 We know current carrying wires produce magnetic field around them. In this video, let's figure out exactly how to calculate the strength of that magnetic field. And it's given by a famous law in magnetism which is called the Biot-Sawar law. That's how you pronounce them, they're Frenchmen. So what does the Biot-Sawar law say? To use this law, we have to consider a tiny element of the wire. You can't just calculate it for the entire wire. So let's say I consider a very tiny piece of that wire, very, very tiny piece of that wire. And let's say that that piece has a length of dL, that's the length of that. And I'm using d because it's very tiny, imagine it's an infinitesimal. And let's say that the current is I. And now imagine I want to calculate what the strength of that magnetic field is at some point over here, some random point over here at distance r from this particular element. So how do I calculate the strength of the magnetic field here? So Biot-Sawar law basically says that the strength of the magnetic field at this point and we'll call that dB, B stands for magnetic field and D because it's a tiny magnetic field here by a tiny piece of wire. So that's going to be vectorially, so it's a vector equation, get ready for this. It's going to be a constant mu naught by 4 pi times I times dL cross r cap divided by r squared. Now I always found vector equations a little scary because you know they have all these cross products and everything. So first thing we'll do is look at a magnitude of this equation and try to make sense of that. So what would be the magnitude of this equation? So if I just look at the magnitude of dB, which I'm just going to write dB, that's going to be, this is a constant, we'll get to that, we'll get to what the mu naught is. So mu naught by 4 pi, I will remain the same, denominator r squared will remain as it is. What is the magnitude of this? How do you take magnitude of A cross B? It will be magnitude of A times magnitude of B times sine of the angle between the two. So magnitude of this, that's just going to be dL times magnitude of this. What is this? What is this r cap? R cap is a unit vector in the direction of r. And since it's a unit vector, its magnitude is 1. So magnitude of this times magnitude of this, which is 1, times the sine of the angle between them. So sine of the angle between the dL vector, what is the direction of dL vector? Well, you choose the direction of the current as the direction of the dL vector. And you choose the direction of r as the direction of r cap. So the angle between the two is going to be this. If I call this theta, then it will be sine theta because a cross product has a sine in it. So sine theta. So that's the magnitude. All right, so what is the equation saying? Well, first of all, there is a constant over here just like how we have constants in Coulomb's law. So there we had 1 over 4 pi epsilon naught. Here we have mu naught by 4 pi. I'll get to that mu naught part in a second, okay, but there is a constant. Then it says that magnetic field depends upon the strength of the current, which makes sense because we know it's the moving charges that create magnetic field. So higher current means more moving charges per second, more magnetic field. That makes sense. We also see that the dB depends upon dL, the length of the current element. Why is that? Well, that's because if you took a longer current element, then you would have more moving charges in them. And as then, so the magnetic field will be due to more moving charges. And so you would expect the magnetic field to increase. So if you have double the dL, you have double the number of moving charges. And so you'll have double the magnetic field. So this also makes perfect sense. What does this say? Oh, it's inversely proportional to r squared. That means that if you go farther away, the magnetic field drops off as 1 over r squared. And that actually makes me feel very comfortable because we've seen this before in Newton's law of gravity. We've seen 1 over r squared in Coulomb's law. And now we're also seeing this in Biosawar law. So that's, that's great. And just like with Coulomb's law, how it only works for point charges, we're seeing that Biosawar only works for point current elements. So we have to only consider very tiny pieces of wire. You can't consider that for the entire length of wire. So so far, you know, everything was very similar to Coulomb's law. But there's one major difference. There's this sine theta coming over here and that's a huge difference. And let me show you how that difference, you know, actually pans out. So if you go back to our charges and electric fields, imagine I consider a circle around a point charge, which is at the center, and I asked you, hey, consider three points a, b, and c. Where do you think the electric field, not magnetic, electric field strength would be higher? Where do you think would it be? The charges at the center, where would it be? Because the distance is the same. It doesn't matter. Everywhere electric field on this circle is going to be the same. One by four pi epsilon naught q by r square q is the same, r is the same everywhere. Okay. But now let's consider replace this charge with a current element, okay, as tiny piece of wire having current is called current element. Now I ask you again, where do you think would the magnetic field be higher? Would it be same everywhere? Would it be different? Can you pause the video? Just look at this sine theta and see if you can figure out out of three places, where it'd be higher, where it'd be lower. Yeah, where it would be. Can you pause and try? All right. So this time you need to be a little bit more careful because we also have to consider the angle theta between the current and the r. So the current is to the right. What is the direction? What's, where's r over here? For this one, our r is going to be from here to here, okay, here to here. And so over here, this is 90 degrees. Oof. Sine 90 is one. That's the maximum value, meaning over here you will get the maximum value of db, magnetic field. Okay. What about at this point? If