Magnetic Field Produced by Straight Currents - Online Tutor, Practice Problems & Exam Prep

Currents in a wire generate a magnetic field, similar to moving charges. The magnetic field's magnitude is given by B1 = μ⁢I / 2πr, where μ is a constant and r is the distance. The right-hand rule determines the field's direction. When combining fields from multiple wires, consider their directions and magnitudes, using subtraction for opposing fields. Understanding these principles is crucial for analyzing electromagnetic interactions.

Magnetic Field Produced by Straight Currents

Hey guys. So in this video, we're going to talk about how currents in a wire will produce a magnetic field. Let's check it out. Alright. So you may remember that if you have a moving charge, that moving charge will produce a new field away from itself. So let's draw a little charge q here, right? In a number of different places away from itself. So, and the magnitude of that magnetic field, if you remember, is μI0qvsin(θ)/4πr2. But this is old news, that's why I said remember. Right?

And what we want to talk about now is how just like moving charges will produce a new field, well, currents will also produce a new field because currents are just charges moving in a wire. Right? So if you have a wire and it's got a current this way, I, you can think of it as, well, there are lots of little q's here that have v's. Right? Lots of little q's with v's. That's what a current is. So if a q moving with v produces a b field, then an I will also produce a B field. Okay?

So currents also produce new magnetic fields away from themselves. Okay? So if you have a current, just like up here, you have a current I, you're also going to have a magnetic field somewhere over here and at any number of distances. The magnitude of that equation, it looks a little cleaner than this one, a little nicer. It is that b=μ0I2πr. And this equation is super important, you absolutely have to know this one. Okay? I mean you should know all the equations but this one is really important; you're going to see this all the time. I should know that this only works for a very long wire, and in most problems, you're just going to assume that the wire is very long, even if it's not stated. Should you have a short wire, you will get a different equation from this one that's a little bit more complicated, but most of you aren't going to have to deal with that. Okay? Remember, μ0 is a constant and this r over here is a distance. Little r is always a distance and never a radius. In fact, if you have a wire, it would be this right here, r. Some people even use the letter d or sometimes a, to make a distinction. Cool.

So that's how you find the magnitude, and then what about the direction? The direction comes from the right-hand rule, and what we're going to do is we're going to grab the wire with our thumb in the direction of the current. This is pretty consistent with everything we've done so far, which is you always want the thumb to be in the direction of the charges. The charges move in the direction of the current, so you're always going to want to use your thumb to point in that direction. Now what's going to happen is, we have a wire, and I'm going to grab the wire in that direction. Right?

If the current is going to the left, I'm going to grab the wire like this. If the current's going to the right, I'm going to grab the wire like this. By the way, you're always going to use the right-hand rule when you have wires. It's always the right-hand rule, never the left-hand rule because the current is always by convention positive. Cool?

And then if you have 2 fields in the same location, and the fields are going in the same direction, we're going to add the magnetic fields. And if they're going opposite directions, we're going to subtract. We'll see this in the second example, so we'll get back to that point. For the first example, we're just looking for the direction. So we're going to get to use this rule here, and I'll show you how it works.

So I want to know, what is the direction of magnetic fields, produced by a current on a very long wire, if the current is oriented up? Meaning, if you have a wire and the current is up, like this, what is the direction of the magnetic field? Remember that means it's going away from you. So if you look at your sheet it's going away from you. Right? Do it yourself, so you follow. And that means that you're looking at the back of the arrow, so you see an x. And this is the symbol for into the page. Okay?

So imagine that this cable is going away from you, like this. Right? Like that. Cool. So what happens with the direction of the magnetic field? Well, we're going to grab the wire with my thumb pointing in the direction of currents. So if I grab the wire with my thumb up, it's going to look like this. Okay? Now this is super important, and I'm actually going to move it over here so you can follow a little bit better. This is super important. Here's what we're going to do:

This is going up, which looks like this. This is the direction of the currents. So I'm going to grab it with my right hand always. I'm going to grab it, and then my hand should look like this. Please do this. Right? Please do this, so that you can see. So what you are going to do here is notice that my fingers are curling into the page here, into the page here. Right? They're going, like in that direction, away from my face, into the page. And when they come back around, they come out, out of the page on the left side. Okay, so please, I'm going to do this really carefully. Hopefully, you can follow. So into the page, here and out of the page. Whenever your thumb is up, you're always going to get that effect.

