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## High school physics

### Course: High school physics > Unit 12

Lesson 3: Series and parallel resistors# Parallel resistors (part 3)

When two resistors are in parallel, the equivalent resistance is the product of the two resistors divided by their sum. When both resistors are the same value, the equivalent parallel resistance is exactly half of the original resistance. Created by Willy McAllister.

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- So, we can see from this video (starting at around6:53) that two resistors in parallel with the same resistance will produce an equivalent resistance that is half of the original resistance. What happens, though, if we keep increasing the number of resistors? Let's say, for example, we have three resistors in parallel with the same resistance. Will the equivalent resistance be one-third of the original resistance?

So what I'm wondering can be summarized like this:

R = resistance of all resistors in circuit

n = number of resistors

Rp = equivalent resistance

So, will Rp = R^n/nR?(10 votes)- With Req = R/N, this means that one could find an integer fraction of a specific resistance given said integer amount of resistors having said specific resistance.

This means if you want a 40 Ohm resistor, you can wire three 120 Ohm resistors in parallel.

40 Ohm = 120 Ohm/3 Resistors(1 vote)

- If you use the Rp=R1*R2/R1+R2 equation presented here how do you solve it if there are more than 2 parallel resistors? I tried Rp=R1*R2*R3/R1+R2+R3 and I didn't come up with the right answer.(4 votes)
- Hello Phelps,

Sorry, the Rp=R1*R2/R1+R2 is only applicable when you have two resistors. When you have more than two you should use 1/Rp = 1/R1 + 1/R2 + .... + 1/Rn

Regards,

APD(13 votes)

- Hello, at6:09you say that (1) the equivalent resistance of a bunch of parallel resistors will always be smaller than the smallest resistance in the bunch. You argument this (2) "because you have two current paths that allow current to go two different ways". I can't really get my head around that, why does (2) imply (1)? How does the current having two different paths (or more) determine a smaller equivalent resistance than the smallest resistance of one of the paths?(6 votes)
- Can I use the part3 equation when I have more than two resistors(3 votes)
- What happened to Parallel resistors (part 2)?(3 votes)
- You are following the Physics AP1 sequence. For some reason the second video on parallel resistors is omitted from the list. I reported this as an issue in the KA Clarification system.

The video appears in the electrical engineering sequence: https://www.khanacademy.org/science/electrical-engineering/ee-circuit-analysis-topic/ee-resistor-circuits/v/ee-parallel-resistors-2(5 votes)

- why is that when the current was split, it became doubled, with a much higher resistor? how does that happened?(3 votes)
- Is their a part 2 or is it mislabeled part 3?(2 votes)
- would there be a difference in the answer if u use the 1/RP FORMULA instead of the RP= R1R2/R1+R2 formula(2 votes)
- i believe at4:25it answers your question(1 vote)

- i do not understand what did you meant by "Because you have two current paths that allow current to go two different ways, the effective resistance is always smaller than the smallest original path, because there's a way for current to go around another way" could you explain more the reason ?(2 votes)
- When you have two resistors in parallel they share the same voltage, and have their own individual currents. You find the current using Ohm's Law, I1 = V/R1, and I2 = V/R2.

We can ask the analytic question, "What is the 'equivalent' resistance?" or asking another way, "If you replace the pair of resistors with a single resistor, what resistance would you chose for Rp such that the total current would be the same? That is, I(Rp) = I1 + I2.

You were asked to derive or memorize an equation to find the precise value of Rp...

Rp = (R1*R2)/(R1+R2)

It is also illuminating also ask a qualitative question, "Is Rp bigger, smaller, or in between R1 and R2?, and why is that?"

If you look at the math it tells you Rp is always smaller than the two original resistors. Can you see why? The current in Rp is Ip = the sum of the currents in R1 and R2. That means Ip is always bigger than both I1 and I2. When you put Ip into Ohm's Law along with the original V, you see that Rp has to be smaller than R1 and R2 to get Ohm's Law to come out true.

Rp = V/Ip = V/(I1 + I2)

That's what the math says. What is the physical reason?

I think of electric current like water in a creek. The tilt of the creek provides the pressure to make water flow, corresponding to the voltage value V. Suppose you have a narrow creek with three stones blocking the water, named Stone1, MiddleStone, and Stone2. If you lift up the stone on the left you will get a current flow proportional to the width of the opening. That's R1 and the flow is I1. If you replace Stone1 and lift up Stone2 on the far right you will see I2 flowing proportional to the width of the opening left by Stone2.

If you lift up both Stone1 and Stone2 there are two openings and total flow is greater than both I1 and I2. That's because the two gaps are open and there is more space for water to flow. There's no arrangement of stones that will make less total water flow.

Resistors in parallel are exactly the same. If you provide more pathways for electric current to flow, the equivalent resistance is always lower than either original resistor.(1 vote)

- Why does this formula for Total Parallel Resistance only work for two resistors?

