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 [2017/08/31 14:42] – dmcpadden | 184_notes:dq [2021/02/13 19:00] – bartonmo | ||
---|---|---|---|
Line 1: | Line 1: | ||
+ | Sections 15.1-15.2 in Matter and Interactions (4th edition) | ||
+ | |||
+ | / | ||
+ | |||
+ | [[184_notes: | ||
+ | |||
===== dQ and the $\vec{r}$ ===== | ===== dQ and the $\vec{r}$ ===== | ||
[[184_notes: | [[184_notes: | ||
- | |||
{{youtube> | {{youtube> | ||
==== dQ - Chunks of Charge ==== | ==== dQ - Chunks of Charge ==== | ||
- | When we are splitting the total charge into small pieces of charge, it helps to write the little bit of charge in terms of that shape. For example, if you have a line of charge, writing the charge in terms of the length is useful. If you have a flat disk of charge, writing the charge in terms of the area is useful. If you have a sphere of charge, writing the charge in terms of the volume is useful. This idea of how much charge is in a particular shape (line, area, or volume) is called **charge density**. | ||
- | {{ 184_notes:dl.png?50}} | + | When we are splitting the total charge into small pieces of charge, it helps to write the little bit of charge in terms of that shape. For example, if you have a line of charge, writing the charge in terms of the length is useful. If you have a flat disk of charge, writing the charge in terms of the area is useful. If you have a sphere of charge, writing the charge in terms of the volume is useful. This idea of how much charge is in a particular shape (line, area, or volume) is called **charge density**.[{{ 184_notes:dldx.png?250|Horizontal " |
- | === Charge on a line === | + | [{{ 184_notes: |
+ | |||
+ | ==== 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: | ||
$$\lambda=\frac{Q_{tot}}{L_{tot}}$$ | $$\lambda=\frac{Q_{tot}}{L_{tot}}$$ | ||
- | {{ 184_notes: | + | |
Once you have the charge density, you can use this to write your little bit of charge in terms of a little bit of length. | Once you have the charge density, you can use this to write your little bit of charge in terms of a little bit of length. | ||
$$dQ=\lambda dl= \lambda dx = \lambda dy$$ | $$dQ=\lambda dl= \lambda dx = \lambda dy$$ | ||
You can write the " | You can write the " | ||
- | {{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 28: | 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 36: | Line 42: | ||
Again, you can write the " | Again, you can write the " | ||
- | We will talk more about dAs, dVs and bigger shapes of charge later, but for now we will focus on lines of charge (or 1D charge distributions). | + | We will talk more about dAs, dVs and bigger shapes |
==== Limits on the integral ==== | ==== Limits on the integral ==== | ||
Line 42: | Line 48: | ||
==== $\vec{r}$ - separation vector ==== | ==== $\vec{r}$ - separation vector ==== | ||
- | {{ 184_notes: | + | [{{ 184_notes: |
In general, we have defined the $\vec{r}$ to be the separation vector that points from the source (q or dQ in this case) to the point of interest. If you pick a general point away from the line of charge, such as Point A in the figure, the separation vector can both a) have very different magnitudes and b) point in very different directions for different dQs along the line of charge. This means that we need to come up with a way to write the separation vector that is true for a variety of points along the line. This generally means writing the separation vector in terms of some variable that changes as you move from one dQ to the next along the line. | In general, we have defined the $\vec{r}$ to be the separation vector that points from the source (q or dQ in this case) to the point of interest. If you pick a general point away from the line of charge, such as Point A in the figure, the separation vector can both a) have very different magnitudes and b) point in very different directions for different dQs along the line of charge. This means that we need to come up with a way to write the separation vector that is true for a variety of points along the line. This generally means writing the separation vector in terms of some variable that changes as you move from one dQ to the next along the line. | ||
Line 48: | 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==== | ||
+ | [[: | ||
+ | |||
+ | [[: |