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Magnetic force between two currents going in opposite directions

Sal shows how to determine the magnetic force between two currents going in opposite directions. Created by Sal Khan.

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  • spunky sam blue style avatar for user Faisal Shahzad
    What is the difference between magnetic filed intensity(H) and magnetic flux density(B) ?
    (23 votes)
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    • blobby green style avatar for user Will Robertson
      You can think of magnetic flux density (B-field) as how many magnetic field lines there are packed in a given cross-sectional area of space that the field is permeating. These SI units are in Tesla (T) or in cgs units, Gauss (10,000 Gauss = 1 Tesla). The magnetic field intensity (H-field) is how the B-field exists within a magnetic material (such a magnet). The H-field is given by:

      H = B/mu + M

      H = magnetic field intensity (in SI units of A/m or cgs units of Oersteds (Oe))
      B = magnetic flux density (in SI units of Tesla (T) or cgs units of Gauss)
      mu = permeability of the magnetic material
      M = magnetization of the material (how the material responds to being exposed to an external applied magnetic field)

      In a vacuum, B = H (the two are indistinguishable), but in magnetic materials, such as a magnet, the two are different and can even point in different directions. Hope that helps!
      (34 votes)
  • blobby green style avatar for user David Carlile
    How can the magnetic fields be going the same direction if the currents are going opposite directions? For current 1 your thumb points upward, but for current 2 it points downward. Should the magnetic field of the area between the wires be into the page and the area outside's come out?
    (16 votes)
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  • piceratops seedling style avatar for user Gaurav Mhatre
    Can you please explain the left hand palm rule ?
    (3 votes)
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    • leaf orange style avatar for user Felipe Moura
      It's the right hand palm rule! And I've just explained up there. I'll paste it here:
      "I prefer to use another right hand's law. Mine is with a wide open hand. I'll try to explain it without a drawing: Take your right hand and positionate it (wide open) in front of your face. Then keep all fingers together pointing up, except your thumb, that will point to the right direction. That's it. This position will be your tool in magnetism.

      Now, how to use it:
      In a given question, you'll have a current direction (or a velocity direction, in case you're using the other formula: F=qBv) and a magnetic field direction. You'll take your thumb in the direction of the current (or the velocity) and the other fingers in the magnetic field direction. Then the force vector will be pointing out of your palm, like it was jumping out of your hand. Amazing and simple."

      But try to memorize JUST ONE OF THEM. Otherwise, it may confuse you ;)
      (9 votes)
  • hopper happy style avatar for user NYLee2501
    so, if there're two wires which has current, the net force for a wire is only affected by the other wire's magnetic field? I mean, a wire is not affected by the magnetic field created by itself?
    (4 votes)
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  • blobby green style avatar for user Ponchit Medina
    Among audiophiles, it is recommended that high voltage cables should intersect low voltage cables at 90 degrees. What is the scientific basis for this?
    (6 votes)
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    • aqualine ultimate style avatar for user Adam Warren
      Imagine a line running north/south, the field induced by the current is in rings perpendicular to the length of wire. If a wire runs through this field in the plane of the field (aka, perpendicular to our original wire) there will be no induced current from the magnetic field. There may be future videos that will accurately explain why, or you could look up Walter Lewin's lectures from MIT (excellent series of videos) for more detailed explanation why. Electromagnetic Flux = B dot product A (B for mag field, A for area), and dot product involves the cos of the angle between the field and length, and as cos(90) = 0, we can see there is no induced current.

