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## Magnetic field created by a current

# 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? Let's do that. 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 1. And then I have another
current carrying wire not too far away. And I will call that
current I2. Now what else do we need
to figure out? Oh, well, let me
just tell you. Let's say that they are
a radius of r 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
this wire to that wire is r. 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? 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 we could say they're floating in space. Let's just focus on
wire 2 for now. This is wire 2, this
is wire 1. What's going to happen
to wire 2? Well, let's think about it. Wire 1-- the current
in it is going to generate a magnetic field. Now what's the shape of
that magnetic field going to look like? Well, we could take our right
hand, do that right hand wrap around 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. So if I 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. So they're going to go
around 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 it's being created by I1. So I1-- its effect keeps going
out to infinity, although it gets much weaker
as we learned. It's inversely proportional
with r. But this is the field of I1. I can 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 top of an arrow, coming out at you. Fair enough. So I1, by going in this
direction, is generating a magnetic field that, at least
where I2 is concerned, that magnetic field is going
into the page. So what was our formula? And this all came from the
first formula we learned about, the effect of a magnetic
field soon. on a moving charge. But what was the formula of the
net magnetic force on a current carrying wire? It was the force-- I'll do it in
blue-- 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 this current, on current 2, right? Caused by this magnetic field,
by magnetic field 1. So it will be equal to I2, 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 wire
is under consideration. So let's say we have
a length of wire. And then of course, if you know
the length of wire and we knew its mass and we knew the
force on it, we could figure out its acceleration
in some directions. So let's say that
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 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 L2, just
so that you know that it deals with wire 2. That's a vector quantity. I could 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. 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 1. So it's magnetic field 1, which
is this magnetic field. So before going into the math,
let's just figure out what direction is this net force
going to be in? So here we say, well, the
current is a scalar, so that's not going to affect
the direction. What's the direction of L2? This is L2. I didn't label it L2
on the diagram. What's the direction of L2? Well, it's up. And then the direction of B1,
the magnetic field created by current 1, is going into
the page here. So here we just do the standard
cross product. Let me see if I can
pull this off. This is actually an
easy one to draw. So I put my index finger
in the direction of L2. And then I put my middle
finger in the direction of the field. So my middle finger's 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 other ways, where they tell you to put your
thumb in the direction of the field, and this and
that, your palm-- those are all valid. They're just different
variations of the same thing. I find this one easier
to remember. Because when I take the cross
product, index finger is the first term of the
cross product. Middle finger is the second
term of the cross product. Thumb is the direction
of the cross product. So anyway, this is the
direction of L2. The magnetic field, we already
know, goes into the page. So 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 its field is-- we know from this wrap
around rule that pops out here and it goes in here-- the effect
that it has on this other wire is that where the
current is going in the same direction, is that it
will be attracted. So the net force you is going
in that direction. We could say the force
from 1 on 2. That's just my convention. Maybe other people would have
written it the force given to 2 by 1. That's the force given
by 1 to 2. That's how I'm writing it. Now what's going to be the force
on current 1 from I2? Well, it's going to be the
current-- well, it's going to be the force there. Well it's going to be
the same thing. Let me draw I2's
magnetic field. You do the wrap around rule,
it's going to look the same. So I2, sure, on this side its
field is going to be going into the page. But what's I2's field going
to be doing here? It's going to be popping out. I just did the wrap around--
take this wrap around, 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 2 on
wire 1 of length L1, from here to here, is equal to current 1
times L1-- which is a vector-- cross the magnetic field
created by current 2. And so we can do the same
cross product here. Put our index finger in
the direction of L1. 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 draw that. So let me draw my hand. And just so you know, before I
do any of these, I actually look at my hand, just
to make sure I'm drawing the right thing. So my index finger in the
direction of I1, my middle finger-- sorry, my 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-- let me make sure
I'm doing this correctly. Oh, no. I was drawing my left hand. See, 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. My index finger going in
the direction of L1. My middle finger's popping
straight up, because the magnetic field created by
I2 is 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
B2 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's going to be
attracted towards that wire, and this wire's going to be
attracted to that wire. They're both going to--
eventually if they were floating in space, they would
slowly get closer and closer to each other and their radiuses
would get closer and closer and they would accelerate
to each other, at ever increasing rates
actually. Anyway, I'm out of time. In the next video I'll do this
same principle, but we'll do it with some numbers. See