Force on Moving Charges & Right Hand Rule - Online Tutor, Practice Problems & Exam Prep

Magnetic forces act on moving charges, described by the Lorentz force equation: $F_B=qvBsin\theta$, where q is charge, v is velocity, B is magnetic field strength, and θ is the angle between vectors. The right-hand rule helps determine the direction of the magnetic force, with fingers indicating the magnetic field, thumb for velocity, and palm for force. For negative charges, the left hand is used instead. Understanding these concepts is crucial for analyzing electromagnetic interactions.

Force on Moving Charges & Right Hand Rule

Hey, guys. So in this video, we're going to talk about forces, magnetic forces on moving charges, and I'm going to introduce the right hand rule. Let's check it out. Alright. So if you have a charge that is moving through an existing magnetic field and it's going to look like this, you have a magnetic field. Remember, fields are usually represented with multiple lines. Magnetic field is B and you have a charge, let's say q, that's moving, let's say this way with a speed v, that charge will experience a magnetic force due to the fact that it's moving in a magnetic field. Okay? The magnitude of that force will be given by this equation:

F B = q V B sin θ

where q is just the charge. V is the speed and it's a vector. Or I guess the velocity, the magnitude of the velocity vector. This is the magnitude of the magnetic field and times sine of theta where theta is the angle between the two vectors. Q is not a vector. V and B are vectors; they have directions. So the angle will be the angle between the two things with directions, between V and B. For example, here, v is going this way and B is going this way. So the angle here is this, which would have been, or which is 90 degrees. Okay? By the way, this is called Lorentz force, named after Mister Lorentz. And you should know that the units for a magnetic field are Tesla, named after Mister Tesla. Cool? And then 1 Tesla is 1 newton divided by 1 ampere times meter, if your professor likes you to memorize these things. Let's do a quick example and calculate some magnitude here.

So here we're saying a 2 coulomb charge, let's write that down. The charge is 2 coulombs, moves perpendicular to a magnetic field. Moves perpendicular to a magnetic field. Perpendicular means 90 degrees. It doesn't tell us, the problem doesn't tell us who's moving where, or which thing points in what direction. So we can just kinda make it up as long as they're 90 degrees apart. So we can say that the velocity is going this way and the magnetic field is this way because we're told that the angle between the two of them has to be 90 degrees. Cool. So let's say here that it's moving with 3 meters per second. So that's our V, 3 meters per second, and it feels a force, that's our magnetic force, of 4 newtons. These are unrealistic numbers but I just wanted to keep it really simple. What must be the magnitude of the magnetic fields? Magnetic field is big B and I want to know what is big B. So the question is, is there an equation that ties these four variables together? And obviously, there is. That's the one we just looked at.

F B = q V B sin θ

Now we have everything but we're looking for B, so all we have to do is solve for B.

B = F qV sin θ Sine of θ is easy. The angle between V and B is 90 and the sine of 90, you should know, is 1. So this whole thing just becomes a 1, so we don't have to worry about it. The force is 4. The charge is 2. The speed is 3. This is 4 over 6 or 2 over 3 or 0.67. We are talking about magnetic field strength. So this is 0.67 Tesla. Cool. That's it. That's all I got to do with that equation.

