Animated Solution for Physics - Magnetic Effects of Current: Consider a galvanometer shunted with $5\Omega$ resistance and $2\%$ of current passes through it. What is the resistance of the given galvanometer ?

Imagine you are building an ammeter. To do this, you take a sensitive galvanometer and connect a very small resistance—called a shunt—in parallel with it. This setup allows the bulk of the current to bypass the delicate galvanometer coil, protecting it from damage while still allowing us to measure the total current.

In our specific problem, we are given a shunt resistance of $R_{sn}=5\Omega$. We are also told that when the main current $I$ enters the junction, only $2\%$ of it flows through the galvanometer. This means the current through the galvanometer is $I_G=0.02I$.

By applying Kirchhoff's Current Law at the junction, we know that whatever current doesn't go through the galvanometer must go through the shunt. Therefore, the current through the shunt is $I_{sn}=I-0.02I=0.98I$.

Because the galvanometer and the shunt resistance are connected in parallel, the fundamental rule of parallel circuits applies: the potential difference across both branches must be exactly the same.

We can write this mathematically as $V_G=V_{sn}$.

Using Ohm's Law ($V=IR$), we can express the voltage across each branch in terms of its current and resistance. This gives us our master equation: where $G$ is the resistance of the galvanometer that we want to find.

Now, let's substitute the values we found into our master equation. We get:

Notice something beautiful here? The main current $I$ appears on both sides of the equation. This means we can completely cancel it out! The resistance of the galvanometer does not depend on the total current flowing through the circuit.

After canceling $I$, we can solve for $G$:

To make the calculation easier, we can multiply the numerator and the denominator by $100$ to remove the decimals:

Dividing $98$ by $2$ gives $49$, so we are left with:

Finally, multiplying $49$ by $5$ gives us our answer: $G=245\Omega$

This is the resistance of the given galvanometer. It's a classic application of parallel circuits and current division!