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In the last video we figured
out that if we had a proton coming into the right at a
velocity of 6 times 10 to the seventh meters per second. So the magnitude of the
velocity is 1/5 the speed of light. And if it were to cross this
magnetic field, we used this formula to figure out that the
magnitude of the force on this proton would be 4.8 times 10
to the negative 12 newtons. And then the direction, we
used our right hand rule because it's a cross product. And we figured out that it would
be perpendicular-- well, it has to be perpendicular to
both, because we're taking the cross product-- and right when
it enters, the net force will be downwards. But then think about
what happens. If you have a downward force
right there, then the particle will be deflected downward a
little bit, so its velocity vector will then look
something like that. But it's still in the magnetic
field, right? And not only is it still in the
magnetic field, but since the particle is still moving
within the plane of your video screen, it's still completely
perpendicular to the magnetic field. And so the magnitude of the
force on the moving particle won't change, just the
direction will. Because if we do the right hand
rule here, but if we just move our hand down a little bit,
if we tilt it down, then our thumb's going to be pointing
in this direction. And that just keeps happening. It gets deflected that
way a little bit. So the magnitude of the velocity
doesn't ever change. It always stays perpendicular to
the magnetic field because it's always staying
in this plane. But the orientation does change
within the plane. And because of that, because
the orientation of the velocity changes,
the orientation of the force changes. So when the velocity is here,
the force is perpendicular. So it acts as kind of a
centripetal force, and so the particle will start moving
in a circle. So let's see if we can break
out our toolkit from what we've learned before in
classical mechanics, and figure out what the radius
of that circle is. And that might seem more
daunting than it really is. Well, what do we know about
centripetal forces and radiuses of circles,
et cetera? So, what is the formula
for centripetal force? And we proved it many, many
videos ago, early in the physics playlist. Well,
centripetal acceleration is the magnitude of the velocity
vector squared over the radius of the circle. And since this is acceleration,
if we want to know the centripetal force,
it's just the mass times acceleration. So it's the mass of the
particle, or the object in question, times the magnitude
of its velocity squared divided by the radius
of the circle. In this case, this is the
radius of the circle. And that's what we're going
to try to solve for. And what do we know about
the centripetal force? What is causing the
centripetal force? Well, it's the magnetic field
and we've figured that out. This is going to be equal to
this, which we figured out is going to be equal to-- at least
the magnitudes-- the magnitude of this is equal
to the magnitude of this. And that magnitude is
4.8 times 10 to the minus 12 newtons. And so the radius is going to
be-- let's see, if we flip both sides of this equation,
we get radius over mass velocity squared is equal
to 1 over 4.8 times 10 to the minus 12. I could just figure out what
that number is, but I won't worry about that now. Then we can multiply both sides
times this mv squared. And we get that the radius of
the circle is going to be equal to the mass of the proton
times the magnitude of its velocity squared divided
by the force from the magnetic field. The centripetal force. 4.8 times 10 to the
minus 12 newtons. And the radius should be in
meters, since everything is kind of in the standard
SI units. And let's see if we can
figure this out. Get our calculator. And this is where that constant
function is useful again, because what is
the mass of a proton? Well, that's something that I personally don't have memorized. But if we go into the built-in
constants on the TI-85-- let's see more. Mass of a proton. This is mass of an electron. This is mass of a proton. So mass of a proton-- that's
what we care about-- times the magnitude of the velocity
squared. What was the velocity? It was 6 times 10 to the seventh
meters per second. So times 6 times 10
to the seventh meters per second squared. And all of that divided
by the magnitude of the centripetal force. Which is the force that's
being generated by the magnetic field. That's 4.8 times 10 to
the negative 12. Divided by 4.8 E minus 12. Let's see. Hopefully we don't get
something funky. There we go. That's actually a pretty
neat number. 1.25 meters. That's actually a number
that we can imagine. So if you have a proton going
in this direction at 1/5 the speed of light through a--
what'd I say it was? It was a 0.5 tesla magnetic
field, where the vectors are pointing out of the video. We have just shown that this
proton will go in a circle of radius 1.25 meters. Which is neat because
it's a number that I can actually visualize. And so this whole business of
magnetic fields making charged particles go into circles, this
is one of the few times that I can actually say has a
direct application into things that you've seen. Namely, your TV. Or at least the old-school TVs.
