okay so this video is all about Kirchhoff's laws there are two of kerkoff's laws the junction rule in the loop rule the junction rule is also sometimes called the current rule tells us that the sum of the currents entering any Junction must equal the sum of the currents leaving that Junction loop rule is sometimes also called the voltage rule and that one we're basically going to travel around in one loop in the circuit and the sum of all of the potential differences or voltages across each thing that we come across like resistors and batteries has to be zero so we're going to go to this example problem here and we need to define a few things before we can do anything with this problem and the things that we're going to define are what the heck a junction is and there's another term called a branch so a junction is any place where two or more wires come together we have two junctions on this chart here on this circuit diagram and I'm going to Marcos with dots so here's our first Junction and there's our second junction okay next definition is for a branch a branch is anything that connects two junctions branch is anything that connects two junctions so this diagram has three branches we have this top branch it goes like that we have this middle branch and we have this bottom branch this circuit is different than anything we've looked at before because there are two sets of batteries separate from each other you'll see the 24 volt battery here the 12 volt battery here and they're separate from each other and there's a resistor in between them and all kinds of other good stuff Kirchhoff's laws can be used for any circuit but they must be used for circuits where there are multiple batteries like this that are separated from each other four circuits like this we're always going to start off with the current rule otherwise known as the junction rule and then we'll do the loop rule second any circuit like this you're going to use both rules to come up with the answer and the goal is always to find the current in each branch or sometimes you may be asked to find like the current through one of the resistors you can't do a combo circuit like what we did before to find that so one general rule the number of branches you have is equal to the number of currents you have so we have three branches here which means we all have three currents every branch gets its own current we will kind of randomly decide which direction those currents are going in and in the end we'll get to find out if our random decision was correct or not you'll see what I mean that sounds a little weird right now so starting off I'm going to choose those three I'm going to call them i1 i2 and i3 and I'm picking a random direction for that so I'm going to say that I 1 goes in this top branch here and it's going to go upward like that so I put an arrow there and I write I 1 and I can even put that on the other side just to remind myself that it's that same current over on the other side I'm going to say I too goes through the middle and I'm going to say that it goes to the right again I'm randomly choosing these directions there's not really a whole lot of rhyme or reason to it and I'm going to say that I three which is in that bottom portion so that it goes downward here which means that over through here would be going upward so that's the first task is um drawing those currents up there and then we're going to go ahead and apply that Junction rule next and so let's look at this first Junction here the one on the left hand side the sign rule is that whatever enters the junction whatever current is pointing towards the junction is positive whatever is pointing away from the junction is negative so i1 is pointing away from the junction so I'm going to write minus i1 negative i1 i2 is pointing away from the junction minus i2 I three is pointing toward the junction plus three equals zero again pointing toward is positive pointing away as negative okay so we're going to look at the other Junction now and with the other Junction I 1 is pointing toward it I 2 is pointing toward it but I 3 is pointing away if you look at those two equations you'll notice that they tell you the exact same thing because if you take the top equation and multiply it by negative one you get the bottom equation so when you have a circuit that looks like this one particularly where there's kind of two chunks there's the top of the bottom as opposed to there being three we'll do a circuit where there's three of them when you have one where there's only two you only need one of the junction rule equations because the other equation doesn't tell you any new information so that's Junction rule now we're going to do loop rule we need the number of equations as we have variables so I have one equation already I have three variables so I'm going to need three equations total I could do three different loop rule equations because I could do the small loop up top the small loop in the bottom and a big loop on the outside but I only need two of those I don't need all three so I am going to do the top small loop and the bottom small loop okay I the thing that I need to decide is if I'm going to go around those loops clockwise or counterclockwise it's just a random decision again it doesn't matter which way you choose but indicate it on your diagram with a little swirl like like I'm about to do so I'm going to go through these loops clockwise and so I indicate the fact that I'm going clockwise with that little swirly okay a couple rules here with batteries if you go from negative to positive if you land on positive you end on positive it's a positive voltage again with batteries if you're going from negative to positive you land on positive it's the last thing you go through it's a positive voltage and then vice versa if you're going positive to negative you end on negative so it's a negative voltage with resistors if you go through the loop the same direction that current is traveling through that loop you have a negative voltage again if you go through the resistor you're going through that loop through the resistor the same direction that current travels through that resistor it is a negative voltage and then the opposite is also true if you go through a resistor in opposite direction the current goes through the resistor it is a positive voltage if you go through a resistor opposite the direction current is traveling through that resistor it is a positive voltage so I'm going to go ahead and do a loop rule I'm going to look at that top loop up here you can start anywhere in the loop as you're writing the loop rule equation I'm going to start at this Junction here this Junction on the left here and again I said I was going clockwise so I'm going to start traveling upward the first thing I encounter is a battery really a set of two batteries but I treat it as one I'm going negative to positive so it's a positive voltage and that is 24 volts and then I go through that resistor this resistor right on top here now I'm traveling around in my loop so I'd be going from the resistor left to right and the current would be going through the resistor left to right so it is minus or a negative voltage and I'm just going to put voltage of the 2 ohm resistor I'll do more with that later and then the same thing goes for this 4 ohm resistor I'm and I'm traveling through my loop I'm going downward through that resistor and the current is traveling downward through that resistor so minus V of the 4 ohm resistor and then to finish my loop I travel through this 3 ohm resistor this time I'm going right to left but the current goes left to right so I'm going in the opposite direction the current travels through that resistor and so I say plus V of the 3 ohm I got back to my starting point I travel through this I'm back at where I started so now I write equals zero ok so now we want an equation that has currents in it we don't want these random V's that are here but we know V equals IR right so I'm going to use that equation V equals IR to write a new equation now I'm going to leave off units you know that that's not common for me to do but it's going to make life a lot easier if I just leave off units in this equation so 24 I'm just going to write 24 not 24 volts we don't do anything to that one V equals IR for this 2 ohm resistor the I've it travels through that is i1 and r is 2 ohms so minus 2 i1 for the 4 ohm resistor it's minus 4 i1 for the three ohm resister it is plus 3 i2 because i2 is now the current that travels through that 3 ohm resistor and then equals 0 at the end to get this equation one more step to do with this equation is to combine like terms minus 2 I 1 minus 4i 1 gives me minus 6i 1 so I can write 24 minus 6i 1 plus 3i 2 equals 0 that's my second equation I'm going to quickly go through the third equation third equation is going to be for my bottom loop here I'm going to start in the top left corner at the same Junction I started at before when I use a different color for this one the first thing that I encounter is I go clockwise around this is this 3 ohm resistor and I'm traveling through it the same direction as current goes through it so I have minus and I'm just going to go ahead and write 3 i2 this time I'm kind of skipping over that first step I did 3 is the resistance I 2 is was traveling through it and then I get to the next Junction I go downward because I'm going in this bottom loop here and so I encounter another resistor it's a 1 ohm resistor and I 3 is traveling through it I'm going through the resistor the same direction this current goes through it so minus 1 I 3 and then also minus 5i 3 that's my next resistor I get two so I don't run out of room and the last thing I come to is this battery