Magnetic field created by a current carrying wire

So not only can a magnetic field exert some force on a moving charge, we're now going to learn that a moving charge or a current can actually create a magnetic field. So there is some type of symmetry here. And as we'll learn later when we learn our calculus and we do this in a slightly more rigorous way, we'll see that magnetic fields and electric fields are actually two sides of the same coin, of electromagnetic fields. But anyway, we won't worry about that now. And I think it's enough to ponder right now that a current can actually induce a magnetic field. And actually, just a moving electron creates a magnetic field. And it does it in a surface of a sphere-- I won't go into all that right now. Because the math gets a little bit fancy there. But what you might encounter in your standard high school physics class-- that's not getting deeply into vector calculus-- is that if you just have a wire-- let me draw a wire. That's my wire. And it's carrying some current I, it turns out that this wire will generate a magnetic field. And the shape of that magnetic field is going to be co-centric circles around this wire. Let me see if I can draw that. So here I'll draw it just like how I do when I try to do rotations of solids in the calculus video. So the magnetic field would go behind and in front and it goes like that. Or another way you can think about it is if-- let's go down here-- is on the left side of this wire. If you say that the wire's in the plane of this video, the magnetic field is popping out of your screen. And on this side, on the right side, the magnetic field is popping into the screen. It's going into the screen. And you could imagine that, right? You could imagine if, on this drawing up here, you could say this is where it intersects the screen. All of this is kind of behind the screen. And all of this is in front of the screen. And this is where it's popping out. And this is where it's popping into the screen. Hopefully that makes a little bit of sense. And how did I know that it's rotating this way? Well, it actually does come out of the cross product when you do it with a regular charge and all of that. But we're not going to go into that right now. And so there's a different right hand rule that you can use. And it's literally you hold this wire, or you imagine holding this wire, with your right hand with your thumb going in the direction of the current. And if you hold this wire with your thumb going in the direction of the current, your fingers are going to go in the direction of the magnetic field. So let me see if I can draw that. I will draw it in blue. So if this is my thumb, my thumb is going along the top of the wire. And then my hand is curling around the wire. Those are my knuckles. Those are the veins on my hand. This is my nail. So as you can see, if I was holding that same wire-- let me draw the wire. So if I was holding that same wire, we see that my thumb is going in the direction of the current. So this is a slightly new thing to memorize. And what is the magnetic field doing? Well, it's going in the direction of my fingers. My fingers are popping out on this side of the wire. They're coming out on this side of the wire. And they're going in, or at least my hand is going in, on that side. It's going into the screen. Hopefully that makes sense. Now, how can we quantify? Well, before we quantify, let's get a little bit more of the intuition of what's happening. It turns out that the closer you get to the wire, the stronger the magnetic field, and the further you get out, the weaker the magnetic field. And that kind of makes sense if you imagine the magnetic field spreading out. I don't want to go into too sophisticated analogies. But if you imagine the magnetic field spreading out, and as it goes further and further out it kind of gets distributed over a wider and wider circumference. And actually the formula I'm going to give you kind of has a taste for that. So the formula for the magnetic field-- and it really is defined with a cross product and things like that, but for our purposes we won't worry about that. You'll just have to know that this is the shape if the current is going in that direction. And, of course, if the current was going downwards then the magnetic field would just reverse directions. But it would still be in co-centric circles around the wire. But anyway, what is the magnitude of that field? The magnitude of that magnetic field is equal to mu-- which is a Greek letter, which I will explain in a second-- times the current divided by 2 pi r. So this has a little bit of a feel for what I was saying before. That the further you go out-- where r is how far you are from the wire-- the further you go out, if r gets bigger, the magnitude of the magnetic field is going to get weaker. And this 2 pi r, that looks a lot like the circumference of a circle. So that gives you a taste for it. I know I haven't proved anything rigorously. But it at least gives you a sense of, look there's a little formula for circumference of a circle here. And that kind of makes sense, right? Because the magnetic field at that point is kind of a circle. The magnitude is equal at an equal radius around that wire. Now what is this mu, this thing that looks like a u? Well, that's the permeability of the material that the wire's in. So the magnetic field is actually going to have a different strength depending on whether this wire is going through rubber, whether it's going through a vacuum, or air, or metal, or water. And for the purposes of your high school physics class, we assume that it's going through air normally. And the value for air is pretty close to the value for a vacuum. And it's called the permeability of a vacuum. And I forget what that value is. I could look it up. But it actually turns out that your calculator has that value. So let's do a problem, just to put some numbers to the formula. So let's say I had this current and it is-- I don't know, the current is equal to-- I'm going to make up a number. 2 amperes. And let's say that I just pick a point right here that is-- let's say that that's 3 meters away from the wire in question. So my question to you is what is the magnitude in the direction of the magnetic field right there? Well, the magnitude is easy. We just substitute in this equation. So the magnitude of the magnetic field at this point is equal to-- and we assume that the wire's going through air or a vacuum-- the permeability of free space-- that's just a constant, though it looks fancy-- times the current times 2 amperes divided by 2 pi r. What's r? It's 3 meters. So 2 pi times 3. So it equals the permeability of free space. So let's see. The 2 and the 2 cancel out over 3 pi. So how do we calculate that? Well, we get out our trusty TI-85 calculator. And I think you'll be maybe pleasantly surprised or shocked to realize that-- I deleted everything just so you can see how I get there-- that it actually has the permeability of free space stored in it. So what you do is you go to second and you press constant, which is the 4 button. It's in the built-in constants. Let's see, it's not one of those. You press more. It's not one of those, press more. Oh look at that. Mu not. The permeability of free space. That's what I need. And I have to divide it by 3 pi. Divide it by 3-- and then where is pi? There it is. It's over the power sign. Divided by 3 pi. It equals 1.3 times 10 to the negative seventh. It's going to be teslas. The magnetic field is going to be equal to 1.3 times 10 to the minus seventh teslas. So it's a fairly weak magnetic field. And that's why you don't have metal objects being thrown around by the wires behind your television set. But anyway, hopefully that gives you a little bit-- and just so you know how it all fits together. We're saying that these moving charges, not only can they be affected by a magnetic field, not only can a current be affected by a magnetic field or just a moving charge, it actually creates them. And that kind of creates a little bit of symmetry in your head, hopefully. Because that was also true of electric field. A charge, a stationary charge, is obviously pulled or pushed by a static electric field. And it also creates its own static electric field. So it's always in the back of your mind. Because if you keep studying physics, you're going to actually prove to yourself that electric and magnetic fields are two sides of the same coin. And it just looks like a magnetic field when you're in a different frame of reference, When something is whizzing past you. While if you were whizzing along with it, then that thing would look static. And then it might look a little bit more like an electric field. But anyway, I'll leave you there now. And in the next video I will show you what happens when we have two wires carrying current parallel to each other. And you might guess that they might actually attract or repel each other. Anyway, I'll see you in the next video.