 Hi, I'm Zor. Welcome to a new Zor education. Today, I would like to continue talking about magnetic properties of electric current. Before, we were talking about very, very long, thin, straight-line electric current and magnetic field around it. Remember? So if this is the current, magnetic field lines are around it. Now, today I will continue talking about this, but today it will be on a little bit more fundamental and slightly more mathematical level. But it will allow to establish certain formula, if you wish, or law which is in the foundation of whatever I was talking about. Straight-line current. And I will use it for determining the magnetic field in the center of an electric current in a loop. So that's what today's research will be directed to. Electric current in a loop, that's our goal. But again, in the beginning, I will talk about more fundamental law, which has actually the name, kind of a difficult to pronounce, French physicist Bue and Savard actually came up with this. So there is a differential form and there is an integral form, and I will talk about both of these. So that's what the program for today. Now, this lecture is part of the whole course called Physics for Teens, and I do recommend you to take the whole course. It's presented on Unisor.com website. If you found this lecture somewhere on YouTube, using some search mechanisms, I do suggest you to go to the website Unisor.com because it's part of the course. This lecture is part of the course. Every lecture has very detailed, textual notes which basically can serve as a textbook. There are certain problems which I'm solving in the course of this material. There are exams for those who would like to challenge themselves, and everything from Unisor.com is free. There are no strings attached. You don't even have to log in. Alright, so let's start from something which is more, I would say, fundamental qualities. We live in a three-dimensional world, and any field we were dealing with before, they were usually, well, at least we started talking about all these fields like gravitational field or electrostatic field. In their simplest form, the source of these fields were a point, so a point charge or a point mass or mass point, whatever. And the field around these point charges or point masses were spherical for obvious reason. It's a three-dimensional, so we have a point as a source, so the field is spreading its influence, if you wish, in all different directions in three-dimensional space. Which means that at any given time the influence of that field is spread around a sphere, around the point which is the source. Which means, again, that the influence of that particular source of the field is spread around the surface of that sphere, and the surface of the sphere is 4 pi r square. r is obviously the radius. Now, that's very important, because r square is in the denominator of all the laws which we were talking about before, like gravitational field, that's the law discovered by Newton for electrostatic field. We also had the same thing, the force of the field is always inversely proportional to r square. And again, it's kind of more inversely proportional to 4 pi r square, but 4 pi is just a multiplier, and it all depends on the units of measurements. But the most important is dependency on 1 over r square. Now, what do we have with magnetic field? Well, we kind of think that it must be something similar to this, and in a way it is. However, there is a very important complication with magnetic fields. You cannot really consider the source of magnetic field as a point. It always has a direction, so it's not just a point, it's a point with a direction. Now, what I mean is, if you have a permanent magnet, it has north and south poles. However small you make this magnetic, this permanent magnet, it will still have a direction, and direction is important, and I will show you why. So, it's not just a point. Now, if you have an electric current as a source of the field, and the field is something like this, around it. Again, it's also direction, there is always direction of the current. However small the piece of the wire is, it still has a direction, and it means it has certain different positions, and depending on the position of this very, very small piece of wire, however small, infinitesimally small, no matter what it is, but there is still a dependency on the position relative to the point where we are measuring the strength of the magnetic field, which is produced by this however small piece of wire. So, that's very important. And right now, I would like to talk about this particular dependency on the direction and how to take it into account. For instance, we have a wire, and we are talking about magnetic field around the electric current, so we need a wire, and let's say this is direction of the current. It's a direct current. Now, we are not actually interested in the whole magnetic field produced by the current. We are interested only in one particular segment. Very small one, well, infinitesimally small one for obvious reasons. And I'm interested in the magnetic field, which is produced only by this particular piece of wire somewhere at point. Now, let's think about it. I think there is a very important analogy. If you take, for instance, the sun rays, they're coming to the Earth. Why do we have such a cold climate on the poles and very hot actually somewhere near equator? Well, because the axis of Earth is at angle to Sun. This is the Sun. So what happens is here the rays are perpendicular to the surface. Here the same rays basically, right? The Sun is much bigger than Earth, so basically it's exactly the same rays. However, it's cold here. Why is it cold? Well, because there is an angle on the square surface, on the square meter, let's say, of surface around the pole. You have much narrower set of rays which are coming to it than here. So what's very important is the angle between the source and this particular surface where the rays are falling on, right? So the same square meter here, if it's turned in such a way that it's almost like parallel to the Sun rays, it doesn't have much Sun's warmth, right? So that's why it's cold. And here we have perpendicular, so the same square meter here gets much more rays. Now, here we have exactly the same story. If this particular piece of wire is directed perpendicularly to this, so from here all the influence are coming basically the same as if it's really coming only from this length. So if this particular length is equal to ds, differential of the length, s is the length, right? So it's actually the active piece of this which influence whatever the position we