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| 184_notes:r_parallel [2020/08/23 20:20] – dmcpadden | 184_notes:r_parallel [2021/03/04 01:38] – bartonmo | ||
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| ==== Node Rule and Current in Parallel ==== | ==== Node Rule and Current in Parallel ==== | ||
| [{{ 184_notes: | [{{ 184_notes: | ||
| - | When two circuit elements are parallel, this means that there are two different paths along the circuit that take you across the same potential difference. For example, consider a circuit with a battery and two resistors (similar to before), but this time there is a split in the wire and the resistors are side-by-side rather than in a row. When we use the node rule, on the this circuit, there are two nodes that are of particular interest (the points where there are three wires coming together in the circuit diagram). The node rule says that the current going into a node should be the same as the current leaving the node to satisfy conservation of charge. For the top node, this means that current from the battery is split when it hits the branches for the $R_1$ and $R_2$ resistors, but **//the sum of the current in each branch should be equal to the total current coming from the battery//**. This is a different mathematical statement of conservation of charge than we had for series elements, but still quite powerful. | + | When two circuit elements are parallel, this means that there are two different paths along the circuit that take you across the same potential difference. For example, consider a circuit with a battery and two resistors (similar to before), but this time there is a split in the wire and the resistors are side-by-side rather than in a row. When we use the node rule, on the this circuit, there are two nodes that are of particular interest (the points where there are three wires coming together in the circuit diagram). The node rule says that the current going into a node should be the same as the current leaving the node to satisfy conservation of charge. For the top node, this means that current from the battery is split when it hits the branches for the $R_1$ and $R_2$ resistors, but **the sum of the current in each branch should be equal to the total current coming from the battery**. This is a different mathematical statement of conservation of charge than we had for series elements, but still quite powerful. |
| $$I_{bat}=I_1+I_2$$ | $$I_{bat}=I_1+I_2$$ | ||
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| $$|\Delta V_1|=|\Delta V_2|$$ | $$|\Delta V_1|=|\Delta V_2|$$ | ||
| - | This tells us that **//circuit elements in parallel have an equal potential difference//**. | + | This tells us that **circuit elements in parallel have an equal potential difference**. |
| $$|\Delta V_{bat}|=|\Delta V_1|=|\Delta V_2|$$ | $$|\Delta V_{bat}|=|\Delta V_1|=|\Delta V_2|$$ | ||
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| Now, from the loop rule, we know that $\Delta V_{bat}=\Delta V_1=\Delta V_2$, so the potentials in this equation will cancel out, leaving: | Now, from the loop rule, we know that $\Delta V_{bat}=\Delta V_1=\Delta V_2$, so the potentials in this equation will cancel out, leaving: | ||
| $$\frac{1}{R_{eq}}=\frac{1}{R_1}+\frac{1}{R_2}$$ | $$\frac{1}{R_{eq}}=\frac{1}{R_1}+\frac{1}{R_2}$$ | ||
| - | Therefore, if we want to replace two parallel resistors with a single resistor, it needs to have a resistance that equal to the inverse sum of the individual resistors. This is an important conclusion because if you actually plug in numbers here you will find that the equivalent resistance is //less than// either of the two individual resistors. This means that **//combining resistors in parallel will reduce the overall resistance//**. 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// | + | Therefore, if we want to replace two parallel resistors with a single resistor, it needs to have a resistance that equal to the inverse sum of the individual resistors. This is an important conclusion because if you actually plug in numbers here you will find that the equivalent resistance is //less than// either of the two individual resistors. This means that **combining resistors in parallel will reduce the overall resistance**. 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 ==== | ||
| [[: | [[: | ||