Animated Solution for Physics - Magnetic Effects of Current: A conducting circular loop of radius $r$ carries a constant current $i$. It is placed in a uniform magnetic field ${\mathbf{B}}_0$ such that ${\mathbf{B}}_0$ is perpendicular to the plane of the loop. The magnetic force acting on the loop is

Imagine you are holding a circular ring of wire, and a steady current is flowing through it. Now, you plunge this ring into a region where a uniform magnetic field exists, pointing straight through the loop. What happens to the loop? Does it fly away? Does it spin? Let's dive into the physics of this classic scenario.

We have a conducting circular loop of radius $r$ carrying a constant current $i$. It is placed in a uniform magnetic field ${\mathbf{B}}_0$. The problem states that ${\mathbf{B}}_0$ is perpendicular to the plane of the loop.

To understand the macroscopic behavior of the loop, we must first look at the microscopic level. We consider a tiny, infinitesimal segment of the wire, represented by the vector $d\mathbf{l}$. The direction of $d\mathbf{l}$ is along the direction of the current $i$.

The magnetic force $d\mathbf{F}$ acting on this tiny current element is given by the fundamental Lorentz force law for currents:

By applying the right-hand rule, we can see that if the magnetic field is pointing out of the page and the current is flowing counter-clockwise, the force $d\mathbf{F}$ on every segment points radially outwards. It seems like the loop wants to expand!

To find the total, net magnetic force ${\mathbf{F}}_{\text{net}}$ acting on the entire loop, we must integrate this force over the closed path of the loop:

Here is where the magic happens. Because the magnetic field ${\mathbf{B}}_0$ is uniform, it is a constant vector everywhere in space. In calculus, constants can be pulled out of the integral. The current $i$ is also constant.

Now, look closely at the term $\oint d\mathbf{l}$. This represents the vector sum of all the tiny displacement vectors around a closed path. If you start walking from one point on a circle and go all the way around until you reach your starting point, what is your net displacement? It is exactly zero!

Substituting this back into our force equation, we get a beautiful and elegant result:

The net magnetic force acting on the loop is zero. This is a profound and universal principle in electromagnetism: The net magnetic force on any closed current-carrying loop placed in a uniform magnetic field is always zero, regardless of the loop's shape.

While the net force is zero (meaning the center of mass of the loop will not accelerate), the individual outward forces still exist. They create a tension within the wire, trying to stretch it. If the magnetic field were not uniform, we couldn't pull ${\mathbf{B}}_0$ out of the integral, and the net force would generally not be zero.