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===== Shapes of Wires and Magnetic Fields ===== | ===== Shapes of Wires and Magnetic Fields ===== | ||
Thus far, we have primarily been talking about the magnetic field from a current in a long, straight wire. However, there are many shapes of wire in the real world that do not correspond to straight wires. Particularly relevant to this class will be a coil of wire and something we will call a solenoid. These notes go more into detail about what coils and solenoids are, and what the magnetic field looks like from both coils and solenoids. | Thus far, we have primarily been talking about the magnetic field from a current in a long, straight wire. However, there are many shapes of wire in the real world that do not correspond to straight wires. Particularly relevant to this class will be a coil of wire and something we will call a solenoid. These notes go more into detail about what coils and solenoids are, and what the magnetic field looks like from both coils and solenoids. | ||
==== Coils ==== | ==== Coils ==== | ||
- | {{ 184_notes: | + | [{{ 184_notes: |
A coil is when a length of wire with current running through it has been wrapped around on itself creating some sort of shape as shown in the figure to the right. (You may also hear a coil referred to as a " | A coil is when a length of wire with current running through it has been wrapped around on itself creating some sort of shape as shown in the figure to the right. (You may also hear a coil referred to as a " | ||
- | {{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 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 | ||
$$\vec{B}_{tot}= \int \frac{\mu_0}{4 \pi}\frac{I \cdot d\vec{l}\times \hat{r}}{r^2}$$ | $$\vec{B}_{tot}= \int \frac{\mu_0}{4 \pi}\frac{I \cdot d\vec{l}\times \hat{r}}{r^2}$$ | ||
- | 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 | + | 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 |
- | {{ 184_notes: | + | [{{ 184_notes: |
[[184_notes: | [[184_notes: | ||
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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: |
We could also calculate the magnetic field for an observation point at the center of the coil, but out of the plane of the ring (shown in the figure to the left). In this case, we would follow the same process as in [[184_notes: | We could also calculate the magnetic field for an observation point at the center of the coil, but out of the plane of the ring (shown in the figure to the left). In this case, we would follow the same process as in [[184_notes: | ||
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Where $R$ is still the radius of the coil, $I$ is the current going around the loop, and $z$ is the distance away from the center of the loop. If you set $z=0$ (in other words you put the observation point back at the center of the coil), you get the same equation as above. //__Note this magnetic field is only true for an observation point at the center of the circular coil, but out of the plane of the ring__// - if you have a different observation point or a different shape of coil, this equation will no longer give you an accurate magnetic field. | Where $R$ is still the radius of the coil, $I$ is the current going around the loop, and $z$ is the distance away from the center of the loop. If you set $z=0$ (in other words you put the observation point back at the center of the coil), you get the same equation as above. //__Note this magnetic field is only true for an observation point at the center of the circular coil, but out of the plane of the ring__// - if you have a different observation point or a different shape of coil, this equation will no longer give you an accurate magnetic field. | ||
- | If you look at many observation points around a circular coil, there is a pattern to the magnetic field produced by that current - the field points through the center of the coil and then wraps around on the outside of the coil. For such a coil, the magnetic field is strongest in the center of the coil and has a smaller magnitude as you move further away from the coil. You could also increase the magnitude of the magnetic field by increasing the number of loops in the coil. Similarly, you increase the amount of current | + | If you look at many observation points around a circular coil, there is a pattern to the magnetic field produced by that current - the field points through the center of the coil and then wraps around on the outside of the coil. This is shown in the figure below. For such a coil, the magnetic field is strongest in the center of the coil and has a smaller magnitude as you move further away from the coil. If you increase the amount of current or decrease the radius of the coil, you can create a stronger magnetic field. You could also increase the magnitude of the magnetic field by increasing the number of loops in the coil (or how many circular loops you have). |
- | You also may have noticed that this pattern to the magnetic field looks similar to the magnetic field from a bar magnet that we talked about previously - this makes the circular coil a powerful tool. Scientists and engineers will often model a bar magnet (or the Earth' | + | 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: |
- | {{ 184_notes: | + | [{{184_notes: |
==== Solenoid ==== | ==== Solenoid ==== | ||
- | {{ 184_notes:10b_solenoid.png?300}} | + | [{{ 184_notes:10b_solenoid_notes.png?300|Magnetic field in and around a solenoid with current |
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 // | ||
- | We'll talk more about the magnetic field in a solenoid next week when we talk about Ampere' | + | We'll talk more about the magnetic field in a solenoid |
==== Examples ==== | ==== Examples ==== | ||
[[: | [[: |