Hey guys, so in this video we're going to talk about moving charges producing a magnetic field. Let's check it out. Alright, so remember that a charge, if a charge moves to an existing magnetic field, it's going to feel a magnetic force. And we've seen this before, you have magnetic fields *B* right here and you have a charge *q* that moves through it with a speed *v*. This charge will feel a force. This is called a Lorentz force and you may even remember the equation is
q
v
B
sin
(
θ
)
. Super important equation. Gotta know it. What I also need you to know and remember is that not only will a moving charge feel a force if it goes to an existing magnetic field, but a moving charge is also going to create its own magnetic fields away from itself.

Okay? So, moving charge is also going to produce a new magnetic field. You need to know that these two things happen and you need to know that they happen at the same time. It's not one or the other, it's both. Now I need you to know that but I also need to have a disclaimer here, big disclaimer that this topic is much less popular than, for example, being asked to find the force that a moving charge feels. Okay. In fact, most professors in most textbooks skip this altogether. And the reason why I'm including this video, across all of the textbooks we cover, even though a lot of the textbooks don't have this topic, is because it's a really big point and I think it's very helpful for you to remember that a moving charge can feel a force and produce a magnetic field. That you have to know no matter what. What you may not need to know is what follows which is how to actually use this equation to solve a problem for this specific kind of question.

Okay? So here's the equation I'll give you. And what you need to do is you need to get this equation, go talk to your professor if you're not sure if you need to know this and ask them, hey, do I need to know this? And if they say no, then problem solved. Or if they look really confused or they don't know where you got this equation from or what are you talking about, then now you know that you don't have to know this, okay? So the equation, it's an ugly one but it's pretty straightforward. It's relatively straightforward to plug stuff in.
B
=
μ
₀
q
v
sin
(
θ
)
/
4
π
r
2
. Here, *μ₀* is a constant, *q* is the charge, *v* is the speed, *sin(θ)* is the sine of the angle *θ*, and *r* is the distance.

Furthermore, these two points here talk about how to properly use this point here, how to figure out *θ* to properly use this equation, and this point talks about how to find the direction of the magnetic field. I'm going to just jump straight into this example because rather than talk about these, it's much better to just show you how they work in action. Okay? So I have a 3 coulomb charge. So, let's draw a little *q* equals 3 coulombs, and it's moving to the right with a speed *v* equals 4 meters per second. And we want to know the magnitude and the direction of the magnetic fields. What is *b* and what is its direction? That a charge produces 2 centimeters directly above itself. So here's a charge, I want to know what is the magnetic field at point *P* at a distance 2 centimeters, distance is *r*, So *r* equals 0.02 meters.

I want to know what is the magnetic field at this point *P* produced by this charge here. And obviously, the equation we're going to use is this *B* equation right here. And I'm going to write that *B* equals *μ₀*, which by the way, if you want, we can already replace with this. So let's go ahead and do that. *μ₀* is going to be 4π times 10 to the negative 7. You always want to write the left version not the right version because the left version, you're going to cancel out with the 4π at the bottom of that equation right away. Cool? And then I have *qv sin(θ)*. *Q* is 3, *v* is 4 and *sin(θ)*. We'll talk about *θ* in just a second, divided by 4π. We already canceled that times *r* squared. So 0.02. You can also write this as 2 times 10 to the negative 2 squared. I think this makes it a little bit easier to manipulate the numbers. So the only question here that's kinda tricky is what is your *θ*? Okay, what is your *θ*? And it says here *θ* is the angle between the *V* vector and the *r* vector. And the *r* vector is a vector between the charge and the location of the produced fields. You can think of the location of the produced field as the target's location. So what the heck does that mean? So *r* is a vector between the charge, the charge is right here. Vector just means an arrow, and the location So I'm going to draw a line from the charge. It says right here, from the charge to the target location. So I'm going to draw this line here. This is my *r* vector. And this *r* vector is only useful, so that I can figure out what is the angle between the blue arrow and the red arrow and this angle is of course 90 degrees. So this is going to be *sin(90)*. A whole lot of work for nothing because *sin(90)* is just 1. But obviously, you have to figure that out. And now we can simplify some stuff here. So I'm going to leave 3 and 4 by themselves times 10 to the negative 7. And then this 2 is going to be squared, which becomes a 4. And then the 10 to the negative 2 squared is 10 to the negative 4. The fours cancel and I end up with 3 times 10 to the negative 3 Tesla because we're talking about the magnetic field. Okay? So that's the field strength, the field magnitude. What about the direction? Now to solve for the direction, it says right here, we want to use the right-hand rule and by the way, we would have used the left-hand rule if *q* was negative. So if *q* is negative, which in this case, it isn't. And what we want to do is we want to grab the line of motion. We want to grab the line of motion. The best thing for me to do here is to just show you how this works. So let's move over here and we're going to do that.

So I want to grab the line of motion. So *Q* is moving this way to the right and the line of motion is just a line formed by the direction of *V*. So what we can do is you can think of the line of motion as sort of like this, right? And what this does, this line of motion does is it separates the page into a top part and a bottom part, okay? By the way, if the *V* was moving up, what it would do is it would separate this line, it would separate the page into left and right. So that's the line of motion, okay. So we got a line of motion there and what we want to do, we want to grab it and we want to grab the line of motion in such a way since we're talking about the right-hand rule, right? In such a way that my thumb points to the right. Why? Because if you remember on the right-hand rule, your thumb is your velocity direction, it's the direction of your velocity. So I'm going to grab this this way and imagine that I can lift this here so that I can grab it and the only way to, there are 2 ways I can grab this, I could grab it like this, right? I'm wrapping my fingers around this line of motion, I can grab it like this or I can grab it like this and we're going to choose to grab it like this because we want our thumb which is our velocity to point to the right so that it's consistent with the problem. When I do this, the way to grab it is to go under, this is super important, is to go under the marker over here and then come back up here. So in the bottom half of this page, I'm going into the plane to grab under the market and then in the top half, I'm coming out of the page and towards myself, towards my face and I have to do that. It's the only way that I can grab in the direction I'm supposed to grab. So what that means is that everything here is going to be into the page and everything here is going to be out of the page popping towards my face. So you can put a bunch of little dots and X's everywhere, anything here, right? Now point *P* was somewhere over here which means that the magnetic fields at point *P* is going to be out of the page. Okay, out of the page and that is the answer for direction.