Now that we have an idea of how magnetic fields work.
We're going to discuss what happens to an electric charge
put in the presence of a magnetic field.
Suppose we do have a charge like this one.
It has some charge, q and it's moving with some velocity, v.
And this field of blue dots that we have here
is actually a magnetic field pointing out of the page
and rather than thinking of the magnetic field
is only existing where the dots are.
You should think of the magnetic field
that sort of permeating that entire space
and again coming out of the page.
What happens with a magnetic field is exactly
because the magnetic field should be produced by moving charges.
Charges only experience a forced from a magnetic field
when they are moving.
So we say that moving charges experience
a forced from magnetic fields.
The value of this force that they experience is magnitude.
The value of the charge that's moving through the field
times its velocity as a vector crossed with the magnetic field B.
So this cross product is something
that can seem sort of complicated.
So what we're going to do is determine the magnitude of the force
and the direction of the force separately.
The cross product is just trying to tell us to the degree
which the velocity and the magnetic field the two terms being crossed here
are perpendicular to each other.
So one key thing to know about this cross product
even if we're not going to get in to all in mathematics of it,
is that any time two vectors are parallel, the cross product is zero.
Anytime the two vectors are perpendicular
the cross product will simply be
the product of the magnitude of the two vectors.
But again to simplify this what we'll do
is just evaluate the magnitude of this expression
separately from its direction which we'll see how to find as well.
So first of all let's just look at the magnitude of this
force from the magnetic field.
If you have a charged q like this one and is moving with the velocity, v,
through a magnetic field
and that magnetic field is that some angle theta with the velocity.
Instead of worrying about how to calculate the cross product
we can calculate the magnitude of the force
as the charge q times its velocity v,
times the strength of the magnetic field B
and then times the sine of the angle of theta.
And the reason we use the sine of theta is the same reason
that we use it when we are talking about Torque.
We only want a perpendicular component.
So for example, a theta is zero.
That means the velocity and magnetic field are pointing
in the same direction and we get no force from this magnetic field.
On the other hand, if theta is 90
if we have a perpendicular angle between the velocity and the magnetic field.
We simply get q times v times B with no mitigation
or mitigating factor from the sine of theta term.
Secondly, now that we have an idea
of how to calculate the magnitude of the force.
We can also calculate the direction of this force.
Unfortunately, we get to use our right hand rule once again.
The way you calculate the direction of the force
when you're using your right hand rule,
is to put your fingers in the direction of the velocity
and then make sure your palm is pointing
in the direction of the magnetic field in this case out of the page.
And when you do this,
your thumb will be pointing in the direction of the force
that your particle will experience.
So give this a try.
I know it's a little bit tricky to visualize and think about it first,
but follow this rules and make sure that you also got a force
that was pointing downwards.
The really interesting thing here is that as your particle moves
experiencing this force downwards that will curved its trajectory,
it's also going to be changing the direction of its velocity.
But if the direction of its velocity changes then the direction of force
will also change because we have to re-implement our right hand rule
at each moment along the motion.
This means that the direction of the force
will always be perpendicular to the velocity where the motion of our particle.
Thinking back to our prior chapters having to do with work
and that work is a force times a distance where the force
and the distance have to be in the same direction.
We can see that it isn't possible for a magnetic field to do work.
In other words, a magnetic field cannot change
the kinetic energy of a particle that is moving through it,
it simply changes the direction of the motion of that particle.
We could also in this scenario,
introduce an electric field as our particle moves in this case
to the right through the magnetic and electric fields.
If this is the case what will happen is this particle will experience
a force from the magnetic field downwards,
given the directions that we already discussed
and it will experience an electric force upwards
in the same direction as the electric field.
In this case, we would write that the total force on this particle
is the sum of both the electric force, q times E,
which we found in earlier chapters plus the magnetic force,
which we saw the magnitude of was, q times v times B.
And actually adding these two together
you should always be careful to take into a count the directions.
So for example, in this picture here,
if we were using this particular example
we have a magnetic force pointing in one direction
but we have the electric force pointing in the other
and they would add negatively rather than simply adding together in total.
We call this total force that a charge experiences moving through electric
and magnetic fields the Lorentz Force.
So the Lorentz Force takes into account all of the electric
and magnetic forces that a moving charge could experience.
There's a very interesting application of this Lorentz Force,
it is called a cyclotron.
What happen is you sense some particle moving through a magnetic field.
And just as we discussed,
there will be some force acting on it always perpendicular
to the direction of motion of our charged particle.
In this case, the particle is going to curve its path
and have a force always towards the center of that path
and it will end up going in a circle.
For this particle to be zipping in a circle
like this as it moves through some magnetic field,
they will have a force that has to be as we saw in earlier chapters
equal to mv squared over r and this just came from a force
that we found for anything moving in uniform circular motion.
But we also know the strength of the magnetic force acting on a particle
because we just found it were the velocity in this case
and the magnetic field are perpendicular.
This is because while the velocity as you can see here is acting exactly to the right,
the magnetic field is acting out of the page
and those are always going to be perpendicular to each other.
And so the magnitude of the magnetic force in this case is simply q times v times B.
The charge times its velocity times the strength of the magnetic field.
Since this is the only force that's pushing the particle
towards the center of this circular path,
it must be the case that these two are the same.
At the force pushing the particle towards the center of the path
is exactly the force keeping it in a circle.
Since we've acquitted these two, we have mv squared over r equals qvB.
This is called the cyclotron equation
because it describes exactly how a particle that's spinning in a circular path
in a magnetic field will behave.
What we can do is simplify this equation slightly,
by solving for the velocity in this equation by a quick rearrangement,
and then asking ourselves how long would it take a particle
to go in a full circular path in this kind of set up.
The time it takes to go in its full circular path
will be the distance divided by velocity.
So just think of the normal equation that we're all used
to which is that a distance is equal to a velocity times the time.
We then know that the time is equal to the distance
divided by the velocity which is what we have here.
In this case since the particle is going in a circle,
the distance is just the circumference of the circle which is 2Pi r.
The velocity is given to us as v and we've already solved for that velocity
in terms of our cyclotron equation.
So what we have to do is insert the equation for this velocity
into the equation for the time it takes
for an object to go in a full circular path.
And we have what's called the cyclotron period.
In other words, how long it takes
or how much time elapses through each cycle
of your object moving in a circle.