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Section 17.3 in Matter and Interactions (4th edition) | Section 17.3 in Matter and Interactions (4th edition) | ||
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===== Moving Charges Make Magnetic Fields ===== | ===== Moving Charges Make Magnetic Fields ===== | ||
Just like we did with electric fields, we will start with magnetic fields by looking at the simplest source: a single moving point charge. When we are talking about this moving charge and the corresponding magnetic field, there are intuitive qualities that we want to make sure our mathematical model includes: (1) the farther away the observation point is from the moving charge, the smaller that we expect the magnetic field to be; (2) the larger the amount of charge, the larger we expect the magnetic field to be; and (3) the faster the charge is moving, the larger we expect the magnetic field to be. The final piece that is perhaps not so intuitive is that the direction of the magnetic field is actually perpendicular to observation point and to the velocity vector. However, each of these demands is consistent with experimental observations of moving charges. These notes will detail the mathematical equation that we use for the magnetic field (only for a single moving point charge) and explain a new tool called the [[184_notes: | Just like we did with electric fields, we will start with magnetic fields by looking at the simplest source: a single moving point charge. When we are talking about this moving charge and the corresponding magnetic field, there are intuitive qualities that we want to make sure our mathematical model includes: (1) the farther away the observation point is from the moving charge, the smaller that we expect the magnetic field to be; (2) the larger the amount of charge, the larger we expect the magnetic field to be; and (3) the faster the charge is moving, the larger we expect the magnetic field to be. The final piece that is perhaps not so intuitive is that the direction of the magnetic field is actually perpendicular to observation point and to the velocity vector. However, each of these demands is consistent with experimental observations of moving charges. These notes will detail the mathematical equation that we use for the magnetic field (only for a single moving point charge) and explain a new tool called the [[184_notes: | ||
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==== Magnetic Field Equation for a Moving Point Charge ==== | ==== Magnetic Field Equation for a Moving Point Charge ==== | ||
- | The general equation for the magnetic field ($\vec{B}$) at some Point P due to a moving charge is given by: | + | The general equation for the magnetic field ($\vec{B}$), with units of Tesla ($T$), |
$$\vec{B}=\frac{\mu_0}{4 \pi}\frac{q\vec{v}\times \hat{r}}{r^2}$$ | $$\vec{B}=\frac{\mu_0}{4 \pi}\frac{q\vec{v}\times \hat{r}}{r^2}$$ | ||
which you may hear referred to as the [[https:// | which you may hear referred to as the [[https:// | ||
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* **Separation Distance** - the $r$ in this equation should feel very familiar from all the work we did with electric fields and potentials. In this equation, this is still the same magnitude of the separation vector that points from the source (the moving charge in this case) to the observation point (P in this case). Mathematically, | * **Separation Distance** - the $r$ in this equation should feel very familiar from all the work we did with electric fields and potentials. In this equation, this is still the same magnitude of the separation vector that points from the source (the moving charge in this case) to the observation point (P in this case). Mathematically, | ||
$$r=|\vec{r}|=|\vec{r_p}-\vec{r_q}|$$ | $$r=|\vec{r}|=|\vec{r_p}-\vec{r_q}|$$ | ||
- | {{ 184_notes: | + | [{{ 184_notes: |
* **Separation Unit Vector** - Similarly the $\hat{r}$ in this equation is the unit vector (has a magnitude of one) that points in the same direction as $\vec{r}$. From before, you know that you can calculate the unit vector using the separation vector and it's magnitude: $$\hat{r}=\frac{\vec{r}}{r}$$ We could also use this unit vector definition to rewrite the magnetic field equation so it is in terms of the $\vec{r}$ rather than $\hat{r}$. $$\vec{B}=\frac{\mu_0}{4 \pi}\frac{q\vec{v}\times \hat{r}}{r^2}=\frac{\mu_0}{4 \pi}\frac{q\vec{v}\times \vec{r}}{r^3}$$ | * **Separation Unit Vector** - Similarly the $\hat{r}$ in this equation is the unit vector (has a magnitude of one) that points in the same direction as $\vec{r}$. From before, you know that you can calculate the unit vector using the separation vector and it's magnitude: $$\hat{r}=\frac{\vec{r}}{r}$$ We could also use this unit vector definition to rewrite the magnetic field equation so it is in terms of the $\vec{r}$ rather than $\hat{r}$. $$\vec{B}=\frac{\mu_0}{4 \pi}\frac{q\vec{v}\times \hat{r}}{r^2}=\frac{\mu_0}{4 \pi}\frac{q\vec{v}\times \vec{r}}{r^3}$$ | ||
- | * **Velocity** - unlike what we did before, the magnetic field equation depends on the velocity of the moving charge. Remember that //velocity is vector//, so it has both a magnitude and direction. The units of velocity here are m/s like normal. Here we have //__assumed that velocity is constant__// | + | * **Velocity** - unlike what we did before, the magnetic field equation depends on the velocity of the moving charge. Remember that //velocity is vector//, so it has both a magnitude and direction. The units of velocity here are m/s like normal. Here we have //__assumed that velocity is constant__// |
* **Direction - Cross Product** - the final piece to this equation is the [[183_notes: | * **Direction - Cross Product** - the final piece to this equation is the [[183_notes: | ||
- | Together, these pieces tell you how the electric | + | Together, these pieces tell you how the magnetic |
==== 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.** | ||
- | {{ 184_notes: | + | [{{ 184_notes: |
However, because we are dealing with the cross product, we need to be able to denote when a vector would be pointing into or out of the page/ | However, because we are dealing with the cross product, we need to be able to denote when a vector would be pointing into or out of the page/ | ||
- | {{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 that is a distance d away from the charge in the $\hat{y}$ direction at the instant the moving change is at the origin. Here, notice that must specific when we want to find the magnetic field as the change before or after that time will be at a different location -- it's moving, remember? We can find the magnetic field by using the equation above: | For example, consider a charge q moving in the $+\hat{x}$ direction. We want to know the magnetic field at point P that is a distance d away from the charge in the $\hat{y}$ direction at the instant the moving change is at the origin. Here, notice that must specific when we want to find the magnetic field as the change before or after that time will be at a different location -- it's moving, remember? We can find the magnetic field by using the equation above: |