FHSST Physics/Electricity/Simple Parallel Circuits

Simple parallel circuits

Let's start with a parallel circuit consisting of three resistors and a single battery:

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The first principle to understand about parallel circuits is that the voltage is equal across all components in the circuit. This is because there are only two sets of electrically common points in a parallel circuit, and voltage measured between sets of common points must always be the same at any given time. Therefore, in the above circuit, the voltage across $R_{1}$ is equal to the voltage across $R_{2}$ which is equal to the voltage across $R_{3}$ which is equal to the voltage across the battery. This equality of voltages can be represented in another table for our starting values:

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Just as in the case of series circuits, the same caveat for Ohm's Law applies: values for voltage, current, and resistance must be in the same context in order for the calculations to work correctly. However, in the above example circuit, we can immediately apply Ohm's Law to each resistor to find its current because we know the voltage across each resistor (9 volts) and the resistance of each resistor:

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At this point we still don't know what the total current or total resistance for this parallel circuit is, so we can't apply Ohm's Law to the rightmost ("Total") column. However, if we think carefully about what is happening it should become apparent that the total current must equal the sum of all individual resistor ("branch") currents:

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As the total current exits the negative (-) battery terminal at point 8 and travels through the circuit, some of the flow splits off at point 7 to go up through $R_{1}$, some more splits off at point 6 to go up through $R_{2}$, and the remainder goes up through $R_{3}$. Like a river branching into several smaller streams, the combined flow rates of all streams must equal the flow rate of the whole river. The same thing is encountered where the currents through $R_{1}$, $R_{2}$, and $R_{3}$ join to flow back to the positive terminal of the battery (+) toward point 1: the flow of electrons from point 2 to point 1 must equal the sum of the (branch) currents through $R_{1}$, $R_{2}$, and $R_{3}$.

This is the second principle of parallel circuits: the total circuit current is equal to the sum of the individual branch currents. Using this principle, we can fill in the $I_{T}$ spot on our table with the sum of $I_{R1}$, $I_{R2}$, and $I_{R3}$:

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Finally, applying Ohm's Law to the rightmost ("Total") column, we can calculate the total circuit resistance:

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Please note something very important here. The total circuit resistance is only 625 $\Omega$: less than any one of the individual resistors. In the series circuit, where the total resistance was the sum of the individual resistances, the total was bound to be greater than any one of the resistors individually. Here in the parallel circuit, however, the opposite is true: we say that the individual resistances diminish rather than add to make the total. This principle completes our triad of "rules" for parallel circuits, just as series circuits were found to have three rules for voltage, current, and resistance. Mathematically, the relationship between total resistance and individual resistances in a parallel circuit looks like this:

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The same basic form of equation works for any number of resistors connected together in parallel, just add as many 1/R terms on the denominator of the fraction as needed to accommodate all parallel resistors in the circuit.