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184_notes:dq [2017/09/04 18:39] – [Examples] tallpaul184_notes:dq [2018/09/12 15:22] – dmcpadden
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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: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 any little piece of charge in the shape should have the same of amount of charge. 
  
-{{  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 "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", $dy$}}]
  
 === Charge on a line === === Charge on a line ===
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 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}}+
 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 ===
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 Again, you can write the "little bit of volume" in a variety of ways, depending on the shape of charge. Again, you can write the "little bit of volume" in a variety of ways, depending on the shape of charge.
  
-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 towards the end of the course, but for now we will focus on lines of charge (or 1D charge distributions). 
  
 ==== Limits on the integral ==== ==== Limits on the integral ====
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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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 ====Examples==== ====Examples====
-[[:184_notes:examples:Week4_charge_ring|Electric Field from a Ring of Charge]] 
- 
 [[: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]]
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  • Last modified: 2021/05/26 13:36
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