So what does that mean when you are drawing it? Well, what it means is that this is a magnetic field. Anywhere to the right of this cable is going to be into the page, and anywhere Okay? So you can draw a bunch of little x's and dots. Cool? Now let's do left. And I actually want you to do left, and just as a hint the top of the wire will be either an x or a dot, and the bottom of the wire will be either an x or a dot. Figure out real quick, pause the video if you have to, figure out real quick which one you think is which. You're going to do the same thing I just did. I'm going to assume you paused the video, and I'm going to do it over here. So my thing is going to be my current is going to be to the right. I'm going to grab it with my right hand which looks like this, right? Notice that my fingers are going into the page out here and out of the page back here. Okay? So what this looks like is into the page here and out of the page here. Cool?

You gotta be good at this stuff. What about here? This is into the page. So again, I think you have a good chance of getting this right. So just pause the video, give it a shot. I'm going to do it over here, and this is going into the page, which means it's going away from you, right? Into the page. Please do this, grab a pen or something and point it into the page, which means my hand is going to grab it like this. I'm actually going to get rid of the pen. My hand is going to grab it like this. So you should have your hand where your thumb is going into the page. And if you do this, look what's happening with my curved fingers: they're going in a clockwise direction. Try it at the angle here. Right? So I'm going to go in a clockwise direction. Please do this yourself. It looks silly but you gotta do it to make sure that you're getting this right. So what that means is that actually the direction the current is going I'm sorry, the direction of the magnetic field is this way.

Okay? What if you had, what if you had the current coming out this way? Well obviously if the current flipped directions then the magnetic field would flip directions as well. Okay? So in this case, you would have your thumb pointing at you, and you can't see from my hand, but if you look at your thumb at your hand with your thumb pointing at you like this, right? Looking really silly right now. You're going to see that B goes counterclockwise. Hope you got that.

Let's look at example 2, which is a computational example. It says 2 wires are shown below 4 meters away from each other. So this distance here is 4 meters. And I want to know what is the magnitude and direction of the magnetic field that is produced at the center between the two. Well, the center between the two is somewhere over here, let's call it point P, and it is a distance of 2 meters away from both of them. And what I want to know is essentially what is the magnetic field at point P. I want the magnitude and direction of the magnetic field. This is really the net magnetic field. It's going to be a combination of 2 because this guy produces a magnetic field, and this guy produces a magnetic field. They both produce magnetic fields there. So really this is you could think of this as the net magnetic field. So it's going to be the magnetic field due to the 1st current, plus the magnetic field due to the second currents, i1 and i2. Except that these guys are going in different directions, so they're going to have different signs. Okay? So we'll get to that in a little bit. But essentially you're adding. It's just that you might be adding a positive to a negative. Cool.

So if you look at current one over here, it's got a dot which means it's coming towards you. And that means that the current is my thumb towards me. Which means that the direction of the magnetic field is going to be counterclockwise. Okay? So it's going to create a magnetic field right at that point that's going to be counterclockwise. So this is going to be b1 is counterclockwise. And then b2 is an x that's going into the page. If it's going into the page it's going to be, look at my thumbs up my fingers over here. If it's going into the page it's going to be clockwise. So please do this yourself. Clockwise which means it's going to go in this, in this sort of direction. B2 is going to be clockwise. They're going in opposite directions, and what we

Find Field due to Two Perpendicular Currents

Hey guys. So in this example, we have two perpendicular wires, and we want to find the magnetic field that they produce at a point. So let's check it out. It says two very long perpendicular wires. Perpendicular, again, remember it means 90 degrees. They intersect at 0,0. So this point right here where they cross is 0, 0. And remember, this is always the x position, comma, y position. The wires carry a current of 2 microamps up, so this current right here, we are going to call it I₁, is 2 microamps. And then this current here is I₂, which is 3 microamps to the left. We want to know what the net magnetic field at point P, located in this position here, is. So I'm going to extend this blue wire a little bit so we can draw it better. Negative 4, negative 9 would be somewhere over here. Negative 4 means you are 4 away from the y-axis, so you're 4 to the left. And negative 9, of course, means that you're 9 down. So here's point P, at (−4, −9) centimeters. The reason it says net magnetic field is that there are two wires, therefore there will be two magnetic fields produced here, and we want to know the combination of those two magnetic fields.

