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| 184_notes:b_current [2018/07/03 14:40] – 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) |
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| [[184_notes:b_sup_comp|Next Page: Using Superposition of Magnetic Field and the Computer]] | /*[[184_notes:b_sup_comp|Next Page: Using Superposition of Magnetic Field and the Computer]] |
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| [[184_notes:perm_mag|Previous Page: Permanent Magnets]] | [[184_notes:perm_mag|Previous Page: Permanent Magnets]]*/ |
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| ===== 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:moving_q|moving charges and how they created magnetic fields]] - since a current is simply many moving charges through a wire. When there are many charged particles that are moving, we could calculate the net magnetic field at a point from each individual charge using superposition. However, this gets tedious very quickly. Instead, we will use an integral to add up over all the charges and the definition of [[184_notes:q_in_wires#conventional_current_vs_electron_current|current]] to re-write the Biot-Savart Law in terms of current and length, rather than charge and velocity. This integral description of the magnetic field is useful when the appropriate anti-derivative is known, but for some situations, we might have to resort to numerical integration (i.e., adding up the contributions of each segment of wire). | Now that we have talked about a single moving charge and permanent magnets, 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:moving_q|moving charges and how they created magnetic fields]] - since a current is simply many moving charges through a wire. When there are many charged particles that are moving, we could calculate the net magnetic field at a point from each individual charge using superposition. However, this gets tedious very quickly. Instead, we will use an integral to add up over all the charges and the definition of [[184_notes:q_in_wires#conventional_current_vs_electron_current|current]] to re-write the Biot-Savart Law in terms of current and length, rather than charge and velocity. This integral description of the magnetic field is useful when the appropriate anti-derivative is known, but for some situations, we might have to resort to numerical integration (i.e., adding up the contributions of each segment of wire). |
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| {{youtube>4cIpX1HK7GM?large}} | {{youtube>4cIpX1HK7GM?large}} |
| We can now use the definition of [[184_notes:q_in_wires#conventional_current_vs_electron_current|current]] as the amount of charge passing a point per second ($I=\frac{dq}{dt}$) to give the Biot-Savart Law in terms of current instead of charge: | We can now use the definition of [[184_notes:q_in_wires#conventional_current_vs_electron_current|current]] as the amount of charge passing a point per second ($I=\frac{dq}{dt}$) to give the Biot-Savart Law in terms of current instead of charge: |
| $$\vec{B}_{tot}= \int \frac{\mu_0}{4 \pi}\frac{I \cdot d\vec{l}\times \hat{r}}{r^2}$$ | $$\vec{B}_{tot}= \int \frac{\mu_0}{4 \pi}\frac{I \cdot d\vec{l}\times \hat{r}}{r^2}$$ |
| **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: |
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| [{{ 184_notes:Week9_5.png?300|B field contribution of a little bit of length ($dl$) on point P}}] | [{{ 184_notes:Week9_5.png?300|B field contribution of a little bit of length ($dl$) on point P}}] |