Animated Solution for Physics - Current Electricity: Shown in the figure adjacent is a meter-bridge set up with null deflection in the galvanometer. The value of the unknown resistor $R$ is

Unlocking Unknown Resistances

Welcome to the fascinating world of electrical instruments! Today, we are going to decode a classic problem involving a Meter Bridge. If you have ever wondered how physicists accurately measure unknown resistances in the lab, this is exactly how they do it.

A meter bridge is essentially a practical, real-world application of the famous Wheatstone Bridge. It consists of a uniform wire that is exactly one meter (or $100\text{ cm}$) long.

In our specific problem, we have a known resistance $P=55\Omega$ connected in the left gap, and an unknown resistance $Q=R$ connected in the right gap. A galvanometer is connected between the central junction of the resistors and a sliding jockey on the wire.

The magic happens when we slide the jockey along the wire until the galvanometer shows zero deflection. This is called the null point. At this exact moment, the bridge is perfectly balanced.

According to the principle of a balanced Wheatstone bridge, the ratio of the resistances in the upper gaps is perfectly equal to the ratio of the resistances of the corresponding wire segments below them. Since the wire is uniform, its resistance is directly proportional to its length ($R\propto l$). Therefore, our master equation becomes:

We are given that the balancing length from the left end is $l_1=20\text{ cm}$. Because the total length of the wire is always $100\text{ cm}$, the remaining length on the right side must be:

Now, let's substitute all our known values into the master equation:

This simplifies beautifully. The fraction on the right side reduces to $\frac{1}{4}$:

By simply cross-multiplying, we can isolate our unknown resistance $R$:

And there we have it! The unknown resistance is exactly $220\Omega$.

While solving this on paper is straightforward, if you are performing this experiment in a real lab, always try to choose your known resistance such that the null point falls near the center of the wire (around $50\text{ cm}$). Balancing near the center minimizes the percentage errors in measuring the lengths $l_1$ and $l_2$, making your final resistance calculation much more robust and accurate!