 Hi, I'm Zor. Welcome to Unisor Education. We will talk today about magnetic field produced by moving charge. We actually approach this from two different sides. One side was if we have electric current, i, amperes, and we are at certain distance r from this current and the first formula which was kind of logically derived, but then obviously confirmed by experiments was that magnetic field at this particular point, magnetic field intensity is equal to mu i divided by 2 pi r. Now mu is magnetic permeability of the media between these. If it's vacuum, it's mu zero. i is amperage and r is distance, basically. Now the magnetic field lines around this straight wire. Well, we assume that wire is infinite, very long, okay, infinite in mathematics. So these magnetic field lines are circles around the wire. Every circle is in the plane which is perpendicular to the wire itself. And obviously there is some circle which is going through this point. That's what magnetic field at this particular point is and the direction is perpendicular to both the magnetic field line and perpendicular to wire and perpendicular to the direction to the wire. So it goes it's a tangential line to magnetic field line. And it would be either directed towards the board or outside of the board perpendicularly. So that was one formula which we have derived. And then we were examining one particular particle moving, charged particle moving in space. And it produced spherical magnetic field lines. And if you are an observer, then the magnetic field intensity in this particular case depends on different, well, we had different formula. The formula is mu times q. q is charge of this particle, electric charge. And then there is a speed vector speed times unit vector along this direction from my observation point to the charge. Now this is speed and divided by 4 pi r square. r is the distance. Variable distance. Now here all these pieces are constants. The current is basically running through the wire, but these are constants. In this case, this r is definitely variable as the particle is moving. Now instead of this, I would rather put v times sine phi, where phi is this. It's the length of this vector times length of this vector, which is unit vector. So it's one times the sine between them. That's the magnitude of the vector product. Now this vector product combines in itself also the direction. And the direction would be the same as in this case perpendicular to the board, either towards the board or outside of the board. Right? So that's basically the formula would be and I will replace it with this. It would be easier for me to prove what I want to prove. Mu, q, v, sine phi divided by 4 pi r square. So my question is now what is the current? Current is the flow of electrons. Now every electron basically can obey these laws. So if I will summarize the magnetic field intensity at this point from all the electrons which are moving along this infinite wire and I will integrate basically all of these together, I shouldn't have this formula. That's kind of a logical thing. If I don't it means something is wrong in my theory. So the purpose of this particular lecture is to prove that the theory is really fine. There is definitely the way from this to this and this way is basically through straightforward mathematics and calculus obviously. And that's exactly what I would like to do right now. So let's think about it. Now I will have to somehow put mathematics into this flow of electrons. So what I have decided to do is this is my infinite wire. So let's start from, let's say it's 0.0. So this is a minus and this is a plus. This part of the length is concerned and I will take only a small piece of the wire from x to x plus dx. Now this is my observation point. Observation point. Now I would like to measure the magnetic field intensity at this point from this infinitesimally small piece of wire. Considering this to be basically a particle, a charged particle it has certain charge, right? Since the electrons are moving, it has a certain number of electrons. I would like to help them. Now the the nucleus of the atom are staying still. So they do not produce any magnetic field because they are at rest but electrons are moving. So electrons have certain charge. So I will have the charge of this particular infinitesimal piece as a particle and I will measure what's my impact, what's my magnetic field intensity at this point relative to relative to this particular piece. And then I will integrate the whole thing from minus infinity to plus infinity, right? Okay, so so this is r. That's my distance to the wire. Same thing as that. So this is x. Now first of all, here is i and here is q and v. We have to somehow relate them to to to together. Now, let's just think about it. What is i? i is amount of electricity per unit of time. However small it is. So I will take only this piece and the time would be the time this particular particles are moving from this point to this point. This is dt. So basically dx is equal to v times dt. Speed times time interval gives me this particular thing. This is the speed of electrons, how they're moving. This is the time takes from this to this and this is the distance from x to x plus dx. Okay, so from this we have derived that. Yes, one more thing. This is the wire and a certain number of electrons per meter. It's the linear density of charge. It will be presented in both cases. So dq is equal to certain delta which is linear density of electric charge. Coulomb's per meter times dx. So dx is piece of the wire. Delta is amount of electricity per unit of length. So their multiplication gives me this. So what do I have from all this? So instead of dt, so i is equal to dq would be delta times dx. dt would be dx times v. dx times v. From this we see that linear density of electric charge per wire, which is constant, obviously, number of electrons per. They're all moving in sync. So the number of electrons or charge of all the electrons per unit of lengths is obviously the same. And v is, as I was saying, speed these electrons are moving. So that would give me i. So instead of i, I will put here delta v. I'm trying to get these two formulas as close as possible. Now sine phi and r are variable. So this is angle phi and this is the distance. As the charge is moving, these are variable. So in this formula everything is constant. The density is constant. The