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| 184_notes:r_series [2019/01/04 03:03] – dmcpadden | 184_notes:r_series [2021/06/28 23:17] (current) – schram45 |
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| Sections 19.2 and 19.3 in Matter and Interactions (4th edition) | Sections 19.2 and 19.3 in Matter and Interactions (4th edition) |
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| [[184_notes:r_parallel|Next Page: Resistors in Parallel]] | /*[[184_notes:r_parallel|Next Page: Resistors in Parallel]] |
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| [[184_notes:cap_in_cir|Previous Page: Capacitors in a Circuit]] | [[184_notes:cap_in_cir|Previous Page: Capacitors in a Circuit]]*/ |
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| ===== Resistors in Series ===== | ===== Resistors in Series ===== |
| ==== Node Rule and Current in Series ==== | ==== Node Rule and Current in Series ==== |
| [{{ 184_notes:Week8_2.png?200|Current remains the same across resistors in series}}] | [{{ 184_notes:Week8_2.png?200|Current remains the same across resistors in series}}] |
| When two circuit elements are in series this means that **//all of the current that goes into the first element also goes into the second element//**, //__assuming a steady state current situation__//. For example, consider two resistors that are in series with a battery. All of the current that comes out of the battery **must** travel through the wire and go through the first resistor. After the first resistor, all the current must then pass through the second resistor - there is no other path for electrons to travel along. Finally, all the current must then return to the battery. Note, we have drawn the conventional current coming from the positive plate of the battery (rather than the electron current which would come from the negative plate). | When two circuit elements are in series this means that **all of the current that goes into the first element also goes into the second element**, //__assuming a steady state current situation__//. For example, consider two resistors that are in series with a battery. All of the current that comes out of the battery **must** travel through the wire and go through the first resistor. After the first resistor, all the current must then pass through the second resistor - there is no other path for electrons to travel along. Finally, all the current must then return to the battery. Note, we have drawn the conventional current coming from the positive plate of the battery (rather than the electron current which would come from the negative plate). |
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| This means that **//in series, circuit elements always have the same current with no alternate paths for current flow//**. This conclusion follows naturally from the [[184_notes:current|node rule]], which is a mathematical statement of the conservation of charge in steady state. In terms of the circuit above, this means that the current that goes through the battery ($I_{bat}$) is equal to the current through the first resistor ($I_1$), which is equal to the current through the second resistor ($I_2$). | This means that **in series, circuit elements always have the same current with no alternate paths for current flow**. This conclusion follows naturally from the [[184_notes:current|node rule]], which is a mathematical statement of the conservation of charge in steady state. In terms of the circuit above, this means that the current that goes through the battery ($I_{bat}$) is equal to the current through the first resistor ($I_1$), which is equal to the current through the second resistor ($I_2$). |
| $$I_{bat}=I_2=I_1$$ | $$I_{bat}=I_2=I_1$$ |
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| ==== Loop Rule and Voltage in Series ==== | ==== Loop Rule and Voltage in Series ==== |
| [{{ 184_notes:Week8_3.png?300|Potential differences (or voltage drops) throughout the circuit}}] | [{{ 184_notes:Week8_3.png?300|Potential differences (or voltage drops) throughout the circuit}}] |
| For resistors in series, we can also say something about the difference in electric potential across each circuit element. Using the [[184_notes:r_energy|loop rule]], we know that the sum of all of the changes in electric potential around a complete loop must equal zero. Consider the same series circuit above with the battery and two resistors. If we think about the electric potential or energy per charge going around the circuit, we know that the highest electric potential will be on the positive side of the battery. As the current moves through the wire, there would be a small drop in electric potential along the wire (from the electron collisions with the nuclei), but a much larger change in potential would occur over the resistor. Then again, there would be another small drop over the next wire with a larger change over the second resistor, with a final small drop over the wire that connects back to the battery. If we move from the negative plate of the battery back to the positive plate, then there is now a large gain in electric potential (instead of a drop), and we have returned to where we started. | For resistors in series, we can also say something about the difference in electric potential across each circuit element. Using the [[184_notes:r_energy|loop rule]], we know that **the sum of all of the changes in electric potential around a complete loop must equal zero.