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| 184_notes:superposition [2020/08/17 17:29] – dmcpadden | 184_notes:superposition [2021/05/26 13:41] (current) – schram45 |
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| 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}} |