Animated Solution for Physics - Magnetic Effects of Current: One of the two identical conducting wires of length $L$ is bent in the form of a circular loop and the other one into a circular coil of $N$ identical turns. If the same current is passed in both, the ratio of the magnetic field at the centre of the loop ($B_L$) to that at the centre of the coil ($B_C$), i.e. $\frac{B_L}{B_C}$ will be

Imagine you have a straight piece of conducting wire of length $L$. If you bend it into a circular loop and pass a current through it, a magnetic field is generated at its center. But what happens if you take an identical wire and wind it tightly into a coil with multiple turns? How does the magnetic field change? Let's dive into the physics and geometry behind this fascinating problem.

The magnetic field at the center of a circular coil with $N$ turns and radius $R$ carrying a current $I$ is given by the fundamental formula derived from the Biot-Savart Law:

This equation tells us that the magnetic field is directly proportional to the number of turns $N$ and the current $I$, but inversely proportional to the radius $R$.

For the first wire, it is bent into a single loop, meaning $N=1$. Let its radius be $R_1$. Since the entire length $L$ of the wire forms the circumference of this single loop, we can write:

Now, substituting this radius into our magnetic field formula, we get the field $B_L$ at the center of the single loop:

Now, let's look at the second wire. It is wound into a coil of $N$ turns. Let its new radius be $R_2$. The total length $L$ now forms $N$ circumferences. This means the wire is shared among $N$ smaller circles:

Notice that the new radius $R_2$ is $N$ times smaller than $R_1$. Substituting this new radius and the $N$ turns into the magnetic field formula, we get the field $B_C$:

Finally, we need to find the ratio of $B_L$ to $B_C$. By dividing the two expressions we just derived, we can see the magic of algebra as the common terms cancel out:

The Big Takeaway: For a fixed length of wire, winding it into $N$ turns doesn't just increase the magnetic field by a factor of $N$. Because the radius also decreases by a factor of $N$, the magnetic field at the center actually shoots up by a factor of $N^2$! This is why electromagnets are made with thousands of tightly wound turns.