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184_notes:r_series [2018/06/26 13:41] – [Equivalent Resistance] curdemma | 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) | ||
- | [[184_notes: | + | /*[[184_notes: |
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+ | [[184_notes: | ||
===== Resistors in Series ===== | ===== Resistors in Series ===== | ||
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{{youtube> | {{youtube> | ||
==== Circuit Diagrams ==== | ==== Circuit Diagrams ==== | ||
- | [{{ 184_notes: | + | [{{ 184_notes: |
Circuit diagrams are a simplified way to represent a circuit. In a circuit diagram, each element is represented by some kind of symbol and the wires are represented by lines. These diagrams are not very good for showing what is happening to surface charges or the electric field, but they do help visualize a circuit particularly for combinations of circuit elements and some of the more macroscopic properties. | Circuit diagrams are a simplified way to represent a circuit. In a circuit diagram, each element is represented by some kind of symbol and the wires are represented by lines. These diagrams are not very good for showing what is happening to surface charges or the electric field, but they do help visualize a circuit particularly for combinations of circuit elements and some of the more macroscopic properties. | ||
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==== Node Rule and Current in Series ==== | ==== Node Rule and Current in Series ==== | ||
[{{ 184_notes: | [{{ 184_notes: | ||
- | 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//**, // | + | 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**, // |
- | 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: | + | 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: |
$$I_{bat}=I_2=I_1$$ | $$I_{bat}=I_2=I_1$$ | ||
==== Loop Rule and Voltage in Series ==== | ==== Loop Rule and Voltage in Series ==== | ||
- | [{{ 184_notes: | + | [{{ 184_notes: |
- | For resistors in series, we can also say something about the difference in electric potential across each circuit element. Using the [[184_notes: | + | For resistors in series, we can also say something about the difference in electric potential across each circuit element. Using the [[184_notes: |
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__// | 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__// | ||
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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: | + | [{{184_notes: |
{{ 184_notes: | {{ 184_notes: | ||
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To simplify series circuits, we will often try to replace the series resistors with a single resistor that has the equivalent resistance as the combination of resistors. This would mean that you get to deal with one resistor instead of two for example. We can do this by comparing the circuit with two resistors to the circuit with the equivalent resistance. In both of these circuits we would want to keep the battery the same (would have the same $\Delta V_{bat}$) and keep the current coming out of the battery as the same (same $I_{bat}$). From the loop rule around the combination circuit, we know that: | To simplify series circuits, we will often try to replace the series resistors with a single resistor that has the equivalent resistance as the combination of resistors. This would mean that you get to deal with one resistor instead of two for example. We can do this by comparing the circuit with two resistors to the circuit with the equivalent resistance. In both of these circuits we would want to keep the battery the same (would have the same $\Delta V_{bat}$) and keep the current coming out of the battery as the same (same $I_{bat}$). From the loop rule around the combination circuit, we know that: | ||
$$|\Delta V_{bat}|=|\Delta V_1| + |\Delta V_2|$$ | $$|\Delta V_{bat}|=|\Delta V_1| + |\Delta V_2|$$ | ||
- | [{{184_notes: | + | [{{184_notes: |
If we //__assume that our resistors are ohmic__//, then we can rewrite the potential changes in terms of the resistance and current: | If we //__assume that our resistors are ohmic__//, then we can rewrite the potential changes in terms of the resistance and current: | ||
$$\Delta V_{bat}=I_1R_1+I_2R_2$$ | $$\Delta V_{bat}=I_1R_1+I_2R_2$$ | ||
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Now, from the node rule, we know that $I_{bat}=I_1=I_2$, | Now, from the node rule, we know that $I_{bat}=I_1=I_2$, | ||
$$R_{eq}=R_1+R_2$$ | $$R_{eq}=R_1+R_2$$ | ||
- | 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// | + | 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// |
==== Examples ==== | ==== Examples ==== | ||
- | [[: | + | * [[: |
+ | * Video Example: Resistors in Series | ||
+ | {{youtube> |