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Lorentz Force

Teacher/Instructor Jonathan Osbourne
Jonathan Osbourne

PhD., University of Maryland
Published author

Jonathan is a published author and recently completed a book on physics and applied mathematics.

Lorentz force is the force on charge in electromagnetic field. Lorentz force is determined by the formula F = qv x B, in which q is the charge, v is the velocity, and B is the magnetic field density. Lorentz force is perpendicular to both velocity and magnetic field. The right hand rule is applied when determining Lorentz force.

Let's talk about Lorentz force, the Lorentz force is name that we give to a force that a charge feels when its moving through a magnetic field. There is a couple of weird things about this that make it very different from an electric field. Remember that an electric field, the force is just equal to charge times electric field so I'm I double the charge I double the force everything is very very simple the force is in the same direction as the electric field. For magnetic fields is very different, we replace that formula with this one f equals charge times the velocity and then this is a cross product which is the type of vector product which we'll talk about in just a second cross the magnetic field.

Alright, now, cross products are strange. What it does is it tells you that I'm going to take these two vectors and I'm going to form from that the single vector that's perpendicular to both of them so let's say the velocity was like that and the magnetic field was like that, well there's a vector that's perpendicular to these two directions and that this vector right here so if I had a magnetic field and a velocity like that the force will be either in this direction or in this direction. In order to determine which one I use the right hand rule. The right hand rule is well simple once you get used to it so let's go ahead and just see how this works.

Supposed that I've got a magnetic field like this, I'm going to have the magnetic field coming out of the board so the magnetic field lines are all pointing like this it's like I've got a north pole back here a south pole here magnetic field lines coming out of the board that's what those dots mean. Now I send a positive charge in to the right alright, so what direction is the force in? Well first of we know because the magnetic field is that way and the charge is moving this way, the force has got to either be up or down because those are the two directions that are perpendicular both to the magnetic field and to the velocity alright, which one do we pick? Well we use our right hand that's why it's called the right hand rule, and we put our thumb in the direction that the charge is moving in we put our fingers in the direction that the magnetic field is in and our palm will now point in the direction of the force. Alright so really simple, what I want to focus on here is, what is the magnitude of this force? Now first I just want to make this statement again very clear; force is perpendicular to both the velocity and to the magnetic field and this is totally different from the way it works in the electric field situation so magnetic fields can't really exist in just two dimensions, I need all three dimensions and that's not really the case for the electric fields magnetic fields inherently three dimensional. Alright, so what's the magnitude of the force? Well the magnitude of this force is equal to the charge times the part of the velocity that's perpendicular to the magnetic field times the magnetic field so only the part of the velocity perpendicular to the magnetic field can contribute so that means that if I had a magnetic field pointed like that and I send the charge in like that no part's perpendicular and that means that there's no force, it will just go straight through along the magnetic field lines so cross products are all about perpendicular that's what you should think soon as you hear the word cross product you should be thinking right hand rule and perpendicular.

Alright let's go ahead and do a problem so suppose oh let's talk about the units first so what is the unit of the magnetic field? We haven't seen that yet well that we've got an expression for force, we can relate the unit of magnetic field which is called the Tesla to our standard units since the force that's Newtons has to be equal to charge that's coulombs times velocity that's meters per second times magnetic field that's Tesla, if we solve for the Teslas then we end up getting 1 Tesla is equal to 1 Newton second per coulomb meter which we could also write as one Newton per ampere meter. Alright a Tesla is a very large magnetic field chances are you've never been around a magnetic field that big unless you've got an M.R.I or something like that so in comparison the earth's magnetic field is only between 30 and 60 micro Tesla millionths of a Tesla, 30 at the equator and 60 at the poles it's stronger near the poles because that's where the field lines are coming together.

Alright let's go ahead and do a problem. So suppose that I've got a 7 micro coulomb charge and it's going to move at 5 kilometers per second at 20 degrees above the horizontal and it's moving that way in a 2 Tesla magnetic field that's directed upwards and I want to know the magnitude of the force that it experiences. Alright, so let's go ahead and look at this, the best thing to do when approaching a problem like this is to make a diagram first I've got the magnetic field pointing up and I've got my velocity which is directed 20 degrees above the horizontal. Alright so we can use the right hand rule really quickly just to get the direction of the force we'll say velocity magnetic field the force is out of the board which we're going to indicate with this dot like that alright so now I want to know the magnitude. Well f is q v perp v alright well in charge is easy enough 7 times 10 to the minus 6 remember we're going to work in SI units because we're going to be good Physicists here alright? So 7 times 10 to the minus 6 what's the part of the velocity that's perpendicular to the magnetic field? Well that would be this part of the velocity so that means that I need to take the speed the hypotenuse of this triangle and multiply by the cosine of 20 degrees because cosine is the adjacent side the side that's helping to make the angle so we'll have 5000 times cosine of 20 degrees alright and that's going to give us the v perp so it will be 5 times 10 to the 3 cosine 20 and then I've got to multiply by the magnetic field which is 2 alright? So if I pull all that in my calculator, all I'll end up with 6.6 times 10 to the -2 Newtons or we could say that 66 million Newtons and that's the force that's the Lorentz force law.