Animated Solution for Physics - Magnetic Effects of Current: Two identical wires A and B, each of length $l$, carry the same current $I$. Wire A is bent into a circle of radius $R$ and wire B is bent to form a square of side $a$. If $B_A$ and $B_B$ are the values of magnetic field at the centres of the circle and square respectively, then the ratio $\frac{B_A}{B_B}$ is

Circle vs Square Magnetic Fields

Imagine you are handed two perfectly identical conducting wires, each of length $l$. You decide to perform a little physics experiment. You bend the first wire into a perfect circle, and the second wire into a neat square. If you pass the exact same current $I$ through both of them, which one produces a stronger magnetic field at its center? And more importantly, what is the exact mathematical ratio of their magnetic fields? Let's dive into the Biot-Savart law and find out!

Let's start with the circular loop, which we will call Wire A. The magnetic field at the center of a circular current-carrying loop of radius $R$ is a standard result derived from the Biot-Savart law:

However, we don't know $R$ directly. What we do know is that the entire wire of length $l$ was used to form the circumference of this circle. Therefore, we can write:

Substituting this expression for $R$ back into our magnetic field equation, we get the magnetic field purely in terms of the given parameters $l$ and $I$:

Now, let's turn our attention to the square loop, Wire B. To find the magnetic field at the center of a square, we must first calculate the magnetic field produced by one of its straight sides and then multiply it by 4 (since all four sides contribute equally and in the same direction by symmetry).

The magnetic field due to a finite straight wire at a perpendicular distance $d$ is given by:

For a square of side $a$, the perpendicular distance from the center to any side is simply half the side length, so $d=\frac{a}{2}$. The angles subtended by the ends of the side at the center are ${\theta }_1={\theta }_2=45^{\circ }$. Plugging these into our formula and multiplying by 4 gives the total field $B_B$:

Just like we did for the circle, we need to express the side length $a$ in terms of the total wire length $l$. Since the perimeter of the square is $l$, we have:

Substituting this into our expression for $B_B$ yields:

We now have the magnetic fields of both shapes expressed in terms of the same variables, ${\mu }_0$, $I$, and $l$. The final step is to find the ratio $\frac{B_A}{B_B}$:

Notice how beautifully the physical constants and variables cancel out! The ${\mu }_0$, $I$, and $l$ terms vanish completely, leaving us with a pure, dimensionless geometric ratio:

This elegant result shows that the magnetic field at the center of a circular loop is slightly stronger than that of a square loop formed from the same length of wire. As a fun challenge, try deriving the general formula for an $n$-sided regular polygon and watch how it converges to the circle's formula as $n$ approaches infinity!