The non-plasma or LCD TVs, your cathode ray TVs,
take advantage of this. Where you essentially
have a beam of not protons but electrons. And a magnet-- if you take
apart a TV, which I don't think you should do, because
you're more likely to hurt yourself because there's a
vacuum in there that can implode, and all that-- but
essentially, you have a magnet that deflects this electron beam
and does it really fast so it scans your entire screen
of different intensities, and that's what forms the image. I won't go into that detail. Maybe one day I'll do a whole
video on how TVs work. So that's one application of a
magnetic field causing a beam of charged particles to curve. And then the other application,
and this is actually one where it's actually
useful to make the particle go in a circle, is
these cyclotrons that you read about, where they take these
protons and they make them go in circles really, really
fast, and then they smash them together. Well, have you ever wondered,
how do they even make a proton go in a circle? It's not like you could
hold it and guide it around in a circle. Well, that's what they do. They pass it through an
appropriate strength magnetic field, and it curves the path
of the proton so that it can keep going through the same
field over and over again. And then they can actually use
those electric fields. I don't claim to have any
expertise in this, but then they can keep speeding it up
using the same devices, because it keeps passing
through the same part of the collider. And then once it collides,
you've probably seen those pictures. You know, that you spend
billions of dollars on supercolliders, and you end
up with these pictures. And somehow these physicists
are able to take these pictures and say, oh, this is
some new particle because of the way it moved. Well, what they're actually
talking about is these are moving at relativistic speeds. And since they're at
relativistic speeds as they move at different velocities,
their mass is changing, and all that. But the basic idea is what
we just learned. They move in circles. They move in circles because
they're going through a magnetic field. But their radiuses are different
because their charges and their velocities
are going to be different. And actually some will move
to the left and some will move to the right. And that might be because
they're positive or negative, and then the radius will be
dependent on their masses. Anyway, I don't want
to confuse you. But I just wanted to show you
that we actually are touching on some physics that
a physicist would actually care about. Now with that said, what would
have happened if this wasn't a proton but if this was an
electron moving at this velocity at 6 times 10 to the
seventh meters per second through a 0.5 tesla
magnetic field popping out of this video. What would have happened? Well, this formula would
have still been safe. The magnitude of the force is
the charge-- but it wouldn't have been the charge of a
proton, it would have been the charge of an electron, times 6
times 10 to the seventh meters per second times 0.5 teslas. So what's the difference between
the charge of a proton and the charge of an electron? Well, the charge of an
electron is negative. So if this was an electron,
then the net force would actually end up being
a negative number. So what does that mean? Well, when we used the right
hand rule with the proton example, we said that the-- at
least when the proton is moving in this direction--
that the net force would be downwards. But now, all of a sudden, if we
reverse the charge, if we say we have a negative charge--
the same magnitude but it's negative,
because it's an electron-- what happens? The force is now in this
direction, using the right hand rule, but it
is a negative. So really it's going to be a
positive force of the same magnitude in this direction. So if we have a proton,
it'll go in a circle in this direction. It'll go like this. But if we have an electron,
it'll go in a circle of the other direction. Now let me ask a question. Is that circle going
to be a tighter circle or a wider circle? Well, the mass of an electron is
a lot smaller than the mass of a proton. And we had the radius is equal
to the mass times the velocity squared divided by the
centripetal force. So this mass is smaller
and the radius is going to be smaller. So the electron's path would
actually move up and it would be a smaller radius. Actually proportional to the
difference in their radiuses is the difference in their
masses, actually. But that would be the path
of the electron. Anyway, I thought you'd be
interested in that, as well. I have run out of time. I will see you in
the next video.