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184_notes:moving_q [2020/10/27 15:24] – dmcpadden | 184_notes:moving_q [2021/07/05 21:51] (current) – schram45 | ||
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{{youtube> | {{youtube> | ||
- | ==== Mathematical Model for Magnetic Field ==== | + | ===== Mathematical Model for Magnetic Field ===== |
- | We have already stated that the magnetic field is a //vector field//, meaning it has both magnitude and direction. As you will read soon, the magnetic field can also be related to the magnetic force. Again, we will start by making the //__point particle assumption - meaning that we will take our charged object and crush it down to single small point that has some mass and charge__//. Only this time, //__we will be considering the case where the point charge is moving with a **constant** velocity__//, | + | We have already stated that the magnetic field is a vector field, meaning it has both //magnitude// and //direction//. As you will read soon, the magnetic field can also be related to the magnetic force. Again, we will start by making the //__point particle assumption - meaning that we will take our charged object and crush it down to single small point that has some mass and charge__//. Only this time, //__we will be considering the case where the point charge is moving with a **constant** velocity__//, |
- | ==== Magnetic Field Equation for a Moving Point Charge ==== | + | ===== Magnetic Field Equation for a Moving Point Charge |
The general equation for the magnetic field ($\vec{B}$), | The general equation for the magnetic field ($\vec{B}$), | ||
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Together, these pieces tell you how the magnetic field from a point charge changes in space. The main take away here is: **the magnetic field is created by //moving// charges, points in a perpendicular direction, and can be calculated for every point in space around the charge**. The examples below show a few instances of how to calculate the magnetic field and how to use the Right Hand Rule to figure out the direction. | Together, these pieces tell you how the magnetic field from a point charge changes in space. The main take away here is: **the magnetic field is created by //moving// charges, points in a perpendicular direction, and can be calculated for every point in space around the charge**. The examples below show a few instances of how to calculate the magnetic field and how to use the Right Hand Rule to figure out the direction. | ||
- | ==== Magnetic Field Vectors ==== | + | ===== Magnetic Field Vectors |
Just like we did before with stationary charges, we will often draw the magnetic field vectors (or just B-field vectors) around the moving charge to help us understand what is happening to the magnetic field. **The magnitude of these vectors represents the magnitude of the magnetic field, and the direction of these vectors points in the direction of the magnetic field.** | Just like we did before with stationary charges, we will often draw the magnetic field vectors (or just B-field vectors) around the moving charge to help us understand what is happening to the magnetic field. **The magnitude of these vectors represents the magnitude of the magnetic field, and the direction of these vectors points in the direction of the magnetic field.** | ||
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[{{ 184_notes: | [{{ 184_notes: | ||
- | For example, consider a charge q moving in the $+\hat{x}$ direction. We want to know the magnetic field at point P, which is a distance d away from the charge in the $\hat{y}$ direction at the instant the moving change is at the origin (see the set up below). Here, notice that we must specific about //when// we want to find the magnetic field. Because the charge is moving, it will be at a different location at different times -- our solution is only accurate for a particular time/ | + | For example, consider a charge q moving in the $+\hat{x}$ direction. We want to know the magnetic field at point P, which is a distance d away from the charge in the $\hat{y}$ direction at the instant the moving change is at the origin (see the set up below). Here, notice that we must be specific about //when// we want to find the magnetic field. Because the charge is moving, it will be at a different location at different times -- //our solution is only accurate for a particular time/ |
$$\vec{B}=\frac{\mu_0}{4 \pi}\frac{q\vec{v}\times \vec{r}}{r^3}$$ | $$\vec{B}=\frac{\mu_0}{4 \pi}\frac{q\vec{v}\times \vec{r}}{r^3}$$ | ||
where our separation vector is $\vec{r}=d \hat{y}$ since it points from the charge to our point of interest. In this case then: | where our separation vector is $\vec{r}=d \hat{y}$ since it points from the charge to our point of interest. In this case then: | ||
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==== Examples ==== | ==== Examples ==== | ||
- | [[: | + | * [[: |
+ | * Video Example: Magnetic Field near a Moving Charge | ||
+ | {{youtube> |