(R + r) = V + Ir$

$V = IR$ is the terminal pd across the load, and $Ir$ is the lost volts. So $V = \varepsilon - Ir$. The larger the current, the larger the lost volts, and the lower the terminal pd. This is why car headlamps dim when the starter motor is used.

By conservation of energy, the energy supplied per coulomb ($\varepsilon$) equals the energy transferred per coulomb in the load ($V$) plus that in the cell ($Ir$).

Work in this order: (1) total resistance $R + r$, (2) current $I = \varepsilon/(R + r)$, (3) terminal pd $V = IR$, (4) lost volts $Ir = \varepsilon - V$.

Power: the total power supplied is $\varepsilon I$, the power in the load is $VI = I^2R$ and the power wasted inside the cell is $I^2r$. The cell is most efficient when $R \gg r$.

Short circuit: if $R \approx 0$, then $I = \varepsilon / r$, which is the largest possible current. The terminal pd is zero and all the power $\varepsilon^2/r$ is dissipated in the cell, so it can overheat.

Cells in a combination: $n$ identical cells in series have emf $n\varepsilon$ and internal resistance $nr$. $n$ identical cells in parallel have emf $\varepsilon$ and internal resistance $r/n$.

Connect the cell in series with an ammeter and a variable resistor, and connect a voltmeter across the cell's terminals. Vary the resistance to obtain a range of currents and record $V$ and $I$ for at least six readings. Disconnect between readings to avoid heating and draining the cell.

Since $V = \varepsilon - Ir$, compare with $y = c + mx$: a graph of $V$ (y-axis) against $I$ (x-axis) is a straight line with intercept $\varepsilon$ and gradient $-r$. Use a large triangle to find the gradient.

A cell with a large internal resistance, or a high-resistance protective resistor in series, keeps the current small and the readings stable.