I were to draw from here to here, the r value remains the same. But look at what happens to theta. It's no longer 90 degrees. It's more than 90. And sine of any angle more than 90, between 90 and some obtuse angle, it's going to be less than one. So db is going to be smaller. What about over here? Well, if I were to draw again, r from here to here, if I were to do that. This time what's the angle? The angle is zero. What is sine zero? Oof. Sine zero is zero. So over here, our db is zero. Look at what you're seeing. Even though we're going at the same distance, the magnetic field is not exactly the same because it not only depends on the distance, it also depends on the angle. That's the most important thing over here. We see that when you are at 90 degrees, whether you are here or you are here somewhere, 90 degrees, you get maximum field. And what we see is that at zero or at 180 degrees, if I go here, this will be a 180 degrees. Sine 180 is also zero, you get zero. And so in between, you will get magnetic field in between the maximum and the minimum value. So it decreases and then increases and then again decreases and then it increases again. So magnetic fields will always be maximum perpendicular to the current element and there will be minimum on the axis of that current element. And that's what this important thing is telling us. All right. Now, a final couple of things. One is what is this mu naught? What's important is that it's a constant for vacuum and most of the times we'll be dealing with vacuum. And the value of that constant is going to be, let me write that down over here. The value of that constant is 4 pi times 10 to the power minus seven. And it'll have some units which you can work out. This is Tesla and you have current. I don't remember the units. I think you can work that out. But it's given a name. It's called per me a oops, you can't, I can't read this, sorry, per me ability of vacuum. And if you think, yeah, that sounds very familiar to what we saw earlier. Yeah. That was called permittivity. This is called permeability. I did not name it. Don't blame me. I know these names are very, very similar to each other. What matters to me is that, hey, I know the value of that, and it's 4 pi times 10 to the power minus seven. And so basically when you look at this whole constant, the value of that constant is just 10 power minus seven because 4 pi would just cancel out. All right, last thing, let's look at the direction of the magnetic field because that's also important. Magnetic field is a vector. It has a direction. How do I figure out the direction of the magnetic field? There are a couple of ways to do that. I like both of them. One is something that we've already seen before, to find the direction of the magnetic field, we can use our right-hand clasp rule. So you take your right hand and you clasp the conductor so that your thumb points in the direction of the current and the encircling fingers will give you the direction of the magnetic field. And then you can use that to figure out what the magnetic field direction would be. So everywhere to the right, doesn't matter where you go, everywhere to the right, the magnetic field is into the screen. So immediately I can say, hey, the magnetic field at point P should also be into the screen. The magnetic field somewhere to the left would be out of the screen. But we have a vector equation. We should also be able to get the same answer just by looking at this vector equation. So let's try that. Let's get rid of this clasp rule. You can always use the clasp rule but using two methods are always better. You can always check yourself. So over here, if you want to get the direction of magnetic field, you have to get the direction of DL cross R. So here's how I like to do. I look at my DL. It's this way. I look at my R. It's this way. So I have to do a cross from DL to R. And how do you do cross from DL to R? You take your right hand and you align it such that your four fingers are along this cross. Here's how I would do it. So if I would show you my hand, I would align it with my DL. And then I cross it this way. And while I do that, look at the direction of my thumb. The thumb gives you the direction of the cross product. The thumb is pointing inwards and therefore the magnetic field over here must be inwards. So both methods, the clasp rule and the cross product rule, we'll give you this same answer. So can you quickly find out what will be the direction of the magnetic field at point A and point B? Can you pause and find that out? All right. So if we use the clasp rule, then we clasp for a conductor such that the thumb points in the direction of the current that gives you the magnetic field and circling fingers gives the magnetic field. I see that at point A, the magnetic field is coming out of the screen. So immediately I understand that the magnetic field over here must be out of the screen and we show out of the screen this way. And over here, everywhere down, the magnetic field should be into the screen. So everywhere below, it should be into the screen. But can we also confirm that using our cross product? Of course. So if I start with, say, at point A, I have to cross from DL to R. So I have to cross this way. So my encircling finger should go this way. And so the way I align my palm is like this, preparing it to cross. And when I cross it with my four fingers, that's how it looks. And so the look at the direction of the thumb, it's coming out of the screen. That's exactly what we predicted. And similarly, if I were to do at point B, this time I have to cross it the other way around. And so I'm going to hold my palm the other way around like this. If I hold it like this, now I cross it in this direction. And so my thumb represents the magnetic field is into the screen. It's exactly what we get over here. And so now that we know how to calculate magnetic field due to tiny pieces of wire, if you want to calculate the total magnetic field due to the entire wire, we just sum them up due to each tiny piece or we have to do an integral. And we'll look at some problems in future videos.