are considering, it's only this piece. So only if perpendicular, and this is also perpendicular obviously, let's call it c. So it's actually AB influence is exactly the same as AC influence to this point p. And the further p is up on this particular picture, the narrower will be this segment AC. So let's call it effective length. So if length of this is AB is ds, then the effective length AC is equal to db times ds. ds times what? What is the length of this? If this is ds, if this is alpha, then this is alpha. So this is AC is AB times sine of alpha. And this is actually the effective length of this particular segment. Now, since we know the lengths, if you remember we were talking about what is actually magnetic field produced by straight line, we were talking about proportionality. The force should be proportional to the electric current, right? And it definitely should be proportional to this length, let's call it s. What else? And obviously it's supposed to be inversely proportional to r square in theory, right? We were talking about the distance and this is kind of, we have come up with a philosophical understanding that the influence must be inversely proportional to the square of the distance. Because we are living in a three-dimensional world and influence of this thing is actually spread around the sphere. So this is the radius r. That what I can say about the strength of magnetic field at that point, it should be proportional to i, obviously, to the electric current. It should be inversely proportional to r square. It should be also proportional to the effective lengths. Now I'm talking about effective lengths which is kind of visible from p. So from p the a-b segment is visible actually like a-c. So that's why I have to multiply it by ds and sin alpha. So this is, let's put it k, coefficient of proportionality and that should be the value of magnetic field intensity at point p. Now, speaking about coefficient k, again it all depends on the measurement units and in c it's mu 0 divided by 4 pi. So we have in the denominator we have a sphere, the surface of the sphere. And mu 0 is called, how is it called? Something related to penetration of the space basically, permeability, something like this. Difficult word. But whatever it is it's the property of the space actually. And again it's in the c units that's what it is. And I wanted to present this as a fundamental property of electromagnetic field produced by a tiny piece of wire with certain electric current running through it. Now why do I need this current piece instead of the whole line? Well, because I would like to actually use it for loop, electric loop rather than straight line. When we were talking about straight line we came up with I would say more intuitive formula of magnetic field around the straight line. If I have this as my more fundamental property of the electromagnetic field produced by the piece of the current, I can actually come up with certain way to derive the formula which we intuitively produced for infinite lengths straight line current. I can produce it mathematically by integrating this. And maybe we can do it as a problem solving kind of exercise. But in this particular case I don't want to use it for the straight line which you will see it's a little bit more difficult than for a loop. And I will use it for a loop. Okay, so let's just remember this formula. Ds is infinitesimal differential of the lengths of the wire. I is the current. Sine is the angle from this piece relative to direction. Again, direction. Remember always we have direction when we have electric current we have a direction. So between this direction and this direction we have an angle so that's what this angle actually is. And r is the distance. Okay, so I will use this formula to derive what's the strength, what's the intensity of magnetic field at the center of the electric current in a loop. So let's just forget about this. And instead of this we will do the loop plus minus. This is the current of electric current running in a loop. And I'm interested in magnetic properties of the space inside at the center. Only at the center it's simpler. Obviously we can do at any point by integrating everything. It's just more difficult and I would like to concentrate on something simpler because it basically explains everything. I don't want to go into technical details because the technical details for a center are very simple. Alright, so let's talk about this. So this is ds. So s is the length. It's from 0 to 2 pi r. r is obviously the distance. The radius of the loop. Now, i is fine. We understand r. ds also we understand. That's this little piece. What's the sign of alpha? Well, let's just think about it. Alpha is an angle between the direction of the ds, which is this one, tangential. This is the direction of the loop basically. And the radius to the point where we are measuring between this and this. And, lo and behold, in case of a loop, it's always 90 degrees because this is the radius and this is tangential. And the sign of 90 degree is 1. So in our case, the formula for a small, actually I should put db here. It's a differential of b. It's infinitesimal because this is differential, so this is differential. So in my case, db would be equal to mu 0 times i times ds divided by 4 pi r squared. Where i is a constant, r is the constant radius and the electric current. And basically, the variable is s. And this is the differential of s. And if I will integrate this, if I will integrate it from 0 to 2 pi r, this is the length of the whole circle. So this is s is equal to 0. And this is s is increasing. And at this point, s is equal to 2 pi r. Well, considering these two points are very close to each other, right? So I have to integrate this. That's more kind of a habitual way to write integral. You have this variable. You have limits of integration. This is the constant. So the constant goes out. And what I have is mu 0 i 4 pi r squared integral from 2 pi r ds. Well, what's integral of ds? Again, if we forgot the calculus, you have to go back to the prerequisite for this physics routine. On the same website, there is a mass routine and there is a very big calculus portion there. So this is the simplest possible integral, obviously. The integral from this ds is just s. And you have to put all these limits. So the formula Newton's Leibniz formula, you have antiderivative, which is s. And you put it at this limit minus this limit. So 2 pi r, substituted to s would be 2 pi r, 0 would be 0, and difference would be 2 pi r. So it's equal to mu 0 i 2 pi r divided by 4 pi r squared equals to mu 0 i divided by 2 pi r. Pi and this, so we have 2 