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| 184_notes:superposition [2018/06/27 19:52] – dmcpadden | 184_notes:superposition [2021/05/26 13:41] (current) – schram45 |
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| Sections 13.5 and 13.6 in Matter and Interactions (4th edition) | Sections 13.5 and 13.6 in Matter and Interactions (4th edition) |
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| [[184_notes:dipole_sup|Next Page: Dipole Superposition Example]] | /*[[184_notes:dipole_sup|Next Page: Dipole Superposition Example]] |
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| | [[184_notes:pc_vefu|Previous Page: Relationships between Force, Field, Potential, and Energy]]*/ |
| ===== Superposition ===== | ===== Superposition ===== |
| Over the last two weeks, you have read about the [[184_notes:pc_efield|electric field]] and [[184_notes:pc_potential|electric potential]] that is created by a single point charge. While a single point charge is definitely the simplest case, most charged objects that you will run into are not single point charges. These notes will talk about the electric field and electric potential due to multiple charges - starting with the next most complicated case: two point charges that are near each other. The idea of adding together the electric field or electric potential for multiple charges is called the **superposition** of the field (for both vector and scalar fields), and it generalizes to any number of charges as you will read soon. | Over the last two weeks, you have read about the [[184_notes:pc_efield|electric field]] and [[184_notes:pc_potential|electric potential]] that is created by a single point charge. While a single point charge is definitely the simplest case, most charged objects that you will run into are not single point charges. These notes will talk about the electric field and electric potential due to multiple charges - starting with the next most complicated case: two point charges that are near each other. The idea of adding together the electric field or electric potential for multiple charges is called the **superposition** of the field (for both vector and scalar fields), and it generalizes to any number of charges as you will read soon. |
| To simplify the situation, we will usually make some sort of assumption. For example, //__we often assume that the charge(s) are fixed in place__// (something is holding them at a particular location, but we don't care what that something is). Or //__we will assume that we are interested in a particular instant in time__// and examine what is happening for that situation (like taking a single frame from a movie or freezing time). | To simplify the situation, we will usually make some sort of assumption. For example, //__we often assume that the charge(s) are fixed in place__// (something is holding them at a particular location, but we don't care what that something is). Or //__we will assume that we are interested in a particular instant in time__// and examine what is happening for that situation (like taking a single frame from a movie or freezing time). |
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| === How useful is this assumption? === | ==== How useful is this assumption? ==== |
| [{{ 184_notes:dipole.png?150|Dipole representation - one positive and one negative charge, separated by a distance d}}] | [{{ 184_notes:dipole.png?150|Dipole representation - one positive and one negative charge, separated by a distance d}}] |
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| [{{ 184_notes:dipole_epoint.png?150|Electric field at a single point (Point P) due to a dipole}}] | [{{ 184_notes:dipole_epoint.png?150|Electric field at a single point (Point P) due to a dipole}}] |
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| [[184_notes:pc_efield|As you have learned]], the electric field from a single //positive charge// at any given point will point away from the charge, and the electric field at any given point from a //negative charge// will point toward the point charge. So what happens to the electric field when you have a positive charge next to a negative charge? The field at any point in space around the two charges will be given by a **net electric field**, which is the [[184_notes:math_review#vector_addition|vector addition]] of the electric field from the positive charge and the electric field from the negative charge. | [[184_notes:pc_efield|As you have learned]], the electric field from a single //positive charge// at any given point will point //away// from the charge, and the electric field at any given point from a //negative charge// will point //toward// the point charge. So what happens to the electric field when you have a positive charge next to a negative charge? The field at any point in space around the two charges will be given by a **net electric field**, which is the [[184_notes:math_review#vector_addition|vector addition]] of the electric field from the positive charge and the electric field from the negative charge. |
| $$\vec{E}_{net}=\vec{E}_{+}+\vec{E}_{-}$$ | $$\vec{E}_{net}=\vec{E}_{+}+\vec{E}_{-}$$ |
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| ==== Superposition of Electric Potential ==== | ==== Superposition of Electric Potential ==== |
| [{{ 184_notes:potentialgraph.jpg?300|Potential vs Distance graph of a positive (blue) and a negative (red) charge with the V=0 reference point at r=$\infty$.}}] | [{{ :184_notes:electricpotential_new_2.png?300|Potential vs Distance graph of a positive (dashed blue) and a negative (solid red) charge with the V=0 reference point at r=$\infty$.}}] |