Rp = (R1*R2)/(R1+R2)

Does it work with three resistors too? Like,

Rp = (R1*R2*R3)/ (R1+R2+R3)(1 vote)- The formula for two resistors in parallel is a
**special case**of the more general equation (the one with all the reciprocal resistor terms). To see the algebra for where this special case comes from, go to the Special Case section of this article. https://www.khanacademy.org/science/electrical-engineering/ee-circuit-analysis-topic/ee-resistor-circuits/a/ee-parallel-resistors

or here,

http://spinningnumbers.org/a/parallel-resistors.html#special-case---2-resistors-in-parallel

The three-resistor equation you suggest cannot be derived from the general equation. It would be cool if it could be, but it's not possible.(1 vote)

## Video transcript

- [Voiceover] In this video
we're gonna talk even some more about parallel resistors. Parallel resistors are resistors that are connected end to end, and share the same nodes. Here's R one and R two,
they share the same nodes. That one and that one. And that means they
share the same voltage. And we worked out an expression for how to replace that
with a single resistor. R parallel, and we found
that one over R parallel is one over R one plus one over R two. So in this video I'm gonna actually start working with this
expression a little bit more. And we'll just change it around
a little bit to an easier, an easy way to remember it, and then we're gonna do a special case, where R one and R two are the same value and we're gonna see what happens. So right now I wanna do
just a little bit of algebra on that expression, one over R p is the same as one over R one plus one over R two. What I wanna end up with
here is an expression that on this side says
R p equals something, and on this side I just
want one expression, not two fractions. So we're gonna go about that by combining these two fractions first. So the least common denominator here, LCD, equals R one times R two, and I'm gonna convert both these to convert this to that proper denominator I have to multiply it by R two over R two, so we'll do it. We'll do all the steps. We'll multiply it by R two over R two. This expression we have to
multiply it by R one over R one. And this equals, of course, one over R p. Continuing on, one over R p equals R two over R one R two plus R one over R one R two, and now I can combine them together. Let's move up here. One over R p equals, we'll keep everything
in numerical order, R one plus R two over R one R two. And now I'm gonna take the reciprocal of both sides of the expression so I get an expression in R p, R p equals, and just flip
over this expression, R one R two over R one plus R two. That is a way you can remember how
to combine parallel resistors. The parallel equivalent resistor, R p, is the product of the two
resistors over the sum. So it's the product over the sum. That's how I remember it. Now this arithmetic, this expression, is exactly the same as the
original one that we had, these are the same, and it's just a question of which one do you want to remember, which one's easier to remember and which one's easiest to calculate. I like to remember this one here. Let's do a quick example using it. I'll move the screen up a little bit. Let's leave that there so we can see it. Let's say we have two parallel resistors. And we'll say the first one is 1,000 ohms, and the second one is 4,000 ohms. And the question is, what
is the parallel combination of those things. How can I replace these with one resistor so the same current flows. And we'll use this expression here. So R p equals a product which is 1,000 times 4,000 divided by the sum 1,000 plus 4,000 and that equals, oh my goodness there's a lot of zeroes here. Four and six zeroes, one, two, three, four, five, six over, that's easy, 5,000. All right? And let's say, let's take out three
zeroes out of this one. Let's knock off three zeroes here, and three zeroes here. And I have 4,000 divided by five and that equals 800 ohms. So that is this and that's how we use this expression. Now just something to notice here, notice that R parallel, the
equivalent parallel resistor, is smaller than both of these. And that happens every time,
that happens every time. The parallel resistance is smaller than the smallest resistor. So here it was 1,000, it's
gonna be smaller than that. And that's a property
of parallel resistors. Because you have two current paths that allow current to
go two different ways, the effective resistance is always smaller than the smallest original path, because there's a way for
current to go around another way. So this is now the expression
for two parallel resistors, and that's a good one to remember. Now I'm gonna show you one special case, and we'll do this in this color. What if R one equals R two? What is R p? And for this special case,
we use the same expression, we say R p, we say R p is the product, we'll just use R because R is, it's the same value. R times R over R plus R, and that's multiplied. And so that is R squared over two R, and one of these Rs cancels, so let's cancel that R and that squared, and we end up with R over two. So for the special case, if R one equals R two, then R p equals R over two. It's just half, and
that should make sense. If we do something like this, if we draw two resistors in parallel, like this. and we say this is 300 ohms and this is 300 ohms, that means the effective, the effective parallel resistor, 150 ohms. And that's got a very
pleasing symmetry to it because these resistors are the same, they have the same voltage, they're going to have the same current, basically twice as much current is gonna flow in this
circuit as would flow if there was just one of these guys, so that's where the
divide by two comes from. So for two resistors in parallel, if the resistors are the same value, the effective parallel
resistance is just half.