A potentiometer wire of length L and a resistance r are connected in series with a battery of e.m.f.  E0 and a resistance r1. An unknown e.m.f. E is balanced at a length l of the potentiometer wire. The e.m.f. E will be given by

The current through the potentiometer wire is I=E0r+r1and the potential difference across the wire is V=Ir=E0rr+r1The potential gradient along the potentiometer wire is k=VL=E0rr+r1LAs the unknown e.m.f. E is balanced against length l of the potentiometer wire,∴  E=kl=E0rr+r1lL