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| Because electric potential is a scalar, it means adding together the electric potentials can be quite a bit simpler than adding electric fields (you don't have to consider direction); however, you must first check that the reference point for each of the individual potentials is the same. This is the same idea that we used with [[183_notes:relative_motion|relative motion]] when we had to choose where the origin was to make measurements of displacements - it doesn't make sense to compare measurements with two different origins. The **reference point** for potential is typically defined as the location where the electric potential is equal to zero. You can find the reference point by setting V = 0 and solving for the position, $r$, or by graphing the electric potential versus distance and finding the location where V=0. **//You can only add potentials that have the same reference point//**. | Because electric potential is a scalar, it means adding together the electric potentials can be quite a bit simpler than adding electric fields (you don't have to consider direction); however, you must first check that the reference point for each of the individual potentials is the same. This is the same idea that we used with [[183_notes:relative_motion|relative motion]] when we had to choose where the origin was to make measurements of displacements - it doesn't make sense to compare measurements with two different origins. The **reference point** for potential is typically defined as the location where the electric potential is equal to zero. You can find the reference point by setting V = 0 and solving for the position, $r$, or by graphing the electric potential versus distance and finding the location where V=0. **//You can only add potentials that have the same reference point//**. |
| Most of the time, //__we assume V=0 is at infinity__//. We actually already made this assumption when we said the [[184_notes:pc_potential|electric potential for a point charge]] is $V=\frac{1}{4\pi\epsilon_0}\frac{q}{r}$ (the only time when $V=0$ is when $r=\infty$). The potential vs distance graph for a positive charge (in blue) and a negative charge (in red) is shown in the figure to right. | Most of the time, //__we assume V=0 is at infinity__//. We actually already made this assumption when we said the [[184_notes:pc_potential|electric potential for a point charge]] is $V=\frac{1}{4\pi\epsilon_0}\frac{q}{r}$ (the only time when $V=0$ is when $r=\infty$). The potential vs distance graph for a positive charge (in blue) and a negative charge (in red) is shown in the figure to right. |
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| [{{ 184_notes:potentialgraphshift.jpg?300|Potential vs Distance of a positive (blue) and a negative (red) charge with the V=0 reference point at a non-infinite reference point.}}] | [{{ :184_notes:electric_potential_non_infinite_reference_new.png?300|Potential vs Distance of a positive (dashed blue) and a negative (solid red) charge with the V=0 reference point at a non-infinite reference point.}}] |
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| However, we could have equally said the voltage for a point charge was $V=\frac{1}{4\pi\epsilon_0}\frac{q}{r} - 2$, which would give us a reference point at $r_{rp}=\frac{1}{4\pi\epsilon_0}\frac{q}{2}$. The electric potential vs distance graphs for this potential/reference point are shown to the right (again, with blue for a positive charge and red for a negative charge). There is nothing wrong with having a different reference point, but we will usually pick a reference point at $r=\infty$ because it makes interpreting voltage numbers easy: a (+) voltage means you are close to a positive charge, a (-) voltage means you are close to a negative charge, and a zero voltage means you are either at $r=\infty$ or somewhere in between a positive and negative charge. //Sometimes it's convenient to set the potential to zero at somewhere other than $r=\infty$, which we will do when we discuss circuits.// | However, we could have equally said the voltage for a point charge was $V=\frac{1}{4\pi\epsilon_0}\frac{q}{r} - 2$, which would give us a reference point at $r_{rp}=\frac{1}{4\pi\epsilon_0}\frac{q}{2}$. The electric potential vs distance graphs for this potential/reference point are shown to the right (again, with blue for a positive charge and red for a negative charge). There is nothing wrong with having a different reference point, but we will usually pick a reference point at $r=\infty$ because it makes interpreting voltage numbers easy: a (+) voltage means you are close to a positive charge, a (-) voltage means you are close to a negative charge, and a zero voltage means you are either at $r=\infty$ or somewhere in between a positive and negative charge. //Sometimes it's convenient to set the potential to zero at somewhere other than $r=\infty$, which we will do when we discuss circuits.// |
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| ==== Examples ==== | ==== Examples ==== |
| [[184_notes:examples:Week3_superposition_three_points|Superposition with Three Point Charges]] | * [[184_notes:examples:Week3_superposition_three_points|Superposition with Three Point Charges]] |
| | * Video Example: Superposition with Three Point Charges |
| [[184_notes:examples:Week3_plotting_potential|Plotting Potential for Multiple Charges]] | * [[184_notes:examples:Week3_plotting_potential|Plotting Potential for Multiple Charges]] |
| | {{youtube>2VLMLuL2N7s?large}} |