Animated Solution for Physics - Magnetic Effects of Current: A galvanometer of resistance $100\Omega$ has 50 divisions on its scale and has sensitivity of $20\mu \text{A/division}$. It is to be converted to a voltmeter with three ranges of 0-2 V, 0-10 V and 0-20 V. The appropriate circuit to do so is

Upgrading the Galvanometer

Imagine you have a delicate, highly sensitive instrument—a galvanometer. It's fantastic at detecting tiny trickles of current, but what if you want to measure the hefty voltage of a car battery or a power supply? If you connect it directly, the massive current will instantly fry its delicate coil!

To transform this fragile current detector into a robust, multi-range voltmeter, we need to give it some armor. In the world of electronics, this armor comes in the form of high resistances connected in series. By adding these resistors, we limit the current flowing through the galvanometer, allowing it to safely measure much higher voltages.

Before we start adding armor, we must understand exactly how much current our galvanometer can handle before its needle hits the maximum limit. This is known as the full-scale deflection current ($I_g$).

We are given two crucial pieces of intel: 1. The scale has $50$ divisions. 2. The sensitivity is $20\mu \text{A}$ per division.

To find the absolute maximum current, we simply multiply these together:

Since $1000\mu \text{A}$ is exactly $1\text{ mA}$, we can write this as:

This $10^{-3}\text{A}$ is our golden constraint. No matter what voltage we measure, the current through the circuit must never exceed this value.

When we connect a galvanometer (with its own internal resistance $G$) in series with an external resistance ($R_{\text{series}}$), the total resistance of the circuit becomes $(R_{\text{series}}+G)$.

According to Ohm's Law, the maximum voltage $V$ this setup can measure is the product of the maximum current and the total resistance:

We will use this master equation to conquer each of our three desired voltage ranges: $2\text{ V}$, $10\text{ V}$, and $20\text{ V}$.

Let's start with the smallest range: $0$ to $2\text{ V}$. Looking at our circuit design, the current for this range will only pass through the first resistor, $R_1$, and the galvanometer.

Plugging our values into the master equation:

To solve for $R_1$, we divide both sides by $10^{-3}$:

So, to safely measure up to $2\text{ V}$, our first line of defense must be a $1900\Omega$ resistor.

Now, we want to measure up to $10\text{ V}$. We need more resistance! For this range, the current will flow through both $R_1$ and a new resistor, $R_2$. The total series resistance is now $(R_1+R_2)$.

Let's set up the equation:

Dividing by $10^{-3}$ gives us the total required resistance:

We already know that $R_1=1900\Omega$. Let's substitute that in:

By adding an $8000\Omega$ resistor in series with our first one, we've successfully upgraded our voltmeter to handle $10\text{ V}$!

The 20V Range

Finally, we want to push our voltmeter to its absolute limit: $20\text{ V}$. For this, the current must traverse the entire gauntlet of resistors: $R_1$, $R_2$, and a final resistor, $R_3$.

Our master equation now looks like this:

We already know the values of $R_1$ and $R_2$. Let's plug them in:

We have successfully engineered our multi-range voltmeter! The required resistances are: $R_1=1900\Omega$ $R_2=8000\Omega$ $R_3=10000\Omega$

Comparing our meticulously calculated values with the given options, we can confidently declare that Option (c) is the correct circuit diagram. Physics and logic prevail once again!