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# Example: Analyzing a more complex resistor circuit

A hairy resistance problem. Created by Sal Khan.

## Want to join the conversation?

• Can we calculate the current in each of the resistors ? • its quite easy infact if you just remember these two basic concepts....
1). current is same across all resistors in series..
2). to find current in a particular resistance (r1) connected in parallel to others (equivalent resistance = R.Eq )...

current through r1 = * [ eq. resistance of all (connected in parallel )except r1 ] X {total current flowing through " R.Eq " ] / [ R.Eq ] *

P.S. - derivation through V = IR or krihoff voltage law..
`code`
``if (x < 0) {    return;}``
(1 vote)
• Why in a real life application would we use multiple resistors in both series and parallel configurations together? • How can we calculate the internal resistance inside a battery? • what i think here is that when the circuit is in parallel the equivalent resistance is less than the least resistance in the parallel circuit .. and when in series... the equi. resistance is greater than the largest resistance in the series circuit.. am i right?? • You are absolutely correct. The series circuit, I need not answer. The parallel circuit, I will prove it intuitively. A wire is also a resistence, and a good example. When we see the electrons bumping into each other and into the atomic nuclei inside the wire, we call that property resistence. Kind of like viscosity in fluids. So if a wire gets longer, it has more chances to bump into something. Also, if a wire gets thicker, an electron will have more chance to AVOID collisions. So a thicker wire means less resistence. In analogy, if one has few wires in parallel, it will have same effects as making one wire thicker. So multiple wires would mean less resistence than the least of one
• can u please make it clear how to determine the direction of the current flow then it would be helpful • The circuit in the video was 2D and therefore could be analyzed step by step. However what would happen if resistors were connected in a 3D circuit. For example, a circuit is made in the shape of a cube where there is a resistor of 1 ohm on each edge of the cube. In this case, what would be the total resistance. • what is e.m.f and terminal voltage of a cell?how are they related to internal resistance?
(1 vote) • EMF is electromotive force, and it's another way to refer to voltage.

Terminal voltage is the voltage that the battery can deliver into a working circuit. If you have a battery, you might think of it as say a 9 volt battery, but that 9v is what it can deliver when it is not having to put out much current - in other words, when the resistance in the rest of the circuit is very high.

When you connect a batter to a circuit where the resistance is low, then you have a situation where the battery is trying to put out a lot of current. If you want to calculate how much current there will actually be, now you can't ignore the resistance that is internal to the battery.

To make the math easy, let's say we have a 10v battery connected to a 10k ohm resistor. The current is going to be very small, only 1 mA, but technically that is only an approximation, because the battery has resistance, too. So, say the battery has internal resistance of 1 ohm. Now the real resistance of the circuit is 10.001k ohm, and you can see, that's not going to make any meaningful difference when you calculate the current. It's still really, really close to 1 mA. The voltage drop throughout the circuit is going to occur almost entirely across the 10k resistor. The battery terminal voltage therefore will be really close to 10 V.

Now, connect the battery instead to a resistor that is only 10 ohm. If you had a batter with no internal resistance, you have 10 V/ 10 ohm = 1 amp. But now you can see that the internal resistance matters. The real resistance around the circuit is not 10, but 11. So the current is closer to 0.9A (I am rounding 10/11) And now you can see that the voltage drop that occurs across the resistor will be 0.9A x 10 ohm = 9V, and that's the terminal voltage of the battery. Where is the other 1 V we expected to see, since the voltage all the way around the loop is supposed to sum to 0 (+10 from battery, -9 lost across resistor, 1 more volt needs to be accounted for). That voltage drop occurs across the internal resistance of the battery itself.
• how do you solve for the current through each individual resistor in the circuit?
(1 vote) • Is the conventional symbol for the resistor a zigzagged line or a kinda like rectangle cause I`m confused.

