# Lorentz Force

by Jared Rovny

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00:01 Now that we have an idea of how magnetic fields work.

00:03 We're going to discuss what happens to an electric charge put in the presence of a magnetic field.

00:09 Suppose we do have a charge like this one.

00:12 It has some charge, q and it's moving with some velocity, v.

00:16 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.

00:26 You should think of the magnetic field that sort of permeating that entire space and again coming out of the page.

00:31 What happens with a magnetic field is exactly because the magnetic field should be produced by moving charges.

00:39 Charges only experience a forced from a magnetic field when they are moving.

00:44 So we say that moving charges experience a forced from magnetic fields.

00:48 The value of this force that they experience is magnitude.

00:54 The value of the charge that's moving through the field times its velocity as a vector crossed with the magnetic field B.

01:01 So this cross product is something that can seem sort of complicated.

01:05 So what we're going to do is determine the magnitude of the force and the direction of the force separately.

01:12 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.

01:22 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.

01:31 Anytime the two vectors are perpendicular the cross product will simply be the product of the magnitude of the two vectors.

01:39 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.

01:47 So first of all let's just look at the magnitude of this force from the magnetic field.

01:53 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.

02:02 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.

02:17 And the reason we use the sine of theta is the same reason that we use it when we are talking about Torque.

02:22 We only want a perpendicular component.

02:25 So for example, a theta is zero.

02:27 That means the velocity and magnetic field are pointing in the same direction and we get no force from this magnetic field.

02:34 On the other hand, if theta is 90 if we have a perpendicular angle between the velocity and the magnetic field.

02:40 We simply get q times v times B with no mitigation or mitigating factor from the sine of theta term.

02:48 Secondly, now that we have an idea of how to calculate the magnitude of the force.

02:53 We can also calculate the direction of this force.

02:55 Unfortunately, we get to use our right hand rule once again.

02:58 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.

03:12 And when you do this, your thumb will be pointing in the direction of the force that your particle will experience.

03:19 So give this a try.

03:20 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.

03:27 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.

03:38 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.

03:47 This means that the direction of the force will always be perpendicular to the velocity where the motion of our particle.

03:54 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.

04:04 We can see that it isn't possible for a magnetic field to do work.

04:08 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.

04:18 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.

04:27 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.

04:41 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.

04:56 And actually adding these two together you should always be careful to take into a count the directions.

05:00 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.

05:13 We call this total force that a charge experiences moving through electric and magnetic fields the Lorentz Force.

05:20 So the Lorentz Force takes into account all of the electric and magnetic forces that a moving charge could experience.

05:28 There's a very interesting application of this Lorentz Force, it is called a cyclotron.

05:34 What happen is you sense some particle moving through a magnetic field.

05:39 And just as we discussed, there will be some force acting on it always perpendicular to the direction of motion of our charged particle.

05:46 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.

05:54 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.

06:10 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.

06:19 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.

06:28 And so the magnitude of the magnetic force in this case is simply q times v times B.

06:33 The charge times its velocity times the strength of the magnetic field.

06:37 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.

06:46 At the force pushing the particle towards the center of the path is exactly the force keeping it in a circle.

06:51 Since we've acquitted these two, we have mv squared over r equals qvB.

06:57 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.

07:06 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.

07:20 The time it takes to go in its full circular path will be the distance divided by velocity.

07:26 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.

07:32 We then know that the time is equal to the distance divided by the velocity which is what we have here.

07:39 In this case since the particle is going in a circle, the distance is just the circumference of the circle which is 2Pi r.

07:46 The velocity is given to us as v and we've already solved for that velocity in terms of our cyclotron equation.

07:53 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.

08:00 And we have what's called the cyclotron period.

08:03 In other words, how long it takes or how much time elapses through each cycle of your object moving in a circle.

The lecture Lorentz Force by Jared Rovny is from the course Magnetism.

### Included Quiz Questions

1. The direction of the force will always be perpendicular to both the magnetic field and charge velocity
2. The magnitude is independent of the charge of the moving particle
3. The direction of the force will always point along the direction of the field
4. The velocity will depend on the magnitude of the field
1. It is the angle between the velocity and magnetic field
2. It is the angle between the charge and magnetic field
3. It is the angle between the charge and velocity
4. It is the angle between the velocity and force
1. The force and direction of motion are always perpendicular
2. The force is perpendicular to the magnetic field
3. The force is always zero in net
4. The total direction always adds to zero
5. The change in kinetic energy is always negative
1. Comparing the Lorentz force and centripetal acceleration
2. Comparing the Lorentz force and Coulomb’s law
3. Comparing Coulomb’s law and centripetal acceleration
4. Comparing Newton’s 2nd law and Coulomb’s law
5. Comparing the centripetal acceleration and Newton’s 2nd law
1. The magnetic and Coulomb forces
2. The magnetic and cyclotron equations
3. The magnetic and centripetal forces
4. The Coulomb and centrifugal forces
5. The Coulomb and Newton’s 2nd law forces

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