 Hi, I'm Zor. Welcome to a new Zor education. We continue talking about a really quite involved part of this electromagnetic field theory about equations, Maxwell equations, and let me repeat again. I'm not really trying to get into these equations in the most general approach. I'm trying to simplify it in as much as possible. So it's not really maybe as rigorous as you would probably get from university. At the same time it gives you a flavor, which is my purpose only. Now this lecture is part of the course called Physics 14's presented in Unisor.com. Part of the course which we are talking about right now is waves and in particular we are talking about electromagnetic waves, which are basically waves of electromagnetic field. And this field obeys certain laws and laws are expressed in equations and James Maxwell was basically the first one who formulated these equations in their differential form. And this is the second lecture, which is about electromagnetic fields, magnetic component. Previous lecture was about electric component and that was the Gauss law and today it's basically Gauss law for magnetic fields. Now I do recommend you to watch this lecture from the Unisor.com and not from any other source like maybe you found it on YouTube or somewhere else. Primarily because the website provides you a course actually, a menu related sequence of lectures. Every lecture has a textual description to it. There are exams, which you can take as many times as you want. The site is totally free. There are no advertisements, so you don't even have to sign on if you don't want to. There is no purpose actually. Now the same website contains math for teens course, which I consider to be, well, not the course itself, but whatever the amount of knowledge is in that course is mandatory for studying physics for teens. Primarily, calculus and vector algebra, algebra. You have to know a lot of things like derivatives, partial derivatives. For the vectors, you have to know what the scalar product, dot product is and vector product, the cross product is, because it's all used in in these particular lectures, for example, and throughout the course. Okay, back to business. First of all, I think it's mandatory for you to review the vector of magnetism part of this course. Before we go to equations of electromagnetic field, that's obvious. And I assume that that part of the course is already covered and obviously, you have to maybe refresh your knowledge, just browsing through the lectures of electromagnetism, which is the previous part of this course. Now we're talking about waves of electromagnetic field. Okay, that's number one. Number two. From this course, I will use certain, well, terminology, concepts, etc. So one of the concepts of which was discussed in the electromagnetism part of this course is magnetic field and its intensity. Magnetic field intensity is usually symbolized with letter b. Now, it's a force in space. So we have a space where there is a magnetic field and at every point there is this particular vector, force is a vector. And what kind of parameters it depends on? Well, time and point coordinate. Obviously, right? Now, at the same time, if this is a vector in a three-dimension space, we can always represent it as sum of three components. It's x-component, it's y-component and z-component, which each one directed along the corresponding x-axis, y-axis and z-axis. Now, these are also vectors along these axes. Well, sometimes it's represented slightly differently without the vector sign here. bx times i plus by times j plus bz times k, where i, j and k are unit vectors along the corresponding x, y and z axes. And these are scholars, not vectors. So these are basically magnitude along the one axis or another. Now, each one of these is obviously, again, a function of all four parameters. Time and all the three coordinates. Okay. So this is the magnetic field intensity. Now, what is important about this? Well, what's important about magnetic field intensity vector is that if there is an electric charge which is moving through this field, then this vector basically acts on it. It's a force and this force acts on moving electric charge. Well, obviously it acts on, for instance, another magnetic field. But that's not what we are interested in. Now, we are talking about electromagnetic interference or something like cooperation sometimes. So we're talking about moving electric charge. So what's important is that if you have a moving charge, electric charge in the magnetic field b, what's important is, and that was already discussed in the electromagnetism part of this course, that the force which is acting on electric charge moving in the magnetic field. Well, this force is actually perpendicular to the vector of speed of that particular electric charge. And it's also perpendicular to the magnetic field intensity vector. So, for example, if magnetic field intensity is uniform, let's say, down and you have an electric, so these are b, you have electric charge which is moving in this direction. Okay, then the vector which is perpendicular to both of them is perpendicular to the plane of this white board. Now direction is determined by direction of this and this. Obviously, if you change this to the opposite, the force change to the opposite. If you change this to the opposite, then the force again change to the opposite. But in any case, it will be along the line which is perpendicular to the white board. So perpendicular to b and perpendicular to v, and that's why it's