Measurement of Internal Resistance of a Cell by Potentiometer

The potentiometer can also directly measure a cell's own internal resistance r. The circuit connections (Figure 2.29) place the potentiometer wire's end C at the positive terminal of a primary battery Bt, with the negative terminal connected to end D through a key $K_1$, forming the primary circuit. The test cell, of emf $\varepsilon$ and unknown internal resistance r, has its positive terminal also connected to end C; its negative terminal connects through a galvanometer and high resistance to a sliding jockey. A resistance box R and a second key $K_2$ are connected directly across the test cell.

With $K_2$ OPEN, the test cell is on open circuit, so its full emf drives the balance: the jockey is adjusted for zero deflection, giving balancing length $l_1$, and $\varepsilon \propto l_1$ (2.62). A suitable resistance (say, 10 $\Omega$) is then dialled into the resistance box, and $K_2$ is CLOSED, so current now flows through both the test cell and R. With internal resistance r, the current through the cell and R is $I=\varepsilon/(R+r)$, so the potential difference actually available across R (which is what the secondary circuit now balances against) is

Rebalancing the jockey for this smaller potential difference gives a new, shorter balancing length $l_2$, with $\varepsilon\dfrac{R}{R+r}\propto l_2$ (2.63). Dividing equation (2.62) by equation (2.63) gives $\dfrac{R+r}{R}=\dfrac{l_1}{l_2}$ (2.64), i.e. $1+\dfrac{r}{R}=\dfrac{l_1}{l_2}$, which rearranges to the working formula

$r = R\left(\dfrac{l_1-l_2}{l_2}\right) \qquad (2.65)$ …