** Consider the same series circuit above with the battery and two resistors. If we think about the electric potential or energy per charge going around the circuit, we know that the highest electric potential will be on the positive side of the battery. As the current moves through the wire, there would be a small drop in electric potential along the wire (from the electron collisions with the nuclei), but a much larger change in potential would occur over the resistor. Then again, there would be another small drop over the next wire with a larger change over the second resistor, with a final small drop over the wire that connects back to the battery. If we move from the negative plate of the battery back to the positive plate, then there is now a large gain in electric potential (instead of a drop), and we have returned to where we started. |
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| Oftentimes, the changes in potential in the wires are so small compared to those over the resistors, that we //__assume the changes in electric potential across the wires are negligible__//. You might also hear this assumption stated as "we assume that the wires are perfect". If we make this assumption, then there are only the three circuit elements that we need to worry about: the battery, the first resistor, and the second resistor. In this case, we could write an equation that states that the gain in electric potential across the battery plus the drops in electric potential across the resistors, should give you zero. | Oftentimes, the changes in potential in the wires are so small compared to those over the resistors, that we //__assume the changes in electric potential across the wires are negligible__//. You might also hear this assumption stated as "we assume that the wires are perfect". If we make this assumption, then there are only the three circuit elements that we need to worry about: the battery, the first resistor, and the second resistor. In this case, we could write an equation that states that the gain in electric potential across the battery plus the drops in electric potential across the resistors, should give you zero. |
| Another way of saying this would be that the energy per charge coming from the battery must be distributed across the two resistors. You can see this in the graph below. | Another way of saying this would be that the energy per charge coming from the battery must be distributed across the two resistors. You can see this in the graph below. |
| $$|\Delta V_{bat}|=|\Delta V_1| + |\Delta V_2|$$ | $$|\Delta V_{bat}|=|\Delta V_1| + |\Delta V_2|$$ |
| This means that **//in a series circuit, the electric potentials add together//**. | This means that **in a series circuit, the electric potentials add together**. |
| [{{184_notes:Week8_4.png?600|Graph of electric potential throughout the circuit. Note that the change in potential along the wires is very small compared to the change in potential across a resistor. }}] | [{{184_notes:Week8_4.png?600|Graph of electric potential throughout the circuit. Note that the change in potential along the wires is very small compared to the change in potential across a resistor. }}] |
| {{ 184_notes:Week8_5b.png?200|Example: loop 1}} | {{ 184_notes:Week8_5b.png?200|Example: loop 1}} |
| Now, from the node rule, we know that $I_{bat}=I_1=I_2$, so the currents in this equation will cancel out, leaving: | Now, from the node rule, we know that $I_{bat}=I_1=I_2$, so the currents in this equation will cancel out, leaving: |
| $$R_{eq}=R_1+R_2$$ | $$R_{eq}=R_1+R_2$$ |
| If we want to replace two series resistors with a single equivalent resistor it needs to have a resistance that is equal to the sum of the individual resistors. Another important conclusion from this is that **//if you put two resistors in series, the total resistance of the circuit will increase//**. Note that one has to be careful with using equivalent resistances for more complex circuits as the current that is computed is that through the equivalent resistor and //not necessarily// that through all the branches of a complex circuit. | If we want to replace two series resistors with a single equivalent resistor it needs to have a resistance that is equal to the sum of the individual resistors. Another important conclusion from this is that **if you put two resistors in series, the total resistance of the circuit will increase**. Note that one has to be careful with using equivalent resistances for more complex circuits as the current that is computed is that through the equivalent resistor and //not necessarily// that through all the branches of a complex circuit. |
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| ==== Examples ==== | ==== Examples ==== |
| [[:184_notes:examples:Week8_resistors_series|Example: Resistors in Series]] | * [[:184_notes:examples:Week8_resistors_series|Example: Resistors in Series]] |
| | * Video Example: Resistors in Series |
| | {{youtube>8EZbxcHjnC0?large}} |