Now one thing we haven't talked about yet, so we talked about the magnitude but we haven't talked about the direction. And direction in magnetism problems are always going to come from the right hand rule, and there are going to be a few variations of the right hand rule. We'll tweak the rule, to make it work for different problems. Right hand rule abbreviated RHR in case you see it around. You know you're getting comfortable with things when you start using, abbreviations. So before we get into the ranking rule, which is massively important, a lot of people get confused here, so we got to go slowly through this, I want to warn you that there's a bunch of different rules. In fact, most books and most professors will use some version like this, which you might have seen in class. Okay. Where it's like you're shooting someone, but then like your middle finger sort of like sticks to the side. It's hard to do this on camera. But this is the most popular one and there's an engineering reason or sort of a more advanced physics reason why this is kind of a clever thing to do. I don't like that version. I use a different version. A long time ago I sort of thought about all the different ones and I've settled on the one I'm going to explain to you guys. It's got a bunch of advantages. You can't fully appreciate the advantages unless I explain the whole thing to you and that's too much. So you just have to kind of trust me or you can use whatever your professor uses or whatever else other valid method you find that you may like. So whatever you do, you have to pick 1 and stick with it. But if what you're doing is different from your professor, different from me, you just have to make sure that they match. What I mean by that is if I'm solving a problem and I got a direction to the left and you're using a different method, your method should still give the same answer. Same thing with your professor. So if you use my method and your professor is using a different method, you have to make sure that you're actually getting the answers that match his answers. Otherwise, something's wrong. Okay. So please be careful. Pick 1 and then go with it. So, when we so the reason we needed the, the right hand rule is because things are now going to be in 3 dimensions. And if you have 2 dimensions, you can be going up or down. That's 1 dimension. The second dimension

Force on Charge Moving at an Angle

Hey guys. So in this example, we are looking for the force on a charge that's moving through a magnetic field in 3 different scenarios. Let's check it out. So we want the magnitude and the direction of the magnetic force. We want the magnetic force on a 3 coulomb charge. So q equals plus 3. It's positive. We're going to use a right-hand rule for direction. We would use the left hand if it was negative. And it's moving with this velocity here, v equals 4. And it has a 5 tesla magnetic field. That's the strength 5 tesla. And that field is directed in the positive x-axis. Okay. So that's the field right there. And we want to know what is this force if the charge is initially moving in these three directions here. So in all three cases, B is going to the right, but the direction of the velocity is different. Here the velocity is going up. Here the velocity because it says positive y-axis. Here the velocity is going to the left because it's negative x-axis. And here it makes 30 degrees with the y-axis. Now the positive y-axis over here, this is a little ambiguous because you can make 30 degrees with the positive y over here. Right? This guy is 30 degrees away from the positive y, but this guy is also 30 degrees away. We'll talk about that when we get there. So the equation we're going to use is the only equation that makes sense. The equation for force on a moving charge, which is qvBθ. I know q, v, B, we're just going to plug those in. So the challenge here is just making sure we find the right angle, the correct angle. So Q is 3, V is 4, B is 5. Those are given they're up here. And the angle we should use is the angle between the two vectors. Between V and B is the angle we should use. V is up, B is to the right. V is directly up. B is directly to the right. So they're exactly perpendicular to each other to make an angle of 90 degrees. So sine of 90 degrees. Sine of 90, by the way, is 1. So the answer is just 60 newtons. Okay. What about the direction? Well, we're going to use the right-hand rule. So remember, my fingers represent multiple lines. So it's my B field, it's going to point up like this and, it's actually going to go like this, right? And my velocity should go up. So it's already up, so this is the direction I should be looking for. Notice that my palm is out, my palm is away from me, and you've got to do this yourself looking at your page. Right, if you put your hand in front of you and you see that your palm is away from you, it's going into the plane or into the page. Okay. So the direction is into the page. Okay. What about here? F_b is going to be the same thing 3×4×5 times sine of theta. But here the angle between v and b is 180, right? They're anti-parallel to each other and the sine of 180 is 0. That means that there is no force at all. Okay. And if there's no force, then there is no direction for you to worry about, right. Now, how can you remember this? One way to remember this is if you look at your right-hand rule, this should serve as a reminder that B and V are supposed to be at 90 degrees. And what I mean by supposed to be is this is the scenario in which you get maximum force. Okay? If your V moves a little bit, now you have less force, you have less than maximum, but you still got some force. And then if you go all the way, right, as you do this you're decreasing the magnetic force all the way to here, and then you get here which is parallel, 0 degrees, right? Parallel, or now you have 0 force. Same thing if you go all the way over here and you are at 180, I can't really do that it hurts, then that's going to be 0 force as well. Cool? Maximum force, a little less force, 0 force. Cool. So let's jump into this one here. Here, we talked about how there are 2 directions, because it's not clear, it's ambiguous, but it actually doesn't matter because the magnitude will be the same. Okay. The magnitude will be the same. If you want, you could have calculated the two different angles. Right? The distance, the angular distance between this red arrow and this blue arrow here is 60. Remember, you don't necessarily use the angle that's given to you. You use the angle depending on the definition. The definition of the angles should be the angle between V and B. So you got to be very careful whenever you see an angle. Okay. So that's one angle you could have used. Let's call it theta 1 or you could have used the angle all the way to this blue arrow here and that theta 2 is 90 degrees right here plus the 30. So 90 plus 30 is 120. You could have used either one of these guys and you would have gotten the same answer because sine, I'm going to use 60. Right? Sine of 60, and you can plug this into double-check, equals sine of 120. Right? So it's the same thing. You get the same answer no matter what. And that answer is 52 newtons. Remember I told you that if you're slightly at an angle, you're going to get less than maximum. This is the maximum force you can get for this arrangement. This is a little less than maximum. Right? And this here is the minimum, which is just 0. Okay. Now what about the direction? Well, B is this way, and you can have your charge even move either moving that way or this way. Right? Either way, for both situations, my palm is away from me. So if you look at your page and your palm is away from you, which means you're looking at the back of your hand, that means that the force is going into the page as well. Okay. So the direction here is into the page for both of those situations. Cool. Let me get out of the way. It's into the page. Cool. That's it for this one guys. Let's keep going.

