Animated Solution for Physics - Current Electricity: $R_1,R_2,R_3$ are different values of $R$. Further, $A,B$ and $C$ are the null points obtained corresponding to $R_1,R_2$ and $R_3$ respectively. For which resistor, the value of $X$ will be the most accurate and why?

The problem presents us with a classic meter bridge setup, a practical application of the Wheatstone bridge. We are given an unknown resistance $X$ in the left gap and a known, variable resistance $R$ in the right gap. The problem states that for three different values of $R$—namely $R_1,R_2$, and $R_3$—we obtain three distinct null points: $A,B$, and $C$, respectively. The core question is: for which of these resistors will the calculated value of $X$ be the most accurate?

To understand the accuracy of our measurement, we first need to revisit the principle of the meter bridge. When the jockey is tapped on the wire and the galvanometer shows zero deflection, the bridge is said to be balanced. At this null point, the ratio of the resistances in the upper arms equals the ratio of the resistances of the corresponding wire segments.

Mathematically, if the null point is found at a distance $l$ from the left end, the resistance of the left segment of the wire is proportional to $l$, and the right segment is proportional to $(100-l)$. The balancing condition is given by:

From this, we can calculate the unknown resistance $X$:

Now, any measurement in physics is subject to errors. In the case of a meter bridge, the accuracy of the calculated value of $X$ depends heavily on the sensitivity of the bridge. A highly sensitive bridge will show a large deflection in the galvanometer even for a very small imbalance in the resistances. This allows us to pinpoint the null point with much greater precision.

But when is a Wheatstone bridge most sensitive? Theoretical analysis and practical experience tell us that a Wheatstone bridge achieves maximum sensitivity when the resistances in all four of its arms are of the same order of magnitude.

Let's apply this condition for maximum sensitivity to our meter bridge. For the resistances of the two wire segments to be of the same order, their lengths must be approximately equal. This means:

Therefore, the bridge is most sensitive, and our measurement is most accurate, when the null point is obtained near the center of the wire (around the 50 cm mark).

Looking at the diagram provided in the question, we can clearly see the positions of the three null points. Point $A$ is towards the left end, point $C$ is towards the right end, and point $B$ is situated right at the center of the wire.

Since point $B$ is at the center, the measurement taken at this null point will be the most accurate. The problem states that null point $B$ corresponds to the resistor $R_2$.

As a bonus tip, avoiding null points near the ends (like $A$ or $C$) is also crucial because of end corrections. The copper strips at the ends of the bridge have some small resistance, and the zero mark of the scale might not perfectly align with the start of the wire. When the null point is near the center, the lengths $l$ and $(100-l)$ are large, making the percentage error introduced by these end corrections negligible!