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| 184_notes:resistors [2019/01/04 02:37] – dmcpadden | 184_notes:resistors [2021/03/04 19:46] (current) – bartonmo |
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| Sections 18.3, 18.8-18.10, and 19.4 in Matter and Interactions (4th edition) | Sections 18.3, 18.8-18.10, and 19.4 in Matter and Interactions (4th edition) |
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| [[184_notes:r_energy|Next Page: Energy in Circuits]] | /*[[184_notes:r_energy|Next Page: Energy in Circuits]] |
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| [[184_notes:current|Previous Page: Current in Wires]] | [[184_notes:current|Previous Page: Current in Wires]]*/ |
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| ===== Resistors in Circuits ===== | ===== Resistors in Circuits ===== |
| To this point, we have talked about what happens inside a wire when connected to two ends of battery - both in the //steady state current// situation and in the initial transient when the circuit is first connected. We found a few important conclusions about the circuit in steady state: | To this point, we have talked about what happens inside a wire when connected to two ends of battery - both in the __//steady state current//__ situation and in the initial transient when the circuit is first connected. We found a few important conclusions about the circuit in //__steady state:__// |
| * The electron current going into a location must be the same as the current leaving that location because [[184_notes:charge|charge is conserved]]. | * The electron current going into a location must be the same as the current leaving that location because [[184_notes:charge|charge is conserved]]. |
| * The electron current moves through the wire because there is a constant electric field inside the wire. This electric field comes primarily from the [[184_notes:q_in_wires|surface charges along the wire]]. | * The electron current moves through the wire because there is a constant electric field inside the wire. This electric field comes primarily from the [[184_notes:q_in_wires|surface charges along the wire]]. |
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| ==== Conservation of Charge in Circuits ==== | ==== Conservation of Charge in Circuits ==== |
| [{{ 184_notes:ThinResistorChargeDistribution.jpg?250|Surface charge distribution around a circuit with a thin section of wire, before steady state is achieved}}] | [{{ 184_notes:ThinResistorChargeDistribution.jpg?250|Surface charge distribution around a circuit with a thin section of wire, BEFORE steady state is achieved}}] |
| === Before steady state === | ==== Before steady state ==== |
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| Just after the circuit is connected, //__before the steady state current is established__//, there would be a constant electric field set up in the wire, which would start to move the electrons in the wire (with some average speed $v_{avg}$) and create the electron current. At this point, the number of electrons per second trying to enter the thin resistor is fairly large, compared to the number of electrons per second that can pass through the resistor. This means that the electrons pile up on one side of the resistor (and there is a similar lack of negative charges on the other side), creating a larger electric field within the resistor than that in the nearby wires. The electrons that pile up generate an electric field that opposes the motion of the electrons attempting to enter the resistor. At this point in time the current going into the resistor //**does not**// equal the current going through the resistor (e.g. the node rule does not apply), but this because we are looking at the situation //__before the circuit is in a steady state__//. (Charge is still conserved in this situation - it's just that some of the charge are piling up on the surface rather than passing through the thin part of the wire.) The electrons will continue to pile up until a steady state current is reached. | Just after the circuit is connected, //__before the steady state current is established__//, there would be a constant electric field set up in the wire, which would start to move the electrons in the wire (with some average speed $v_{avg}$) and create the electron current. At this point, the number of electrons per second trying to enter the thin resistor is fairly large, compared to the number of electrons per second that can pass through the resistor. This means that the electrons pile up on one side of the resistor (and there is a similar lack of negative charges on the other side), creating a larger electric field within the resistor than that in the nearby wires. The electrons that pile up generate an electric field that opposes the motion of the electrons attempting to enter the resistor. At this point in time the current going into the resistor //**does not**// equal the current going through the resistor (e.g. the node rule does not apply), but this because we are looking at the situation //__before the circuit is in a steady state__//. (Charge is still conserved in this situation - it's just that some of the charge are piling up on the surface rather than passing through the thin part of the wire.) The electrons will continue to pile up until a steady state current is reached. |
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| === In steady state === | ==== In steady state ==== |
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| Once a //__steady state current is reached__//, we know that the current in the resistor (or thin part of the wire) must be the same as the current in the thick part of the wire. In the steady state, charge has already built up on the ends of the resistor to create a larger electric field and no more charge is added to the surface at this point. If the charges aren't moving to the surface, conservation of charge says that all of the charges passing through the large part of the wire must also pass through the thin part of the wire. | Once a //__steady state current is reached__//, we know that the current in the resistor (or thin part of the wire) must be the same as the current in the thick part of the wire. In the steady state, charge has already built up on the ends of the resistor to create a larger electric field and no more charge is added to the surface at this point. If the charges aren't moving to the surface, conservation of charge says that all of the charges passing through the large part of the wire must also pass through the thin part of the wire. |
| $$E_{thin}=\frac{A_{thick}}{A_{thin}}E_{thick}$$ | $$E_{thin}=\frac{A_{thick}}{A_{thin}}E_{thick}$$ |
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| [{{ 184_notes:thinresistorefield.jpg?200|Electric field around a circuit in steady state with a thin wire}}] | [{{ :184_notes:thinresistorefield_new_.png?300|Electric field and surface charges around a circuit in steady state with a thin wire}}] |
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| Since $A_{thick}>A_{thin}$, this means that the electric field will be bigger in the thin section of wire compared to the thick wires. Because $v_{avg}=uE$, this also means that the average speed of the electrons in the thin wire must also be bigger than the average speed in the thick wires. Ultimately, this means that in the steady state situation, the largest surface charge gradient occurs over the resistor, with only small gradients in the thick wires. | Since $A_{thick}>A_{thin}$, this means that the electric field will be bigger in the thin section of wire compared to the thick wires. Because $v_{avg}=uE$, this also means that the average speed of the electrons in the thin wire must also be bigger than the average speed in the thick wires. Ultimately, this means that in the steady state situation, the largest surface charge gradient occurs over the resistor, with only small gradients in the thick wires. |