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# Kirchhoff's current law

## Video transcript

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 also we've 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 going to we're going to work out kirchoff'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 then it 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 I 1 equals i2 that's that seems pretty clear from our argument about about charge now let me add something else here we'll add another resistor to our node like that and this now there's going to be some current going this way let's call that i3 and now this doesn't work anymore this I 1 and I 2 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 I 1 equals I 2 plus I 3 that seems pretty reasonable and in general what we have here is it if we take all the current flowing in it equals all the current flowing out and that's 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 summation sign is the summation of I out that's one expression of kirchoff's current law so now I want to generalize I want to generalize this a little bit let's let's say we have a node and we have some wires going into it here's some wires connected up to a node and there's current going into each one I'm going to define the current arrows this this is 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 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 expressed 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 1 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 the same thing works some 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 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 kirchoff's current law it's basically we reasoned through it from first principles because everything that comes in has to leave by some route and when we talked about it that way we ended up with this expression for kirchoff'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 defined them all going out this is kirchoff's current law and we'll use this all the time when we do circuit analysis
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