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184_notes:q_b_force [2017/10/10 20:37] – created dmcpadden | 184_notes:q_b_force [2018/07/03 13:53] – [Magnetic Force Equation] curdemma | ||
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- | ===== Magnetic Force on Moving Charges ===== | + | Section 20.1 in Matter and Interactions (4th edition) |
+ | [[184_notes: | ||
+ | [[184_notes: | ||
+ | |||
+ | ===== 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. | ||
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+ | {{youtube> | ||
==== Magnetic Force Equation ==== | ==== Magnetic Force Equation ==== | ||
- | The magnetic force on a moving charge from an // | + | Mathematically, |
$$\vec{F}_{B \rightarrow q} = q \vec{v} \times \vec{B}$$ | $$\vec{F}_{B \rightarrow q} = q \vec{v} \times \vec{B}$$ | ||
+ | where: | ||
+ | *$\vec{F}_{B \rightarrow q}$ is the force //on// the charge //by// the external magnetic field (units: $N$). | ||
+ | *q is the charge of the moving object (units: $C$). | ||
+ | *$\vec{v}$ is the velocity that the charge is moving with (units: $\frac{m}{s}$). Note that this is the velocity, not the speed, so this includes the direction. | ||
+ | *$\vec{B}$ is the external magnetic field, both the magnitude and direction (units: $T$). | ||
+ | |||
+ | The last piece that is missing here is the cross product, which tells us about the direction of the magnetic force. [[184_notes: | ||
+ | |||
+ | In terms of calculating the magnetic force, there are a couple of ways that we can go about the math. If you know the vector components of the velocity and magnetic field, one method you can use is the general [[183_notes: | ||
+ | |||
+ | === Magnitude of the Magnetic Force === | ||
+ | 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: | ||
+ | $$F = q v B sin(\theta)$$ | ||
+ | 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**. | ||
+ | |||
+ | === Direction of the Magnetic Force === | ||
+ | Just like we did with the [[184_notes: | ||
+ | [{{184_notes: | ||
+ | [{{ 184_notes: | ||
+ | |||
+ | 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 find that the force on the charge points up ($+\hat{y}$ direction). | ||
+ | ==== Examples ==== | ||
+ | [[: |