Let's look at what a magnetic force does
to a single moving charge. So there's our
charge, it's moving with some velocity,
which means it's going to experience a
force if there's a magnetic field. Let's
say there is a magnetic field, let's have it
pointing up, out of the screen. So what
direction is the force going to be?
Alright, so we need the direction of the
velocity - so right hand pointing the
direction of the fingers [towards V] and then I need to have my fingers swing [anti-clockwise] around to the B,
so that means I have to swing up and
that means that my force has to be in
this direction. Note that there's no way
you can have the force going in the
direction of the velocity or even
backwards. It always has to be at right
angles. Now what kind of motion do you
get if your force is never in the
direction of velocity? It means it never
does any work, it means it never changes
the kinetic energy of that particle, so
that particle is never going to change
speed. It's going to keep on going at
speed (the magnitude of V) the whole time.
But what's going to happen because of this force? What's going to happen is it's
going to change direction and then as
soon as it changes direction a little
bit, the force is going to move, and so it
changes direction. So after a little amount
of time you'll find that this
velocity has changed and then you'll
find that the force changes, and so on
and so on. And so what's going to happen
is that we're going to have this charge
move in a circle and we can figure out
how big a circle, because we know that
the centripetal acceleration has to be
provided by the magnetic force. And so
the magnetic force is going to be equal
to the mass of that particle, times its
velocity squared, divided by the radius
of that circle and we also know that
that force is being generated by the
magnetic field and so we know how big
that is. So we know that the force is
also equal to the charge, times the
velocity and because everything's at
right angles, it's the simple version so
it's just the magnetic field. And so
therefore these two things must be the
same and so we can equate them.
And then solve for  R, so the stronger
the magnetic field, that tighter the
circle you can swing it around and the
faster it's going, the bigger the circle
will be. So that all makes sense. It may come as a
surprise but actually this phenomenon is
something that you owe your very life to
and indeed probably all life on Earth.
Because coming from the Sun is an enormous
stream of charged particles, high-energy
protons and electrons and ions. And if they were to hit the earth, they would do
enormous damage to anything that was
living. So we owe our lives to the fact
that the Earth has a very strong
magnetic field. If a charged particle
comes flying from the Sun as though it
was going to hit the Earth, it would hit
these magnetic fields and then they
would start spiraling around the
magnetic field lines, just going in
circles and they in fact form quite
complex behavior and they end up getting
trapped in belts around the Earth. And
here's a zoom in - there's sort of loosely two belts, there's an outer belt an inner belt.
They each are complicated and changing
as the solar winds change and as various
events happen to our magnetosphere. But
this is a big area of highly charged
particles. If those particles hit us, we'd
be sad. In fact if they hit satellites,
the satellites tend to cook too, so you
have to be very careful if you're gonna
have satellites flying through these
regions. These regions of charged
particles trapped by the Earth's
magnetic field are known as the Van
Allen belts. The Sun is so hot and
energetic because of the fusion
reactions going on inside it. When we try
and build fusion reactors, we also need
to find a way to control incredibly hot
plasmas of high-energy charged particles
and all modern designs work very much
like the Van Allen belts in that they
generate very strong magnetic fields in
order to try and trap the charged
particles going in circles around the
field lines. And you can also use a known
magnetic field to figure out what
charged a particle has, because the
different charged particles will move
with different radii. And this is used
extensively in a device called a cloud
chamber. So in a cloud chamber you have a
supersaturated gas,  so it's full of a
supersaturated gas. And if you give it a
little jiggle then it tends to form a
little cloud. And so what you do to learn
about particles is you pass them through
this cloud chamber and wherever they go,
they tend to form a little line of clouds
and you can see here we have a particle
that's formed a line of clouds, and
it's moving in a curved trajectory and
the reason it's moving in a curved
trajectory is that it's in a magnetic
field. And from the direction of the
magnetic field and the direction that
the particle is moving, you can tell what
charge it has and indeed this is a
famous cloud chamber picture, because
this is the picture in which they saw
something that looked just like an
electron, except it was curving the wrong
direction, so it had the wrong sign of
charge. And so this trajectory here is a
trajectory of the positron. It was the
first discovery of the positron. Which is
just like an electron, except it has a
positive rather than a negative charge.
