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184_notes:examples:week9_current_segment [2017/10/20 01:16] – [Solution] tallpaul | 184_notes:examples:week9_current_segment [2017/10/20 02:13] (current) – [Magnetic Field from a Current Segment] 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?200 |Segment of Current}} |
===Facts=== | ===Facts=== | ||
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Below, we show a diagram with a lot of pieces of the Biot-Savart Law unpacked. We show an example d→l, | Below, we show a diagram with a lot of pieces of the Biot-Savart Law unpacked. We show an example d→l, | ||
- | {{picture}} | + | {{ 184_notes: |
For now, we write d→l=⟨dx,dy,0⟩ | For now, we write d→l=⟨dx,dy,0⟩ | ||
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d→l×→r=⟨0,0,dx(L+x)−(−dx)(−x)⟩=⟨0,0,Ldx⟩=Ldxˆz | d→l×→r=⟨0,0,dx(L+x)−(−dx)(−x)⟩=⟨0,0,Ldx⟩=Ldxˆz | ||
r3=(x2+(L+x)2)3/2 | r3=(x2+(L+x)2)3/2 | ||
- | The last thing we is the bounds on our integral. Our variable of integration is x, since we chose to express everything in terms of x and dx. 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:// | + | 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 dx. 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} &= \int \frac{\mu_0}{4 \pi}\frac{I \cdot d\vec{l}\times \vec{r}}{r^3} \\ | \vec{B} &= \int \frac{\mu_0}{4 \pi}\frac{I \cdot d\vec{l}\times \vec{r}}{r^3} \\ |