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184_notes:r_parallel [2018/06/26 14:11] – [Node Rule and Current in Parallel] curdemma | 184_notes:r_parallel [2020/08/23 20:20] – dmcpadden | ||
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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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===== Resistors in Parallel ===== | ===== Resistors in Parallel ===== | ||
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{{youtube> | {{youtube> | ||
==== Node Rule and Current in Parallel ==== | ==== Node Rule and Current in Parallel ==== | ||
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- | 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 (Points A and B 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 Point A, 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// | + | 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 |
$$I_{bat}=I_1+I_2$$ | $$I_{bat}=I_1+I_2$$ | ||
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==== Loop Rule and Voltage in Parallel ==== | ==== Loop Rule and Voltage in Parallel ==== | ||
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For a parallel circuit, using the loop rule becomes a little more involved, because now there are actually three different loops with the circuit. This means that we need to look at energy conservation around every loop in the circuit. To start, we will again mark which side of the resistors and battery would have a higher potential and which side would have a lower potential. For the first loop going clockwise, we see that there is a gain in potential from the battery and then a drop of potential across the resistor: | For a parallel circuit, using the loop rule becomes a little more involved, because now there are actually three different loops with the circuit. This means that we need to look at energy conservation around every loop in the circuit. To start, we will again mark which side of the resistors and battery would have a higher potential and which side would have a lower potential. For the first loop going clockwise, we see that there is a gain in potential from the battery and then a drop of potential across the resistor: | ||
$$+|\Delta V_{bat}| - |\Delta V_1| = 0$$ | $$+|\Delta V_{bat}| - |\Delta V_1| = 0$$ | ||
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==== Equivalent Resistance ==== | ==== Equivalent Resistance ==== | ||
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To simplify parallel circuits, we will again try to replace the resistors in parallel with a single resistor that has an equivalent resistance. We will still do this by comparing a circuit with two resistors in parallel to a circuit with a single equivalent resistor, keeping the $\Delta V_{bat}$ and $I_{bat}$ the same for both circuits. From the node rule in the two-resistor circuit, we know that: | To simplify parallel circuits, we will again try to replace the resistors in parallel with a single resistor that has an equivalent resistance. We will still do this by comparing a circuit with two resistors in parallel to a circuit with a single equivalent resistor, keeping the $\Delta V_{bat}$ and $I_{bat}$ the same for both circuits. From the node rule in the two-resistor circuit, we know that: |