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184_notes:b_shapes [2018/07/03 15:00] – curdemma | 184_notes:b_shapes [2021/06/16 19:23] – [Coils] bartonmo | ||
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===== Shapes of Wires and Magnetic Fields ===== | ===== Shapes of Wires and Magnetic Fields ===== | ||
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[{{184_notes: | [{{184_notes: | ||
- | In a circular coil, the current through the wire would run either clockwise or counter-clockwise around the loop. If we wanted to calculate the magnetic field from that current, we could use the Biot-Savart Law | + | In a circular coil, the current through the wire would run either |
$$\vec{B}_{tot}= \int \frac{\mu_0}{4 \pi}\frac{I \cdot d\vec{l}\times \hat{r}}{r^2}$$ | $$\vec{B}_{tot}= \int \frac{\mu_0}{4 \pi}\frac{I \cdot d\vec{l}\times \hat{r}}{r^2}$$ | ||
where the $d\vec{l}$ now would need to represent a small piece of the circular loop, rather than a small piece of the straight wire. This is shown in the figure above, where the circular coil now is drawn in the xy-plane and the z-direction points straight through the center of the coil. | where the $d\vec{l}$ now would need to represent a small piece of the circular loop, rather than a small piece of the straight wire. This is shown in the figure above, where the circular coil now is drawn in the xy-plane and the z-direction points straight through the center of the coil. | ||
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$$\vec{B}=\frac{\mu_0 I}{2R} \hat{z}$$ | $$\vec{B}=\frac{\mu_0 I}{2R} \hat{z}$$ | ||
- | Which depends only on the current going through the coil ($I$) and the radius of the coil ($R$). The direction of the magnetic field comes from the cross product of the $\vec{r}$ with each $d\vec{l}$ in the coil, which we can check with the right hand rule. If you point your fingers in the direction of the current at any point on the coil and curl them toward the center of the circular coil, your thumb will always be pointing up in the $+\hat{z}$ direction. //__Note this equation is only true for the magnetic field at the center of a circular coil__// - if you have a different observation point or a different shape of coil, this equation will no longer give you an accurate magnetic field. | + | Which depends only on the current going through the coil $I$ and the radius of the coil $R$. The direction of the magnetic field comes from the cross product of the $\vec{r}$ with each $d\vec{l}$ in the coil, which we can check with the right hand rule. If you point your fingers in the direction of the current at any point on the coil and curl them toward the center of the circular coil, your thumb will always be pointing up in the $+\hat{z}$ direction. //__Note this equation is only true for the magnetic field at the center of a circular coil__// - if you have a different observation point or a different shape of coil, this equation will no longer give you an accurate magnetic field. |
[{{ 184_notes: | [{{ 184_notes: | ||
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You also may have noticed that this pattern to the magnetic field looks similar to the magnetic field from a bar magnet that we [[184_notes: | You also may have noticed that this pattern to the magnetic field looks similar to the magnetic field from a bar magnet that we [[184_notes: | ||
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==== Solenoid ==== | ==== Solenoid ==== | ||
- | [{{ 184_notes: | + | [{{ 184_notes: |
A solenoid is similar to a coil, but instead of having multiple loops of wire one on top of another, the loops are spread out (kind of like a slinky). An example of a solenoid is shown in the figure on the right. As the solenoid is composed of multiple, connected loops, this creates a strong magnetic field in the center of the solenoid, which becomes weaker and wraps around the outside of the solenoid. Because of the superposition of the magnetic field, the field inside the solenoid is actually // | A solenoid is similar to a coil, but instead of having multiple loops of wire one on top of another, the loops are spread out (kind of like a slinky). An example of a solenoid is shown in the figure on the right. As the solenoid is composed of multiple, connected loops, this creates a strong magnetic field in the center of the solenoid, which becomes weaker and wraps around the outside of the solenoid. Because of the superposition of the magnetic field, the field inside the solenoid is actually // |