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184_notes:examples:week4_charge_ring [2017/09/15 15:56] – [Solution] tallpaul | 184_notes:examples:week4_charge_ring [2018/02/03 21:37] – [Solution] tallpaul | ||
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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 as $z = 0$ (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? |
===Facts=== | ===Facts=== | ||
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* Electric field at $P$. | * Electric field at $P$. | ||
* $\text{d}Q$ and $\vec{r}$ | * $\text{d}Q$ and $\vec{r}$ | ||
- | + | | |
- | ===Approximations & Assumptions=== | + | |
- | | + | |
- | * The thickness of the ring is infinitesimally small, and we can approximate it as a circle. | + | |
- | * The ring is not discharging -- $Q$ is constant. | + | |
===Representations=== | ===Representations=== | ||
- | * The electric field from the point charge can be written as $$\vec{E} = \frac{1}{4\pi\epsilon_0}\frac{q}{r^3}\vec{r}$$ | + | We can represent the ring and $P$ as follows, with coordinates chosen conveniently. We choose cylindrical coordinates because we will be integrating over the length of the ring, and being able to represent its radius as constant will simplify calculations. |
- | * We can represent the ring and $P$ as follows, with coordinates chosen conveniently. We choose cylindrical coordinates because we will be integrating over the length of the ring, and being able to represent its radius as constant will simplify calculations. | + | |
{{ 184_notes: | {{ 184_notes: | ||
- | * We can represent $\text{d}Q$ and $\vec{r}$ for our ring as follows: | ||
- | {{ 184_notes: | ||
====Solution==== | ====Solution==== | ||
- | Just like before, we want to write our $\text{d}Q$ in terms of the little bit of length; however this time, our $\text{d}Q$ is spread over a little bit of the ring. We could try to write this little bit of length in terms of $\text{d}x$ or $\text{d}y$, | + | <WRAP TIP> |
+ | === Approximation === | ||
+ | We begin with an approximation, | ||
+ | * The thickness of the ring is infinitesimally small, and we can approximate it as a circle. | ||
+ | </ | ||
+ | We also make a plan to tackle the integrating, | ||
+ | |||
+ | <WRAP TIP> | ||
+ | === Plan === | ||
+ | We will use integration to find the electric field from the ring. We'll go through the following steps. | ||
+ | * Reason about the geometry of the ring to find and $\text{d}l$ and $\lambda$. | ||
+ | * Write an expression for $\text{d}Q$. | ||
+ | * Assign a variable location to the $\text{d}Q$ piece, and then use that location to find the separation vector, $\vec{r}$. | ||
+ | * Write an expression for $\text{d}\vec{E}$. | ||
+ | * Figure out the bounds of the integral, and integrate to find electric field at $P$. | ||
+ | </ | ||
+ | |||
+ | Just like before, we want to write our $\text{d}Q$ in terms of the little bit of length; however this time, our $\text{d}Q$ is spread over a little bit of the ring. We could try to write this little bit of length in terms of $\text{d}x$ or $\text{d}y$, | ||
+ | {{ 184_notes: | ||
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:// | ||
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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? | ||
+ | |||
+ | 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: |