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# Example: Analyzing a more complex resistor circuit

A hairy resistance problem. Created by Sal Khan.

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.