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## Ohm's law and circuits with resistors

# Kirchhoff's current law

## Video transcript

- [Voiceover] Up to now we've talked about resistors and capacitors
and other components, and we've connected them up and learned about Ohm's
law, for resistors, and we've also learned some
things about series resistors, like we show here. The idea of Kirchhoff's Laws, these are basically common sense laws that we can derive from
looking at simple circuits, and in this video we're gonna work out Kirchhoff's Current Law. Let's take a look at these
series resistors here. There's a connection point right there, and that's called a node, a junction. And one of the things we know is that when we put current through this, let's say we put a current through here. And we know that current
is flowing charge, so we know that the charge
does not collect anywhere. So that means it comes
out of this resistor and flows into the node, and that goes across and
it comes out on this side, all the current that comes in comes out. That's something we know, that's the conservation of charge, and we know that the charge
does not pile up anywhere. We'll call this current i1. And we'll call this current i2. And we know, we can just write
right away, i1 equals i2. That seems pretty clear from
our argument about charge. Now let me add something else here, we'll add another resistor to our node. Like that. And this now, there's gonna be
some current going this way. Let's call that i3. And now this doesn't work anymore, this i1 and i2 are not
necessarily the same. But what we do know is
any current that goes in has to come out of this node. So we can say that i1 equals i2 plus i3. That seems pretty reasonable. And in general, what we have here isn't, if we take all the current flowing in, it equals all the current flowing out. And that's Kirchhoff's Current Law. That's a one way to say it, in mathematical notation,
we would say i in, the summation of currents going in, this is the summation sign, is the summation of i out. That's one expression of
Kirchhoff's Current Law. So now I want to generalize
this a little bit. Let's say we have a node, and we have some wires going into it, here's some wires connecting up to a node. And there's current going into each one. I'm gonna define the current arrows, this looks a little odd,
but it's okay to do. All going in. And what Kirchhoff's Current Law says is that the sum of the currents going into that node has to be equal to zero. Let's work out how that works. Let's say this is one
amp, and this is one amp, and this is one amp. And the question is, what is this one? What's that current there? If I use my Kirchhoff's
Current Law, express this way, it says that one plus one plus one plus i, whatever this i
here, has to equal zero. And what that says is
that i equals minus three. So that says, minus three amps flowing in is the same exact thing as
plus three amps flowing out. So one amp, one amp, one amp comes in, three amperes flows out. Another way we could do it, equally valid, this is just three ways to
say exactly the same thing. I have a bunch of wires going
to a junction, like this. And this time I define
the currents going out, let's say I define them all going out. And this same thing works. The sum of the currents, this time going out, I'll go back over here, I'll write in, all the currents going in. That has to equal zero as well. And you can do the same exercise, if I make all these one amp, and ask, what is this
one here, what is i here, outgoing current, it's one plus one plus one plus one, those are the four that I know, and those are the ones going out, so what's the last one going
out, it has to equal zero. The last one has to be
minus four equals zero. So this is a current
of minus four amperes. So that's the idea of
Kirchhoff's Current Law. It's basically, we've reasoned through
it from first principles, because everything that comes in has to leave by some route, and when we've talked about it that way, we ended up with this expression for Kirchhoff's Current Law. And we can come up with a slightly smaller mathematical expression, if we say, let's define all the
currents to be pointing in. Some of them may turn out to be negative, but then that's another way to write Kirchhoff's Current Law. And in the same way, if we define all the currents going out, and you actually have
your choice of any of these three any time
you want to use these. If we define them all going out. This is Kirchhoff's Current Law, and we'll use this all the time
when we do circuit analysis.