Potentiometer: Principle

A potentiometer consists of a long (often several metres), uniform resistance wire, through which a steady current is maintained by a driver cell (of EMF higher than any PD to be measured) connected in series with a rheostat and key -- this constitutes the primary circuit. Because the wire is of uniform cross-section and material, and carries a constant current, the potential falls perfectly UNIFORMLY along its length; the potential drop per unit length, called the potential gradient, $k$, is constant along the whole wire:

where $V$ is the potential difference across the whole length $L$ of wire (as maintained by the primary circuit). This constant, known potential gradient is what makes the instrument useful: the potential difference between the wire's starting end and ANY point a distance $l$ along it is simply $kl$, continuously and precisely adjustable just by choosing where along the wire to make contact.

Principle (null-point / balance method). A secondary circuit, containing the EMF or PD to be measured (through a galvanometer and a sliding jockey contact on the wire), is connected so that this unknown EMF OPPOSES the potential drop along the portion of primary wire between the fixed end and the jockey. As the jockey is slid along the wire, the point is found where the galvanometer shows exactly zero deflection -- the balance point, at length $l$ from the fixed end -- at which the potential drop along that length of wire, $kl$, exactly equals the unknown EMF or PD:

$\varepsilon \ (\text{or } V) = k\,l$ …