      This of course is using perfectly ideal straight wires, which you probably don't have, but getting them perpendicular will at least minimize the resultant interference.
      (1 vote)
  • blobby green style avatar for user Henry Zhang
    you did not calculate F12. Does it always equal to F21?
    (4 votes)
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  • blobby green style avatar for user chinmaya datt
    In the question posed in this video shouldn't we take account of Newton's third law . If wire 1 is exerting a force on wire2 then wire2 must be exerting an equal and opposite force wire1 ?
    (3 votes)
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  • starky tree style avatar for user rjclemens
    This video only made me more confused. he drew his left hand, not his right hand. If he drew his right hand the force should be pointing to the left, right?
    (1 vote)
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  • purple pi purple style avatar for user Twishampati Ghosh
    Is there any DIRECT formula to calculate the force exerted by a current carrying wire on another current carrying wire?
    (1 vote)
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  • aqualine ultimate style avatar for user jaj_sentina
    when you say mu, permeability of a material, why air, vacuum or the space around it? i know that we assume our space is air but shouldn't it be the wire, copper or the material it self? its it the material or the space around it that affect the magnetic field?
    (1 vote)
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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? Would they attract or repel each other? And you can probably guess that, but let's go through the exercise. Because I realize that last time I did it, I got a little bit messy. And I'll do it a little bit cleaner this-- I don't have to draw as many magnetic field lines. So let's say that's wire 1. That's wire 2. And I'll just make the currents go in opposite directions. So this is I1. And this is I2. So what would the magnetic field created by current 1 look like? Well, let's do the wrap around 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. If you put your thumb up like that. Your right hand, always use your right hand. And then you'll get that type of magnetic 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 it'll get weaker and weaker. But even here, this magnetic field is going into the page. Now we know, just as a little bit of review, the force created by current 1 on current 2-- that's just the convention I'm using, you wouldn't always put the 1 first-- is equal to current 2 times some length-- let's call that length 2-- along the wire. This is going to be a vector because it's a magnitude of length and a direction. And it goes in the same direction as the current. So let's say that that is L2. 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 1. Now it all seems pretty complicated, but you can just take your right hand rule and figure out the direction. So we put our index finger-- I'm doing it right now, you can't see it-- you put your index finger in the direction of L2. You can write the 2 down here, instead of writing a big 2 up there. Put your index finger in the direction of L2. I keep redoing it just to make sure I'm drawing it right. Put your middle finger in the direction of-- so this is L2. This is this. Goes in the direction of the middle finger. 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, because the field is going into the page, 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 in the direction of the net force. So your thumb is going to go like that. So there you have it. This is the top of your hand. You have your little veins or tendons, whatever those are, 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 might want to try it yourself, but the force on current 1 or on wire 1, or some length of wire 1, caused by the magnetic field due to current 2, is also going to be outwards. So here, if you want to think about it little bit, or have a little bit of intuition, if the current's going in the same direction they will attract, and if currents are going in opposite directions they will repel each other. So anyway, let's apply some numbers. Let's apply some numbers to this problem. Let's do it with the opposite current direction. So let's say that current 1-- I'm just going to make up some numbers-- is 2 amperes. Current 2 is, I don't know, 3 amperes. What else do you need to know? We need to know how far apart they are. So let's say that this distance right here is, I don't know, let's say it's small. Let's try to get a respectable number. Let's say that they're 1 millimeter apart. But we want everything in our standard unit, so that all the units work out. So let's convert it to meters. So that equals 1 times 10 to the minus 3 meters. So they're pretty close apart. Now let's figure out the-- well, let's do 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 the length in question, L1, is equal to-- I don't know, let's make it 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 I2. I drew this hand too big, took up too much space. So the magnetic field created by current 2, worried about the magnitude of it, that is equal--. And we saw before, we're assuming that these are-- it's in air. So we can use the permeability of a vacuum. So it's equal to that constant, the permeability of a vacuum. Times I2. Just a magnitude. Now 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 the magnitude of the magnetic field is equal to-- well, we'll just keep that that-- I2, I said is 3 amperes-- times 3 amperes divided by 2 pi times 1 times 10 to the minus 3. And let's see, that answer will be in teslas. All right. There, we already have the permeability of a vacuum there. So let's write that down. Permeability of a vacuum times 3 divided by 2 second pi times 1E minus 3. And I get-- the answer will be in teslas-- 6E minus 4 teslas. So the magnitude of the magnetic field there created by current 2 is equal to 6 times 10 to the minus 4 teslas. Now what's the direction of that magnetic field? So here we use our wrap around 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 E like this, and my knuckles. The fingers are going to come out on that end. So the magnetic field caused by current 2 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. Fair enough. So now we can figure out what the net force on this first wire is. Let me erase some of this, just so I have some free space. Let's see. We already used 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 anyway, I'll leave it there, just so we remember what the whole problem was. So what's the net force on wire 1? The net force on wire 1-- so we could say caused by wire 2 on wire 1, is equal to the current in wire 1-- so that's 2 amperes-- times the vector-- well, this is L1. I'll write L1 right now. Cross 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 L1 is going upwards. So that's the direction of our index finger. B is going into the page. This is B2. 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 the direction of I1. My middle finger is pointed downwards, so you can't see it's pointing into the page. And my other two fingers do what they need to do. And so my thumb will point in the direction of the net force. That's the top of my hand. So the net force is in that direction. 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-- 2 amperes-- times the magnitude of the distance-- times 10 meters-- times the magnitude of the magnetic field. That's 6 times 10 to the minus 4 teslas. And then when you take the cross product, you take the sine of the theta between these two vectors. But they're perpendicular. The magnetic field is 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 1. So when things are perpendicular you don't even 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 20 times 6-- 120 times 10 to the minus 4. That's the same thing as 1.2 times 10 to the minus 2. So the magnitude of the force is 1.2 force from current 2 on wire 1. The magnitude is 1.2 times 10 to the minus 2 Newtons. Because we used 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, and you would know how fast it's accelerating at that moment outwards. Of course, as it gets further and further away, 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. 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.