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# Magnetic force between two currents going in the same direction

## Video transcript

we've now learned that a current or a stream of moving charges can be affected by a magnetic field and we've also learned that it can induce a magnetic field so that begs the question what is the effect of one current carrying wire on another current carrying wire so let's do that let's let's draw my first current carrying wire in green that's the first current carrying wire and let's say that the current is it's in magenta and we'll call this current one current one and then I have another current carrying wire not too far away not too far away I have another current carrying wire and I will call that current i2 now what else do we need to figure out oh well let's let me just tell you let's say that they are a radius of our apart and I say radius because we learned in the last video that the magnetic field created by a current carrying wire is kind of a you know they're kind of these circular cylinders around the wire so let's say the distance from I from this wire to that wire is our that distance is R and so my question to you is well first just before we break into the math what's going to happen if I well we don't know the magnitudes of the currents or anything just yet but what's going to happen what will be the net effect on let's say this wire let's say for some reason this wire is fixed or well we could say they're floating in space but let's just say let's just focus on Y or two for now this is Y or - this is Y or one what's going to happen to Y or two well let's think about it Y or one the current in it is going to generate a magnetic field and what's the shape of that magnetic field going to look like well we could take our right hand do that right hand wraparound rule it's a little different than the cross-product rule although it's kind of a byproduct so that's my right hand and I'm wrapping it around right so if I put point my thumb in the direction of the current so that's the direction of the current just like I did then the magnetic field goes in the direction of my fingers right so they're going to go around around this this this wire and so if I were to just draw the magnetic field where it intersects with this screen on the right hand side it will go into the screen so we'll just see the rear ends of the magnetic field line and I'll draw it in the same color as the current so you know that's been created by i1 so i1 and it you know but its effect keeps going out to infinity although it gets much weaker as we learned it's proportional with inversely proportional with r but this is the field of i1 i could draw these i don't want to crowd my page up too much and then on this side of i1 what happens well on this side you can see the fingers come back around so it pops out when it intersects with your video monitor so on this side the vectors this is the the top of an arrow coming coming out at you alright fair enough so i won by going in this direction is generating a magnetic field that at least where I 2 is concerned that magnetic field is going into into the page right so what was our what was our formula and this was all this all came from the for the first form we learned about the effect of a magnetic field on moving charge what was a formula of the net magnetic force on a current carrying wire oh he it was the force I'll do it in blue the force it's a vector has a magnitude and direction is equal to the current well in this case we want to know the force on on this current on current 2 right caused by this magnetic field by magnetic field 1 so it will be equal to I 2 the magnitude of this current times L where L is because you can't just say oh well what is the effect on this wire you have to know how much wires to consideration so let's say we have a length of wire and then of course if we know a length of wire and we knew its mass and we do the force on it we could figure out its acceleration in some direction so let's say that this this distance is L and it's a vector L goes in the same direction as the current that's just the convention we're using it makes things simple so that's L so the force on this on this on this wire or at least the length L of this wire is going to be equal to current 2 times L we could call that even L l2 just so that you know that it deals with wire 2 and that's a that's a vector quantity I can make it a full arrow doesn't matter it's just a notation I've seen professors do it either way I've seen it written either way as well cross the magnetic field that it's in well what's the magnetic field that it's in the magnetic field I'll do it in magenta because it's the magnetic field created by current one so it's magnetic field magnetic field one which is this magnetic field right so before like going into the math let's just figure out what's going on what what direction is this net force going to be in so here we just use we say well the current is a scalar so that's not going to affect the direction what's the direction of L 2 this is L 2 let me I didn't label it L 2 on the diagram what's the direction of L 2 well it's up and then the direction of B 1 the magnetic field created by current one is going into the page here so here we just do the standard cross-product where let me see if I can pull this off this is actually an easy one to draw so I put my index finger the index finger in the direction of L 2 and then I put my middle finger in the direction of the field so my middle finger is going to point straight down into this page my other fingers just do what they would naturally do and then my thumb would go in the direction of the net force this is just the cross-product you'll see teachers teach the cross-product to other ways where they tell you to put your thumb in the direction of the field this and that your palm was in it those are all valid they're just different variations the same thing I find this one easier to remember cuz when I take the cross-product index finger is the first term in the cross-product fingers the second term in the cross product thumb is the direction of the cross product right so anyway this is the direction of L to the magnetic field we already know goes into the page so my my index finger is going in oh it's my middle finger is going into the page and my thumb is in the direction of the force on the magnetic field so that's the direction of the force so there you have it if this current is moving in this direction and it's field is we know from this wraparound rule that pops out here and it goes in here the effect that it has on this other wire is that's that where the current is going in the same direction is that it will be attractive so the net force the net force is going in that direction we can say the force of from one on to that's just my convention maybe these other people written it force you know given to two by one but that's a force on one I'm sorry that's the force given by one to two that's how I'm writing it now what's going to be the force on current one from I to what's going to be the current what's going to be the force there well it's going to be the same thing right let me draw I twos magnetic field you do the wraparound rule it's going to look the same so I - sure on this side its field is going to be going into the page but what's ITU's feel going to be doing here it's going to be popping out all right I just did the wraparound take this wraparound wrap it around that wire so that's the field of i2 so then we can write down that the force and let's take I don't know this is some distance let's call that l1 so the force from current - on wire one of length l1 from here to here is equal to current one current one times l1 which is a vector cross the magnetic field created by current - the magnetic field created by current - and so we can do the same cross-product here put our index finger in the direction of of l1 right that's what you do with the first element of the cross-product and then you put your middle finger in the direction of b2 and then your thumb is going to tell you what the net force is going to be so let me let me draw that so let me draw my hand and just so you know before I do any of these actually look at my hand just to make sure I'm drawing the right thing so my index finger in the cut direction of i1 my middle finger I'm sorry the index finger in the direction of l1 which is the same as i1 and then my middle finger is going to do what the magnetic field is doing so my middle finger is actually going to point straight up and then my other fingers are just going to do what they do and so now you're looking at the palm of my hand and my thumb my thumb let me make sure I'm doing this correctly oh no no no no no I'm that is I was drawing my left hand see that's a that's an error you don't want to draw your left hand when you're doing the right hand rule with cross products so let me draw it down here so my index finger going in the direction of l1 right my middle finger is popping straight up right because the magnetic field created by i2 is caught popping straight out of the page here so my middle finger goes straight up and my other fingers do what they need to do looking at the palm and then my thumb will go in that direction so the cross product of L with B - popping out of this page the net force is going to be in this direction so there's a little bit of symmetry here this wire is going to be attracted towards that wire and this wire is going to be attracted to that wire they're both going to eventual if they were floating in space they would slowly get closer and closer to each other and the radiuses would get closer and closer and they would start you know they would accelerate to each other I've ever increasing rates actually anyway I'm out of time in the next video I'll do this same principle but well I'll do it with some numbers see you soon