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# Magnetic force between two currents going in opposite directions

## Video transcript

in the last video we saw that if we have two currents or two wires carrying current and the current is going in the same direction that they'll attract each other now what would happen before we break into the numbers what would happen if the two currents are going in opposite directions what would would they attract or repel each other and you could probably guess that but let's go through the exercise it's because I realize the last time I did it I did it I got a little bit messy and I'll do a little bit cleaner this I don't have to draw as many magnetic field lines so let's say that's why one that's wire - it's my wire - and let's make the currents go go in opposite direction so this is i1 i1 and this is i2 so what would the magnetic field created by current one look like well let's do our wraparound rule put our thumb in the direction of the current and then the magnetic field will wrap around it'll go into the page here and it'll go out of the page here right if you point your thumb up like that your right hand always use your right hand and then you'll get that type of magnet field and of course it's going into the page into the video screen all the way out to infinity it gets weaker and weaker it's inversely proportional to the radius away from the wire so get weak and weaker but even here this magnetic field is going into the page now we know just does a little bit of review the force on this the force on the force created by current one on current - that's just the convention I'm using you wouldn't always put the one first is equal to current to current - times some length let's call that length - along the wire this is going to be a vector because it's a magnitude of length and a direction if it goes in the same direction as as the current so that's let's say that that is l2 all right so we're talking about from here to here that's the current cross product that with the magnetic field I'll switch back to that the magnetic field created by one and all it seems pretty complicated but you can just take your right hand rule and figure out figure out the direction so we put our index finger on doing it right now you can't see it you put your index finger in the direction let me see you put your index finger there you go in the direction of L 2 I can write there 2 down here instead of writing a big 2 right there put your index finger in the direction of L 2 I keep redoing it just to make sure I'm drawing it right put your middle finger in the direction of so this is L 2 this is this goes in the direction of the middle finger put your sorry the index finger your middle finger is going to go in the direction of the field so it's going to be pointing downwards right because the field is going into the page to the right on this side of this wire and then your other hands are going to do what they will and then your thumb is going to go into the direction of the net force so your thumb is going to go like that so there you have this is the top of your hand you have your little veins or tendons whatever those are that's your nail that's your nail so in this situation when the current is going in opposite direction the net force is actually going to be outward on this wire the net force is outward and then if you don't believe me you could you might want to try it yourself but the the force on current one or on wire one or some length of wire one caused by the magnetic field to due to current to is also going to be outwards so here if you want to you know think about a little bit or have a little bit of intuition if the currents going in the same direction they will attract and if the currents are going in opposite directions they will repel each other so anyway let's let's apply some numbers right let's apply some numbers to this problem let's do it with the the opposite current direction so let's say that current one I'm just going to make up some numbers is two amperes current two is I don't know three amperes what else do we know we know the how how far apart they are so let's say that this distance right here is I don't know let's say it's let's it small let's let's try to get a respectable number let's say this is one let's say that they're one millimeter apart one millimeter but we want everything in kind of our standard you so that all the units work out so let's convert to meter so that equals 1 times 10 to the minus 3 meters so they're pretty close apart and let's figure out the well let's let's do let's do the the force on wire 1 due to current 2 just so that we can see that this is also repelled so let's say that we're the the length in question L 1 is equal to I don't know let's make it a long a long wire 10 meters all right so how do we do this so first let's figure out the magnitude of the magnetic field created by I 2 so the magnitude of the magnetic field created by I 2 I drew this hand too big to cup too much space so the magnetic field created by current too worried about the magnitude of it that is equal to we saw it before we're assuming that these are you know it's an air so we can use the permeability of a vacuum so it's equal to that constant the permeability of a vacuum times I to just a magnitude and remember we figure out the direction by wrapping our hand around it we'll do that in a second divided by 2 pi times the radius so 2 pi radius so let's see so let me just we can just put it in the so the magnitude of the magnetic field is equal to well we'll just keep that that I to I said is 3 amperes times 3 amperes divided by 2 pi times times 1 times 10 to the minus 3 and let's see that answer will be in Tesla's all right now we already have the permeability of a vacuum there so let's write that down permeability of a vacuum times 3/2 second pie times 1 e minus 3 and I get the answer will be in Tesla's 6 e minus 4 Tesla's so the magnitude of the magnetic field there created by current 2 is equal to 6 times 10 to the minus 4 Tesla's now what's the direction of that magnetic field so here we use our wraparound rule take your right hand wrap it around the wire in the direction of the current and then you'll get the shape of the magnetic field so I took my right hand my thumb goes in the shape of the current my hand is going to look something like this and my knuckles the fingers are going to come out on that end right so the magnetic field caused by current too is going to look something like that so on this side of the wire where it intersects with the plane it'll be popping out and on this side it'll be popping in all right so on this side it'll be popping in fair enough so now we can figure out what the net force on this first wire is let me let me erase some of this just so I have some free space let's see we already know we already know we already use the 3 amperes we already used all of that we already used all of that in fact we just need to know that this magnetic field that's popping out of the page we just need to know its magnitude we actually could even get rid of this whole drawing because now we just know that this has created a magnetic field and now we just worry about the magnetic field and this wire but I'll leave it there just so we remember what the whole problem was so what's the net force on this on wire 1 so the net force on Y or 1 so we could say caused by wire 2 on wire 1 is equal to the current in Y or 1 so that's 2 amperes times the vector well this is L 1 L 1 I'll write L 1 right now very cross the magnetic field across the magnetic field and really we just worry about the magnitude because the direction we can figure out what the right well let's figure out the direction first of all so L 1 is going upwards so that's the direction of our index finger B is going into the page right this is B - that's the magnetic field created by this wire so it's going into the page so if we use the right hand rule what happens my index finger is going in the direction of the current in direction of i1 my middle finger is pointed downward so you can't see it's pointing into the page of my other two fingers what they need to do and so my thumb will point in the direction of the net force all right that's the top of my hand so the net force is in that direction right so we don't have to worry about the vectors too much anymore because we know the end direction of the net force so what's the magnitude so the magnitude of the force is equal to the current to amperes times the magnitude of the times the magnitude of the distance times ten meters times the magnitude of the magnetic field that's six times ten to the minus four Tesla's and then we take the cross-product you take the sine of the theta between these two vectors right but they're perpendicular the magnetic field is put going into the page while the direction vector of the wire the length of the wire is going along the page so they are perpendicular so the sine of theta just comes out to be one so you can just so when things are perpendicular don't have to worry about that sine theta you can almost just multiply the terms and then use your right hand rule for the direction so anyway this gives us twenty times six 120 times ten to the minus four that's the same thing as what 1.2 times ten to the minus two so the magnitude of the force is 1.2 force from current to unwire 1 the magnitude is 1.2 times 10 to the minus 2 newtons because we use all the right units and the direction is outward and so if we knew the mass of this we would you know you just divide the force by the mass you would know how fast it's accelerating at that moment outwards of course as it gets further and further away that the magnetic field is going to get weaker so the net force is going to get weaker so it'll start accelerating at a slower and slower speed I'm sorry at a slower and slower rate but of course you're still accelerating so you are going to continue to move away faster and faster anyway all out of time see you in the next video