Magnetic field due to a wire

Our canonical example for the magnetic field will be a long straight wire with a current I running through it. In this case, we will consider that the wire is a bit thicker than the average wire, so that it has a current density $J=I/A$, which is uniform. That is, at every point in the wire the same amount of charge per unit time per unit area exists. This will help us understand the power of Ampere's Law. The figure below shows a cross-section of the wire.

Add figure with cross-section of wire

As with the electric field and Gauss' Law, the structure and shape of the magnetic field is very important to understanding Ampere's Law. So we will start with the left-hand side of the equation,

$$\oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 I_{enc}$$

to understand what it means and how we can calculate it. In doing this, we will always check our work with the result for a long wire we obtained from using the Biot-Savart law, by adding up all the contributions.

As you have seen, the magnetic field circulates around a moving charge. Below is a simulation that shows this for a series of moving point charges.

As you can see the size of the magnetic field vector (the arrows) gets larger as the charges get closer to the observation locations and smaller as the pass it and move away. The direction is always around the path of the charges. Now, imagine a constant stream of particles very close together, that is, a current of electrons moving in a wire. In that case the direction of the magnetic field is still around the wire, but the magnitude stays constant. That is, for a steady current, the magnetic field at any one point is a constant in time. Furthermore, for cylindrical wires (the ones we will focus on), the magnitude of magnetic field is constant for a given distance from the wire. You saw this with the long wire when we solved the problem using Biot-Savart.

The structure (or symmetry) of this field is very useful for Ampere's Law. That is, the fact that it always circulate around the wire and is the same magnitude for a given distance makes this problem an excellent candidate for using Ampere's Law. In fact, this symmetry holds inside the wire as well. That is, there will be a constant magnitude magnetic field circulating around every point inside the wire as well! This is a really important results that we can explore with Ampere's Law, but that can be a real challenge to deal with using Biot-Savart.

You have seen these kinds of dot products before, i.e., when we defined electric potential. But in that case, we understood this to be a measure work per unit charge. In the case of the magnetic field there is no such analog. That integral of the magnetic field along a path is not an work. As you will see the magnetic force has important properties that make it such that the magnetic field can do no work. So what is this thing?

This integral formulation comes from the mathematical model that the magnetic field seems to obey in every instance we observe it. Namely, how much the magnetic field curls around the wire depends on how much current there is creating the magnetic field. This measure, which is beyond the scope of this course, called the curl of the magnetic field is an important property of the field and is an essential quality of what distinguishes it from the electric field, which can have no curly shape when it is created by static charges.

So think of $\mathbf{B}\cdot d\mathbf{l}$ as the little measure of how much the magnetic field curls around its source (the current). Our job is to add up all those little contributions to find out how curl the magnetic field is at a given distance from the current,

$$\oint \mathbf{B} \cdot d\mathbf{l}.$$

This gives us the full left-hand side of the equation, it is measure of how curly the magnetic field is at some distance from the current.

As we will work exclusively with cylindrically shaped wires, the loop you follow the magnetic field around will always be a circle with the wire at the center. This loop isn't real, but rather, it's like the Gaussian surface that we had before. It's a construct that helps guide our work with Ampere's Law. We refer to it as the “Amperian Loop.”

Figure of amperian loop

It is around this loop that we compute the integral, checking how much of $\mathbf{B}$ lines up with our little $d\mathbf{l}$ and adding it up. Formally, we are doing a path or line integral around a loop with the magnetic field, but for most cases this integral will simplify quite a bit. Also, it's ok if this idea is a bit abstract now, we will put it all together with an example in next set of notes.