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184_notes:q_b_force [2017/10/11 15:19] – dmcpadden | 184_notes:q_b_force [2018/07/03 13:53] – [Magnetic Force Equation] curdemma | ||
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+ | Section 20.1 in Matter and Interactions (4th edition) | ||
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===== Magnetic Force on Moving Charges ===== | ===== Magnetic Force on Moving Charges ===== | ||
We'll start thinking about the magnetic force in terms of the simplest case: a single moving charge through an external magnetic field. The source of the external magnetic field could be another moving charge, a current, a bar magnet or any combination of those. Most of the time though, we will concern ourselves with how the charge interacts with the field (whatever it may be) and we will not care as much about what produces that magnetic field. These notes will describe how we can calculate the force from the magnetic field, including the magnitude and direction of that force. | We'll start thinking about the magnetic force in terms of the simplest case: a single moving charge through an external magnetic field. The source of the external magnetic field could be another moving charge, a current, a bar magnet or any combination of those. Most of the time though, we will concern ourselves with how the charge interacts with the field (whatever it may be) and we will not care as much about what produces that magnetic field. These notes will describe how we can calculate the force from the magnetic field, including the magnitude and direction of that force. | ||
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==== Magnetic Force Equation ==== | ==== Magnetic Force Equation ==== | ||
Mathematically, | Mathematically, | ||
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where: | where: | ||
- | *$\vec{F}_{B \rightarrow q}$ is the force //on// the charge //by// the external magnetic field (units: N). | + | *$\vec{F}_{B \rightarrow q}$ is the force //on// the charge //by// the external magnetic field (units: |
- | *q is the charge of that is moving (units: C). | + | *q is the charge of the moving |
- | *$\vec{v}$ is the velocity that the charge is moving with (units: m/s). Note that this is the velocity, not the speed, so this includes the direction. | + | *$\vec{v}$ is the velocity that the charge is moving with (units: |
- | *$\vec{B}$ is the external magnetic field, both the magnitude and direction (units: T). | + | *$\vec{B}$ is the external magnetic field, both the magnitude and direction (units: |
The last piece that is missing here is the cross product, which tells us about the direction of the magnetic force. [[184_notes: | The last piece that is missing here is the cross product, which tells us about the direction of the magnetic force. [[184_notes: | ||
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We can find the magnitude of any general cross product using $|\vec{a} \times \vec{b} |= |\vec{a}| |\vec{b}| sin(\theta)$ where $\theta$ is the angle between $\vec{a}$ and $\vec{b}$. In terms of the magnetic force then, we can find the magnitude by using: | We can find the magnitude of any general cross product using $|\vec{a} \times \vec{b} |= |\vec{a}| |\vec{b}| sin(\theta)$ where $\theta$ is the angle between $\vec{a}$ and $\vec{b}$. In terms of the magnetic force then, we can find the magnitude by using: | ||
$$F = q v B sin(\theta)$$ | $$F = q v B sin(\theta)$$ | ||
- | where F is the magnitude of the force, q is the charge, v is magnitude of the velocity (speed), and B is the magnitude of the magnitude field. $\theta$ then is angle between the velocity of the charge and the magnetic field. This equation is often much easier to use and think about, but it does not tell us anything about the direction of the force - only the magnitude. | + | where F is the magnitude of the force, |
=== Direction of the Magnetic Force === | === Direction of the Magnetic Force === | ||
- | Just like we did with the [[184_notes: | + | Just like we did with the [[184_notes: |
- | + | [{{184_notes: | |
- | FIXME Add Picture | + | [{{ |
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- | For example, if the charge is moving to the right ($+\hat{x}$ direction) through a magnetic field that points into the page ($-\hat{z}$ direction), you should | + | |
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- | ==== Path of a Charge through a Magnetic Field ==== | + | |
- | FIXME Add figure | + | |
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- | Let's start by considering a charge moving to the left ($-\hat{x}$ direction) in a magnetic field that points into the page ($-\hat{z}$ direction). Using the right hand rule, we can figure out that the force on this charge is down ($-\hat{y}$ direction). So what does this mean for the path of our charge? We can use the [[183_notes: | + | |
- | $$\Delta \vec{p} = \vec{p}_f - \vec{p}_i = \vec{F}_{net, | + | |
- | where momentum is simply $\vec{p}=m\vec{v}$ (meaning momentum points in the same direction as velocity). We can rewrite the momentum principle into it's " | + | |
- | $$\vec{p}_f | + | |
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- | FIXME Add Figure(s) | + | |
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- | This means that if our magnetic force is pushing the charge down for a short time, the velocity of the charge | + | |
- | === Work Done === | + | For example, |
- | You may be wondering | + | |
- | $$W_{total} = \int_i^f \vec{F}\cdot d\vec{r}$$ | + | |
- | In our case, the force is the magnetic force, which we have already said points | + | |
- | $$W_{B}=\int_i^f \vec{F}_{B}\cdot d\vec{r}$$ | + | |
- | Since the dot product is between the magnetic force and the path direction, and because those vectors are perpendicular, | + | |
- | $$W_{B}=0$$ | + | |
- | This is an important result because this tells us that there is no change in the energy | + | |
- | This also means that we do not have a way to define a magnetic potential or a magnetic potential energy in the same way that we did with electric fields. It turns out that we can define a [[https:// | + | ==== Examples ==== |
+ | [[:184_notes: |