So, the magnetic fields we're looking for will be a combination of magnetic field B₁, which comes from current 1, and B₂ which comes from current 2, and we will figure out a way to combine the two. So, the equation is B equals, remember, mu naught I divided by 2 pi r. But if I am looking for B₁, it's I₁ and r₁ where r is the distance. And for B₂ is 2 pi r₂. Now we're just going to plug all the numbers. So, mu naught is 4pi × 10 to the power of −7. The current for 1 is 2 × 10 to the power of −6 (because it's micro), divided by 2 pi the distance. The distance is the part that you have to pause on and make sure you get the right number. B₁ is coming from I₁. So what we have to look at is the parallel distance or the shortest distance between wire 1 and point P. The shortest distance is going to be right here. So this is going to be the distance which is 4. Now I know it's negative 4 but the distance between those two points is just going to be a positive 4. And that's 4 centimeters, so it's 0.04 meters. If you do this, you will notice that the pi cancels. There's a 4 here that's going to cancel with this 4, so this becomes a 1. This becomes 0.01. And then the twos also cancel. So you're left with 10⁷ × 10⁻⁷ × 10⁻⁶ divided by 10⁻² down here. You can combine all this, and you get 10⁻¹¹. So, or 1 × 10⁻¹¹ Tesla.

Let's calculate them, and then we're going to talk about direction to see how we can combine. Same thing I'm going to do here. 4 pi × 10⁻⁷. The current is 3 microamps and 2 pi. The distance is 9. Again that distance is just 0.09 meters because it's 9 centimeters. And if you do this, I have it here, you get 0.67 × 10⁻¹¹ Tesla. Now, how they combine depends on their directions. We're going to grab wire, the first wire in the direction of the current. Notice how my fingers, and you have to do this yourself, don't just look at me. Or I guess look at me and do it yourself as well to make sure that you get it right because sometimes it looks weird on the camera. Right? So you want to grab the wire and when you grab the wire your fingers are going to be going into the page which is away from you. My fingers are going towards you but you need to look at your fingers. How they're going to go away from you on this side. Right? With my thumb up. But when they come back around, 'cause P is over here. Right? When they come back around, they are coming towards you. So they're coming out of the page. Wire 1 is going to produce a B₁ on this side everywhere. That's into the page and then a B₁ right here everywhere that is out of the page. So at this point, B₁ is going to be out of the page. Out of page. Now I want you to do the same thing for B₂. What is the direction of B₂? I'm going to keep rolling here. I'm going to grab this wire, with my thumb to the right with my thumb to the right, and this is a little bit easier to see. On the top of the wire, my fingers are going into the page, away from my face. It's towards you because it's my fingers, right? And I am facing you. But if you do it yourself, it's into the page. And then when it comes back around because P is below here. Right? When it comes back around, it comes towards this side, B₂ is going to be into the page. But everywhere over here below this line, B₂ is going to be out of the page. P is on this side, so B₂ below the line, so B₂ is also out of the page. Out of the page. And because both of these are in the same direction, 6.7 × 10⁻¹¹ Tesla. 6.7 × 10⁻¹¹ Tesla. And that is the final answer. Cool? That's it for this one. Let's keep going.

Find Zero Magnetic Field

Hey guys, so in this example we want to find where between two wires the magnetic field is 0. Let's check it out. We got 2 horizontal wires that are 6 meters away. They're both horizontal, which means they're parallel. I'm gonna make the first one red and the second one blue. It says here that the currents are 4 amps at the bottom. I'm gonna call that I2, 4 amps, and 5 amps at the top. And they're both going to the right. I1 equals 5 amps, so it looks like this. They are a distance of 6 meters away from each other. We will know where, or how far from the bottom wire, the net magnetic field is going to be equal to 0.

First, let's look at the direction of these magnetic fields, that will be produced by these currents. If you have a wire and it's pointing to the right, you're gonna grab your right hand. Right? Your right hand right here. And you're gonna grab it and it's going to be pointing, your thumb is gonna be pointing in the direction of the current which means it has to