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# Kirchhoff's current law

Kirchhoff's Current Law says the currents flowing into a node must add up to zero. Created by Willy McAllister.

## Want to join the conversation?

• what is a node??
(10 votes)
• its a point where the current will split / join.

often it is the connection between two or more conductors
(46 votes)
• Isn't current just the rate at which the electric charge flows? So when it crosses the resistor, wouldn't the charge flow slower? But it doesn't necessarily mean that charge isn't conserved because it's just the rate at which it flows that has decreased, no?
Thanks very much
(7 votes)
• The best way to see that Kirchhoff must be right is to think about what it would mean if he was wrong. If you have more current flowing into a point than out of it (for example if the point is somewhere the charge starts to flow slower), then the charges would start building up in in front of the point. Eventually there would be so much charge packed together that lightning would start shooting out of the circuit (assuming you could keep the current going that long). The fact that we don't see lighting shooting out of resistors indicates that the current flows at the same rate before and after the resistor.

Now why is it so? The reason is that electrons repel each other. The electrons do get slowed down by the resistor. In fact, that is what a resistor is: a "brake" slowing down electrons by having them collide with the atoms it's made of, turning some of their kinetic energy into heat. But when an electron slows down in the resistor, the electron right behind it bumps into it, and since electrons repel, it also gets slowed down. This electron then slows down the electron behind it again, and so it goes around the entire circuit. You can think of the electrons in the circuit as marbles packed tightly in a tube. If you slow down one of them, you slow down all of them.

So yes, a resistor makes the charge flow slower, but it makes the charge flow slower in the entire circuit it is part of, not just after the resistor.
(25 votes)
• is it necessary that in every condition this law is applied ? like if i change the physical conditions like temperature does this law will still be applied??
(9 votes)
• Hello DeepShankarPratap,

Kirchhoff's current and voltage laws are ALWAYS true.

On a related note, Ohm's law is also true. However, many physical systems have resistance / impedance that changes with temperature.

Regards,

APD
(6 votes)
• Ok I am confused around , where that one current equals -3. I get how current in general can be negative, but I am confused on how the current going in is equal to zero and if the current going on/out is always equal to zero.
(6 votes)
• Another way of putting it is all the current going in is equal to all the current going out. So if you have 3A of current going into a node, then there must be 3A of current going out of the node. The reason Sal says that i = -3A is because he defined the direction of current i to be going in, so negative would mean the direction is going in the opposite direction as it was defined.
(15 votes)
• At , does the expression for Kirchhoff's current law equal zero?
(5 votes)
• Iin = Iout, here in the two examples Iin = 0 then Iout = 0. In the other example, Iout = 0, then Iin =0. So the general expression for kirchoff's current law is Iin = Iout (current in, current out)
(1 vote)
• The Kirchhoff's current law only works for Direct Current (DC), right?
(8 votes)
• How to find the current between two nodes ?
(3 votes)
• Hello Pranjal,

It depends on the circuit configuration. Series circuits are easy - the current is the same everywhere in the circuit. Things get complicated from here when we start to add parallel plus series branches. The tools of choice are mesh analysis, nodal analysis, and superposition.

Please look for these tools here on the Khan Academy EE section.

Regards,

APD
(4 votes)
• why cant we take i1+i3=i2
(3 votes)
• Sorry for the late response.

If you switch a number with the plus side to the other side it will switch to negative so you can do: i1 - i3 = i2, but you can't do i1 + i3 = i2. That could also depend: maybe you switch the names around then you can do that otherwise it is ILLEGAL in physics.
(4 votes)
• Why is the sum of the current equal to 0?
(3 votes)
• because charge can't be created or destroyed, so however much charge goes into that junction has to also come out
(4 votes)
• What is Mutual inductance?
(3 votes)
• Hi MD,

Mutual inductance describes the operation of a transformer.

Here we have two coils of wire. A changing magnetic field in one coil causes a voltage change in the other.

Regards,

APD
(3 votes)

## 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.