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# Parallel resistors (derivation continued)

Learn how to calculate the equivalent resistance for multiple parallel resistors. This method simplifies complex resistor networks by replacing them with a single equivalent parallel resistor, streamlining circuit analysis and improving efficiency. Grasp the concept of current splitting among resistors and understand that all parallel resistors share the same voltage, making this technique applicable to various electrical engineering scenarios. Created by Willy McAllister.

## Want to join the conversation?

• So, if resistors are connected by a common point, they're immediately considered in parallel?
• Hello Levi,

Correct, it's like two people holding hands:

Left to left plus right to right is parallel.

Left to right plus right to left is still parallel as resistors generally do not care which direction they are installed.

Regards,

APD
• is there another analogie for parallel resistors
• Hello Ethan,

Do you recall those problems involving pumps? For example if a pump can empty a swimming pool in 1/2 hour and another pump can empty the pool in 1/3 hour, how long does it take both pumps operating together to empty the pool?

Here is another: A car gets 30 miles to the gallon and another car gets 20 miles to the gallon. What is there combined millage if they travel together.

I hope these analogies help. It's all about flow of material...

Regards,

APD
• If I will connect 10000 resisters to a circuit consisting of a battery which is a 9v battery. will all these 10000 resisters get the same 9volts?
• If all 10,000 resistors are connected in parallel, yes, each one will have 9v across its terminals.

The ideal parallel connection assumes the wires connecting the resistors are ideal (zero resistance) wires. If you try this experiment in real life you have to use real wires, which have a very small (but not 0) resistance. With that many resistors you might end up with a small extra resistance from the wire. "small" means less than 1 ohm.
• how do we know that voltage for both the R1 and R2 resistor is the same meaning why is it that if they share the same nodes then they are going to have the same voltage?
• So what you’re basically saying is...
Series circuits- current stays the same and voltage changes ( voltage drops )
Parallel circuits- voltage stays the same, and the current changes ( split current )? Is that correct?
• Yes that is correct @Dionna Huett
(1 vote)
• Each current calculation for each of the resistors is just using Ohm's Law, yeah?
(1 vote)
• That is correct. Once you find the voltage across the parallel connection the individual currents are found with Ohm's Law, i = v/R.
• When we are using the formula 1/R(p)=1/R1+1/R2, will the resistor R(p) be a series resistor? It is because there is one resistor connected to one battery.
(1 vote)
• You are asking about two resistors in parallel near the beginning of the video. On the left, R1 and R2 are connected in parallel with each other. They can be replaced by a single Rp as shown on the right. The replacement resistor, Rp, ends up connected in series with the battery.
• Hello, I would like to know how we calculate the total resistance for a circuit that has both parallel and series resistors
(1 vote)
• Why do all the resistors in parallel share the same voltage? How does he know that?
(1 vote)
• Because of the definition of what it means to be in parallel. In a parallel connection the terminal wires of all the resistors are connected to the same two nodes. There can only be one voltage because there are only two nodes.
(1 vote)
• How come the current, that is entering the node, "knows" how to "split itself" into currents such that Ohm's law gives equal voltages? In other words, how does the current know what kind of resistor is ahead? Would it be something similar to road traffic, when a slow traffic would mean a lot of cars and fast traffic means few cars in terms of density?
• This is a good question. I compare this electrical situation with what happens to the flow of water in a shallow creek. When you drop a rock into the water the current splits and flows around either side of the rock. The water molecules don't "know" or "decide" how to split. There is no choosing which side to go on, it just happens.

It's best not to think about the speed of current or electrons. This splitting happens no matter how fast the electrons or water is moving.

## Video transcript

- [Voiceover] In the last video, we introduced the idea of parallel resistors. These two resistors are in parallel with each other because they share nodes and they have the same voltage across them. So that configuration is called the parallel resistor. And we also showed that these two resistors could be replaced by a single resistor we labeled this one R1, this is R2. We showed that we can replace R1 and R2 by an equivalent parallel resistor with this expression here for two resistors. RP ..one over RP equals one over R1 plus one over R2. So that's how you calculate the equivalent resistance for two parallel resistors. Now you can ask and it's a good thing to ask what if there's more resistors? What if there's more resistors in parallel here? What if I have R3 and R4 and RN all connected up here. What happens in this expression? So like we did before we had a current here and we know that current comes back here. The first current split, some current goes down through R1, some current goes through R2 and if we had more resistors some goes down through R3, and some goes through RN. So the current basically is coming down here and splitting amongst all the resistors. Now all the resistors share the same all the resistors share the same voltage. So that's just V, let's label V. That's just V, they all share the same V and they all have a different current. Assuming they all have a different resistance value. So we do exactly the same analysis that we did before. Which was, we know that I here here has to be the sum, there's the summation symbol of all the I's. I1, plus I2, plus I3 plus IN, that's as many as we have. So we know that's true. And we also know that the current, we also know that the current in each individual resistor IN is equal to one over that resistor times V, and V is the same for every one of them. So now we substitute this equation into here for I For I. We get the big I, the overall I is equal to voltage times, it's gotta be a big expression, one over R1 plus one over R2 plus one over R3, plus as many resistors as we have One over RN, like that. And we do the same thing as we did before which was we say that this expression here is equivalent to one parallel resistor we're going to make the equal to one parallel resistor. So this whole guy here is gonna become one over RP That gives us a way to simplify any number of resistors down to a single parallel resistor and I'll write that over here. So for N resistors, multiple resistors one over RP the equivalent parallel resistor is equal to the same thing. One over R1 plus one over R2 plus dot, dot, dot, one over RN. So this tells you how to simplify any number of parallel resistors down to one equivalent parallel resistor.