speed of electrons is constant. r is the distance, shortest distance from observer to the wire. Everything is constant. Here these are variables. I would like to express all these variables through something which will allow me to integrate the whole thing. Well, x basically. So now instead of q I actually have to put, I will be talking only about this particular piece of the wire, right? So it's dq. It's an infinitesimal piece, which is, I know, is delta times dx. Okay, fine. Now what is r? r, by Pythagorean theorem, is r square plus x square square root. What is sine of phi? Now this is phi. This is phi. The sine of this angle is r divided by hypotenuse, right? So I have expressed everything through x and all you need right now is to substitute into this formula and integrate from minus infinity to plus infinity and check if I will have this result. Okay, let's write it down. So I have an integral from minus infinity to plus infinity times q which is delta times dx times v times sine of f, sine of phi which is r divided by square root of r square plus x square times 4 pi and times r square which is r square plus x square. That's what on the right side. On the left side I have this. Well, let's just think about if this is equal to this, well, my task is accomplished. So let's check it out. And now this is the pure mathematics which I hope we don't have any problems with calculus. This is a simple integral actually. So first of all, instead of proving that this is equal to this, let me do a little bit simplification. So I have to prove that this is equal to mu delta v divided by 2 pi r. I have to prove it, right? Well, I don't have to prove this. I don't have to prove this. I told you I don't care about density because it cancels. I don't have to prove this. I don't care about the speed either, you see. Now I don't have to care about pi. Now this is an even function which means it's symmetrical relatively to zero. Left part from minus infinity to zero is exactly the same as from zero to plus infinity because it's all x square, right? So instead of integrating from minus infinity to plus infinity, it would be easier for me if I would integrate it from zero to infinity and multiply by 2, okay? Now 2 and 4, this is 2, 2 divided by 4. So I don't care about this, don't care about this, don't care about this. So from now on, I will just have to prove. Now I will multiply both by r. So it would be r square which is 1. So I have to prove this integral equals to 1. Right? All right. So let me write it down. Integral from zero to infinity. What do I have left? r square dx divided by r square plus x square. Now this and this would be power of three seconds, right? Okay. And I have to prove that this is equal to 1. That seems to be a relatively easy thing and actually it is. A couple of substitutions will give me the good result. So the first substitution, obviously, is too many r's. So I will have to do this. I will have to have x divided by r is equal to y. So x is r times y dx is equal to r times dy. So that's now x from zero to infinity is exactly the same as y from zero to infinity. So I don't have to change my limits of integration. It's still from zero to infinity by y. r square dx is r dy. r square plus x square is r square y square three seconds equals. Now think about this. If I will have r square, if I will factor out what I will have would be r square times 1 plus y square. And all this in three seconds, right? So r square in the power of three seconds will give me what? It will give me r cube. And this is r cube. So we cancel it out. So I have only things which does not depend on r and it's not supposed to. So this is the integral. I have to prove that it's equal to 1, right? Okay. The standard substitution in this case would be y is equal to tangent of z. Now y is from zero to infinity, zero would be from zero to pi over two, right? As angle is changing from zero to pi over two, its tangent is moving from zero to infinity, okay? Now dy is equal to first derivative of tangent. I think I wrote it down somewhere. Yes. It's dz over cosine square of z. Now I do not remember this formula. It's very easy to derive it because it's sine over cosine and I do remember that sine derivative is cosine and cosine derivative is minus sine. So anyway, that's the result of this. It's very easy. What else do we need? Yes. One plus tangent z square. This is sine over square over cosine square and it will be one over cosine square z, right? That's another formula. So since this is, so it will be integral of zero to pi over two dz divided by cosine square z, that's dy. And this would be one over cosine square z to the power of three second, which is what? Well, that would be one over cosine cube, which is equal to integral from zero to pi over two. Cosine goes there. So this is cosine cube, this is cosine cube, it cosines second degree. So it will be cosine z dz. You see how simple it is? Now, what is the integral of cosine? That's a sine. So from zero to pi over two, sine z, pi over two, it will be one. At zero, it will be zero. So the result is zero. So we have proven that going from a point charge and doing all the integral integrated calculations, we can derive the formula for a straight line current running in a wire. And that would actually kind of give much more, I don't know, solid feeling that whatever reasonable logical kind of statements we came up with by deriving these formulas in both cases for straight line current and from moving particle, they do have some reason behind it, some logic behind it. I mean, I feel much more satisfied that these two relatively independent solutions to some problems are really in sync with each other. That's what usually happens if you will, let's say, solve the problem in one way and then completely different way and come up with the same result. That gives you assurance that whatever you're doing is right. So now we have assurance that both formulas, which I presented in the very beginning, are actually fine and there are no logical contradictions between them. I do suggest you to read the notes for this lecture. Ghostinunisor.com, Ghost of Physics 14 scores. You choose the electromagnetism and among them there is a magnetic field produced by electric current and there you will find this lecture. It's called problem number three in moving charges. That's it, thank you very much and good luck.