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184_notes:b_current [2018/07/03 13:27] – curdemma | 184_notes:b_current [2021/06/16 18:53] – [Magnetic field from Many Charges] bartonmo | ||
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Sections 17.2 and 17.6-17.8 in Matter and Interactions (4th edition) | Sections 17.2 and 17.6-17.8 in Matter and Interactions (4th edition) | ||
- | [[184_notes: | + | /*[[184_notes: |
- | [[184_notes: | + | [[184_notes: |
===== Currents Make Magnetic Fields ===== | ===== Currents Make Magnetic Fields ===== | ||
- | Now that we have talked about a single moving charge, the next source of magnetic fields that we are going to consider is currents (either comprised of electrons or some other charged particle). This builds on what we learned about [[184_notes: | + | Now that we have talked about a single moving charge |
{{youtube> | {{youtube> | ||
==== Magnetic field from Many Charges ==== | ==== Magnetic field from Many Charges ==== | ||
- | {{ 184_notes: | + | [{{ 184_notes: |
If we consider a straight wire with a steady current, there would be many moving charges everywhere in wire that would all contribute to the magnetic field outside of the wire. If we take a " | If we consider a straight wire with a steady current, there would be many moving charges everywhere in wire that would all contribute to the magnetic field outside of the wire. If we take a " | ||
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Since we have many small charges that we are adding the field contributions from, we can turn the summation into an integral and the individual charges $q_i$ into $dq$: | Since we have many small charges that we are adding the field contributions from, we can turn the summation into an integral and the individual charges $q_i$ into $dq$: | ||
$$\vec{B}_{tot}= \int \frac{\mu_0}{4 \pi}\frac{dq \cdot \vec{v}\times \hat{r}}{r^2}$$ | $$\vec{B}_{tot}= \int \frac{\mu_0}{4 \pi}\frac{dq \cdot \vec{v}\times \hat{r}}{r^2}$$ | ||
- | Again this equation just says that we are going to add together the magnetic field contributions at a point from every charge (dq) that is moving in the wire. | + | Again this equation just says that we are going to add together the magnetic field contributions at a point from every charge ($dq$) that is moving in the wire. |
Now we can rewrite the velocity in terms of the differential length and time: $\vec{v}=\frac{d\vec{l}}{dt}$. In other words, velocity is simply the change in displacement (or length) over a change in time. | Now we can rewrite the velocity in terms of the differential length and time: $\vec{v}=\frac{d\vec{l}}{dt}$. In other words, velocity is simply the change in displacement (or length) over a change in time. | ||
$$\vec{B}_{tot}= \int \frac{\mu_0}{4 \pi}\frac{dq \cdot \frac{d\vec{l}}{dt}\times \hat{r}}{r^2}$$ | $$\vec{B}_{tot}= \int \frac{\mu_0}{4 \pi}\frac{dq \cdot \frac{d\vec{l}}{dt}\times \hat{r}}{r^2}$$ | ||
- | Since dt represents a small amount of time, dl represents a small amount of length, and dq represents a small amount of charge, we will treat these as independent and rewrite (much to the chagrin of our mathematician friends): | + | Since $dt$ represents a small amount of time, $dl$ represents a small amount of length, and $dq$ represents a small amount of charge, we will treat these as independent and rewrite (much to the chagrin of our mathematician friends): |
$$dq \cdot \frac{d\vec{l}}{dt} = \frac{dq \cdot d\vec{l}}{dt}=\frac{dq}{dt}\cdot d\vec{l}$$ | $$dq \cdot \frac{d\vec{l}}{dt} = \frac{dq \cdot d\vec{l}}{dt}=\frac{dq}{dt}\cdot d\vec{l}$$ | ||
So our magnetic field equation then becomes: | So our magnetic field equation then becomes: | ||
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We can now use the definition of [[184_notes: | We can now use the definition of [[184_notes: | ||
$$\vec{B}_{tot}= \int \frac{\mu_0}{4 \pi}\frac{I \cdot d\vec{l}\times \hat{r}}{r^2}$$ | $$\vec{B}_{tot}= \int \frac{\mu_0}{4 \pi}\frac{I \cdot d\vec{l}\times \hat{r}}{r^2}$$ | ||
- | **Note that $I$ here is the //conventional// current**, not the electron current. Otherwise many of the pieces of this equation would be what you expected: | + | **Note that $I$ here is the conventional current, not the electron current**. Otherwise many of the pieces of this equation would be what you expected: |
- | {{ 184_notes: | + | [{{ 184_notes: |
*The constant is the same as before (we haven' | *The constant is the same as before (we haven' | ||
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We will go into detail about how to put the pieces of this equation together in an example; however, it is important to realize that this equation doesn' | We will go into detail about how to put the pieces of this equation together in an example; however, it is important to realize that this equation doesn' | ||
==== Magnetic Field from a Very Long Wire ==== | ==== Magnetic Field from a Very Long Wire ==== | ||
- | {{ 184_notes: | + | [{{ 184_notes: |
Let's look at a particular example of finding the magnetic field a distance $s$ away from a very long wire with some // | Let's look at a particular example of finding the magnetic field a distance $s$ away from a very long wire with some // | ||
$$\vec{B}_{tot}= \int \frac{\mu_0}{4 \pi}\frac{I \cdot d\vec{l}\times \hat{r}}{r^2}$$ | $$\vec{B}_{tot}= \int \frac{\mu_0}{4 \pi}\frac{I \cdot d\vec{l}\times \hat{r}}{r^2}$$ |