perpendicular to the whole plane where these two are located. But this is a simple case. I mean, sometimes the obviously doesn't really have to be uniform. It can change, and it can change with time, it can change with the position, etc. So there are many different cases. But at any point in time and space there is a vector, and at the same point in time and space, if at the same point in time and space there is a moving charge, so there is a vector b, and there is a vector b, and there is a vector which is perpendicular to both of them. Now, just as a couple of examples, maybe, magnetic field can be created, for instance, artificially, or you can actually take the magnetic field created by permanent magnets. Now, what happens with artificially created? Well, for example, if you have the electric wire, which has direction of electricity by convention from plus to minus, then there is a concentric lines of magnetic field, which are around this field. Which means what? Well, which means the direction of the force b, let's say, at this point is this. Direction of b at this point is this, and obviously the intensity depends on how much electricity goes through this wire per unit of time. So, basically, the magnitude of vector b is related to current, and related to r, which is distance from the current. Because there are bigger and smaller circle, so there are many different magnetic field lines around it, but they are all cyclic. That's very, very important, and I would like you to pay attention to this. They do not have the beginning and end. You see, with electricity, if you have, let's say, a negative, then the electric lines go from this source of electricity, radial, for instance, in this case. I mean, they can actually end up in some positive thing. They can, but not necessarily. Maybe this doesn't exist, and we exist only this one. Then there is no ending, basically. There is beginning. In case of magnetic field, force lines, there is no beginning or end. They are always closed in some kind of a loop. I mean, not necessarily a real circle, like in this particular case. It can be any kind of a shape, but it's always closed in the loop. Let me just give another example. Let's have a solenoid. This is electric wire, which goes like this, and there is current running in it. Now, what kind of magnetic field will be there? Magnetic field lines would actually be like this. Obviously, they will be weaker the further we are. They will be almost parallel inside and still closed in some kind of loops. Again, magnetic field lines do not have beginning and end. Okay. Now, as far as mathematical representation of the fact that the force is the force which magnetic field acts on moving collector charges, always perpendicular to both, is expressed as the following. This is the force which is basically caused by the magnetic field on the moving charge. And it's actually proportional to the charge. And it's, as I was saying, it's perpendicular to both the intensity vector and the speed, the velocity vector. And what's important is it's proportional to both. So, the stronger magnetic field, the intensity, the stronger will be the force. And the faster the electric charge is moving, the stronger will be the force on it. And here, there is a vector product of these two vectors. That's what actually gives me proportionality to their magnitude and perpendicularity. Again, if you remember from the course of mathematics what vector product is, vector product is always perpendicular to both. So, this is one vector, this is another that would be perpendicular to this and perpendicular to this in three-dimensional space. It's always three-dimensional. So, that's a vector product, which is also a vector. Vector product of two vectors is a vector perpendicular to both, proportional to magnitude of each of them. And then there is some angle, basically, depending, et cetera. But that's very interesting. It's proportional to the charge, magnitude of the charge, magnitude of the speed of that charge in the magnetic field and proportional to the intensity of the magnetic field itself. So, this is the formula which mathematically express this force, which acts on a moving part. And that was covered in electromagnetism part of this course. Okay, fine. Let's forget about this. Magnetic lines are always closed in some kind of a cycle or a loop or whatever. And that is extremely important because this is basically what Gauss' law for magnetic field is, that magnetic lines are always closed in some kind of a loop or cycle. Now, what does it mean? It means the following. And that's what the difference between magnetic field and electric field is. Since magnetic field lines are always looping, if you will take any kind of a value, any kind of element of space, a value, then there are lines which are coming and lines which are going. And the number of lines which are coming, which basically characterizes the flux of the magnetic field, is the same as number of lines which are going out. And each line kind of corresponds to a unit of intensity. The more intense field we usually describe with more densely drawn magnetic lines. But no matter what it is, the number of lines coming in, because there is a direction on every magnetic line, it's a loop and it has a certain direction. And that's exactly how the B vector is working. So, what it means mathematically is that the flow through any closed surface, let's say this is a cube or it can be a sphere or anything, the amount of magnetic field lines going in is the same as