I drew that on a test and failed(HORRIBLY)

ANY HELP WOULD BE APPRECIATED,THANKS!
(1 vote) • May I know what is the difference between rheostat and resistor?
And where exactly is rheostat used in real life?
(1 vote) ## Video transcript

Let's see if we can apply what we've learned to a particularly hairy problem that I have constructed. So let me see how I can construct this. So let's say in parallel, I have this resistor up here. And I try to make it so the numbers work out reasonably neat. That is 4 ohms. Then I have another resistor right here. That is 8 ohms. Then I have another resistor right here. That is 16 ohms. And then, I have another resistor here, that's ohms. Actually, I'm now making it up on the fly. I think the numbers might work out OK. 16 ohms. And let's say that now here in series, I have a resistor that is 1 ohm, and then in parallel to this whole thing-- now you can see how hairy it's getting-- I have a resistor that is 3 ohms. And let's say I have a resistor here. Let's just make it simple: 1 ohm. And just to make the numbers reasonably easy-- I am doing this on the fly now-- that's the positive terminal, negative terminal. Let's say that the voltage difference is 20 volts. So what I want us to do is, figure out what is the current flowing through the wire at that point? Obviously, that's going to be different than the current at that point, that point, that point, that point, all of these different points, but it's going to be the same as the current flowing at this point. So what is I? So the easiest way to do this is try to figure out the equivalent resistance. Because once we know the equivalent resistance of this big hairball, then we can just use Ohm's law and be done. So first of all, let's just start at, I could argue, the simplest part. Let's see if we could figure out the equivalent resistance of these four resistors in parallel. Well, we know that that resistance is going to be equal to 1/4 plus 1/8 plus 1/16 plus 1/16. So that resistance-- and now it's just adding fractions-- over 16. 1/4 is 4/16 plus 2/16 plus 1 plus 1, so 1/R is equal to 4 plus 2 is equal to 8/16-- the numbers are working out-- is equal to 1/2, so that equivalent resistance is 2. So that, quickly, we just said, well, all of these resistors combined is equal to 2 ohms. So let me erase that and simplify our drawing. Simplify it. So that whole thing could now be simplified as 2 ohms. I lost some wire here. I want to make sure that circuit can still flow. So that easily, I turned that big, hairy mess into something that is a lot less hairy. Well, what is the equivalent resistance of this resistor and this resistor? Well, they're in series, and series resistors, they just add up together, right? So the combined resistance of this 2-ohm resistor and this 1-ohm resistor is just a 3-ohm resistor. So let's erase and simplify. So then we get that combined resistor, right? We had the 2-ohm that we had simplified and then we had a 1-ohm. So we had a 2-ohm and a 1-ohm in series, so those simplify to 3 ohms. Well, now this is getting really simple. So what do these two resistors simplify to? Well, 1 over their combined resistance is equal to 1/3 plus 1/3. It equals what? 2/3. 1/R is equal to 2/3, so R is equal to 3/2, or we could say 1.5, right? So let's erase that and simplify our drawing. So this whole mess, the 3-ohm resistor in parallel with the other 3-ohm resistor is equal to one resistor with a 1.5 resistance. And actually, this is actually a good point to give you a little intuition, right? Because even though these are 3-ohm resistors, we have two of them, so you're kind of increasing the pipe that the electrons can go in by a factor of two, right? So it's actually decreasing the resistance. It's giving more avenues for the electrons to go through. Actually, they're going to be going in that direction. And that's why the combined resistance of both of these in parallel is actually half of either one of these resistances. I encourage you to think about that some more to give you some intuition of what's actually going on with the electrons, although I'll do a whole video on resistivity. OK so we said those two resistors combined-- I want to delete all of that. Those two resistors combined equal to a 1.5-ohm resistor. That's 1.5 ohms. And now all we're left with is two resistors in parallel, so the whole circuit becomes this, which is the very basic one. This is a resistor: 1.5 ohms, 1 ohm in series. Did I say parallel just now? No, they're in series. 1.5 plus 1, that's 2.5 ohms. The voltage is 20 volts across them. So what is the current? Ohm's law. V is equal to IR. Voltage is 20 is equal to current times our equivalent resistance times 2.5 ohms. Or another way to write 2.5 five is 5/2, right? So 20 is equal to I times 5/2. Or I is equal to 2/5 times 20, and what is that? 2/5 is equal to I is equal to 8. 8 amperes. That was not so bad, I don't think. Although when you saw it initially, it probably looked extremely intimidating. Anyway, if you understood that, you can actually solve fairly complicated circuit problems. I will see you in future videos.