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184_notes:loop [2018/03/29 19:58] – dmcpadden | 184_notes:loop [2018/07/24 14:00] – [What is $\vec{B}\cdot d\vec{l}$?] curdemma | ||
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Section 21.6 in Matter and Interactions (4th edition) | Section 21.6 in Matter and Interactions (4th edition) | ||
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+ | [[184_notes: | ||
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+ | [[184_notes: | ||
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===== Magnetic field due to a wire ===== | ===== Magnetic field due to a wire ===== | ||
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==== What is the " | ==== What is the " | ||
- | Since we already calculated the B-field from a long straight wire using the [[184_notes: | + | Since we already calculated the B-field from a long straight wire using the [[184_notes: |
=== Magnetic field circulates around moving charges === | === Magnetic field circulates around moving charges === | ||
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=== 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' |
We'll start by thinking about the magnetic field outside a wire and the mathematical representation Ampere' | We'll start by thinking about the magnetic field outside a wire and the mathematical representation Ampere' | ||
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{{youtube> | {{youtube> | ||
- | ==== What is $\vec{B}\cdot d\vec{l}$? ==== | + | ==== What is $\vec{B}\bullet |
- | You have seen these kinds of dot products before, i.e., when we defined [[184_notes: | + | You have seen these kinds of dot products before, i.e., when we defined [[184_notes: |
This integral formulation comes from the [[https:// | This integral formulation comes from the [[https:// | ||
- | 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, | + | So think of $\vec{B}\bullet |
- | $$\oint \vec{B} \cdot d\vec{l}.$$ | + | $$\oint \vec{B} \bullet |
This gives us the full left-hand side of the equation, it is measure of how curly the magnetic field is at some distance from the current. | This gives us the full left-hand side of the equation, it is measure of how curly the magnetic field is at some distance from the current. | ||
=== How do we compute it? === | === How do we compute it? === | ||
- | {{ 184_notes: | + | [{{ 184_notes: |
- | Since we get to pick what kind of loop we want, we want to choose a loop that will make our math easy. If we look at the $$\oint \vec{B} \cdot d\vec{l}$$ - there are a couple of steps that will make our calculation MUCH simpler (just like with [[184_notes: | + | Since we get to pick what kind of loop we want, we want to choose a loop that will make our math easy. If we look at the $$\oint \vec{B} \bullet |
- We want $\vec{B}$ to be parallel to $d\vec{l}$ (because then the dot product turns into multiplication) | - We want $\vec{B}$ to be parallel to $d\vec{l}$ (because then the dot product turns into multiplication) | ||
- We want the magnitude of $\vec{B}$ to be constant so we can pull it out of the integral | - We want the magnitude of $\vec{B}$ to be constant so we can pull it out of the integral | ||
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//It's important to note that this loop isn't real//, but rather, it's like the [[184_notes: | //It's important to note that this loop isn't real//, but rather, it's like the [[184_notes: | ||
- | {{ 184_notes: | + | [{{ 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. | 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. | ||
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\begin{align*} | \begin{align*} | ||
- | \oint \vec{B} \cdot d\vec{l} &= \oint |\vec{B}||d\vec{l}| \\ | + | \oint \vec{B} \bullet |
&= \oint B dl \\ | &= \oint B dl \\ | ||
&= B \oint dl \\ | &= B \oint dl \\ | ||
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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 $l$, which is the length of the loop or just 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: | 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 $l$, which is the length of the loop or just 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: | ||
- | $$\oint \vec{B} \cdot d\vec{l} = B(2 \pi R)$$ | + | $$\oint \vec{B} \bullet |
Remember that the $B$ in this equation is the magnitude of the B-field on that loop (so this would be the magnetic field at a distance $R$ away from the straight wire). | Remember that the $B$ in this equation is the magnitude of the B-field on that loop (so this would be the magnetic field at a distance $R$ away from the straight wire). |