Animated Solution for Physics - Magnetic Effects of Current: A long wire carries a steady current. It is bent into a circle of one turn and the magnetic field at the centre of the coil is $B$. It is then bent into a circular loop of $n$ turns. The magnetic field at the centre of the coil will be

Have you ever wondered how electromagnets in junkyards can lift entire cars, or how MRI machines generate such immensely powerful magnetic fields? It all comes down to the clever geometry of the wire. In this problem, we explore a fascinating phenomenon: how simply reshaping a wire can drastically amplify the magnetic field it produces.

Imagine a long, straight wire carrying a steady current $I$. We take this wire and bend it into a single, perfect circular loop. Let's denote the radius of this loop as $r$. From Biot-Savart's law, the magnetic field at the exact center of this single loop is given by the standard formula:

Before we proceed, we must establish a fundamental constraint: the total length of the wire is conserved. Since the wire forms a single circle, its total length $l$ is simply the circumference of the circle. Therefore, we can write:

This simple geometric fact allows us to express our original radius $r$ in terms of the total length $l$ as $r=\frac{l}{2\pi }$.

Now for the interesting part. We take that exact same wire—meaning the total length $l$ is strictly conserved—and we bend it again. But this time, instead of one big loop, we wind it tightly into a smaller coil with $n$ identical turns. Let's call the radius of this new, smaller coil $r^{\prime }$.

Because we used the same wire, the total length $l$ must now equal $n$ times the circumference of one small loop. Mathematically, this is:

By equating our two expressions for the length $l$, we can find out exactly how small this new radius $r^{\prime }$ is:

This is a crucial insight: when you wind the wire into $n$ turns, the radius shrinks by a factor of $n$.

Now, let's calculate the new magnetic field at the center of this $n$-turn coil, which we will call $B^{\prime }$. The formula for the magnetic field at the center of a coil with $n$ turns is:

Notice that the number of turns $n$ is directly multiplying the field here. Now, we substitute the value of our new radius $r^{\prime }=\frac{r}{n}$ into this equation:

Look closely at the algebra. We have an $n$ in the numerator, and an $r/n$ in the denominator. The $n$ in the denominator of the denominator flips up and multiplies with the $n$ already in the numerator, giving us $n^2$:

But wait, the term inside the parentheses, $\frac{{\mu }_{0}I}{2r}$, is exactly our original magnetic field $B$! Therefore, the new magnetic field $B^{\prime }$ is:

The field hasn't just increased by a factor of $n$; it has been amplified by $n^2$! This is a beautiful "double effect": one factor of $n$ comes from having more turns (more current loops contributing to the field), and the other factor of $n$ comes from the radius being smaller (bringing the current closer to the center, where the field is measured).