r. So this is the formula for intensity of the magnetic field inside the loop, in the center of the loop. Now, obviously as a separate problem, which we might consider in the future, I don't have to really... I can actually calculate not only in the center of the loop, but on the axis, on any point, which is projected into the center of this loop, on a certain distance. And we can do it. Again, it's just simple integration. We can do it at any other point inside the loop, not only in the center. That's a little bit more difficult integration. But again, it's all possible. Everything comes from this, from the differential form of this view subar law. Okay, so we have come up with this formula, very simple formula. Music constant i is your current and r is the radius. So that's the intensity. In Teslas, by the way, in C system of units, it's measured in Teslas. So this is... the answer is in Teslas. So this is in amperes, and this is our best meters. And mu 0 is the permeability constant. Okay, now, what if our loop is... consists not only from one basic loop, but few loops. So we are just... let's say we have certain cylinder. And then we are looping the wire around and around and around. Basically, each circle which we are making is an independent loop, because the current goes inside this loop and inside that loop and inside that loop, etc. etc. So obviously the intensity should be added together. And if you have n circles of the wire making a thicker loop, if you wish, then you just have to multiply it by n, where n is the number of loops. So that's a simple thing. Now, what else I did not cover? And it's related to this law of Bu Savar. I have actually wrote the formula in the following form. What else? ds sin alpha divided by 4 pi r square. Now, this is a magnitude of the vector, right? So if you have this ds and this is the p... Oops, that's a p. And wrote p. This is the angle. Now, this is a magnitude. Now, but we know that intensity is actually a vector. It has magnitude and direction. And we also know from experimental fact, if you wish, I don't think I can explain it right now. But usually you know that if you have certain current and certain direction, at that point the magnetic field produced by moving electrons on this distance is a vector which has this magnitude and direction which is perpendicular to both direction of the current and direction towards the point where we are measuring. So these two vectors, vector ds and vector r, are vectors. Now, this is the magnitude of intensity. And what is the direction? Well, let's think about it. Direction is supposed to be perpendicular to this and perpendicular to that. Now, these two are lines on the surface of this board. Now, if you want to have perpendicular to two lines on a plane, that's a perpendicular to the whole plane. So it should be perpendicular to the board. So at this point, the direction of magnetic intensity force would be perpendicular to my board. And this is obviously the magnitude of this. But if this is true, we can use a very convenient way to represent. If you remember, again from the vector algebra, we were using something which is called vector product or cross product. If you have two vectors, a and b, it's the vector which is perpendicular to both, to a and to b, and whose magnitude is equal to magnitude of a times magnitude of b times sine of the angle between them from a to b in this particular case. So we do have basically exactly the same thing here. Let's imagine that you have a unit vector. I'll use lowercase r, unit. It has only one length, it is equal to one unit. Then this formula can be expressed as in a vector form. Vector db is equal to mu zero i vector ds, vector product r divided by four pi r squared. So r is a unit vector. So when you multiply it by r from the strength, from the magnitude perspective, you just multiply by one. But you also multiply by sine between this and this, sine of this alpha. So this basically contains the same information as this. However, why this formula is a little bit better? Because it contains both the magnitude and the direction. Because now we know the direction of magnetic intensity field would be perpendicular to ds and this unit vector. And then there is another representation which personally I don't like. You see we have to introduce this unit vector lowercase r. Sometimes it's maybe preferable to use the whole vector r. But if we will write r here, let me just write it again. db is equal to mu i ds vector r divided by four pi r squared. Now if I, oh sorry, cross product. Now if I want to use the original vector r, what happens here? I'm adding a multiplier into this formula which is equal to the value of r. So I have one r more in the numerator than I need. So I have to compensate it by putting three here instead of two. So then the magnitude of r would just cancel out and that will have exactly the same thing as here. So r is the unit vector, capital R is basically the whole vector to the point. So these are equivalent. And this is basically the form which this law of use of r actually takes. And again it's a differential form. You see we are talking about infinitesimal piece. You can obviously say that okay what if I would like to have the magnetic field of an entire wire and the wire is like this. Well what you have to do, you have to integrate this along the lengths of this wire. And obviously all is changing here. Obviously this sign will be changing because you have a different direction of ds all the time. So if you have a point p somewhere here and ds here, well this is one angle, ds is here, it's another angle. So it's a whole different ball game and much more difficult. So if you will integrate this along the wire line whatever the mathematical representation of this curve is. So you have to do the integral along the curve which is more difficult. So here we stop. That's exactly the form which I wanted to use for this law of use of r. And as a very practical implementation of that law we have come up with the magnitude of the vector of intensity. And the middle in the center of electric current loop. Okay that's it. I do suggest you to go to the website unizor.com if you will choose physics 14's course. Then it will go to electromagnetism and then you will find the magnetic magnetism of electric current. And it's one of those lectures current in the loop where you can find the lecture itself, the video and the textual representation basically. Which might be a little bit better than whatever I'm talking right now. And yes I do recommend you to take the whole course of physics 14's. So that's it for today. Thank you very much and good luck.