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# Magnetic force on a charge

Video transcript

We know a little bit
about magnets now. Let's see if we can study it
further and learn a little bit about magnetic field and
actually the effects that they have on moving charges. And that's actually really how
we define magnetic field. So first of all, with any field
it's good to have a way to visualize it. With the electrostatic fields
we drew field lines. So let's try to do the same
thing with magnetic fields. Let's say this is
my bar magnet. This is the north pole and
this is the south pole. Now the convention, when we're
drawing magnetic field lines, is to always start at the north
pole and go towards the south pole. And you can almost view it as
the path that a magnetic north monopole would take. So if it starts here-- if a
magnetic north monopole, even though as far as we know they
don't exist in nature, although they theoretically
could, but let's just say for the sake of argument
that we do have a magnetic north monopole. If it started out here, it would
want to run away from this north pole and would try
to get to the south pole. So it would do something,
its path would look something like this. If it started here, maybe
its path would look something like this. Or if it started here, maybe
its path would look something like this. I think you get the point. Another way to visualize it is
instead of thinking about a magnetic north monopole and the
path it would take, you could think of, well, what if
I had a little compass here? Let me draw it in a
different color. Let's say I put the
compass here. That's not where I
want to do it. Let's say I do it here. The compass pointer will
actually be tangent to the field line. So the pointer could
look something like this at this point. It would look something
like this. And this would be the north pole
of the pointer and this would be the south pole
of the pointer. Or you could-- that's how north
and south were defined. People had compasses, they said,
oh, this is the north seeking pole, and it points
in that direction. But it's actually seeking
the south pole of the larger magnet. And that's where we got into
that big confusing discussion of that the magnetic geographic
north pole that we're used to is actually the
south pole of the magnet that we call Earth. And you could view the last
video on Introduction to Magnetism to get confused
about that. But I think you see
what I'm saying. North always seeks south the
same way that positive seeks negative, and vice versa. And north runs away
from north. And really the main conceptual
difference-- although they are kind of very different
properties-- although we will see later they actually end up
being the same thing, that we have something called an
electromagnetic force, once we start learning about Maxwell's
equations and relativity and all that. But we don't have to worry
about that right now. But in classical electricity and
magnetism, they're kind of a different force. And the main difference--
although you know, these field lines, you can kind of view
them as being similar-- is that magnetic forces always come
in dipoles, soon. while you could have electrostatic
forces that are monopoles. You could have just a positive
or a negative charge. So that's fine, you say,
Sal, that's nice. You drew these field lines. And you've probably seen it
before if you've ever dropped metal filings on top
of a magnet. They kind of arrange themselves along these field lines. But you might say, well,
that's kind of useful. But how do we determine the
magnitude of a magnetic field at any point? And this is where it
gets interesting. The magnitude of a magnetic
field is really determined, or it's really defined, in terms of
the effect that it has on a moving charge. So this is interesting. I've kind of been telling you
that we have this different force called magnetism that
is different than the electrostatic force. But we're defining magnetism in
terms of the effect that it has on a moving charge. And that's a bit of a clue. And we'll learn later, or
hopefully you'll learn later as you advance in physics,
that magnetic force or a magnetic field is nothing but an
electrostatic field moving at a very high speed. At a relativistic speed. Or you could almost view it as
they are the same thing, just from different frames
of reference. I don't want to confuse
you right now. But anyway, back to what I'll
call the basic physics. So if I had to find a magnetic
field as B-- so B is a vector and it's a magnetic field-- we
know that the force on a moving charge could be an
electron, a proton, or some other type of moving
charged particle. And actually, this is the basis
of how they-- you know, when you have supercolliders--
how they get the particles to go in circles, and how they
studied them by based on how they get deflected by
the magnetic field. But anyway, the force on a
charge is equal to the magnitude of the charge-- of
course, this could be positive or negative-- times, and this is
where it gets interesting, the velocity of the charge
cross the magnetic field. So you take the velocity of the
charge, you could either multiply it by the scalar first,
or you could take the cross product then multiply
it by the scalar. Doesn't matter because
it's just a number, this isn't a vector. But you essentially take the
cross product of the velocity and the magnetic field, multiply
that times the charge, and then you get the
force vector on that particle. Now there's something that
should immediately-- if you hopefully got a little bit of
intuition about what the cross product was-- there's something interesting going on here. The cross product cares about
the vectors that are perpendicular to each other. So for example, if the
velocity is exactly perpendicular to the magnetic
field, then we'll actually get a number. If they're parallel, then the
magnetic field has no impact on the charge. That's one interesting thing. And then the other interesting
thing is when you take the cross product of two vectors,
the result is perpendicular to both of these vectors. So that's interesting. A magnetic field, in order to
have an effect on a charge, has to be perpendicular
to its you velocity. And then the force on it is
going to be perpendicular to both the velocity of the charge
and the magnetic field. I know I'm confusing you at
this point, so let's play around with it and
do some problems. But before that, let's figure
out what the units of the magnetic field are. So we know that the cross
product is the same thing as-- so let's say, what's the
magnitude of the force? The magnitude of the
force is equal to? Well, the magnitude of the
charge-- this is just a scalar quantity, so it's still just
the charge-- times the magnitude of the velocity times
the magnitude of the field times the sine of the
angle between them. This is the definition of a
cross product and then we could put-- if we wanted the
actual force vector, we can just multiply this times the
vector we get using the right-hand rule. We'll do that in a second. Anyway we're just focused
on units. Sine of theta has no units so
we can ignore it for this discussion. We're just trying to figure
out the units of the magnetic field. So force is newtons-- so we
could say newtons equals-- charge is coulombs, velocity is
meters per second, and then this is times the-- I don't know
what we'll call this-- the B units. We'll call it unit sub B. So let's see. If we divide both sides by
coulombs and meters per second, we get newtons
per coulomb. And then if we divide by meters
per second, that's the same thing as multiplying
by seconds per meter. Equals the magnetic
field units. So the magnetic field in SI
terms, is defined as newton seconds per coulomb meter. And that might seem a little
disjointed, and they've come up with a brilliant name. And it's named after a deserving
fellow, and that's Nikolai Tesla. And so the one newton second
per coulomb meter is equal to one tesla. And I'm actually running out of
time in this video, because I want to do a whole
problem here. But I just want you to sit and
think about it for a second. Even though in life we're used
to dealing with magnets as we have these magnets-- and they're
fundamentally maybe different than what at least we
imagine electricity to be-- but the magnitude or actually
the units of magnetism is actually defined in terms of the
effect that it would have on a moving charge. And that's why the unit-- one
tesla, or a tesla-- is defined as a newton second
per coulomb. So the electrostatic charge
per coulomb meter. Well, I'll leave you
now in this video. Maybe you can sit
and ponder that. But it'll make a little bit
more sense when we do some actual problems with
some actual numbers in the next video. See