going out, which means that the flux through this closed surface is equal to zero. And that's how I meant, not max, I meant mag, magnetic field. So, this is the flux. What is the flux? I mean, in a simple case, flux is B times S. Well, S is actually characterized by a normal to a surface. So, if you have a surface and you have vectors B, so you multiply basically B times, well, you need a normal to this particular surface to multiply these two vectors. So, whenever, and this is by the way a scalar product of normal to the S, and the B. So, I should actually put instead of S, I should put normal vector. So, this is the flux. It's basically amount of magnetic field which is going through the surface. But if surface is enclosed, whatever comes in goes out and the total will always be zero. This is how it's formulated basically in integral form, so to speak. Now, but Maxwell did, he basically expressed it using the differential form. And in the differential form, it looks very much like we were talking about in a previous lecture about electric field flux, the Gauss law for electric field. Now, if you remember, the Gauss law for electric field was actually looking like this. So, rho is density of electric charge inside infinitesimally small closed piece of space. And in that piece of space, we assume that there is some kind of a charge, and that's the density of this charge. Which is basically like amount of electricity per unit of volume. But again, we are talking about infinitesimal piece. And E x, E y, and E z are components of the electric field intensity vector, right? Now, this same law in case of magnetic field actually is very close to this, with very small difference, zero. Why is it zero? Because you cannot put a source of magnetic energy into enclosed infinitesimal piece of space in such a way that magnetic lines are only going out and nothing goes in. It's precisely because all magnetic lines are cyclic, any kind of enclosed volume will have total flux which is equal to zero. And in the differential form it looks exactly like this. So, this is basically the Gauss law for magnetic field, zero here. Now, as in case of electric field, there is some kind of a more convenient notation which smart physicists or mathematicians, I don't know, came up with. You see what they're saying. Since B x, B y, and B z are components of the vector, so we can always say that B is equal to B x, B y, B z. These are components of this vector along each axis, right? Now, the Nabla symbol is absolutely artificial symbol. It doesn't really mean that this is a vector, but you can actually write it as it's kind of, it reminds me, it reminds the vector. It has three components. So, basically it's three operators of differential, partial differentiation, differentiation by variable x, y, and z. And if this represents, well, purely formal kind of notation of a vector, this is a real notation of the vector encoded form. Then, if you take their dot product, the scalar product, what will be? The scalar product is first times first plus second times second plus third times. Now, again, what doesn't mean times. This is operator and this is basically a constant, right? Well, it means, again, purely artificially, they put differential operator to the B x. Well, there is, I mean, in scalar product it's actually a multiplication. These are two components of vectors. In this case, it's just purely formal notation plus D to D, y, B y plus D to D z, B z. So, using this kind of notation, you can have this law looks like this, which is kind of nice, very short one. And it's very easy to remember, basically. All you have to do is, you have to remember that NABLA is an artificial pseudo vector which contains three components, which are partial differentiation by x, by y, and by z. I mean, yeah, it's convenient, that's all. So, again, not a vector. It's a purely artificial construction, which we can call pseudo vector or whatever. And multiplication means actually applying this operator. This is operator differentiation. So, this times B x is actually, this multiplication means we are actually differentiating B x by D, B x by D x, and B y by D y, etc. And the way how you write it, this way, where you put B x like close to D, doesn't really matter. And, again, it's just a notation. But this is a convenient notation, which represents the Gauss law for magnetic field. In a really nice and concise form, which is a contemporary representation of the second Maxwell equation, which is Gauss law for magnetic field. Well, that's it for today. It's kind of easy thing. Tomorrow, next one will be much more difficult, the third equation. But this is kind of easy. And, again, it's based on purely practical, experimental kind of fact, that we do not have a single source of magnetic, well, energy, if you will. We always have north and south, so to speak. And magnetic fields are going from north to south, and then from south to north. They're always a loop around. And you can't really do anything about it. If you have, for instance, a permanent magnet north and south, you divide it in half, each half will have its own north and south. So, we do not know if there are these single sources of magnetic, energy, magnetic monopoles, as they're saying. Okay, that's it. I recommend you to read the text for this particular lecture on Unisor.com. You go to Physics for Teens. Waves is the part, and it's the field waves, basically, topic. And it has all these equations, which we are talking about. So, thank you very much, and good luck.