Measurement of Potential Difference Using a Potentiometer

To measure an unknown potential difference $V$ (for instance, the PD across a resistor carrying current in some external circuit) using a potentiometer, the wire's potential gradient $k$ must first be established using a source of KNOWN EMF (a standard cell, or a cell whose EMF has been independently measured), connected in the secondary circuit in place of the unknown PD, and its balance length $l_0$ found. Then

With the primary circuit current left undisturbed (so $k$ stays fixed), the unknown potential difference $V$ is then connected into the secondary circuit in its place, and its OWN balance length $l$ is found in the usual way. The unknown PD is then

$V = k\,l = \varepsilon_{\text{known}}\,\frac{l}{l_0}$ …