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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? 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.