Differences
This shows you the differences between two versions of the page.
Both sides previous revision Previous revision Next revision | Previous revision Next revisionBoth sides next revision | ||
184_notes:dq [2018/09/12 15:22] – dmcpadden | 184_notes:dq [2021/05/25 14:17] – schram45 | ||
---|---|---|---|
Line 1: | Line 1: | ||
Sections 15.1-15.2 in Matter and Interactions (4th edition) | Sections 15.1-15.2 in Matter and Interactions (4th edition) | ||
- | [[184_notes: | + | /*[[184_notes: |
- | [[184_notes: | + | [[184_notes: |
===== dQ and the $\vec{r}$ ===== | ===== dQ and the $\vec{r}$ ===== | ||
Line 15: | Line 15: | ||
[{{ 184_notes: | [{{ 184_notes: | ||
- | === Charge on a line === | + | ==== Charge on a line ==== |
For a **1D uniform charge density** (such as lines of charge), we use the variable $\lambda$, which has units of $\frac{C}{m}$ (coulombs per meter). You can calculate $\lambda$ by taking the total charge that is spread over the total length: | For a **1D uniform charge density** (such as lines of charge), we use the variable $\lambda$, which has units of $\frac{C}{m}$ (coulombs per meter). You can calculate $\lambda$ by taking the total charge that is spread over the total length: | ||
Line 26: | Line 26: | ||
[{{ 184_notes: | [{{ 184_notes: | ||
- | === Charge on a surface === | + | ==== Charge on a surface |
For a **2D uniform charge density** (such as sheets of charge), we use the variable $\sigma$, which has units of $\frac{C}{m^2}$ (coulombs per meter squared). You can calculate $\sigma$ by taking the total charge that is spread over the total area: | For a **2D uniform charge density** (such as sheets of charge), we use the variable $\sigma$, which has units of $\frac{C}{m^2}$ (coulombs per meter squared). You can calculate $\sigma$ by taking the total charge that is spread over the total area: | ||
Line 34: | Line 34: | ||
You can write the " | You can write the " | ||
- | === Charge in a volume === | + | ==== Charge in a volume |
Similarly, for a **3D uniform charge density** (such as a sphere of charge), we use the variable $\rho$, which has units of $\frac{C}{m^3}$ (coulombs per meter cubed). You can calculate $\rho$ by taking the total charge that is spread over the total volume: | Similarly, for a **3D uniform charge density** (such as a sphere of charge), we use the variable $\rho$, which has units of $\frac{C}{m^3}$ (coulombs per meter cubed). You can calculate $\rho$ by taking the total charge that is spread over the total volume: | ||
Line 54: | Line 54: | ||
Because we talk about lines of charge, we usually pick some length variable like " | Because we talk about lines of charge, we usually pick some length variable like " | ||
- | For the picture shown, we can find the $\vec{r}$ by splitting it into components. | + | For the picture shown, we can find the $\vec{r}$ by using the same separation vector equation that we were using before: |
+ | $$ \vec{r} = \vec{r}_{observation}-\vec{r}_{source}$$ | ||
+ | First, we need to pick a coordinate system - so lets pick the $(0,0)$ location to be at the bottom of the tape with +x being to the right and +y being up like normal. | ||
+ | $$\vec{r}_{obs} | ||
+ | The source location | ||
+ | $$\vec{r}_{source}= \langle 0, y, 0 \rangle$$ | ||
+ | The x-component of $\vec{r}_{source}$ here is zero because we have the tape located on the y-axis, which would be true no matter where on the tape our dQ is located. When we combine these pieces, we get the total separation | ||
+ | $$\vec{r}=\langle -d, L, 0 \rangle - \langle 0, y, 0 \rangle$$ | ||
$$\vec{r}=\langle -d, L-y, 0 \rangle$$ | $$\vec{r}=\langle -d, L-y, 0 \rangle$$ | ||
- | This way of writing | + | This equation for the $\vec{r}$ works for any spot along the piece of tape, and functions like any other vector (we can find its magnitude, unit vector, etc.). Because $\vec{E}$ and $V$ rely heavily on $\vec{r}$ and the $|r|$, we will use this method and reasoning when we are dealing with lines of charge (though this works more generally for planes, spheres, or blobs too). |
====Examples==== | ====Examples==== | ||
- | [[: | + | Written Example: |
- | [[: | + | Video Example: |
+ | {{youtube> | ||
+ | Written out work: [[: |