Hi. It's Mr. Andersen and this AP Physics essentials video 31. It is on the magnetic
force. Imagine we have a charged particle, like a proton moving through space. And since
there is no force acting on it, no field acting on it, it is just going to keep going in that
direction. But if we have a charged particle that is moving by a magnetic field, like it
is rushing from the sun and it is coming by the earth and the earth acts as a big magnet.
What it is going to do is it is going to apply a force to that. It is going to spiral into
the planet and it causes the Northern Lights. And as these protons and electrons hit our
atmosphere they are giving off ionization of colors. And so as we move into magnetic
forces, we really are moving into a third dimension. And so we have to have a way to
deal with that. So if we have just two dimensions it is easy to do that on a piece of paper,
but how do you do it with three dimensions, a magnetic field for example coming at you
or going away from you? The way I remember it is like an arrow. If an arrow is going
through a hole you are just going to see that point on the end. And so if we see concentric
circles like that or a bunch of them that means the magnetic field is coming towards
you. And the the tail end of the arrow is going to look like that. And so if we see
a plus sign or an x inside the circle, that means it is going away from you. And that
will make sense when you see diagrams like this. So we have a magnetic field. You can
see that it is coming out at us. And let's say we have a charged particle that has no
velocity. It is just sitting there. Is there a magnetic force acting on that charged particle?
The answer is no. If it is not moving, there is no magnetic force. But as it moves it has
a velocity vector. And if you are ever moving as a charged particle through a magnetic field
then you are going to have a force applied to you. We call that the magnetic force or
F sub n. Now things that you need to remember are that it is a perpendicular line between
the magnetic force and the velocity. And it is also perpendicular between the magnetic
force and the magnetic field. But the velocity vector and the magnetic field vector do not
have to be perpendicular to each other. How do we calculate the magnetic force? Well there
are only three things that really deal with that. First one is going to be Q. That is
going to be the charge of that particle. In this case it would just be the elementary
charge. We then have the velocity vector, or the speed of the particle. And then we
are going to cross product that with the magnetic field. What is a cross product? Well these
two vectors, when we multiply those together, we are going to get a third vector that is
going to move in a different dimension. And I have an animation that is going to show
you that. Now depending on that angle we can get different amounts of this magnetic force.
And so we use sine theta to figure out how much that is going to error over time. And
so let's say we have a magnetic field like this. You can see that it is coming at us.
To figure out where that magnetic force is acting you have to use the right hand rule.
And so it takes a second to get this figured out, but I am going to hold my fingers out
like this. So I have my index finger pointed out. That is going to be the vector of the
velocities. In other words where that charged particle is moving. I then have my middle
finger moving at 90 degrees to that. And that is going to be the magnetic field. And then
the force is going to be your thumb. And so you try to make these three dimensions with
your finger. And you can solve really intense problems like this. You might look like an
idiot as you are doing that, but it is worth it. And so let's say we have a charged particle
that has no velocity and it is in this magnetic field. How big is our force going to be? Well,
we are not going to have a force. Again, if it is not moving there is no force. And so
let me give you a harder problem. Let's say we have a magnetic field like this and we
have a charged particle, a positive particle, that is moving up and to the left. And so
who do you figure out where that magnetic force is going to be? Well first of all make
your hand like this. And then you are going to point your index finger in the direction
of that moving charge. So I am moving it up and to the left. Then you want to make sure
that your middle finger, that magnetic field, is pointed towards you. Because you can see
on this diagram that the field is coming towards us. And so once I have that in order, my thumb
is going to show me the magnetic force. And so that is going to go up and to the right.
So that was pretty easy to solve. Let's go to another one. Now we have a different magnetic
field. And let's say that that charged particle, that proton is moving from right to left.
How do you solve this one? Again, stick your finger towards the left. That is the movement
of the particle. Now the magnetic field is not coming towards you, it is going away from
you. So I have turn my hand like this. And now where is the magnetic force? Well you
can see that that is going to be acting down. What if we have one like this? So we have
a different magnetic field. What if we have an electron moving? Okay, so if you have an
electron moving there are two ways you can go at this. Number one you could just use
your right hand and then just turn the force around when you are done. Or you have a left
hand. And a left hand is going to work great. So I am going to put my finger, point my index
finger in the direction of the movement of the electron. The magnetic field, my middle
finger is this case is going to come at me. And so where is the force? Now the force is
going to be to the left. And so if you know the magnetic field and you know the direction
you can always figure out where that magnetic force is going to be. Now the angle, as I
said, between the velocity and the magnetic field is not always 90 degrees. And so we
call that theta. And depending on what that is, we are going to get a different amount
of that magnetic force. And so those two are perpendicular but the third one is not going
to be perpendicular. And so here is our equation. It is a little scary but it is not that bad.
q is going to be the charge. v is going to be the velocity. And then this is a cross
product between the two. And so let's watch the animation before we get to the sine of
theta. And so if they are pointed in this same direction, velocity and magnetic field,
we have no force. But watch what happens when they become perpendicular. We have a greater
force, or magnetic force. If it moves to 180 degrees what is our magnetic force at this
point? It is going to be 0. As I move it back to 90 degrees, then it is going to be at its
maximum at this time. And so by using this sine theta in the middle we can figure out
how much this cross product, in other words multiplying the velocity times the magnetic
field, is affecting the overall magnetic force. And so the parts of this formula are q, which
is charge measured in coulombs. v is going to be the vector velocity. That is going to
be in meters per second. And then we have B which is going to be our magnetic field.
And we measure that is teslas. And then as we solve for the magnetic force, that is going
to be measured in Newtons. And so let's get to that theta and the sine of theta. Where
is that coming from? Well let's say that our magnetic field and our velocity are perpendicular
to each other. We know that that is going to be the maximum amount of magnetic force
that we can get. What if it is this direction? They are in opposite directions. Do you remember
what our magnetic force is going to be here? It is zero. And if they are both moving in
the same direction it is also going to be at zero. And so if we just choose that angle
to be theta and then we have sine of theta, look what numbers we are going to get. So
if it is a 0 degree angle, sine of 0 is going to be 0. There is going to be no magnetic
force. Likewise sine of 180 is going to be 0. But if we are at 90 degrees that is going
to be 1. And the AP folks say that you should really understand 0 , 90 degrees and 180.
But the sine makes sense to me and it should make sense to you as well. What does our angle
look like right here? Well that is around a 45 degree angle. So what is the sine of
45, it is just going to be 0.70. It is going to be somewhere between 1 and 0. And so if
we do a problem like this, calculate the magnetic force acting on a proton traveling at 3.0
times 10 to the 5th meters per second perpendicular to a 0.32 tesla magnetic field, how could
you figure out the magnetic force? Well you just start with your equation, q v sine theta
of B. What do I know? Well I know the magnetic field. That is 0.32 teslas. I know the velocity
3.0 times 10 to the 5th. I know what the sine of theta is going to be, because that is a
90 degree angle. So that is going to 1. The only thing I do not have, it seems like is
q. That is going to be the elementary charge. So we just write that out like this. And then
we solve. And we are going to have a really small force that is acting on that charged
particle, that proton. And so did you learn to apply mathematical routines to express
the force exerted on a moving charged particle in a magnetic field? Always use your right
hand, and you will probably need a calculator to do the rest. And I hope that was helpful.