Q.To find the resistance of a gold bangle, two diametrically opposite points of the bangle are connected to the two terminals of the left gap of a metre bridge. A resistance of 4 \Omega is introduced in the right gap. What is the resistance of the bangle if the null point is at 20 cm from the left end?

The two main sources of error in a meter bridge are end resistance at the copper strips and non-uniformity of the bridge wire; both are corrected by calibration/measurement technique rather than by circuit change.

(1) End resistance (end correction): The resistance of the copper strips at the two ends of the meter bridge wire, and of the connections/jockey contact, adds an unknown extra resistance not accounted for by the measured length alone, causing a small error in the balance-point reading.

Minimisation: Determine the end correction experimentally (by interchanging known resistances in the two gaps and comparing the calculated vs. actual balance lengths), and add this correction to every observed balancing length; alternatively, keep the unknown and known resistances comparable so the balance point falls near the middle of the wire, where end effects influence the percentage error least.

(2) Non-uniform cross-section / material of the wire: If the meter-bridge wire does not have a strictly uniform cross-sectional area or uniform resistivity along its length, the assumption $R \propto l$ (resistance proportional to length) breaks down, introducing error. …

Meter bridge balance condition: $\dfrac{X}{R} = \dfrac{l}{100-l}$.

In a meter bridge, with unknown resistance $X$ in the left gap and known resistance $R$ in the right gap, if the null (balance) point is at length $l$ (cm) from the left end, the balance (Wheatstone bridge) condition gives

$\frac{X}{R} = \frac{l}{100-l}$

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