Comparison of EMFs of Two Cells Using a Potentiometer

To compare the EMFs $\varepsilon_1$ and $\varepsilon_2$ of two cells using a potentiometer, each cell is connected in turn into the secondary circuit (usually via a two-way key that allows quickly switching between them without disturbing the primary circuit's current, and hence without disturbing the potential gradient $k$), and its own balance length is found:

Dividing the two equations, the (unknown) potential gradient $k$ cancels out completely, leaving

So the ratio of the two EMFs is obtained directly from the ratio of their two balance lengths, without ever needing to know the value of $k$, the exact resistance of the potentiometer wire, or the current in the primary circuit -- all that is required is that the primary current (and hence $k$) stays exactly the same between the two balance readings, which is why the primary circuit's rheostat setting must NOT be touched while switching between the two cells. …