# Magnetic field created by a current carryingÂ wire

## Video transcript

So not only can a magnetic field
exert some force on a moving charge, we're now going
to learn that a moving charge or a current can actually
create a magnetic field. So there is some type
of symmetry here. And as we'll learn later when
we learn our calculus and we do this in a slightly more
rigorous way, we'll see that magnetic fields and electric
fields are actually two sides of the same coin, of
electromagnetic fields. But anyway, we won't worry
about that now. And I think it's enough to
ponder right now that a current can actually induce
a magnetic field. And actually, just
a moving electron creates a magnetic field. And it does it in a surface of a
sphere-- I won't go into all that right now. Because the math gets a little
bit fancy there. But what you might encounter in
your standard high school physics class-- that's not
getting deeply into vector calculus-- is that if you
just have a wire-- let me draw a wire. That's my wire. And it's carrying some current
I, it turns out that this wire will generate a magnetic
field. And the shape of that magnetic
field is going to be co-centric circles
around this wire. Let me see if I can draw that. So here I'll draw it just like
how I do when I try to do rotations of solids in
the calculus video. So the magnetic field would go
behind and in front and it goes like that. Or another way you can think
about it is if-- let's go down here-- is on the left
side of this wire. If you say that the wire's in
the plane of this video, the magnetic field is popping
out of your screen. And on this side, on the right
side, the magnetic field is popping into the screen. It's going into the screen. And you could imagine
that, right? You could imagine if, on this
drawing up here, you could say this is where it intersects
the screen. All of this is kind of
behind the screen. And all of this is in
front of the screen. And this is where it's
popping out. And this is where it's popping
into the screen. Hopefully that makes a
little bit of sense. And how did I know that it's
rotating this way? Well, it actually does come out
of the cross product when you do it with a regular
charge and all of that. But we're not going to go
into that right now. And so there's a different
right hand rule that you can use. And it's literally you hold
this wire, or you imagine holding this wire, with your
right hand with your thumb going in the direction
of the current. And if you hold this wire with
your thumb going in the direction of the current, your
fingers are going to go in the direction of the
magnetic field. So let me see if I
can draw that. I will draw it in blue. So if this is my thumb, my thumb
is going along the top of the wire. And then my hand is curling
around the wire. Those are my knuckles. Those are the veins
on my hand. This is my nail. So as you can see, if I was
holding that same wire-- let me draw the wire. So if I was holding that same
wire, we see that my thumb is going in the direction
of the current. So this is a slightly new
thing to memorize. And what is the magnetic
field doing? Well, it's going in the
direction of my fingers. My fingers are popping out
on this side of the wire. They're coming out on this
side of the wire. And they're going in, or
at least my hand is going in, on that side. It's going into the screen. Hopefully that makes sense. Now, how can we quantify? Well, before we quantify, let's
get a little bit more of the intuition of what's
happening. It turns out that the closer
you get to the wire, the stronger the magnetic field, and
the further you get out, the weaker the magnetic field. And that kind of makes sense
if you imagine the magnetic field spreading out. I don't want to go into too
sophisticated analogies. But if you imagine the magnetic
field spreading out, and as it goes further and
further out it kind of gets distributed over a wider and
wider circumference. And actually the formula I'm
going to give you kind of has a taste for that. So the formula for the magnetic
field-- and it really is defined with a cross product
and things like that, but for our purposes we won't
worry about that. You'll just have to know that
this is the shape if the current is going in
that direction. And, of course, if the current
was going downwards then the magnetic field would just
reverse directions. But it would still
be in co-centric circles around the wire. But anyway, what is the
magnitude of that field? The magnitude of that magnetic
field is equal to mu-- which is a Greek letter, which I will
explain in a second-- times the current divided
by 2 pi r. So this has a little bit
of a feel for what I was saying before. That the further you go out--
where r is how far you are from the wire-- the further you
go out, if r gets bigger, the magnitude of the magnetic
field is going to get weaker. And this 2 pi r, that
looks a lot like the circumference of a circle. So that gives you
a taste for it. I know I haven't proved
anything rigorously. But it at least gives you a
sense of, look there's a little formula for circumference of a circle here. And that kind of makes
sense, right? Because the magnetic
field at that point is kind of a circle. The magnitude is equal at an
equal radius around that wire. Now what is this mu, this thing
that looks like a u? Well, that's the permeability
of the material that the wire's in. So the magnetic field is
actually going to have a different strength depending on
whether this wire is going through rubber, whether it's
going through a vacuum, or air, or metal, or water. And for the purposes of your
high school physics class, we assume that it's going
through air normally. And the value for air
is pretty close to the value for a vacuum. And it's called the permeability
of a vacuum. And I forget what
that value is. I could look it up. But it actually turns
out that your calculator has that value. So let's do a problem,
just to put some numbers to the formula. So let's say I had this current
and it is-- I don't know, the current is equal
to-- I'm going to make up a number. 2 amperes. And let's say that I just pick
a point right here that is-- let's say that that's
3 meters away from the wire in question. So my question to you is what
is the magnitude in the direction of the magnetic
field right there? Well, the magnitude is easy. We just substitute
in this equation. So the magnitude of the magnetic
field at this point is equal to-- and we assume that
the wire's going through air or a vacuum-- the
permeability of free space-- that's just a constant, though
it looks fancy-- times the current times 2 amperes
divided by 2 pi r. What's r? It's 3 meters. So 2 pi times 3. So it equals the permeability
of free space. So let's see. The 2 and the 2 cancel
out over 3 pi. So how do we calculate that? Well, we get out our trusty
TI-85 calculator. And I think you'll be maybe
pleasantly surprised or shocked to realize that-- I
deleted everything just so you can see how I get there--
that it actually has the permeability of free
space stored in it. So what you do is you go to
second and you press constant, which is the 4 button. It's in the built-in
constants. Let's see, it's not
one of those. You press more. It's not one of those,
press more. Oh look at that. Mu not. The permeability
of free space. That's what I need. And I have to divide
it by 3 pi. Divide it by 3-- and
then where is pi? There it is. It's over the power sign. Divided by 3 pi. It equals 1.3 times 10 to
the negative seventh. It's going to be teslas. The magnetic field is going to
be equal to 1.3 times 10 to the minus seventh teslas. So it's a fairly weak
magnetic field. And that's why you don't have
metal objects being thrown around by the wires behind
your television set. But anyway, hopefully that gives
you a little bit-- and just so you know how it
all fits together. We're saying that these moving
charges, not only can they be affected by a magnetic field,
not only can a current be affected by a magnetic field
or just a moving charge, it actually creates them. And that kind of creates a
little bit of symmetry in your head, hopefully. Because that was also true
of electric field. A charge, a stationary charge,
is obviously pulled or pushed by a static electric field. And it also creates its own
static electric field. So it's always in the
back of your mind. Because if you keep studying
physics, you're going to actually prove to yourself that
electric and magnetic fields are two sides
of the same coin. And it just looks like a
magnetic field when you're in a different frame of reference,
When something is whizzing past you. While if you were whizzing along
with it, then that thing would look static. And then it might look a
little bit more like an electric field. But anyway, I'll leave
you there now. And in the next video I will
show you what happens when we have two wires carrying current parallel to each other. And you might guess that they
might actually attract or repel each other. Anyway, I'll see you
in the next video.