Animated Solution for Physics - Current Electricity: Consider two identical galvanometers and two identical resistors with resistance $R$. If the internal resistance of the galvanometers $R_c<\frac{R}{2}$, which of the following statement(s) about any one of the galvanometers is (are) true?

Imagine you are given a toolkit containing two identical galvanometers and two identical resistors. Your mission is to wire them up in ways that push the boundaries of what a single galvanometer can measure. Specifically, we want to find the configurations that yield the maximum possible voltage range and the maximum possible current range.

To tackle this, we need to revisit the fundamental principles of how a galvanometer is converted into a voltmeter and an ammeter.

A voltmeter is essentially a galvanometer with a high resistance connected in series. The voltage $V$ that the combination can measure is given by Ohm's law: where $I_g$ is the full-scale deflection current of the galvanometer and $R_{eq}$ is the total equivalent resistance of the circuit.

Since $I_g$ is a fixed property of the galvanometer, the only way to maximize the voltage range $V$ is to maximize the equivalent resistance $R_{eq}$.

Now, think about how resistors combine. When you connect components in series, their resistances add up. When you connect them in parallel, the equivalent resistance drops below the smallest individual resistance. Therefore, to get the absolute maximum resistance out of our toolkit, we must connect all four components in series.

In this configuration, the total resistance is $R_{eq}=R_c+R_c+R+R$, which is the highest possible value. Thus, the maximum voltage range is obtained when all components are in series.

An ammeter, on the other hand, is a galvanometer with a very small resistance (called a shunt) connected in parallel. The total current $I$ that the ammeter can measure is the sum of the current through the galvanometer and the current through the shunt: where $S_{eq}$ is the equivalent resistance of the shunt path.

To maximize the current range $I$, we need to make the term $\frac{R_c}{S_{eq}}$ as large as possible. This means we must minimize the shunt resistance $S_{eq}$.

How do we get the smallest possible resistance from our components? By connecting them all in parallel! If we place the second galvanometer and both resistors in parallel with the first galvanometer, we create the path of least resistance for the bypass current.

In this configuration, the equivalent shunt resistance is given by: This yields the absolute minimum resistance, allowing the maximum amount of current to bypass the measuring galvanometer. Thus, the maximum current range is obtained when all components are connected in parallel.

By simply applying the rules of series and parallel combinations to the formulas for voltmeter and ammeter ranges, we can confidently conclude that: 1. Maximum voltage range requires a series connection. 2. Maximum current range requires a parallel connection.