The length of a potentiometer wire is 'L'. A cell of e.m.f. 'E' is balanced at a length $\frac{L}{4}$ from the positive end of the wire. If the length of the original wire is increased by $\frac{L}{3}$, then using the same cell null point is obtained at

The potentiometer principle establishes that cell e.m.f. is proportional to balancing length under uniform potential gradient. For a wire of original length $L$ with applied potential $V_p$, the initial gradient is $k_1 = \frac{V_p}{L}$.

With cell e.m.f. $E$ balanced at $x_1 = \frac{L}{4}$, we have $E = k_1 \cdot x_1 = \left(\frac{V_p}{L}\right) \cdot \frac{L}{4} = \frac{V_p}{4}$.

After increasing the wire length by $\frac{L}{3}$, the new length becomes $L' = \frac{4L}{3}$. The potential gradient becomes $k_2 = \frac{V_p}{L'} = \frac{3V_p}{4L}$.

Equating the e.m.f. expressions $\frac{V_p}{4} = \left(\frac{3V_p}{4L}\right) \cdot x_2$ yields $x_2 = \frac{L}{3}$.

The new balancing length is $\frac{L}{3}$.