Here’s what students ask on this topic:

What is the Lorentz force equation and how is it used to calculate the force on a moving charge in a magnetic field?

The Lorentz force equation is used to calculate the force experienced by a moving charge in a magnetic field. The equation is given by:

$F{B}_{}=qvBsin\theta$

where $q$ is the charge, $v$ is the velocity, $B$ is the magnetic field strength, and $\theta$ is the angle between the velocity and the magnetic field vectors. This equation helps determine the magnitude of the force. The direction of the force is found using the right-hand rule for positive charges and the left-hand rule for negative charges.

How do you use the right-hand rule to determine the direction of the magnetic force on a moving charge?

The right-hand rule is used to determine the direction of the magnetic force on a moving charge. Follow these steps:

1. Point your four fingers in the direction of the magnetic field $B$.
2. Point your thumb in the direction of the velocity $v$.
3. The direction your palm faces represents the direction of the magnetic force $F{B}_{}$.

For negative charges, use the left hand instead. This method helps visualize the three-dimensional nature of the force direction.

What is the significance of the angle $\theta$ in the Lorentz force equation?

The angle $\theta$ in the Lorentz force equation represents the angle between the velocity vector $v$ and the magnetic field vector $B$. The equation is:

$F{B}_{}=qvBsin\theta$

The sine function accounts for the component of the velocity that is perpendicular to the magnetic field. When $\theta$ is 90 degrees, $sin\theta$ is 1, and the force is maximized. When $\theta$ is 0 degrees, $sin\theta$ is 0, and there is no force.

What are the units of the magnetic field strength in the Lorentz force equation?

The units of magnetic field strength in the Lorentz force equation are Tesla (T). One Tesla is defined as:

$1 T=1 N / A m$

where $N$ is Newton, $A$ is Ampere, and $m$ is meter. This unit measures the strength of the magnetic field that exerts a force on a moving charge.

How does the right-hand rule change when dealing with negative charges?

When dealing with negative charges, the right-hand rule is modified by using the left hand instead. The steps are:

1. Point your four fingers in the direction of the magnetic field $B$.
2. Point your thumb in the direction of the velocity $v$.
3. The direction your palm faces represents the direction of the magnetic force $F{B}_{}$.

This adjustment accounts for the opposite charge, ensuring the correct direction of the magnetic force is determined.