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| 184_notes:line_fields [2018/09/12 15:11] – dmcpadden | 184_notes:line_fields [2021/02/13 18:58] (current) – [Building Electric Field for Lines of Charge] bartonmo |
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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) |
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| [[184_notes:dq|Next Page: dQ and the $\vec{r}$]] | /*[[184_notes:dq|Next Page: dQ and the $\vec{r}$]] |
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| | [[184_notes:comp_super|Previous Page: Superposition and the Computer]]*/ |
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| ===== Electric Field and Potential for Lines of Charge ===== | ===== Electric Field and Potential for Lines of Charge ===== |
| [[184_projects:s18_project_3|In your project last week]], you modeled a line of charge as seven point charges and used [[184_notes:superposition|superposition]] in your code to show what the electric field would look like around the line of charge. The notes this week will show how we can do a similar process for a line of charge analytically (using ideas from calculus) rather than using code. We will also talk about how these ideas ideas extend to large distributions of charge (including 2D sheets of charge and 3D volumes of charge). This page of notes will start by focusing on how we conceptually think of the electric field or electric potential for a line of charge (and we will go into the mathematical details in the next few pages). | In the previous notes, we talked about how to add fields using [[184_notes:superposition|superposition]], which can be greatly aided by the [[184_notes:comp_super|use of a computer]] (especially if there are many charges). We can also do a similar process for a line of charge analytically (using ideas from calculus) rather than using code. We will also talk about how these ideas ideas extend to large distributions of charge (including 2D sheets of charge and 3D volumes of charge). This page of notes will start by focusing on how we conceptually think of the electric field or electric potential for a line of charge (and we will go into the mathematical details in the next few pages). |
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| {{youtube>phR9OP9j46M?large}} | {{youtube>phR9OP9j46M?large}} |
| Say we have a line of charge (e.g. a piece of tape), and we are interested in what the electric field looks like at some distance d away from the piece of the tape (Point A). How could we find the electric field at Point A? We cannot simply use $\vec{E}=\frac{1}{4\pi\epsilon_0}\frac{Q_{tot}}{r^2}\hat{r}$ because the equation was built on the //__assumption that the charge was a point charge__//. However, we do know that the electric field follows **superposition** - meaning that electric field at a given point is the total (or sum) of all the electric fields from whatever sources are nearby. | Say we have a line of charge (e.g. a piece of tape), and we are interested in what the electric field looks like at some distance d away from the piece of the tape (Point A). How could we find the electric field at Point A? We cannot simply use $\vec{E}=\frac{1}{4\pi\epsilon_0}\frac{Q_{tot}}{r^2}\hat{r}$ because the equation was built on the //__assumption that the charge was a point charge__//. However, we do know that the electric field follows **superposition** - meaning that electric field at a given point is the total (or sum) of all the electric fields from whatever sources are nearby. |
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| So one way we could model the piece of tape would be to model it as two point charges - each point charge with half the charge. In this case the electric field at Point A would be given by: $\vec{E}_{tot}=\vec{E}_1+\vec{E}_2$. Here we have to calculate the electric field twice (find the different $\vec{r}$, find the magnitude of r, and calculate $\vec{E}$), but we get a better model. Now this might not be a great model for a line - but it's better than one point. We could make this model even better if we divided it into 4 point charges, spread out over the length of the tape, each with an amount of charge Q/4. Then the field would be given by $\vec{E}_{tot}=\vec{E}_1+\vec{E}_2+\vec{E}_3+\vec{E}_4$. We have to do more work computationally, but we also get a much better model. But still, we could make this model better if we split the charge into 8 points, or even 16 points, or better yet 32 points. However, the more points you split the charge into, the more work you have to do to compute the electric field (lucky for us, [[184_notes:comp_super|a computer is very good at doing these repetitive computations]]). | So one way we could model the piece of tape would be to model it as two point charges - each point charge with half the total charge of the tape. In this case the electric field at Point A would be given by: $\vec{E}_{tot}=\vec{E}_1+\vec{E}_2$. Here we have to calculate the electric field twice (find the different $\vec{r}$, find the magnitude of r, and calculate $\vec{E}$), but we get a better model. Now this might not be a great model for a line - but it's better than one point. We could make this model even better if we divided it into 4 point charges, spread out over the length of the tape, each with an amount of charge Q/4. Then the field would be given by $\vec{E}_{tot}=\vec{E}_1+\vec{E}_2+\vec{E}_3+\vec{E}_4$. We have to do more work computationally, but we also get a much better model. But still, we could make this model better if we split the charge into 8 points, or even 16 points, or better yet 32 points. However, the more points you split the charge into, the more work you have to do to compute the electric field (lucky for us, [[184_notes:comp_super|a computer is very good at doing these repetitive computations]]). |
| [{{ 184_notes:dqandr.png?300|$dQ$ and $r$ for point P}}] | [{{ 184_notes:dqandr.png?300|$dQ$ and $r$ for point P from a little bit charge along the line of tape}}] |
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| However, to make the best model of this line of charge, we would need to split the line into extremely small pieces of charge or infinitesimally small pieces of charge, which in calculus notation, we would write as dQ. We can then find the electric field at Point A due to only that small piece of charge - this would be "a little bit of electric field" since it comes from "a little bit of charge", so we would write this as dE. Now if we have an extremely small piece of a line of charge, this **__DOES__** look exactly like a point of charge. We know how to handle points of charge. [[184_notes:pc_efield|Using the equation for electric field of a point charge]], we can say that the little bit of electric field that comes from our little bit of charge would be: | However, to make the best model of this line of charge, we would need to split the line into extremely small pieces of charge or infinitesimally small pieces of charge, which in calculus notation, we would write as dQ. We can then find the electric field at Point A due to only that small piece of charge - this would be "a little bit of electric field" since it comes from "a little bit of charge", so we would write this as dE.// Now if we have an extremely small piece of a line of charge, this DOES look exactly like a point of charge.// We know how to handle points of charge. [[184_notes:pc_efield|Using the equation for electric field of a point charge]], we can say that the little bit of electric field that comes from our little bit of charge would be: |
| $$\vec{dE}=\frac{1}{4\pi\epsilon_0}\frac{dQ}{r^2}\hat{r}=\frac{1}{4\pi\epsilon_0}\frac{dQ}{r^3}\vec{r}$$ | $$\vec{dE}=\frac{1}{4\pi\epsilon_0}\frac{dQ}{r^2}\hat{r}=\frac{1}{4\pi\epsilon_0}\frac{dQ}{r^3}\vec{r}$$ |
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