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184_notes:loop [2017/10/06 14:55] – dmcpadden | 184_notes:loop [2022/04/04 12:40] – hallstein | ||
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- | ===== Magnetic field due to a wire ===== | + | Section 21.6 in Matter and Interactions (4th edition) |
- | Our canonical example for the magnetic field will be a long straight wire with a current I running | + | / |
- | FIXME Add figure with cross-section of wire | + | [[184_notes: |
- | ==== What is the " | + | ===== Magnetic |
- | As with the [[184_notes: | + | For the context 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, |
- | $$\oint \vec{B} \cdot d\vec{l} | + | ==== What is the " |
- | + | Since we already calculated | |
- | 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 [[184_notes: | + | |
=== Magnetic field circulates around moving charges === | === Magnetic field circulates around moving charges === | ||
- | As you have seen, the [[184_notes: | + | As you have seen before, the [[184_notes: |
- | {{url>http:// | + | {{url>https:// |
- | 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, | + | 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, |
=== Symmetry is critical === | === Symmetry is critical === | ||
- | The structure (or symmetry) of this field is very useful for Ampere' | + | The structure (or symmetry) of this field is very useful for Ampere' |
- | ==== What is $\vec{B}\cdot d\vec{l}$? ==== | + | We'll start by thinking about the magnetic field outside a wire and the mathematical representation Ampere' |
- | You have seen these kinds of dot products before, i.e., when we defined [[184_notes: | + | $$\oint |
- | This integral formulation comes from the [[https:// | + | Again, |
- | 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, | + | {{youtube> |
- | $$\oint \vec{B} \cdot d\vec{l}.$$ | + | ==== What is $\vec{B}\bullet d\vec{l}$? ==== |
+ | |||
+ | You have seen these kinds of dot products before - when we defined [[184_notes: | ||
+ | |||
+ | This integral formulation comes from the [[https:// | ||
+ | |||
+ | **So think of $\vec{B}\bullet d\vec{l}$ 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 along a closed path to find the curl of the magnetic field at a given distance from the current, | ||
+ | |||
+ | $$\oint \vec{B} \bullet | ||
This gives us the full left-hand side of the equation, it is measure of how curly the magnetic field is at some distance from the current. | This gives us the full left-hand side of the equation, it is measure of how curly the magnetic field is at some distance from the current. | ||
=== How do we compute it? === | === How do we compute it? === | ||
+ | [{{ 184_notes: | ||
- | As we will work primarily with cylindrically shaped wires, the loop you choose will generally be a circle with the wire at the center because | + | First, **we want to pick a Amperian loop (or the imaginary path) so that the observation point is somewhere along the path**. For example, if we wanted to find the magnetic field a distance of $R$ away from the wire, we would want to pick a loop that has a radius of $R$ around |
- | FIXME Figure | + | Then, since we get to pick what kind of loop we want, we want to choose a loop that will make our math easy. If we look at the $$\oint \vec{B} \bullet d\vec{l}$$ - there are a couple of steps that will make our calculation MUCH simpler: |
+ | - **We want $\vec{B}$ to be parallel to $d\vec{l}$ because then the dot product turns into multiplication** $$\oint \vec{B} \bullet d\vec{l} = \oint |\vec{B}||d\vec{l}| $$ | ||
+ | - **We want the magnitude of $\vec{B}$ to be constant so we can pull it out of the integral** $$\oint |\vec{B}||d\vec{l}| = |\vec{B}| \oint |dl|$$ | ||
- | It is around this loop that we compute the integral, checking how much of the $\vec{B}$ lines up with our little | + | This means we want to choose a closed |
- | ==== What do we get for the integral? ==== | + | **It's important to note that this loop isn't real** - there is not a wire or anything physically around |
- | If the magnetic field points in the same direction as our Amperian loop and it has the same magnitude along that loop, then the calculation of this integral is relatively straight-forward. Both of these conditions are satisfied for any Amperian loop that is centered on a wire with a uniform distribution of current. | + | [{{ 184_notes: |
+ | |||
+ | Formally, we are doing a path or line integral around a loop with the magnetic field, but for most cases this integral will simplify quite a bit. Also, it's ok if this idea is a bit abstract now, we will put it all together with an example in next set of notes. | ||
+ | |||
+ | ==== Long Wire Example ==== | ||
+ | |||
+ | Both of these conditions are satisfied if we pick a circular Amperian loop that is centered on a wire with a uniform distribution of current: 1) the magnetic field points in the same direction as our Amperian loop and 2) it has the same magnitude along that loop. This makes the calculation of this integral is relatively straight-forward. | ||
For an Amperian loop of radius $R$ centered on a wire with uniform current, we find that this integral is, | For an Amperian loop of radius $R$ centered on a wire with uniform current, we find that this integral is, | ||
- | $$\oint \vec{B} \cdot d\vec{l} = B\oint dl = B 2\pi R$$ | + | \begin{align*} |
+ | \oint \vec{B} \bullet | ||
+ | & | ||
+ | &= B \oint dl \\ | ||
+ | &= B l | ||
+ | \end{align*} | ||
+ | |||
+ | where the first two steps listed come from the fact that $\vec{B}$ points in the direction of $d\vec{l}$ everywhere on the loop (so the dot product simplifies to multiplication) and the last two steps listed come from the fact that the B-field is the same magnitude along that loop (so you can pull it out of the integral). The resulting integral of $dl$ gives just $l$. Because this length comes from the integration of the $dl$, **the length $l$ here represents the length of the Amperian loop**. For our example, the length of the loop would just be the circumference of the loop. Since we said that the loop would have a radius R, this means $l=2\pi R$, so we get: | ||
+ | $$\oint \vec{B} \bullet d\vec{l} | ||
- | where the last two steps come from the fact that $\vec{B}$ is the direction | + | Remember that the $B$ in this equation |