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===== Example: Electric Field from a Ring of Charge ===== | ===== Example: Electric Field from a Ring of Charge ===== | ||
Suppose we have a ring with radius $R$ that has a uniform charge distribution with total charge $Q$. What is the electric field at a point $P$, which is a distance $z$ from the center of the ring, along a line perpendicular to the plane of the ring? What happens to the electric field if $z = 0$ (i.e., when $P$ is in the center of the ring rather than above it)? Why? | Suppose we have a ring with radius $R$ that has a uniform charge distribution with total charge $Q$. What is the electric field at a point $P$, which is a distance $z$ from the center of the ring, along a line perpendicular to the plane of the ring? What happens to the electric field if $z = 0$ (i.e., when $P$ is in the center of the ring rather than above it)? Why? | ||
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We begin with an approximation, | We begin with an approximation, | ||
* The thickness of the ring is infinitesimally small, and we can approximate it as a circle. | * The thickness of the ring is infinitesimally small, and we can approximate it as a circle. | ||
+ | * The ring is in a perfect circle. | ||
</ | </ | ||
We also make a plan to tackle the integrating, | We also make a plan to tackle the integrating, | ||
- | |||
- | This example is complicated enough that it's worthwhile to make a plan. | ||
<WRAP TIP> | <WRAP TIP> | ||
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So the $\text{d}Q$ in our representation takes up a small angle out of the whole circle, which we can call $\text{d}\phi$. The length of our $\text{d}Q$ is therefore $R\text{d}\phi$ (which comes from the [[https:// | So the $\text{d}Q$ in our representation takes up a small angle out of the whole circle, which we can call $\text{d}\phi$. The length of our $\text{d}Q$ is therefore $R\text{d}\phi$ (which comes from the [[https:// | ||
$$\text{d}Q=\lambda\text{d}l=\frac{Q}{2\pi R}R\text{d}\phi=\frac{Q\text{d}\phi}{2\pi}$$ | $$\text{d}Q=\lambda\text{d}l=\frac{Q}{2\pi R}R\text{d}\phi=\frac{Q\text{d}\phi}{2\pi}$$ | ||
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+ | <WRAP TIP> | ||
+ | ===Assumption=== | ||
+ | The charge is evenly distributed along the ring. This also assumes the ring is a perfect conductor where charges will distribute evenly along the conductor. If this were not true, the charge density along the ring would not be constant. | ||
+ | </ | ||
To find an expression for $\vec{r}$, we can also consult the representation. $\vec{r}$ points from the location of $\text{d}Q$ to the point $P$. The location of $\text{d}Q$ is $\vec{r}_{\text{d}Q}=R\hat{s}$. This unit vector $\hat{s}$ may be unfamiliar, since we are used to working in Cartesian coordinates. $\hat{s}$ is the unit vector that points along the radius of a cylinder centered on the $z$-axis in our cylindrical coordinate system. In fact, $\hat{s}$ actually depends on $\phi$, and is more appropriately written as a function in terms of $\phi$, or $\hat{s}(\phi)$. We do not acknowledge the $\phi$-dependence in some of our expressions here, because as you will soon see, all terms containing $\hat{s}$ will disappear. | To find an expression for $\vec{r}$, we can also consult the representation. $\vec{r}$ points from the location of $\text{d}Q$ to the point $P$. The location of $\text{d}Q$ is $\vec{r}_{\text{d}Q}=R\hat{s}$. This unit vector $\hat{s}$ may be unfamiliar, since we are used to working in Cartesian coordinates. $\hat{s}$ is the unit vector that points along the radius of a cylinder centered on the $z$-axis in our cylindrical coordinate system. In fact, $\hat{s}$ actually depends on $\phi$, and is more appropriately written as a function in terms of $\phi$, or $\hat{s}(\phi)$. We do not acknowledge the $\phi$-dependence in some of our expressions here, because as you will soon see, all terms containing $\hat{s}$ will disappear. | ||
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There are still a couple issues to sort out before we proceed. First, what are the limits of integration? | There are still a couple issues to sort out before we proceed. First, what are the limits of integration? | ||
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+ | Before we dive into the second issue, it's worth updated the plan, since this ended up a bit more complicated than anticipated. | ||
+ | |||
+ | <WRAP TIP> | ||
+ | === Plan === | ||
+ | We need to update the plan. Here are the steps we will take. We just now finished setting up the integral. | ||
+ | * We are not sure how to take an integral when $\hat{s}$ is involved. | ||
+ | * Zoom out. Try to figure out what the electric field might look like qualitatively. | ||
+ | * Try to simplify the integral to match our expectations for what the result will be. | ||
+ | </ | ||
The second issue has to do with the $\hat{s}$ terms, since $\hat{s}$ depends on $\phi$. Before we proceed, let's split up the integral for clarity: | The second issue has to do with the $\hat{s}$ terms, since $\hat{s}$ depends on $\phi$. Before we proceed, let's split up the integral for clarity: |