Potentiometer Principle

A potentiometer consists physically of a long wire AB, of length L, resistance R and uniform cross-sectional area A, connected across a driving cell of emf $\varepsilon$ and internal resistance r (together with a key and a rheostat, used to set a convenient current). When the circuit is switched on, a steady current

flows the length of the wire. The potential difference across the whole wire is then, by Ohm's law,

and dividing this by the wire's own length L gives the potential difference PER UNIT LENGTH of the wire,

As long as the driving emf $\varepsilon$ (and hence the current I) stays constant, this quantity $V_{AB}/L$ stays constant too. It is called the potential gradient of the wire, usually denoted K, and can be defined simply as the potential difference per unit length of the potentiometer wire.

Because K is constant along a uniform wire carrying a steady current, the potential difference between the starting point A and ANY other point on the wire is directly proportional to the length of wire in between. If C is a point on the wire at a distance $l$ from A, then

$V_{AC} = Kl \qquad \text{i.e.} \qquad V_{AC} \propto l$ …