E.m.f. and Internal Resistance

The $\text{e.m.f.}$ $\epsilon$ is the total energy the cell gives each coulomb. But the charge must travel through the cell itself, whose internal resistance $r$ opposes it, so some energy is dissipated there as heat.

What reaches the external circuit is the $\text{terminal potential difference}$ $V$, the energy per coulomb actually delivered. The difference, $Ir$, is called the $\text{lost volts}$.

Applying conservation of energy round the loop: the e.m.f. equals the p.d. across the external resistance plus the p.d. across the internal resistance.

The lost volts $Ir$ grow in direct proportion to the current, so drawing more current means more energy dissipated inside the cell and less available outside.

At the extremes the behaviour is clear. With no current drawn — an open circuit, or a voltmeter of very high resistance — there are no lost volts and the terminal p.d. equals the e.m.f. exactly. That is why a voltmeter across an unconnected cell reads its e.m.f.

At the other extreme, a short circuit makes $R\approx 0$, so the current is limited only by the internal resistance: $I_{\text{max}}=\frac{\epsilon }{r}$. All the energy is then dissipated inside the cell, which is why a shorted battery heats up dangerously.

Rearranging $V=\epsilon -Ir$ into the form $y=mx+c$ shows that a graph of terminal p.d. against current is a straight line.

The $\text{intercept}$ on the $V$-axis is $\epsilon$, since that is the terminal p.d. when no current flows. The $\text{gradient}$ is $-r$, so the internal resistance is the magnitude of the (negative) gradient.

This is the standard experimental method: vary the external resistance, record $V$ and $I$ at each setting, plot $V$ against $I$, and read both quantities off one graph.

The cell supplies total power $\epsilon I$. Of that, $I^{2}R$ is delivered usefully to the external circuit and $I^{2}r$ is wasted as heat inside the cell.

The efficiency is therefore $\frac{R}{R+r}$, which approaches 1 when the external resistance greatly exceeds the internal resistance. Low internal resistance means an efficient source.

A subtlety worth knowing: maximum $\text{power transfer}$ to the external circuit occurs when $R=r$, but the efficiency is then only 50%. Maximum power and maximum efficiency are different objectives, and which you want depends on the application.