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- | ===== Magnetic Force on Moving Charges ===== | + | Section 20.1 in Matter and Interactions (4th edition) |
+ | / | ||
- | ==== Magnetic Force Equation | + | [[184_notes: |
- | The magnetic force on a moving charge from an // | + | |
+ | ===== 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. | ||
+ | |||
+ | {{youtube> | ||
+ | ===== Magnetic Force Equation ===== | ||
+ | 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 ==== | ||
+ | [[: |