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| 184_notes:superposition [2018/05/29 14:23] – [Superposition of Electric Potential] curdemma | 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}}] | [{{ 184_notes:dipole.png?150|Dipole representation - one positive and one negative charge, separated by a distance d}}] |
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| It turns out that this assumption is fairly useful. In the case of charges fixed in place, this often true for metals and insulators that are experiencing a fixed external electric field. The electrons redistribute, but do not continue moving because they have either moved as far as they can ([[184_notes:charge_and_matter#Types_of_Matter|in the case of conductors (metals)]]) or are strongly bound to their nuclei ([[184_notes:charge_and_matter#Types_of_Matter|in the case of insulators]]). This assumption also works if we rub some excess charge off onto an insulator as the excess charge stays fairly localized to the place where it was rubbed off. If we are interested in the distribution at a given time, that is an even stronger assumption as we are taking a snapshot of the distribution at a specific instant in time, so the electrons are fixed in the snapshot. | It turns out that this assumption is fairly useful. In the case of charges fixed in place, this often true for metals and insulators that are experiencing a fixed external electric field. The electrons redistribute, but do not continue moving because they have either moved as far as they can ([[184_notes:charge_and_matter#Types_of_Matter|in the case of conductors (metals)]]) or are strongly bound to their nuclei ([[184_notes:charge_and_matter#Types_of_Matter|in the case of insulators]]). This assumption also works if we rub some excess charge off onto an insulator as the excess charge stays fairly localized to the place where it was rubbed off. If we are interested in the distribution at a given time, that is an even stronger assumption as we are taking a snapshot of the distribution at a specific instant in time, so the electrons are fixed in the snapshot. |
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| ==== Superposition of Electric Field ==== | ==== Superposition of Electric Field ==== |
| [{{ 184_notes:dipole_epoint.png?150|Electric field at one point in space due to a dipole}}] | [{{ 184_notes:dipole_epoint.png?150|Electric field at a single point (Point P) due to a dipole}}] |
| [[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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| [{{ 184_notes:dipole_field.png?200|Electric field vectors in space surrounding a dipole}}] | [{{ 184_notes:dipole_field.png?200|Electric field vectors in space surrounding a dipole}}] |
| If you repeat this process for many observation points all around the dipole, you will end up with the electric field shown in the next figure to the right. Close to the individual charges the electric field looks much like it would for a single point charge (the electric field points away from the positive charge and toward the negative charges); however in the middle of the two charges the field bends to point from the positive charge to the negative charge. | If you repeat this process for many observation points all around the dipole, you will end up with the electric field shown in the next figure to the right. Close to the individual charges, the electric field looks much like it would for a single point charge (the electric field points away from the positive charge and toward the negative charges); however in the middle of the two charges the field bends to point from the positive charge to the negative charge. |
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| If you have more than two charges, the net electric field is simply found by adding up the individual fields from each point charge. | If you have more than two charges, the net electric field is simply found by adding up the individual fields from each point charge. |
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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}}] | [{{ :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//**. |
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| 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 for a 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 above to the left (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.// | [{{ :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.// |
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| If the reference points for the individual electric potentials are the same, you can find the **total electric potential** at a given location by summing all of the individual electric potentials at that point. | If the reference points for the individual electric potentials are the same, you can find the **total electric potential** at a given location by summing all of the individual electric potentials at that point. |
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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}} |