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184_notes:dq [2018/06/05 13:37] – [dQ - Chunks of Charge] curdemma184_notes:dq [2021/05/26 13:36] (current) schram45
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 Sections 15.1-15.2 in Matter and Interactions (4th edition) Sections 15.1-15.2 in Matter and Interactions (4th edition)
  
-[[184_notes:linecharge|Next Page: Line of Charge]]+/*[[184_notes:linecharge|Next Page: Line of Charge]]
  
-[[184_notes:line_fields|Previous Page: Building Electric Field and Potential for a Line of Charge]]+[[184_notes:line_fields|Previous Page: Building Electric Field and Potential for a Line of Charge]]*/
  
 ===== dQ and the $\vec{r}$ ===== ===== dQ and the $\vec{r}$ =====
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 {{youtube>2HuJ0GiDFoM?large}} {{youtube>2HuJ0GiDFoM?large}}
 ==== 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**.  For the purposes of this class, we will //__assume that the charge density is uniform__//, which means that every little piece of charge in the shape should have the same of amount 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:dldx.png?250|Horizontal "little bit of length", $dx$}}]  For the purposes of this class, we will //__assume that the charge density is uniform__//, which means that every little piece of charge in the shape should have the same of amount of charge.
  
 [{{  184_notes:dl.png?50|Vertical "little bit of length", $dy$}}] [{{  184_notes:dl.png?50|Vertical "little bit of length", $dy$}}]
  
-=== 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:
 $$\lambda=\frac{Q_{tot}}{L_{tot}}$$ $$\lambda=\frac{Q_{tot}}{L_{tot}}$$
-[{{  184_notes:dldx.png?250|Horizontal "little bit of lenght" $dx$}}]+
 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 "little bit of length" in a variety of ways, depending on how you define your coordinate system or what variables you wish to use. (For example, a little bit of vertical length is usually written as "dy" or a little bit of horizontal length is usually "dx".)  If you check the units of this equation you get $C=\frac{C}{m}*m$, so this equation seems to be giving us what we want - a little bit of charge written in terms of a little bit of length. You can write the "little bit of length" in a variety of ways, depending on how you define your coordinate system or what variables you wish to use. (For example, a little bit of vertical length is usually written as "dy" or a little bit of horizontal length is usually "dx".)  If you check the units of this equation you get $C=\frac{C}{m}*m$, so this equation seems to be giving us what we want - a little bit of charge written in terms of a little bit of length.
  
-{{184_notes:da.png?200  }}+[{{  184_notes:da.png?200|"Little piece of area", $dA$}}]
  
-=== 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:
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 You can write the "little bit of area" in a variety of ways, depending on the shape of charge. You can write the "little bit of area" in a variety of ways, depending on the shape of charge.
  
-=== 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:
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 ==== $\vec{r}$ - separation vector ==== ==== $\vec{r}$ - separation vector ====
-{{  184_notes:sepvec.png?350}}+[{{  184_notes:sepvec.png?350|Separation vector for a vertical line of charge, broken into its components}}]
  
 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. 
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 Because we talk about lines of charge, we usually pick some length variable like "L", "x" or "y". **You always want this variable to match the "little bit of length" variable that you chose for your dQ.** So if you choose a "dL", you want to use "L" as your variable; if you choose "dx", you want to use "x"; etc.  This will prevent you from referring to the same length with two different variable names. Because we talk about lines of charge, we usually pick some length variable like "L", "x" or "y". **You always want this variable to match the "little bit of length" variable that you chose for your dQ.** So if you choose a "dL", you want to use "L" as your variable; if you choose "dx", you want to use "x"; etc.  This will prevent you from referring to the same length with two different variable names.
  
-For the picture shown, we can find the $\vec{r}$ by splitting it into components. 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. The x component here will always have magnitude of length d; no matter which r vector we pick, the $|r_x|=d$. However, it points in the -x direction, so $r_x=-d \hat{x}$The y-component here is a little more tricky. If we start by saying we want the magnitude of $r_y$then we can focus first on the length then deal with the direction later. Remember we want to find the y component of r, which points from the dQ to the point of interest. So we define our variable, lets say "y" because the tape is hanging vertically, to be the length between zero and where ever the dQ is. Then $r_y$ is the length of tape that is //not// covered by y, so $|r_y|=L-y$ where L is the total length of the tape. This component of the vector should always point up, so then we get $\vec{r}_y=(L-y\hat{y}$. When we combine these components, we get the total vector:+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. In this coordinate system, the observation location is at a height of L and distance of d in the -x direction. So we get: 
 +$$\vec{r}_{obs} \langle -d, L, 0 \rangle$
 +The source location is a little more tricky because our dQ could be any on the piece of tapeFor example, dQ could be at the bottom of the piece of tapeat the top, or somewhere in the middle - so we ideally we want to write the location of dQ (aka the $\vec{r}_{source}$) so that it is in terms of some variable that we can adjust. Since the tape is oriented vertically, we'll pick a variable $y$ for the height of the dQ. This means we can write $\vec{r}_{source}$ as: 
 +$$\vec{r}_{source}\langle 0, y, 0 \rangle$$ 
 +The x-component of $\vec{r}_{source}$ here is zero because we have the tape located on the y-axis, which would be true no matter where on the tape our dQ is located. When we combine these pieces, we get the total separation vector: 
 +$$\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 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).+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====
-[[:184_notes:examples:Week4_tilted_segment|A Tilted Segment of Charge]]+  * [[:184_notes:examples:Week4_tilted_segment|A Tilted Segment of Charge]] 
 +  * [[:184_notes:examples:Week4_two_segments|Two Segments of Charge]] 
 +    *  Video Example: Two Segments of Charge 
 +{{youtube>BiqTwMrD774?large}}
  
-[[:184_notes:examples:Week4_two_segments|Two Segments of Charge]] 
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  • Last modified: 2018/06/05 13:37
  • by curdemma