184_notes:loop

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For the purpose of our explanation, we will use a long straight wire with a current $I$ running through it as our example. We'll start by talking about a thin wire and eventually build up to talking about a thick wire. In this class, we will also make the typical assumption that the current in the wire is uniform and in a steady state (upper-division courses may address non-uniform currents). That is, at every point in the wire the same amount of charge per unit time exists.

Since we already calculated the B-field from a long straight wire using the Biot-Savart law (by adding up all the contributions), we already have some insight into to what we think the magnetic field should be from the wire - it should curl around the wire. (We will eventually use the Biot-Savart result at the end to verify what we find using Ampere's Law.) As with the electric field and Gauss' Law, the structure, shape, and symmetry of the magnetic field is very important to understanding how to use Ampere's Law.

Starting with the mathematical representation Ampere's Law:

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

The first thing we will consider will be the shape of the magnetic field for the situation - which is related to the left-hand side of Ampere's Law - namely the $\oint \vec{B} \cdot d\vec{l}$ part.

Magnetic field circulates around moving charges

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 sizes of the magnetic field vectors (the arrows) get larger as the charges get closer to the observation locations and get smaller as they pass and move away. The direction of the magnetic field is always around the path of the charges. Now, imagine a constant stream of particles very close together, that is, a current of many 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.

Symmetry is critical

The structure (or symmetry) of this field is very useful for Ampere's Law. That is, the fact that the B-field always circulates 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 result 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 $\vec{B}\cdot d\vec{l}$ 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 (the E-field cannot have a curly shape when it is created by static charges).

So think of $\vec{B}\cdot d\vec{l}$ 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 the curl of the magnetic field at a given distance from the current,

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

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.

How do we compute it?

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

It is around this loop that we compute the integral, checking how much of the $\vec{B}$ lines up with our little $d\vec{l}$ 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.

If the magnetic field points in the same direction as our Amperian loop and it has the same magnitude along that loop, then the calculation of this integral is relatively straight-forward. Both of these conditions are satisfied for any Amperian loop that is centered on a wire with a uniform distribution of current.

For an Amperian loop of radius $R$ centered on a wire with uniform current, we find that this integral is,

$$\oint \vec{B} \cdot d\vec{l} = B\oint dl = B 2\pi R$$

where the last two steps come from the fact that $\vec{B}$ is the direction of $d\vec{l}$ (so the dot product simplifies to multiplication) and the B-field is the same magnitude along that loop (so you can pull it out of the integral). The resulting integral of $dl$ gives just $l$, which is the length of the loop or just the circumference of the loop.

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