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184_notes:examples:week9_current_segment [2017/10/20 00:36] – [Magnetic Field from a Current Segment] tallpaul | 184_notes:examples:week9_current_segment [2017/10/20 02:13] – tallpaul | ||
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You may have read about how to find the [[184_notes: | You may have read about how to find the [[184_notes: | ||
- | {{ 184_notes:9_current_segment.png?400 |Segment of Current}} | + | {{ 184_notes:9_current_segment_bare.png?400 |Segment of Current}} |
===Facts=== | ===Facts=== | ||
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====Solution==== | ====Solution==== | ||
- | We begin by cracking open the Biot-Savart Law. In order to find magnetic field, we will need to take a cross product | + | Below, we show a diagram with a lot of pieces of the Biot-Savart Law unpacked. We show an example |
- | \begin{align*} | + | |
- | \vec{v}\times \vec{r}_1 &= 0 \\ | + | |
- | \vec{v}\times \vec{r}_2 &= 1 \text{ | + | |
- | \vec{v}\times \vec{r}_3 &= 1 \text{ m}^2\text{s}^{-1} \hat{z} | + | |
- | \end{align*} | + | |
- | Note that the first cross-product is 0 because location 1 is situated perfectly | + | |
- | Next, we find the magnitudes of $r^3$, since that is another quantity we need to know in the Biot-Savart Law. | + | {{ 184_notes: |
- | \begin{align*} | + | |
- | {r_2}^3 &= 0.125 \text{ m}^3 \\ | + | |
- | {r_3}^3 &= 0.354 \text{ m}^3 | + | |
- | \end{align*} | + | |
- | + | ||
- | We don't including location 1 above since we already know the magnetic field is 0 at that location! Below, we give the magnetic field at all three locations. | + | |
+ | For now, we write $$\text{d}\vec{l} = \langle \text{d}x, \text{d}y, 0 \rangle$$ | ||
+ | and $$\vec{r} = \vec{r}_{obs} - \vec{r}_{source} = 0 - \langle x, y, 0 \rangle = \langle -x, -y, 0 \rangle$$ | ||
+ | Notice that we can rewrite $y$ as $y=-L-x$. This is a little tricky to arrive at, but is necessary to figure out unless you rotate your coordinate axes, which would be an alternative solution to this example. If finding $y$ is troublesome, | ||
+ | $$\text{d}\vec{l} = \langle \text{d}x, -\text{d}x, 0 \rangle$$ | ||
+ | $$\vec{r} = \langle -x, L+x, 0 \rangle$$ | ||
+ | Now, a couple other quantities that we see will be useful: | ||
+ | $$\text{d}\vec{l} \times \vec{r} = \langle 0, 0, \text{d}x(L+x) - (-\text{d}x)(-x) \rangle = \langle 0, 0, L\text{d}x \rangle = L\text{d}x \hat{z}$$ | ||
+ | $$r^3 = (x^2 + (L+x)^2)^{3/ | ||
+ | The last thing we need is the bounds on our integral. Our variable of integration is $x$, since we chose to express everything in terms of $x$ and $\text{d}x$. Our segment begins at $x=-L$, and ends at $x=0$, so these will be the limits on our integral. Below, we write the integral all set up, and then we evaluate using some assistance some [[https:// | ||
\begin{align*} | \begin{align*} | ||
- | \vec{B}_1 &= \frac{\mu_0}{4 \pi}\frac{q\vec{v}\times \vec{r}_1}{{r_1}^3} = 0 \\ | + | \vec{B} & |
- | \vec{B}_2 & | + | & |
- | \vec{B}_3 | + | &= \frac{\mu_0}{2 \pi}\frac{I}{L}\hat{z} |
\end{align*} | \end{align*} |