The magnetic field of a current loop is one of the problems that is easily solved using Biot-Savart Law. This simple derivation is done in most college physics textbooks, but I had a difficult time locating it online, (although I’m sure it exists somewhere.)
Let start with stating the Biot-Savart Law.
Quick reminder of the terms used here. is the permeability of free space, which you should look up and use in the appropriate units if you will be doing calculations. I is the magnitude of current flowing through the loop. The little circle in the integral symbol means that it’s a closed loop integral, so we need to make sure that we integrate dl, the wire length differential, all the way around the loop so that it connects back to itself as a closed loop. r is the distance from the wire to the point of interest we would like to know the magnetic field. However, is the direction that r is pointing. Make sure you understand the difference between these!
This integral may look a bit daunting, but it simplifies very nicely. First, we will convert the cross product to it’s sine equivalent.
I used 90 degrees as the angle for the sine function because the direction of dl is always perpindicular to the direction of as we integrate around the loop. That’s a little hard to visualize so maybe stare at the diagram for a bit until you are convinced. We are left with,
Now we want to focus on the z-axis component of the B-field. We only worry about that part because the other components of the B-field will cancel out with each other due to the symmetry of the loop.
From trig we can see the can be replaced in terms of r and R.
We need to get rid of this r and get it into terms of something we are more comfortable with. From pythagorean theorem, we see that r is just,
All that is left is to finish the integral dl, which is a circle. So, it’s just the circumference of the circle, !
Nice! Notice if we set z=0, then we get the B-field at the center of the loop.